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[{"Name":"Aether","URL":"aether","Title":"Aether","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{},"Description":"In the classical worldview, people believed that electromagnetic (EM) radiation, described by \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e’s equations (which represent one of the crowning achievements of the classical era), propagated throughout space via the \u003cstrong\u003eluminiferous aether\u003c/strong\u003e — some kind of mysterious, all-pervasive substance that provided a \u003cem\u003ephysical model\u003c/em\u003e for the phenomena described by Maxwell’s equations. The classical worldview was thus dominated by the intuitively satisfying notion that local, deterministic physical laws, operating autonomously through some kind of real physical medium, could produce the observed behavior of nature. This is essentially identical to the \u003ca href=\"cellular-automaton\" target=\"_blank\"\u003ecellular automaton\u003c/a\u003e (CA) framework described above.","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eAether\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-1\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn the classical worldview, people believed that electromagnetic (EM) radiation, described by \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e’s equations (which represent one of the crowning achievements of the classical era), propagated throughout space via the \u003cstrong\u003eluminiferous aether\u003c/strong\u003e — some kind of mysterious, all-pervasive substance that provided a \u003cem\u003ephysical model\u003c/em\u003e for the phenomena described by Maxwell’s equations. The classical worldview was thus dominated by the intuitively satisfying notion that local, deterministic physical laws, operating autonomously through some kind of real physical medium, could produce the observed behavior of nature. This is essentially identical to the \u003ca href=\"cellular-automaton\" target=\"_blank\"\u003ecellular automaton\u003c/a\u003e (CA) framework described above.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOne harbinger of the end of the classical field model was the famous Michelson-Morley experiment of 1887, which is widely regarded as disproving the existence of the aether. This experiment used patterns of interference from light beams traveling in different directions to test for any differences in the speed of light as a function of the relative motion of the Earth through the aether. The idea was that if the aether is a fixed medium for light, the Earth must be moving in some direction relative to this fixed medium (as a result of its orbit around the Sun, and the Sun through the galaxy, etc), and this difference should thus be measurable in terms of the differential speed of light in different directions. The experiment revealed no such differences — light always travels at the same speed in every direction \u003cspan class=\"math inline\"\u003e\\((c \\approx 3.0*10^8)\\)\u003c/span\u003e.\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eHowever, it remains remarkably under-appreciated to this day that special relativity is \u003cem\u003eentirely compatible\u003c/em\u003e with the notion of a luminiferous aether, and indeed provides exactly the right explanation for why the Michelson-Morley experiment failed to detect it: because the speed of light is a constant, the lengths of objects must actually contract in their direction of relative motion, and time dilates, so that even if you are racing very close to the speed of light, almost keeping up with a speeding light ray, you measure the speed of this light to be the same as someone standing still.\u003c/p\u003e\u003cp id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSpecifically, because your measuring devices (rulers) have all shrunk in the direction of motion, distances appear longer, and time dilation causes measured time intervals to appear shorter, with the net result that a moving observer obtains the same measured distance per unit time (i.e., speed) that someone standing still would measure. This \u003cem\u003eLorentz transformation\u003c/em\u003e was already well established prior to Einstein’s 1905 paper on special relativity, based on measurements of electromagnetic phenomena.\u003c/p\u003e\u003cp id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAs shown by the \u003ca href=\"klein-gordon\" target=\"_blank\"\u003eKlein-Gordon\u003c/a\u003e equation, a simple wave equation with a mass term results in waves that can travel at any velocity below the speed of light, in proportion to the wavelength of the wave. This relationship between wavelength and speed is exactly as required by the Lorentz contraction (and the core quantum relationship between momentum and frequency), where distances would be measured as a function of these contracting wavelengths.\u003c/p\u003e\u003cp id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThus, we only need to modify our understanding of the properties of the aether, in accordance with the Lorentz transformation, to reconcile the appealing classical world view with the observed facts. But there are two obvious problems with such an approach. First, the aether becomes essentially unmeasurable, and thus a belief in its existence would seem to be outside the scope of objective science. Second, the framework of special relativity has no need for such a thing, and relativity provides such a nice compelling and self-contained world view, that there is no motivation to retain this clunky, outdated notion of the aether.\u003c/p\u003e\u003cp id=\"text-6\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eHowever, if we can develop a compelling and accurate physical model of electrodynamics based on the CA framework, with the central property that the discrete CA state cells provide the basis for the discrete massive particles in the pilot-wave framework, then at least there would be no basis to reject such a framework outright.\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cstretch id=\"content-buttons-go-167\" style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1ch;min-height:1em;font-weight:thin;text-align:start\"\u003e\u003c/stretch\u003e\u003ca id=\"content-buttons-go-168\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"cellular-automaton\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M686-450H160v-60h526L438-758l42-42 320 320-320 320-42-42 248-248Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Cellular automaton","URL":"cellular-automaton","Title":"Cellular automaton","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{"figure":["figure_ca2d","figure_cubes"]},"Description":"The computational modeling approach here is essentially a complex version of a \u003cstrong\u003ecellular automaton (CA)\u003c/strong\u003e, which has been investigated as a basis for fundamental physics modeling since the 1950s. A CA consists of a regular, uniform division of space into discrete \u003cem\u003ecells\u003c/em\u003e, each of which has one or more \u003cem\u003estate\u003c/em\u003e values, and each cell interacts only with its nearest neighbors (i.e., locally) to update its state value over time (\u003ca href=\"cellular-automaton#figure_ca2d\" target=\"_blank\"\u003eFigure 1\u003c/a\u003e; \u003ca href=\"cellular-automaton#figure_cubes\" target=\"_blank\"\u003eFigure 2\u003c/a\u003e).","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eCellular automaton\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-2\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"figure_ca2d\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_ca2d\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 1:\u003c/b\u003e \u003cp\u003eIllustration of a simple 2-dimensional cellular automaton: space is divided into regular square cells (a uniform, regular tiling of space), and neighboring states interact by influencing the state update. Time updates synchronously, setting the fastest rate of propagation as cell width / time update.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"figure_cubes\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_cubes\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 2:\u003c/b\u003e \u003cp\u003eNeighborhood interactions in regular cubic tiling of space in three-dimensions — these interactions are used to compute the wave equation locally.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cp id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe computational modeling approach here is essentially a complex version of a \u003cstrong\u003ecellular automaton (CA)\u003c/strong\u003e, which has been investigated as a basis for fundamental physics modeling since the 1950s. A CA consists of a regular, uniform division of space into discrete \u003cem\u003ecells\u003c/em\u003e, each of which has one or more \u003cem\u003estate\u003c/em\u003e values, and each cell interacts only with its nearest neighbors (i.e., locally) to update its state value over time (\u003ca href=\"cellular-automaton#figure_ca2d\" target=\"_blank\"\u003eFigure 1\u003c/a\u003e; \u003ca href=\"cellular-automaton#figure_cubes\" target=\"_blank\"\u003eFigure 2\u003c/a\u003e).\u003c/p\u003e\u003cp id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSuch a system was first described by Stanislaw Ulam in 1950, and has been popularized in its two-dimensional form in “the game of Life” by John Conway (described by \u003ca href=\"ref://Gardner70\" target=\"_blank\"\u003eGardner, 1970\u003c/a\u003e). In this CA (widely available as a screensaver), there is a two-dimensional grid of square cells, with each cell having a single binary state value (0 = “dead” and 1 = “alive”). This state value updates in discrete, simultaneous steps as a function of the state values in the 8 neighbors of each cell. If the sum of the neighbors’ states is \u003e 3 or \u003c 2, then the cell is dead (0) on the next time step (from “overcrowding” or “loneliness”, respectively). Otherwise if it has exactly 3 live neighbors and is currently dead, then it is “born” and goes to 1, and if it was already “alive” then it remains so if it has 2-3 living neighbors. As anyone who has seen this system in operation knows, it is capable of producing remarkable complexity from such simple, local, deterministic rules.\u003c/p\u003e\u003cp id=\"text-6\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe CA framework provides the simplest kinds of answers to fundamental questions about space, time, and the basic nature of physical laws (\u003ca href=\"ref://VonNeumannBurks66\" target=\"_blank\"\u003eVon Neumann \u0026 Burks, 1966\u003c/a\u003e; \u003ca href=\"ref://Zuse70\" target=\"_blank\"\u003eZuse, 1970\u003c/a\u003e; \u003ca href=\"ref://FredkinToffoli82\" target=\"_blank\"\u003eFredkin \u0026 Toffoli, 1982\u003c/a\u003e; \u003ca href=\"ref://Feynman82\" target=\"_blank\"\u003eFeynman, 1982\u003c/a\u003e; \u003ca href=\"ref://Fredkin90\" target=\"_blank\"\u003eFredkin, 1990\u003c/a\u003e; \u003ca href=\"ref://Hooft15\" target=\"_blank\"\u003eHooft, 2015\u003c/a\u003e). Space is \u003cem\u003ereal\u003c/em\u003e and fundamental in the form of the underlying cells — it isn’t just an empty vacuum or a mathematical continuum. The discretization of space, as contrasted with a true continuum, can be motivated by the levels of infinities associated with the Cantor sets: a discrete space corresponds to the lowest level of infinity associated with the integer number line, and thus represents the simplest way of representing space.\u003c/p\u003e\u003cp id=\"text-7\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOne still has an infinity to deal with, and this is plenty mind-blowing all by itself: space and time continuing infinitely in all directions, forever. But at least the further difficulty of an infinity of space or time \u003cem\u003ewithin\u003c/em\u003e any given segment, which is required for a truly continuous dimension, can be avoided. One could reasonably argue that the infinity of space and time is more plausible than the notion of an edge, as in the old flat Earth models and the end of the world.\u003c/p\u003e\u003cp id=\"text-8\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe physical phenomenon of discrete point-like elementary particles also suggests the need for an \u003cem\u003eultraviolet cutoff\u003c/em\u003e at some distance scale: infinitely small point-like particles predict infinite field amplitudes in their immediate neighborhood. Instead, a discrete lattice with a given grid spacing distance provides a tractable, non-divergent mechanism for such discrete particles, ad discussed in \u003ca href=\"stochastic-particles\" target=\"_blank\"\u003estochastic particles\u003c/a\u003e.\u003c/p\u003e\u003cp id=\"text-9\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eTime emerges naturally in its unique unidirectionality within the CA framework, simply as a discrete rate of change in the state values. Furthermore, the ratio of discrete spatial cell width to discrete rate of state update provides a natural upper limit to the rate at which anything can propagate within this system: i.e., the \u003cem\u003espeed of light\u003c/em\u003e in a vacuum. Thus, this principal postulate of special relativity that light has a fixed upper speed limit emerges as a necessary consequence of more fundamental assumptions about the nature of space and time in the CA framework.\u003c/p\u003e\u003cp id=\"text-10\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFurthermore, as we will see in the subsequent chapters, the basic wave equation can be computed using a simple local neighborhood interaction among cells in a CA-like system, and \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e’s equations for the electromagnetic field and \u003ca href=\"dirac\" target=\"_blank\"\u003eDirac\u003c/a\u003e’s equation for the quantum wave function of an electron can be computed using primarily this basic wave equation. We discuss the more detailed features of special relativity in relationship to the CA framework next, but the main conclusion is that this framework predicts all of the features of special relativity, from first principles based on the discretization of space and time, together with wave dynamics.\u003c/p\u003e\u003cp id=\"text-11\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eNote that these wave-based equations do require real-valued state variables, which is a departure from the simplest form of CA that only employs simple discrete state values. Thus, the CA framework provides a potential answer to a central question for the pilot-wave model: what is the ontological basis for the quantum wave function? It is just another set of state variables updating according to local wave equations, just like the EM field.\u003c/p\u003e\u003cp id=\"text-12\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn addition, we may require stochastic processes such as randomly choosing the next location for a discrete particle to move based on the local wave field gradients. Ultimately, the model needs to \u003cem\u003ework\u003c/em\u003e to explain the available data, and our intuitions about “mechanistic” level plausibility are secondary concerns: if these intuitions align with reality in a way that makes everything work, it is obviously great, but you cannot let them stand in the way of making progress.\u003c/p\u003e\u003cp id=\"text-13\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAnother affordance of the CA framework is that it unambiguously establishes the position basis as primary for representing discrete massive particles, consistent with the pilot-wave framework. In addition, the \u003ca href=\"pauli-exclusion-principle\" target=\"_blank\"\u003ePauli exclusion principle\u003c/a\u003e is strongly suggestive of a discrete CA-like state. This principle posits that only one \u003cem\u003efermion\u003c/em\u003e (electron, quark, etc, with a quantum spin of \u003csup\u003e1\u003c/sup\u003e⁄\u003csub\u003e2\u003c/sub\u003e) can occupy the same quantum state, including position, at a time.\u003c/p\u003e\u003cp id=\"text-14\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThus, the underlying CA state representation only needs to be able to hold one of each particle type, which eliminates the difficult problem of having to represent a variable number of such particles at each location. In other words, the “memory allocation” for each cell is constant. Note that this is not the case for \u003cem\u003eboson\u003c/em\u003e particles which obey no such exclusion principle, and would thus require an indefinite number of memory slots to represent (along with all the other difficulties involved in the particle picture for force fields).\u003c/p\u003e\u003cp id=\"text-15\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe notion of \u003cem\u003eautonomy\u003c/em\u003e in a CA is also particularly important as a physical model: the CA is entirely self-contained and can just plug away forever, running the same exact local laws every time step. By contrast, most calculational tools used in physics require a specific setup and different computational steps depending on exactly what situation is being modeled: they are far from “autonomous” in the sense of a CA.\u003c/p\u003e\u003cp id=\"text-16\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWhen you look at the examples of plausible physical models (\u003ca href=\"tools-vs-models\" target=\"_blank\"\u003etools vs models\u003c/a\u003e), they all have this same autonomous character: e.g., general relativity and Maxwell’s equations in the Lorenz gauge can just be configured with a starting state and then everything can evolve autonomously from there.\u003c/p\u003e\u003cp id=\"text-17\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn summary, the CA framework is simple, elegant, and consistent with the most basic facts of physics. If one could develop a viable physical theory within the general confines of this framework, it would provide a uniquely simple and satisfying model of how nature works.\u003c/p\u003e\u003cp id=\"text-18\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eHowever, two important objections are typically raised about such a framework: isn’t it just like the \u003ca href=\"aether\" target=\"_blank\"\u003eaether\u003c/a\u003e that was so famously rejected by the Michelson-Morley experiment; and if it has purely local interactions, how could it possibly account for the apparent \u003ca href=\"non-locality\" target=\"_blank\"\u003enon-locality\u003c/a\u003e of QM?\u003c/p\u003e\u003ch2 id=\"quantum-cellular-automata-and-random-walks\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eQuantum cellular automata and random walks\u003c/h2\u003e\u003cp id=\"text-20\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThere is a growing literature on quantum cellular automata (QCA) and the quantum random walk (QW) model, going back to the influential papers by \u003ca href=\"ref://Feynman82\" target=\"_blank\"\u003eFeynman (1982)\u003c/a\u003e, \u003ca href=\"ref://AharonovDavidovichZagury93\" target=\"_blank\"\u003eAharonov et al. (1993)\u003c/a\u003e, and \u003ca href=\"ref://Bialynicki-Birula94\" target=\"_blank\"\u003eBialynicki-Birula (1994)\u003c/a\u003e. This approach has become somewhat more active recently due to the practical application of these models for designing quantum algorithms (\u003ca href=\"ref://ChildsCleveDeottoEtAl03\" target=\"_blank\"\u003eChilds et al., 2003\u003c/a\u003e; \u003ca href=\"ref://ChiribellaDArianoPerinotti11\" target=\"_blank\"\u003eChiribella et al., 2011\u003c/a\u003e; \u003ca href=\"ref://DAriano17\" target=\"_blank\"\u003eD’Ariano, 2017\u003c/a\u003e; \u003ca href=\"ref://Kempe09\" target=\"_blank\"\u003eKempe, 2009\u003c/a\u003e). Although this work shares many essential properties in common with the present approach, it is not directly applicable, due to some differences in basic assumptions.\u003c/p\u003e\u003cp id=\"text-21\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe central idea behind the random walk model proposed originally by \u003ca href=\"ref://AharonovDavidovichZagury93\" target=\"_blank\"\u003eAharonov et al. (1993)\u003c/a\u003e is that a shift operator that translates the quantum state in either the left or right direction along a discrete 1D lattice can be driven by a quantum measurement process on a \u003cem\u003edifferent\u003c/em\u003e element of the quantum state (e.g., the spin), which thus would produce a random sample according to the usual probabilistic quantum theory. This “internal” source of randomness will thus produce an emergent random walk dynamic as the state evolves (translates) over time, in the same way that thermal noise produces Brownian random walk motion as originally analyzed by Einstein.\u003c/p\u003e\u003cp id=\"text-22\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe results from these analyses show that standard wave functions such as the \u003ca href=\"dirac\" target=\"_blank\"\u003eDirac\u003c/a\u003e, \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e and Weyl functions emerge from these random walk processes. A closely related analysis in 3D by \u003ca href=\"ref://Bialynicki-Birula94\" target=\"_blank\"\u003eBialynicki-Birula (1994)\u003c/a\u003e showed the emergence of the Weyl and Maxwell wave functions in the context of a cellular automaton update rule with \u003cem\u003eunitary\u003c/em\u003e update rules.\u003c/p\u003e\u003cp id=\"text-23\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe basic intuition is that these unitary update rules cause the propagation of a wave-like pattern at the speed of light (one unit cell per unit time), and if multiple internal cell states are present, this propagation can also include the \u003ca href=\"spin\" target=\"_blank\"\u003espin\u003c/a\u003e property where the state rotates through these internal states as it also propagates. This is the essential feature of the Weyl wave functions, which describe a “pure spin” particle such as a massless neutrino that travels at the speed of light while spinning in one fixed helical rotation.\u003c/p\u003e\u003cp id=\"text-24\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eCritically, these approaches do \u003cem\u003enot\u003c/em\u003e provide a model of a discrete particle moving with graded momentum (velocities) in an isotrophic manner along a cubic grid, which is what we develop in \u003ca href=\"stochastic-particles\" target=\"_blank\"\u003estochastic particles\u003c/a\u003e. That work builds on the approach originated by \u003ca href=\"ref://Nelson66\" target=\"_blank\"\u003eNelson, 1966\u003c/a\u003e in analyzing single-particle Brownian motion, using equations of motion initially developed by \u003ca href=\"ref://Sciarretta18\" target=\"_blank\"\u003eSciarretta (2018)\u003c/a\u003e.\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"aether\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"m274-450 248 248-42 42-320-320 320-320 42 42-248 248h526v60H274Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003ePrevious\u003c/p\u003e\u003c/a\u003e\u003cstretch id=\"content-buttons-go-167\" style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1ch;min-height:1em;font-weight:thin;text-align:start\"\u003e\u003c/stretch\u003e\u003ca id=\"content-buttons-go-168\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"complex-kg\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M686-450H160v-60h526L438-758l42-42 320 320-320 320-42-42 248-248Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Complex KG","URL":"complex-kg","Title":"Complex KG","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{"eq":["eq_kg-complex","eq_kg-charge","eq_kg-current"],"figure":["figure_vgrad"],"sim":["sim_cc"]},"Description":"In \u003ca href=\"klein-gordon\" target=\"_blank\"\u003eKlein-Gordon\u003c/a\u003e we saw that a major problem with the scalar KG equation is that it doesn’t represent any kind of conserved value: you cannot compute some constant, unchanging value from the \u003cspan class=\"math inline\"\u003e\\(\\varphi\\)\u003c/span\u003e state variables under this equation. This is a problem if you want to develop a probabilistic interpretation of the wave, as in \u003ca href=\"schrodinger\" target=\"_blank\"\u003eSchrodinger\u003c/a\u003e’s equation for the standard \u003ca href=\"copenhagen\" target=\"_blank\"\u003eCopenhagen\u003c/a\u003e interpretation. But it is also a problem for other interpretations as well.","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eComplex KG\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-3\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn \u003ca href=\"klein-gordon\" target=\"_blank\"\u003eKlein-Gordon\u003c/a\u003e we saw that a major problem with the scalar KG equation is that it doesn’t represent any kind of conserved value: you cannot compute some constant, unchanging value from the \u003cspan class=\"math inline\"\u003e\\(\\varphi\\)\u003c/span\u003e state variables under this equation. This is a problem if you want to develop a probabilistic interpretation of the wave, as in \u003ca href=\"schrodinger\" target=\"_blank\"\u003eSchrodinger\u003c/a\u003e’s equation for the standard \u003ca href=\"copenhagen\" target=\"_blank\"\u003eCopenhagen\u003c/a\u003e interpretation. But it is also a problem for other interpretations as well.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn particular, the KG waves seem to actually represent \u003cstrong\u003ewaves of charge\u003c/strong\u003e (i.e., \u003ca href=\"matter-waves\" target=\"_blank\"\u003ematter waves\u003c/a\u003e), because charge is also strictly conserved, and it comes in both positive and negative varieties, which the KG waves produce. By contrast, the Schrödinger wave only represents a positive-valued quantity, which fits better with the probabilistic interpretation. Indeed, the authors that do write extensively about the KG equation adopt this interpretation (\u003ca href=\"ref://Greiner00\" target=\"_blank\"\u003eGreiner, 2000\u003c/a\u003e; \u003ca href=\"ref://Gingrich04\" target=\"_blank\"\u003eGingrich, 2004\u003c/a\u003e; \u003ca href=\"ref://MandlShaw13\" target=\"_blank\"\u003eMandl \u0026 Shaw, 2013\u003c/a\u003e), and this idea was pursued initially by Schrödinger in 1926.\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFurthermore, we will see that this charge interpretation fits naturally with the coupling of this KG equation to the electromagnetic (EM) field (\u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e’s equations), where the conserved charge value acts just like the electric charge in driving the field. However, to fully accomplish this coupling in a physically accurate way, the wave function needs \u003ca href=\"spin\" target=\"_blank\"\u003espin\u003c/a\u003e, which is what the \u003ca href=\"dirac\" target=\"_blank\"\u003eDirac\u003c/a\u003e wave function adds on top of the KG waves. This coupling of charge waves and EM waves has been pursued more recently in neoclassical self-coupled field theory (\u003ca href=\"ref://JaynesCummings63\" target=\"_blank\"\u003eJaynes \u0026 Cummings, 1963\u003c/a\u003e; \u003ca href=\"ref://CrispJaynes69\" target=\"_blank\"\u003eCrisp \u0026 Jaynes, 1969\u003c/a\u003e; \u003ca href=\"ref://BarutVanHuele85\" target=\"_blank\"\u003eBarut \u0026 Van Huele, 1985\u003c/a\u003e; \u003ca href=\"ref://BarutDowling90\" target=\"_blank\"\u003eBarut \u0026 Dowling, 1990\u003c/a\u003e; \u003ca href=\"ref://Crisp96\" target=\"_blank\"\u003eCrisp, 1996\u003c/a\u003e; \u003ca href=\"ref://FinsterSmollerYau99a\" target=\"_blank\"\u003eFinster et al., 1999\u003c/a\u003e; \u003ca href=\"ref://Radford03\" target=\"_blank\"\u003eRadford, 2003\u003c/a\u003e; \u003ca href=\"ref://MasielloDeumensOhrn05\" target=\"_blank\"\u003eMasiello et al., 2005\u003c/a\u003e).\u003c/p\u003e\u003cp id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe first step toward a more complete wave function for something like an \u003ca href=\"electron\" target=\"_blank\"\u003eelectron\u003c/a\u003e is to introduce a wave state with \u003ca href=\"complex-numbers\" target=\"_blank\"\u003ecomplex numbers\u003c/a\u003e (\u003cspan class=\"math inline\"\u003e\\(\\phi\\)\u003c/span\u003e) instead of scalars (\u003cspan class=\"math inline\"\u003e\\(\\varphi\\)\u003c/span\u003e), which then supports the computation of a conserved quantity across the two complex state values. When we translate this complex wave function into two separate second-order wave equations without an \u003cspan class=\"math inline\"\u003e\\(i\\)\u003c/span\u003e imaginary number factor, the KG equation is identical to computing two separate KG equations on each of the two scalar values represented by the complex state variable (i.e., \u003cspan class=\"math inline\"\u003e\\(\\varphi_a\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(\\varphi_b\\)\u003c/span\u003e). Note that this was not true of Schrödinger’s wave equation, which is first order and has an \u003cspan class=\"math inline\"\u003e\\(i\\)\u003c/span\u003e term that causes the \u003cspan class=\"math inline\"\u003e\\(a\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(b\\)\u003c/span\u003e terms to intermix as the wave unfolds.\u003c/p\u003e\u003cdiv id=\"inline-container-5\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_kg-complex\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_kg-complex\"\u003e\u003cb\u003eEq 1:\u003c/b\u003e Klein-Gordon on complex state\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-6\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2 \\phi}{\\partial t^2} = (\\nabla^2 - m_0^2) \\phi\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-7\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eBecause differentiation operates independently on the two separate scalar variables in a complex number, this is equivalent to two parallel scalar KG equations, which we can write as:\u003c/p\u003e\u003cp id=\"text-8\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[d\n\\frac{\\partial^2 \\varphi_a}{\\partial t^2} = \\left( \\nabla^2 - m_0^2 \\right) \\varphi_a\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-9\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2 \\varphi_b}{\\partial t^2} = \\left( \\nabla^2 - m_0^2 \\right) \\varphi_b\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-10\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhere again the \u003cspan class=\"math inline\"\u003e\\(\\varphi_a\\)\u003c/span\u003e indicates a scalar state variable representing the real \u003cspan class=\"math inline\"\u003e\\(a\\)\u003c/span\u003e component of \u003cspan class=\"math inline\"\u003e\\(\\phi\\)\u003c/span\u003e, and \u003cspan class=\"math inline\"\u003e\\(\\varphi_b\\)\u003c/span\u003e represents the imaginary \u003cspan class=\"math inline\"\u003e\\(b\\)\u003c/span\u003e value.\u003c/p\u003e\u003cp id=\"text-11\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOk, so how do you get a charge out of that, so to speak? As with Schrödinger’s equation, the procedure involves multiplying by the complex conjugate (\u003cspan class=\"math inline\"\u003e\\(\\phi^* = \\varphi_a - i \\varphi_b\\)\u003c/span\u003e), which generally produces the overall magnitude or length of the vector represented by the two components of the complex number: \u003cspan class=\"math inline\"\u003e\\(\\varphi_a^2 + \\varphi_b^2\\)\u003c/span\u003e.\u003c/p\u003e\u003cp id=\"text-12\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSee this page for all of the details regarding the \u003ca href=\"conservation\" target=\"_blank\"\u003econservation\u003c/a\u003e properties of wave functions. In summary, if you compute the sum of the complex conjugate across all of space (actually an integral, using continuous equations), and set it equal to zero (so that it never changes), you end up with an expression for the density and motion (current) of a quantity that is conserved (i.e., the charge).\u003c/p\u003e\u003cp id=\"text-13\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe resulting expression for computing the density of charge (typically written as \u003cspan class=\"math inline\"\u003e\\(\\rho\\)\u003c/span\u003e, which is the Greek letter “rho”), which is to say, the amount of charge per cubic state unit, is:\u003c/p\u003e\u003cdiv id=\"inline-container-15\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_kg-charge\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_kg-charge\"\u003e\u003cb\u003eEq 2:\u003c/b\u003e Klein-Gordon charge density\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-16\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\rho \\equiv \\frac{i \\hbar e}{2m_0c^2} \\left( \\phi^* \\frac{\\partial \\phi}{\\partial t} - \\phi \\frac{\\partial \\phi^*}{\\partial t} \\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-17\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhere the \u003cspan class=\"math inline\"\u003e\\(e\\)\u003c/span\u003e value is a constant (1.6e-19 in SI units) that converts natural units into proper units of charge.\u003c/p\u003e\u003cp id=\"text-18\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis can be expressed in terms of the underlying scalar state variables that make up the complex state (\u003cspan class=\"math inline\"\u003e\\(\\phi = \\varphi_a + i \\varphi_b\\)\u003c/span\u003e), and their first temporal derivatives (\u003cspan class=\"math inline\"\u003e\\(\\dot \\varphi_a\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(\\dot \\varphi_b\\)\u003c/span\u003e), and using natural units where \u003cspan class=\"math inline\"\u003e\\(c=\\hbar=1\\)\u003c/span\u003e, as:\u003c/p\u003e\u003cp id=\"text-19\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\rho_i = \\frac{e}{m_0} ({\\varphi_b}_i \\dot {\\varphi_a}_i - {\\varphi_a}_i \\dot {\\varphi_b}_i)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-20\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis is directly computable for each cubic cell \u003cspan class=\"math inline\"\u003e\\(i\\)\u003c/span\u003e in the system.\u003c/p\u003e\u003cp id=\"text-21\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIt becomes clear when explicitly written out in this manner that charge represents a coupling of the two otherwise independently-updated variables in the complex number, and this suggests why a single scalar number cannot represent a conserved charge value.\u003c/p\u003e\u003cdiv id=\"sim_cc\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"inline-container-23\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"sim_cc\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"sim_cc\"\u003e\u003cb\u003eSim 1:\u003c/b\u003e Complex charge\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003ccollapser id=\"collapser-1\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"frame-0\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.1em;font-weight:thin;text-align:start\"\u003e\u003cinput id=\"toggle\" style=\"color:var(--primary-color);display:flex;flex-direction:row;justify-content:center;align-items:center;font-weight:thin;text-align:center;border-radius:8px\" type=\"checkbox\" value=\"false\"\u003e\u003c/input\u003e\u003cp id=\"text-1\" style=\"font-weight:normal;line-height:1.5;text-align:start\"\u003eComplex charge\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"frame-1\" style=\"display:none;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ceditor id=\"editor-0\" style=\"background:var(--surface-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:10em;padding-top:0.5em;padding-right:0.5em;padding-bottom:0.5em;padding-left:0.5em;font-weight:thin;line-height:1.3;text-align:start;border-radius:16px\"\u003e\u003c/editor\u003e\u003c/div\u003e\u003c/collapser\u003e\u003cdiv id=\"frame-2\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cplot id=\"plot-0\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:512px;min-height:384px;width:512px;height:384px;font-weight:thin;text-align:start\"\u003e\u003c/plot\u003e\u003cp id=\"text-1\" style=\"min-width:80ch;width:80ch;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eCharge: 0.5 \u003c/b\u003e\u003c/p\u003e\u003cp id=\"text-2\" style=\"min-width:40ch;width:40ch;font-weight:normal;line-height:1.5;text-align:start\"\u003eb phase: 90\u003c/p\u003e\u003cinput id=\"slider-3\" style=\"background:#E1E2EB;display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:20em;min-height:1em;width:20em;height:1em;padding-top:8px;padding-right:8px;padding-bottom:8px;padding-left:8px;font-weight:thin;text-align:start;border-radius:1e+09px\" type=\"range\" value=\"90\"\u003e\u003c/input\u003e\u003cp id=\"text-4\" style=\"min-width:40ch;width:40ch;font-weight:normal;line-height:1.5;text-align:start\"\u003emass: 0.1\u003c/p\u003e\u003cinput id=\"slider-5\" style=\"background:#E1E2EB;display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:20em;min-height:1em;width:20em;height:1em;padding-top:8px;padding-right:8px;padding-bottom:8px;padding-left:8px;font-weight:thin;text-align:start;border-radius:1e+09px\" type=\"range\" value=\"0.1\"\u003e\u003c/input\u003e\u003cp id=\"text-6\" style=\"min-width:40ch;width:40ch;font-weight:normal;line-height:1.5;text-align:start\"\u003ehbar: 0.5\u003c/p\u003e\u003cinput id=\"slider-7\" style=\"background:#E1E2EB;display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:20em;min-height:1em;width:20em;height:1em;padding-top:8px;padding-right:8px;padding-bottom:8px;padding-left:8px;font-weight:thin;text-align:start;border-radius:1e+09px\" type=\"range\" value=\"0.5\"\u003e\u003c/input\u003e\u003c/div\u003e\u003c/div\u003e\u003cp id=\"text-25\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003ca href=\"complex-kg#sim_cc\" target=\"_blank\"\u003eSim 1\u003c/a\u003e demonstrates how this works, in terms of two simple \u003ca href=\"harmonic-oscillator\" target=\"_blank\"\u003eharmonic oscillator\u003c/a\u003e variables \u003cem\u003ea\u003c/em\u003e and \u003cem\u003eb\u003c/em\u003e, which are set to be a specific phase apart from each other (+90 degrees shifts \u003cem\u003eb\u003c/em\u003e to the \u003cem\u003eleft\u003c/em\u003e (earlier) relative to \u003cem\u003ea\u003c/em\u003e, while -90 shifts to the right, due to the trigonometric convention of 0 degrees being at 1,0 and proceeding counter-clockwise from there). Regardless of the phase relationship, the computed charge value remains constant across the cycles of oscillation. However, critically, the value of the charge is directly a function of this phase relationship, with a maximum of 0.5 when the \u003cem\u003eb\u003c/em\u003e value is +90 degrees in relation to the \u003cem\u003ea\u003c/em\u003e value, and a minimum of -0.5 for -90 degrees, and zero for 0 or 180 degrees. These relationships are fairly obvious once you appreciate the relationship between velocity and position for each of the variables (which are 90 degrees out of phase with each other, always), and how they enter into the charge equation.\u003c/p\u003e\u003cp id=\"text-26\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eCritically, complex numbers are \u003cem\u003ealways\u003c/em\u003e 90 degrees out of phase with each other by the very nature of the complex plane. Thus, even though the separate real-valued wave functions are independently updated, it is critical that these two wave states are \u003cem\u003einitialized\u003c/em\u003e with the 90 degree phase relationship appropriate for complex numbers, which will then determine the sign of the charge value represented.\u003c/p\u003e\u003cp id=\"text-27\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ePerhaps the most important feature of this charge equation is that it can be either positive or negative, as a function of the phase relationship. This is not true of the corresponding expression for Schrödinger’s equation, which is “definitely positive”, or, in mathematical terminology, “positive definite”. This is one of the major reasons why standard quantum physics has strongly embraced Schrödinger’s equation, and not KG: KG does not fit with the standard probabilistic framework, where the wave describes a probability, and a probability is always positive.\u003c/p\u003e\u003cp id=\"text-28\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eInterestingly, the Dirac equation, which standard physics has adopted as a model of the electron (and we’ll cover later), also produces negative “probabilities”, but these have been (correctly, in our framework) reinterpreted as representing antiparticles (i.e., particles with an opposite charge). The antiparticle of the electron is the \u003cstrong\u003epositron\u003c/strong\u003e, and it is just like an electron, except it has the opposite charge. Historically, this antiparticle nature of the Dirac equation was regarded as a major problem, until positrons were subsequently discovered, and then Dirac looked like a genius for having made such a bold prediction. Nevertheless, there seems to be some residual discomfort in all this, and many treatments of quantum electrodynamics marginalize the Dirac equation in favor of a largely particle-based treatment. We return to these issues later.\u003c/p\u003e\u003cp id=\"text-29\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWe can also derive an expression for the motion of charge over space, which is the \u003cem\u003echarge current density\u003c/em\u003e \u003cspan class=\"math inline\"\u003e\\(\\vec{J}\\)\u003c/span\u003e:\u003c/p\u003e\u003cdiv id=\"inline-container-31\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_kg-current\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_kg-current\"\u003e\u003cb\u003eEq 3:\u003c/b\u003e Klein-Gordon current density\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-32\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{J} \\equiv - \\frac{i \\hbar e}{2m_0} \\left( \\phi^* \\vec{\\nabla} \\phi - \\phi \\vec{\\nabla} \\phi^* \\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-33\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn terms of the underlying scalar state variables (and again for natural units), this is:\u003c/p\u003e\u003cp id=\"text-34\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{J} = \\frac{e}{m_0} (\\varphi_a \\vec{\\nabla} \\varphi_b - \\varphi_b \\vec{\\nabla} \\varphi_a)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-35\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis value indicates how much charge is moving in each of the three different coordinate directions; the \u003cspan class=\"math inline\"\u003e\\(\\vec{}\\)\u003c/span\u003e symbol on top of \u003cspan class=\"math inline\"\u003e\\(\\vec{J}\\)\u003c/span\u003e indicates that this is a vector, containing a separate real scalar value for each direction: \u003cspan class=\"math inline\"\u003e\\(\\vec{J} = (J_x, J_y, J_z)\\)\u003c/span\u003e. As noted earlier the \u003cspan class=\"math inline\"\u003e\\(\\vec{\\nabla}\\)\u003c/span\u003e symbol is the vector gradient operator, which computes the rate of change of the values along each dimension:\u003c/p\u003e\u003cp id=\"text-36\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{\\nabla} \\equiv \\left(\\frac{\\partial}{\\partial x}, \\frac{\\partial}{\\partial y}, \\frac{\\partial }{\\partial z}\\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cdiv id=\"figure_vgrad\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_vgrad\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 1:\u003c/b\u003e \u003cp\u003eThe gradient of a scalar field (\u003cspan class=\"math inline\"\u003e\\(\\vec{\\nabla}\\)\u003c/span\u003e), which produces a vector field describing the slope at each point in space. These gradient vectors point in the direction of maximum ”downhill” slope. In this example the scalar field is a circularly-symmetric bump.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cp id=\"text-39\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis just means that this \u003cspan class=\"math inline\"\u003e\\(\\vec{\\nabla}\\)\u003c/span\u003e takes a three-dimensional field, in this case the field of our wave value \u003cspan class=\"math inline\"\u003e\\(\\varphi_a\\)\u003c/span\u003e or \u003cspan class=\"math inline\"\u003e\\(\\varphi_b\\)\u003c/span\u003e distributed over space, and computes how steeply this field is changing in each of the three different directions \u003ca href=\"complex-kg#figure_vgrad\" target=\"_blank\"\u003eFigure 1\u003c/a\u003e. If we assume that this value is actually computed in our model, then we’ll need a way of computing the gradient \u003cspan class=\"math inline\"\u003e\\(\\vec{\\nabla}\\)\u003c/span\u003e in discrete space-time. This is covered in the next section.\u003c/p\u003e\u003cp id=\"text-40\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eBefore proceeding, we look ahead to the next major development. We have ways of computing the density and current of charge (\u003cspan class=\"math inline\"\u003e\\(\\rho\\)\u003c/span\u003e, \u003cspan class=\"math inline\"\u003e\\(\\vec{J}\\)\u003c/span\u003e), which drive the electromagnetic field. Thus, we need to think of these variables as physically real values, which can be computed directly from the underlying wave state variables, that give rise to long-range electromagnetic forces, through which our particle-waves interact. The next step is to see how the electromagnetic fields can push our particle waves around.\u003c/p\u003e\u003cp id=\"text-41\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSee \u003ca href=\"discrete-gradient\" target=\"_blank\"\u003ediscrete gradient\u003c/a\u003e for how to compute the gradient in the discrete CA framework.\u003c/p\u003e\u003cp id=\"text-42\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSo, in the end, the computation of the current, which is a vector having three separate components (\u003cspan class=\"math inline\"\u003e\\(J_x, J_y, J_z\\)\u003c/span\u003e), looks like this:\u003c/p\u003e\u003cp id=\"text-43\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nJ_x = \\frac{e}{m_0} \\left[ \\varphi_a \\left( \\sum_{j \\in N_{X}} k_j {(\\varphi_b}_{j+} - {\\varphi_b}_{j-}) \\right) - \\varphi_b \\left( \\sum_{j \\in N_{X}} k_j ({\\varphi_a}_{j+} - {\\varphi_a}_{j-}) \\right) \\right]\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-44\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nJ_y = \\frac{e}{m_0} \\left[ \\varphi_a \\left( \\sum_{j \\in N_{Y}} k_j ({\\varphi_b}_{j+} - {\\varphi_b}_{j-}) \\right) - \\varphi_b \\left( \\sum_{j \\in N_{Y}} k_j ({\\varphi_a}_{j+} - {\\varphi_a}_{j-}) \\right) \\right]\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-45\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nJ_z = \\frac{e}{m_0} \\left[ \\varphi_a \\left( \\sum_{j \\in N_{Z}} k_j ({\\varphi_b}_{j+} - {\\varphi_b}_{j-}) \\right) - \\varphi_b \\left( \\sum_{j \\in N_{Z}} k_j ({\\varphi_a}_{j+} - {\\varphi_a}_{j-}) \\right) \\right]\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-46\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAgain, it does not look as simple as before, but nevertheless it is necessary to have a current to be able to drive the magnetic field in an manner consistent with known physics. Specifically, the electromagnetic field equations require both \u003cspan class=\"math inline\"\u003e\\(\\rho\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(\\vec{J}\\)\u003c/span\u003e values as their sources (see \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e). In addition, this gradient operation is necessary for several other computations in our model, so, like the laplacian, it can be thought of as one of just a few basic operations that take place over the neighborhood of cells.\u003c/p\u003e\u003ch2 id=\"minimal-coupling-of-charge-waves-with-electromagnetic-fields\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eMinimal Coupling of Charge Waves with Electromagnetic Fields\u003c/h2\u003e\u003cp id=\"text-48\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAt this point, we have a charged wave that can generate an electromagnetic field according to the charge density \u003cspan class=\"math inline\"\u003e\\(\\rho\\)\u003c/span\u003e and current density \u003cspan class=\"math inline\"\u003e\\(\\vec{J}\\)\u003c/span\u003e, and we know how this electromagnetic field propagates according to wave equations. Now, we need to have that electromagnetic field interact with the charge wave to produce actual forces on our model. This occurs by introducing new terms into the complex KG wave equation, which, intuitively speaking, act as external driving forces on this charge wave, in much the same way that the charge and current act as driving forces on the electromagnetic wave equations.\u003c/p\u003e\u003cp id=\"text-49\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn the electromagnetic field equations, the driving force from charge \u003cspan class=\"math inline\"\u003e\\(\\rho\\)\u003c/span\u003e adds into the second-order temporal derivative \u003cspan class=\"math inline\"\u003e\\(\\frac{\\partial^2 {}}{\\partial t^2}\\)\u003c/span\u003e (\u003ca href=\"maxwell#eq_scalar-pot-chg\" target=\"_blank\"\u003eEq 2\u003c/a\u003e in \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e):\u003c/p\u003e\u003cp id=\"text-50\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2 {A_0}}{\\partial t^2} = \\nabla^2 A_0 + \\frac{1}{\\epsilon_0} \\rho\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-51\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eHowever, in Schrödinger’s equation, external forces enter as a potential (\u003cspan class=\"math inline\"\u003e\\(V\\)\u003c/span\u003e), in the first-order derivative \u003cspan class=\"math inline\"\u003e\\(\\frac{\\partial {}}{\\partial t}\\)\u003c/span\u003e:\u003c/p\u003e\u003cp id=\"text-52\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\ni \\hbar \\frac{\\partial {\\phi}}{\\partial t} = \\frac{\\hbar^2}{2m} \\nabla^2 \\phi + V \\phi\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-53\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis makes sense, because force is the derivative of a potential, so potential is a first-order factor, and force is a second-order factor.\u003c/p\u003e\u003cp id=\"text-54\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOur KG (Klein-Gordon) charge wave equation is a second-order equation, expressed in terms of \u003cspan class=\"math inline\"\u003e\\(\\frac{\\partial^2 {}}{\\partial t^2}\\)\u003c/span\u003e, and therefore we need to include external driving forces, not potentials. However, for various reasons, it is necessary to derive such an equation starting from the potential. To do this, we can re-derive a second-order wave equation by replacing the first-order derivative with the first-order derivative with the external driving potential:\u003c/p\u003e\u003cp id=\"text-55\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\partial^\\mu \\rightarrow \\partial^\\mu - \\frac{e}{c} {A}^\\mu\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-56\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWe can then substitute this first-order four-vector derivative into the four-vector version of the KG wave equation (in natural units), which is (\u003ca href=\"klein-gordon#eq_kg-4vec\" target=\"_blank\"\u003eEq 4\u003c/a\u003e):\u003c/p\u003e\u003cp id=\"text-57\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\partial_\\mu \\partial^\\mu \\phi = - m_0^2 \\phi\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-58\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ebecause, as we’ve noted before:\u003c/p\u003e\u003cp id=\"text-59\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\partial_\\mu \\partial^\\mu = \\frac{\\partial^2 {}}{\\partial t^2} - \\nabla^2\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-60\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSo the compact form of the KG wave equation with minimal coupling is therefore:\u003c/p\u003e\u003cp id=\"text-61\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\left( \\partial_\\mu - e {A}_\\mu \\right) \\left(\\partial^\\mu - e {A}^\\mu \\right) \\phi = -m_0^2 \\phi\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-62\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eTo get all the units right, and perhaps add some conceptual clarity, we can do the same thing with the four-momentum version of the wave equation, which is:\u003c/p\u003e\u003cp id=\"text-63\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\hat{p}^\\mu \\hat{p}_\\mu \\phi = m_0^2 c^2 \\phi\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-64\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhere the momentum operator is essentially just the four-derivative, plus the pesky \u003cspan class=\"math inline\"\u003e\\(i\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(\\hbar\\)\u003c/span\u003e factors:\u003c/p\u003e\u003cp id=\"text-65\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\hat{p}^\\mu = i \\hbar \\partial^\\mu\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-66\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSo now we can say that the electromagnetic potential pushes directly on the momentum of the wave-particle:\u003c/p\u003e\u003cp id=\"text-67\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\hat{p}^\\mu \\rightarrow i \\hbar \\partial^\\mu - \\frac{e}{c} {A}^\\mu\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-68\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eand the same goes for the covariant forms:\u003c/p\u003e\u003cp id=\"text-69\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\hat{p}_\\mu \\rightarrow i \\hbar \\partial_\\mu - \\frac{e}{c} {A}_\\mu\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-70\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis notion of the potential pushing directly on the momentum of the particle hopefully makes good intuitive sense, even if all the associated mathematics does not. In any case, the resulting KG wave equation becomes:\u003c/p\u003e\u003cp id=\"text-71\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\left({\\hat{p}^\\mu} - \\frac{e}{c}{A}^\\mu \\right) \\left({\\hat{p}_\\mu} - \\frac{e}{c}{A}_\\mu \\right)\\phi = m_0^2 c^2 \\phi\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-72\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhich can also just be written more compactly as a squared expression:\u003c/p\u003e\u003cp id=\"text-73\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\left(i \\hbar \\partial_\\mu - \\frac{e}{c}{A}_\\mu \\right)^2\\phi = m_0^2 c^2 \\phi\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-74\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWhen this equation is crunched through to produce separate time and space derivatives (as detailed in a subsequent section), we get a standard second-order wave update equation plus a few extra terms (in natural units):\u003c/p\u003e\u003cp id=\"text-75\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2 {\\phi}}{\\partial t^2} = \\nabla^2 \\phi - m_0^2\\phi - 2 i e \\left(A_0 \\frac{\\partial\n {\\phi}}{\\partial t} + \\vec{A} \\cdot \\vec{\\nabla} \\phi \\right) + e^2 \\phi \\left(A_0^2 - \\vec{A}^2\\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-76\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIncluding all of the various constants, it is:\u003c/p\u003e\u003cp id=\"text-77\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2 {\\phi}}{\\partial t^2} = c^2 \\left( \\nabla^2 - \\frac{m_0^2 c^2}{\\hbar^2} \\right) \\phi -\n \\frac{2 i e}{\\hbar} \\left(A_0 \\frac{\\partial {\\phi}}{\\partial t} + c \\vec{A} \\cdot \\vec{\\nabla} \\phi \\right) + \\frac{e^2 \\phi}{\\hbar^2} \\left(A_0^2 - \\vec{A}^2\\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-78\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis equation amounts to the basic KG wave equation, plus terms that involve the interaction between the charge wave and the electromagnetic field potentials \u003cspan class=\"math inline\"\u003e\\(A_0\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(\\vec{A}\\)\u003c/span\u003e. For example, in the second term of this equation, the vector potential \u003cspan class=\"math inline\"\u003e\\(\\vec{A}\\)\u003c/span\u003e “pushes” on the gradient of the wave function \u003cspan class=\"math inline\"\u003e\\(\\vec{\\nabla} \\phi\\)\u003c/span\u003e, and the scalar potential \u003cspan class=\"math inline\"\u003e\\(A_0\\)\u003c/span\u003e pushes on the temporal derivative \u003cspan class=\"math inline\"\u003e\\(\\frac{\\partial {\\phi}}{\\partial t}\\)\u003c/span\u003e. Notice that these interactions are all first-order, in terms of the potentials and first-order derivatives of the wave equations. The second-order electromagnetic force fields \u003cspan class=\"math inline\"\u003e\\(\\vec{E}\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(\\vec{B}\\)\u003c/span\u003e do not appear at all! This is despite the fact that these are widely regarded as the primary observables of electromagnetic force. Also, the electromagnetic terms introduce a coupling between the two components of the complex variable \u003cspan class=\"math inline\"\u003e\\(\\phi\\)\u003c/span\u003e, because of the presence of the imaginary number \u003cspan class=\"math inline\"\u003e\\(i\\)\u003c/span\u003e in this term. Therefore, it is only the “free” particle (without electromagnetic forces) that has the completely uncoupled scalar components.\u003c/p\u003e\u003cp id=\"text-79\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eBefore we pause to reflect more, we need to take two more steps. First, because \u003cspan class=\"math inline\"\u003e\\(\\phi\\)\u003c/span\u003e is a complex variable, we need to further compute the separate real-valued update equations for \u003cspan class=\"math inline\"\u003e\\(\\varphi_a\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(\\varphi_b\\)\u003c/span\u003e to simulate this in our model. Second, we need to update the charge and current equations for this new version of the KG wave equations.\u003c/p\u003e\u003cp id=\"text-80\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe resulting CA model update equations (including all the relevant \u003cspan class=\"math inline\"\u003e\\(c\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(\\hbar\\)\u003c/span\u003e factors) are:\u003c/p\u003e\u003cp id=\"text-81\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\ddot \\varphi_a^{t+1} = c^2 \\left(\\nabla^2 \\varphi_a - \\frac{m_0^2c^2}{\\hbar^2} \\varphi_a \\right) + \\frac{2 e}{\\hbar} \\left(A_0 \\dot \\varphi_b + c \\vec{A} \\cdot \\vec{\\nabla} \\varphi_b \\right) + \\frac{e^2}{\\hbar^2} \\varphi_a \\left(A_0^2 - \\vec{A}^2 \\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-82\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eand:\u003c/p\u003e\u003cp id=\"text-83\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\ddot \\varphi_b^{t+1} = c^2 \\left(\\nabla^2 \\varphi_b - \\frac{m_0^2c^2}{\\hbar^2} \\varphi_b \\right) - \\frac{2 e}{\\hbar} \\left(A_0 \\dot \\varphi_a + c \\vec{A} \\cdot \\vec{\\nabla} \\varphi_a \\right) + \\frac{e^2}{\\hbar^2} \\varphi_b \\left(A_0^2 - \\vec{A}^2 \\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-84\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e(where for simplicity the right hand side variables are implicitly taken at time \u003cspan class=\"math inline\"\u003e\\(t\\)\u003c/span\u003e for cell \u003cspan class=\"math inline\"\u003e\\(i\\)\u003c/span\u003e, and the discrete versions of \u003cspan class=\"math inline\"\u003e\\(\\nabla^2\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(\\vec{\\nabla}\\)\u003c/span\u003e presented earlier are assumed). Note that we again need the first-order spatial gradient operator \u003cspan class=\"math inline\"\u003e\\(\\vec{\\nabla}\\)\u003c/span\u003e as a basic computation in our model, but otherwise all the variables are local to the system.\u003c/p\u003e\u003cp id=\"text-85\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe \u003ca href=\"conservation\" target=\"_blank\"\u003econserved\u003c/a\u003e charge and current values computed by this equation must also be updated. The coupling with the electromagnetic field has introduced additional factors here, which depend on the electromagnetic potentials \u003cspan class=\"math inline\"\u003e\\(A_0\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(\\vec{A}\\)\u003c/span\u003e. For the charge density \u003cspan class=\"math inline\"\u003e\\(\\rho\\)\u003c/span\u003e, we have:\u003c/p\u003e\u003cp id=\"text-86\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\rho = \\frac{i \\hbar e}{2m_0c^2} \\left( \\phi^* \\frac{\\partial {\\phi}}{\\partial t} - \\phi \\frac{\\partial {\\phi^*}}{\\partial t} \\right) - \\frac{e^2}{m_0 c^2} A_0 \\phi \\phi^*\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-87\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIf you compare with the original charge equation for the complex KG equation (\u003ca href=\"complex-kg#eq_kg-charge\" target=\"_blank\"\u003eEq 2\u003c/a\u003e), it is the same except for the last term. Similarly, the current density \u003cspan class=\"math inline\"\u003e\\(\\vec{J}\\)\u003c/span\u003e is:\u003c/p\u003e\u003cp id=\"text-88\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{J} \\equiv - \\frac{i \\hbar e}{2m_0} \\left( \\phi^* \\vec{\\nabla} \\phi - \\phi \\vec{\\nabla} \\phi^* \\right) - \\frac{e^2}{m_0 c} \\vec{A} \\phi \\phi^*\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-89\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eagain with an extra term at the end relative to \u003ca href=\"complex-kg#eq_kg-current\" target=\"_blank\"\u003eEq 3\u003c/a\u003e.\u003c/p\u003e\u003cp id=\"text-90\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThese equations need to be converted into computational expressions in terms of the separate complex components, as before:\u003c/p\u003e\u003cp id=\"text-91\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\rho = \\frac{\\hbar e}{m_0c^2} (\\varphi_b \\dot \\varphi_a - \\varphi_a \\dot \\varphi_b) - \\frac{e^2}{m_0 c^2} A_0 (\\varphi_a^2 + \\varphi_b^2)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-92\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{J} = \\frac{\\hbar e}{m_0} (\\varphi_a \\vec{\\nabla} \\varphi_b - \\varphi_b \\vec{\\nabla} \\varphi_a) - \\frac{e^2}{m_0c} \\vec{A} (\\varphi_a^2 + \\varphi_b^2)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-93\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAt this point, we have reached an important milestone — if you take the equations just presented above, this describes a particle as a distributed wave of charge that gets pushed around by the electromagnetic field potentials (\u003cspan class=\"math inline\"\u003e\\(A_0\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(\\vec{A}\\)\u003c/span\u003e). Furthermore, this wave of charge produces electromagnetic fields, in terms of charge and current densities \u003cspan class=\"math inline\"\u003e\\(\\rho\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(\\vec{J}\\)\u003c/span\u003e. Thus, we finally have a complete system of equations that can potentially simulate charged particles whizzing around and interacting with each other. In other words, we finally have the potential to make direct contact with observable physics! Indeed, you can explore the behavior of this system in the model, by using the Complex Coupled KG wave equations setting.\u003c/p\u003e\u003ch2 id=\"numerical-issues-with-coupling-symmetry-breaking\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eNumerical Issues with Coupling: Symmetry Breaking\u003c/h2\u003e\u003cp id=\"text-95\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWhen you actually simulate these equations on the computer, something very interesting (and initially distressing) happens — they blow up! As you run the equations over time in the presence of a fixed electromagnetic field, the total charge value, far from being a constant, increases steadily, and eventually you end up with numbers approaching infinity. This is not because the math is wrong (after very thorough checking!), but because of the coupling between the two elements of the complex variable that occurs in the update equation. Specifically, the update of \u003cspan class=\"math inline\"\u003e\\(\\varphi_a\\)\u003c/span\u003e depends on \u003cspan class=\"math inline\"\u003e\\(\\dot \\varphi_b\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(\\vec{\\nabla} \\varphi_b\\)\u003c/span\u003e, and vice-versa. This interdependency creates numerical instabilities when we simply substitute in the discrete computed values at each time step. In particular, because the change in \u003cspan class=\"math inline\"\u003e\\(\\varphi_a\\)\u003c/span\u003e depends on the change in \u003cspan class=\"math inline\"\u003e\\(\\varphi_b\\)\u003c/span\u003e, this cycle of dependency can get out of whack.\u003c/p\u003e\u003cp id=\"text-96\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIntuitively, the electromagnetic potentials drive a rotation through the \u003cspan class=\"math inline\"\u003e\\(\\varphi_a\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(\\varphi_b\\)\u003c/span\u003e variables, which is evident in the fact that they subtract from \u003cspan class=\"math inline\"\u003e\\(\\varphi_b\\)\u003c/span\u003e but add to \u003cspan class=\"math inline\"\u003e\\(\\varphi_a\\)\u003c/span\u003e — these opposite signs are the signature of a rotation (and incidentally are caused by the presence of the imaginary \u003cspan class=\"math inline\"\u003e\\(i\\)\u003c/span\u003e numbers in the equations). To the extent that the potentials are pushing the \u003cspan class=\"math inline\"\u003e\\(\\varphi_a\\)\u003c/span\u003e variable up, there should be an equal and opposite pushing of the \u003cspan class=\"math inline\"\u003e\\(\\varphi_b\\)\u003c/span\u003e variable down, causing the rotation. However, if the \u003cspan class=\"math inline\"\u003e\\(\\varphi_b\\)\u003c/span\u003e variable only has the “old” data from the previous time step about how much \u003cspan class=\"math inline\"\u003e\\(\\varphi_a\\)\u003c/span\u003e got pushed up, then it doesn’t compensate correctly in how much it gets pushed down. Thus, you end up with a “leak” in the system, where instead of rotating nicely in place, the system starts to fly out of control, spinning wider and wider circles each time.\u003c/p\u003e\u003cp id=\"text-97\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe solution to this problem is to \u003cem\u003ebreak the symmetry\u003c/em\u003e between \u003cspan class=\"math inline\"\u003e\\(\\varphi_a\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(\\varphi_b\\)\u003c/span\u003e in these update equations. Instead of updating each of them at the same time, based on the prior values of the other, we choose one variable (\u003cspan class=\"math inline\"\u003e\\(\\varphi_a\\)\u003c/span\u003e, arbitrarily) and update its values first. Then, when we compute \u003cspan class=\"math inline\"\u003e\\(\\varphi_b\\)\u003c/span\u003e, we use the \u003cem\u003ecurrent value\u003c/em\u003e of \u003cspan class=\"math inline\"\u003e\\(\\dot \\varphi_a\\)\u003c/span\u003e in the update equation for \u003cspan class=\"math inline\"\u003e\\(\\varphi_b\\)\u003c/span\u003e. This prevents the rotation between these variables from getting out of whack, and restores numerical stability to the system.\u003c/p\u003e\u003cp id=\"text-98\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003ccode\u003e\\begin{figure}\u003c/code\u003e\n\u003ccode\u003e \\centering\\includegraphics[height=2in]{fig.dirac_invr5_50.eps} \\caption{\\small Total charge for updating of the charge wave equation over 100,000 time steps in the presence of a fixed $1/r$ potential with magnitude .5, in a universe of $50^3$ cubes. The\u003c/code\u003e\n\u003ccode\u003e instantaneous total charge varies considerably over time, but the average across time is constant, demonstrating that total charge is conserved on average, but not at each moment. If no potential is present, then charge is identically conserved from one time step to the next.}\u003c/code\u003e\n\u003ccode\u003e \\label{fig.kg_invr5_50} \\end{figure}\u003c/code\u003e\u003c/p\u003e\u003cp id=\"text-99\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eHowever, if you run this equation in a static electromagnetic field, it is clear that the total amount of charge in the model at any one time changes over time (Figure~\\ref{fig.kg_invr5_50}). This is flies in the face of the fancy math that says that these equations conserve charge! Somewhat amazingly, however, if you run the system long enough, it becomes clear that the \u003cem\u003eaverage\u003c/em\u003e amount of charge never changes.\u003c/p\u003e\u003cp id=\"text-100\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOverall, we have now happened upon a very interesting situation. The breaking of symmetry between the two variables in the complex wave state, forced upon us by implementational considerations, actually fits at least qualitatively with a known and otherwise very puzzling property of physics. The weak force also breaks symmetry in a very similar way: there is a preferred direction of rotation in the weak force. Although it is not yet clear (to me at least) that this preferred rotational direction in the weak force maps identically onto this preferred rotation direction, it is nevertheless a tantalizing possibility. The weak force has been integrated with the electromagnetic force, as the \u003cem\u003eelectroweak\u003c/em\u003e force — it is possible that the effects described by the electroweak force correspond in some way to the oscillations in charge value that are observed in our model. We will return to these issues later.\u003c/p\u003e\u003cp id=\"text-101\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eMeanwhile, we nee need to introduce just a bit more complexity into our KG wave equation before we have a fully satisfactory model of a fundamental particle of nature: the electron (and its antiparticle, the positron). This extra bit of complexity extends the phenomenon of rotation that we’ve just been discussing, to account for the strange quantum mechanical property of \u003cstrong\u003espin\u003c/strong\u003e. The resulting equation goes by the name of the second-order Dirac equation. Once we have that, we will have a complete system that, if all the math is correct, should make direct and numerically accurate contact with observable phenomena!\u003c/p\u003e\u003cp id=\"text-102\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eBefore moving on to spin, there are two remaining loose ends for the present set of equations. First, there is an interesting interpretation of the way that the electromagnetic potential interacts with our charge wave, called \u003cstrong\u003elocal gauge invariance\u003c/strong\u003e, which provides a template for exploring the other two forces of nature: the weak and strong forces (which we will not cover further in this paper). Second, we have the actual mechanics of deriving the above equations.\u003c/p\u003e\u003ch2 id=\"local-gauge-invariance\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eLocal Gauge Invariance\u003c/h2\u003e\u003cp id=\"text-104\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIf you look at it in the right way, the electromagnetic field can be seen as a way of canceling out an extra degree of freedom present in the complex KG wave field equations. As discussed in \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e, the Lorenz gauge is an example of a situation where we introduced some extra constraints on the field variables, and this eliminated a degree of freedom in the electromagnetic equations, and also made them appear a lot simpler than they otherwise would.\u003c/p\u003e\u003cp id=\"text-105\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis notion of a \u003cem\u003egauge\u003c/em\u003e is very general: it just means that whenever you have some unconstrained values in your equations (i.e., values that can change without changing the observable results that you can measure in physics experiments), then you need to apply some kind of gauge to fix these variables. In the Lorenz gauge example, the extra degree of freedom comes from the fact that the observable variables are the force vectors \u003cspan class=\"math inline\"\u003e\\(\\vec{E}\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(\\vec{B}\\)\u003c/span\u003e, which are essentially derivatives of the underlying potentials \u003cspan class=\"math inline\"\u003e\\(A_0\\)\u003c/span\u003e, \u003cspan class=\"math inline\"\u003e\\(\\vec{A}\\)\u003c/span\u003e. Thus, the raw values of the potentials can be moved up or down, and it won’t change the slope of the fields, and therefore it won’t change the observable force vectors.\u003c/p\u003e\u003cp id=\"text-106\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eHowever, it is not clear exactly how to reconcile such an argument with the fact that the potentials appear directly in our coupling equations, and also are observable in terms of the Arahnov-Bohm effect, as discussed elsewhere.\u003c/p\u003e\u003cp id=\"text-107\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eNevertheless, here is the argument for the electromagnetic coupling being a form of local gauge invariance. If you just take our basic complex KG wave equation, you can get exactly the same overall behavior if you multiply the thing by a “phase transformation” (it is often said that gauge invariance should really be phase invariance) which is basically just a rotation along the complex axes. This is exactly the kind of rotation discussed above. The generic form of a rotation in complex numbers is to multiply by an exponential term:\u003c/p\u003e\u003cp id=\"text-108\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\phi(x) \\rightarrow \\exp \\left(\\frac{ie}{\\hbar c} \\chi \\right) \\phi(x)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-109\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWhere the \u003cspan class=\"math inline\"\u003e\\(\\chi\\)\u003c/span\u003e term is the amount that you’re rotating (the rest are just convenient constants for getting \u003cspan class=\"math inline\"\u003e\\(\\chi\\)\u003c/span\u003e into the right units) — think of it as some number of degrees of rotating the underlying \u003cspan class=\"math inline\"\u003e\\(\\varphi_a\\)\u003c/span\u003e value into \u003cspan class=\"math inline\"\u003e\\(\\varphi_b\\)\u003c/span\u003e (and vice-versa) for the complex number \u003cspan class=\"math inline\"\u003e\\(\\phi\\)\u003c/span\u003e.\u003c/p\u003e\u003cp id=\"text-110\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIf \u003cspan class=\"math inline\"\u003e\\(\\chi\\)\u003c/span\u003e is independent of location (\u003cspan class=\"math inline\"\u003e\\(x\\)\u003c/span\u003e), then it is just a constant and nothing happens. This is a \u003cem\u003eglobal\u003c/em\u003e gauge/phase transformation, and it is not very interesting. However, if \u003cspan class=\"math inline\"\u003e\\(\\chi\\)\u003c/span\u003e is now itself a function of location (i.e., \u003cspan class=\"math inline\"\u003e\\(\\chi(x)\\)\u003c/span\u003e ), this is a \u003cem\u003elocal\u003c/em\u003e gauge transformation, and this is where the electromagnetic coupling comes in. If you have such a local variable, and you take the derivative of the resulting overall system that includes this locally-varying thing, you get this extra term for the derivative of \u003cspan class=\"math inline\"\u003e\\(\\chi(x)\\)\u003c/span\u003e with respect to x:\u003c/p\u003e\u003cp id=\"text-111\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\partial_\\mu \\phi(x) \\rightarrow \\exp \\left( \\frac{ie}{\\hbar c} \\chi(x) \\right) \\left(\\partial_\\mu + \\frac{ie}{\\hbar c} \\partial_\\mu \\chi(x) \\right) \\phi(x)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-112\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSo now your nice wave equation is a mess, and it varies from one place to another as a function of this \u003cspan class=\"math inline\"\u003e\\(\\partial_\\mu \\chi(x)\\)\u003c/span\u003e term. So here is the trick: you basically just add this annoying term into the overall EM potential field (which is OK because such additions do not change the gradients and therefore do not affect observable EM field vectors):\u003c/p\u003e\u003cp id=\"text-113\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n{A}_\\mu(x) \\rightarrow {A}_\\mu(x) + \\partial_\\mu \\chi(x)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-114\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAnd then you just subtract this whole thing back out from your messy equation, and this gives you something just slightly less messy:\u003c/p\u003e\u003cp id=\"text-115\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\left(i \\hbar \\partial_\\mu - \\frac{e}{c} {A}_\\mu\\right)\\phi\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-116\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSo, this ends up being the same thing as the minimal coupling described earlier. Somehow, this whole process seems like a rather contrived way to end up with something that already made quite a bit of sense before hand. However, as noted earlier, the true payoff of such a procedure appears to come in addressing the weak and strong forces, which we leave for a future refinement of the model.\u003c/p\u003e\u003ch2 id=\"summary\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eSummary\u003c/h2\u003e\u003cp id=\"text-118\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWe now have a full electrodynamic system with bidirectional interactions between a wave of charge and the electromagnetic force field. The wave equation remains at the core of both the charge wave and the electromagnetic field equations, but we did have to add in a few extra first-order spatial gradient computations here and there. All of this was necessary to get our electrodynamics working according to known physical laws.\u003c/p\u003e\u003cp id=\"text-119\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIt may not be quite as elegant as our simplest system of a pure wave field for forces, and a simple scalar KG wave equation for our particle, but elegance cannot override physical facts. Still, all the broad implications of those core wave equations (special relativity, Newtonian-like dynamics, quantum physics, etc) all hold for our more elaborated equations.\u003c/p\u003e\u003cp id=\"text-120\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ePerhaps the most surprising development here is that we had to break the symmetry between the two components of the complex state variable, and introduce a preferred direction of rotation, in order to avoid numerical instability. This numerical instability results when variables are coupled, and the rotation through the \u003cspan class=\"math inline\"\u003e\\(\\varphi_a\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(\\varphi_b\\)\u003c/span\u003e variables produced by the interaction with the electromagnetic field is manifest as such a coupling. This break in symmetry and preferred direction of rotation bears a tantalizing resemblance to features of the weak force, suggesting a possible explanation for an otherwise very strange aspect of nature.\u003c/p\u003e\u003cp id=\"text-121\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOur overall model at this point consists of a small handful of locally computable equations, which can be readily simulated on a computer. As these equations play out, they deterministically, locally, and automatically generate physics that should be largely consistent with what we know about the world. However, as we noted before, our equations are missing one critical piece, which is \u003cem\u003espin\u003c/em\u003e.\u003c/p\u003e\u003cp id=\"text-122\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eInterestingly, the introduction of spin in the \u003ca href=\"dirac\" target=\"_blank\"\u003eDirac\u003c/a\u003e function creates a parity in the number of variables participating in the electromagnetic field equations (four) with those in the particle charge wave, which is currently two, but will now double to four. Overall, the elegance of having four variables each in two systems of interacting wave equations, which unfold in a four-dimensional space-time, seems suspiciously neat.\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"cellular-automaton\"\u003e\u003csvg id=\"icon\" 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style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Complex numbers","URL":"complex-numbers","Title":"Complex numbers","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{"figure":["figure_complex"]},"Description":"The symbol \u003cspan class=\"math inline\"\u003e\\(\\phi\\)\u003c/span\u003e (another variant of the Greek symbol “phi”, like \u003cspan class=\"math inline\"\u003e\\(\\varphi\\)\u003c/span\u003e) is used to represent a complex-valued state variable:","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eComplex numbers\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-4\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"figure_complex\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_complex\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 1:\u003c/b\u003e \u003cp\u003eComplex numbers are just a way of representing two real values with one number, where these two values are aligned along two separate orthogonal dimensions. The imaginary number \u003cspan class=\"math inline\"\u003e\\(i\\)\u003c/span\u003e, where \u003cspan class=\"math inline\"\u003e\\(i^2 = -1\\)\u003c/span\u003e, is what keeps these two values orthogonal — the first value \u003cspan class=\"math inline\"\u003e\\(a\\)\u003c/span\u003e is along the real axis, and the second value \u003cspan class=\"math inline\"\u003e\\(b\\)\u003c/span\u003e is along the imaginary axis. The complex conjugate, \u003cspan class=\"math inline\"\u003e\\(c^*\\)\u003c/span\u003e, is simply subtracting the imaginary part instead of adding it (i.e., it represents a reflection along the imaginary dimension). Multiplying \u003cspan class=\"math inline\"\u003e\\(c c^*\\)\u003c/span\u003e gives the squared magnitude of the vector, which is a single real-valued scalar number. It is the (squared) length of the hypotenuse of the vector.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe symbol \u003cspan class=\"math inline\"\u003e\\(\\phi\\)\u003c/span\u003e (another variant of the Greek symbol “phi”, like \u003cspan class=\"math inline\"\u003e\\(\\varphi\\)\u003c/span\u003e) is used to represent a complex-valued state variable:\u003c/p\u003e\u003cp id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\phi = a + i b\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n= \\varphi_a + i \\varphi_b\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSo, \u003cspan class=\"math inline\"\u003e\\(\\phi\\)\u003c/span\u003e is composed of two separate real-valued numbers, designated \u003cspan class=\"math inline\"\u003e\\(a\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(b\\)\u003c/span\u003e (or \u003cspan class=\"math inline\"\u003e\\(\\varphi_a\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(\\varphi_b\\)\u003c/span\u003e, to indicate that they are scalar state variables). A complex number is really just a way of representing two separate real valued numbers, aligned along orthogonal dimensions, in an efficient and compact manner (\u003ca href=\"complex-numbers#figure_complex\" target=\"_blank\"\u003eFigure 1\u003c/a\u003e). It is essential to appreciate that, despite the presence of the imaginary number \u003cspan class=\"math inline\"\u003e\\(i\\)\u003c/span\u003e (where \u003cspan class=\"math inline\"\u003e\\(i^2 = -1\\)\u003c/span\u003e or \u003cspan class=\"math inline\"\u003e\\(i = \\sqrt{-1}\\)\u003c/span\u003e), \u003cem\u003eall you ever really have is two real-valued numbers.\u003c/em\u003e There is nothing “imaginary” or mysterious or spooky about the second number in a complex number: all the \u003cspan class=\"math inline\"\u003e\\(i\\)\u003c/span\u003e does is keep these two values separate from each other.\u003c/p\u003e\u003cp id=\"text-6\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn the end, we will deconstruct all of our complex numbers into their real-valued components, and write purely real-valued expressions that determine their update rules. These expressions will be more complicated than the ones using complex numbers, but they are required for actually implementing the equations on the computer, and they also provide a more explicit and obvious indication of exactly what drives each value.\u003c/p\u003e\u003cp id=\"text-7\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eHere’s a few interesting facts about complex numbers:\u003c/p\u003e\u003cp id=\"text-8\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eTo do algebra on them, you just have to remember to \u003cem\u003ekeep the real-values sorted separately from the imaginary ones,\u003c/em\u003e but otherwise treat them just like a pair of numbers:\u003c/p\u003e\u003cp id=\"text-9\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\ny = a + ib\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-10\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nz = c + id\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-11\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\ny + z = (a + c) + i(b + d)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-12\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\ny z = a c + i a d + i b c - b d\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-13\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n= (ac - bd) + i (ad + bc)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-14\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eNotice that this multiplication rule is the same as \u003cspan class=\"math inline\"\u003e\\((a + b)(c + d) = ac + ad + bc + bd\\)\u003c/span\u003e, where you just multiply everything through, except that you end up with these \u003cspan class=\"math inline\"\u003e\\(i\\)\u003c/span\u003e terms crossing over, and when you multiply \u003cspan class=\"math inline\"\u003e\\(ib\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(id\\)\u003c/span\u003e, you end up with \u003cspan class=\"math inline\"\u003e\\(i^2bd\\)\u003c/span\u003e, at which point the \u003cspan class=\"math inline\"\u003e\\(i^2\\)\u003c/span\u003e disappears into a \u003cspan class=\"math inline\"\u003e\\(-1\\)\u003c/span\u003e (i.e., it crosses over from the imaginary world into the real one).\u003c/p\u003e\u003cp id=\"text-15\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAs you should expect from \u003ca href=\"complex-numbers#figure_complex\" target=\"_blank\"\u003eFigure 1\u003c/a\u003e, adding two complex numbers is like adding two vectors, and multiplication is just like multiplying vectors. Complex numbers really are just a compact way of writing vectors!\u003c/p\u003e\u003cp id=\"text-16\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eMultiplication by \u003cspan class=\"math inline\"\u003e\\(i\\)\u003c/span\u003e: If you multiply a complex number by \u003cspan class=\"math inline\"\u003e\\(i\\)\u003c/span\u003e, then you basically switch the real and imaginary parts: the real moves to the imaginary position, and the imaginary becomes real:\u003c/p\u003e\u003cp id=\"text-17\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\ni(a + ib) = ia - b = -b + ia\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-18\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eGeometrically, this is equivalent to rotating a vector by 90 degrees! If you do this four times, you’ll end up back where you started (as you would expect by doing a 360).\u003c/p\u003e\u003cp id=\"text-19\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe \u003cstrong\u003ecomplex conjugate\u003c/strong\u003e of a complex number is just that number with imaginary dimension inverted:\u003c/p\u003e\u003cp id=\"text-20\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\ny^* = a - i b\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-21\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe primary use of such a thing is to find the magnitude of a complex number (i.e., the length of the vector that it represents), as:\u003c/p\u003e\u003cp id=\"text-22\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n|y|^2 = y y^* = (a + i b)(a - i b)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-23\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n= a^2 - iab + iab + b^2\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-24\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n= a^2 + b^2\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-25\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis should be recognizable as simply the pythagorean theorem for the squared length of the hypotenuse of a right triangle (\u003cspan class=\"math inline\"\u003e\\(a^2 + b^2 = c^2\\)\u003c/span\u003e). Again, complex numbers have no mystery: they just represent a two-valued vector.\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"complex-kg\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"m274-450 248 248-42 42-320-320 320-320 42 42-248 248h526v60H274Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003ePrevious\u003c/p\u003e\u003c/a\u003e\u003cstretch id=\"content-buttons-go-167\" style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1ch;min-height:1em;font-weight:thin;text-align:start\"\u003e\u003c/stretch\u003e\u003ca id=\"content-buttons-go-168\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"configuration-space\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M686-450H160v-60h526L438-758l42-42 320 320-320 320-42-42 248-248Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Configuration space","URL":"configuration-space","Title":"Configuration space","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{},"Description":"\u003cstrong\u003eConfiguration space\u003c/strong\u003e is a critical but perhaps generally underappreciated element of standard quantum mechanics, in most of its various formulations (e.g., in the \u003ca href=\"hilbert-space\" target=\"_blank\"\u003eHilbert space\u003c/a\u003e formulation). It is the space defined by the \u003cstrong\u003emultiparticle\u003c/strong\u003e configuration of all the elements of relevance to a given experimental setup being analyzed.","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eConfiguration space\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-5\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cstrong\u003eConfiguration space\u003c/strong\u003e is a critical but perhaps generally underappreciated element of standard quantum mechanics, in most of its various formulations (e.g., in the \u003ca href=\"hilbert-space\" target=\"_blank\"\u003eHilbert space\u003c/a\u003e formulation). It is the space defined by the \u003cstrong\u003emultiparticle\u003c/strong\u003e configuration of all the elements of relevance to a given experimental setup being analyzed.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eBecause it describes the \u003cem\u003econfiguration\u003c/em\u003e of these elements, it is \u003cstrong\u003eexponential\u003c/strong\u003e in size, with a different space corresponding to each combination of such elements, and manifestly non-local. Thus, it is an entirely implausible, highly problematic element of standard quantum mechanical approaches, including the existing \u003ca href=\"pilot-wave\" target=\"_blank\"\u003epilot-wave\u003c/a\u003e models.\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe need for configuration space at a mathematically deep level arises because the equations being used are \u003cem\u003elinear\u003c/em\u003e, so they cannot represent any kind of actual interaction among different particles. Without configuration space, every particle would fully superpose on every other particle — they would just slip on past each other. This is in fact how \u003cem\u003ebosons\u003c/em\u003e (e.g., photons) behave, but not how \u003cem\u003efermions\u003c/em\u003e like electrons behave.\u003c/p\u003e\u003cp id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe \u003ca href=\"pilot-wave\" target=\"_blank\"\u003epilot-wave\u003c/a\u003e approach has been (perhaps unfairly) criticized for using configuration space, because it posits that the wave function is actually a “real” thing, thus exposing the implausibility of this otherwise purely \u003ca href=\"tools-vs-models\" target=\"_blank\"\u003ecalculational tool\u003c/a\u003e. See \u003ca href=\"ref://NorsenMarianOriols15\" target=\"_blank\"\u003eNorsen et al., 2015\u003c/a\u003e for an analysis of the contributions of configuration space to the pilot-wave results. They concluded that indeed the configuration space contains a large amount of “redundant” information, and that even the simplest approximation for the inter-particle interaction terms does a reasonable (yet imperfect) job of capturing the behavior of the full configuration-space model. Exploration of higher-order terms in this approximation are ongoing (\u003ca href=\"ref://Norsen22\" target=\"_blank\"\u003eNorsen, 2022\u003c/a\u003e).\u003c/p\u003e\u003cp id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIf the underlying dynamics of the system are \u003cem\u003enonlinear\u003c/em\u003e, and in particular involve interactions between \u003ca href=\"stochastic-particles\" target=\"_blank\"\u003estochastic particles\u003c/a\u003e and wave functions, then it is possible that these nonlinear interactions end up producing all of the relevant dynamics that are otherwise captured via the configuration space calculational tool. This is the approach taken here.\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"complex-numbers\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"m274-450 248 248-42 42-320-320 320-320 42 42-248 248h526v60H274Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003ePrevious\u003c/p\u003e\u003c/a\u003e\u003cstretch id=\"content-buttons-go-167\" style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1ch;min-height:1em;font-weight:thin;text-align:start\"\u003e\u003c/stretch\u003e\u003ca id=\"content-buttons-go-168\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"conservation\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M686-450H160v-60h526L438-758l42-42 320 320-320 320-42-42 248-248Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Conservation","URL":"conservation","Title":"Conservation","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{"eq":["eq_conserve-int","eq_schrodinger","eq_charge-current-4vec"]},"Description":"A critical property of quantum wave functions is that they exhibit strict \u003cstrong\u003econservation\u003c/strong\u003e of some overall quantity over time. In the case of the \u003ca href=\"schrodinger\" target=\"_blank\"\u003eSchrodinger\u003c/a\u003e wave function, this quantity is interpreted as the overall \u003cstrong\u003eprobability\u003c/strong\u003e of finding a particle in a given location, according to the standard \u003ca href=\"copenhagen\" target=\"_blank\"\u003eCopenhagen\u003c/a\u003e interpretation. For the \u003ca href=\"klein-gordon\" target=\"_blank\"\u003eKlein-Gordon\u003c/a\u003e and \u003ca href=\"dirac\" target=\"_blank\"\u003eDirac\u003c/a\u003e functions, this quantity makes more sense as the \u003cstrong\u003echarge\u003c/strong\u003e of an associated particle such as an \u003ca href=\"electron\" target=\"_blank\"\u003eelectron\u003c/a\u003e, because the conserved quantity can be either positive or negative valued, just like charge (i.e., the anti-particle of the electron, the \u003cem\u003epositron\u003c/em\u003e, has positive charge).","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eConservation\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-6\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eA critical property of quantum wave functions is that they exhibit strict \u003cstrong\u003econservation\u003c/strong\u003e of some overall quantity over time. In the case of the \u003ca href=\"schrodinger\" target=\"_blank\"\u003eSchrodinger\u003c/a\u003e wave function, this quantity is interpreted as the overall \u003cstrong\u003eprobability\u003c/strong\u003e of finding a particle in a given location, according to the standard \u003ca href=\"copenhagen\" target=\"_blank\"\u003eCopenhagen\u003c/a\u003e interpretation. For the \u003ca href=\"klein-gordon\" target=\"_blank\"\u003eKlein-Gordon\u003c/a\u003e and \u003ca href=\"dirac\" target=\"_blank\"\u003eDirac\u003c/a\u003e functions, this quantity makes more sense as the \u003cstrong\u003echarge\u003c/strong\u003e of an associated particle such as an \u003ca href=\"electron\" target=\"_blank\"\u003eelectron\u003c/a\u003e, because the conserved quantity can be either positive or negative valued, just like charge (i.e., the anti-particle of the electron, the \u003cem\u003epositron\u003c/em\u003e, has positive charge).\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eHere, we derive the relevant mathematical properties of this conserved quantity, which depends in general on the use of \u003ca href=\"complex-numbers\" target=\"_blank\"\u003ecomplex numbers\u003c/a\u003e. The essential, intuitive reason for this is that the conserved quantity is like the \u003cstrong\u003ehypotenuse\u003c/strong\u003e of a triangle, i.e., the \u003cstrong\u003eradius\u003c/strong\u003e on a circle that the state of the system is continuously rotating through. If you only have one number, then that number will oscillate like the sine or cosine as the system oscillates. If you instead have the two numbers within a complex number, then you can compute this hypotenuse / radius as the square of these two numbers, which is what the \u003cstrong\u003ecomplex conjugate\u003c/strong\u003e function computes:\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n|y|^2 = y y^* = (a + i b)(a - i b) = a^2 + b^2\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn addition to computing the total conserved value, we are also interested in computing the local \u003cstrong\u003edensity\u003c/strong\u003e and \u003cstrong\u003ecurrent\u003c/strong\u003e flow of this conserved value, through the use of a \u003cstrong\u003econtinuity equation\u003c/strong\u003e, which in the case of a conserved charge value then allows one to couple to the electromagnetic wave functions (\u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e).\u003c/p\u003e\u003cp id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eStarting at the most general level, the mathematical definition of a conserved quantity is that the sum total (i.e., integral) of its value across all of space does not change. For the case of a complex-valued wave state \u003cspan class=\"math inline\"\u003e\\(\\phi\\)\u003c/span\u003e, this is the complex conjugate: \u003cspan class=\"math inline\"\u003e\\(\\phi \\phi^*\\)\u003c/span\u003e. Therefore, the appropriate integral is:\u003c/p\u003e\u003cp id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\int_{-\\infty}^{+\\infty} \\phi(t,\\vec{x}) \\phi^*(t,\\vec{x}) dx\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-6\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhere we have expressed the state value as a continuous function of both time and spatial coordinates, which we subsequently drop for convenience.\u003c/p\u003e\u003cp id=\"text-7\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eTo determine the conserved quantity, we set the rate of change of this integral to zero, which means that its value cannot change over time:\u003c/p\u003e\u003cp id=\"text-8\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial }{\\partial t}\\left[\\int_{-\\infty}^{+\\infty} \\phi \\phi^* dx\\right] = 0\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-9\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eby propagating the temporal derivative into the integral, you get:\u003c/p\u003e\u003cdiv id=\"inline-container-11\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_conserve-int\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_conserve-int\"\u003e\u003cb\u003eEq 1:\u003c/b\u003e Conserved value\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-12\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\int_{-\\infty}^{+\\infty} \\left( \\frac{\\partial \\phi^*}{\\partial t} \\phi + \\phi^* \\frac{\\partial \\phi}{\\partial t} \\right) dx = 0\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-13\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFor Schrödinger’s equation, we have a definition of this first-order time differential directly within the wave function itself:\u003c/p\u003e\u003cdiv id=\"inline-container-15\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_schrodinger\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_schrodinger\"\u003e\u003cb\u003eEq 2:\u003c/b\u003e Schrödinger's equation\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-16\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\ni \\hbar \\frac{\\partial {\\phi}}{\\partial t} = -\\frac{\\hbar^2}{2 m_0} \\nabla^2 \\phi + V \\phi\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-17\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eTherefore, the right-hand-side of this equation can be substituted directly, resulting in this definition of a \u003cstrong\u003eprobability gradient\u003c/strong\u003e, which is the directional (vector) flow of the conserved probability across space and time:\u003c/p\u003e\u003cp id=\"text-18\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{J} \\equiv - \\frac{i \\hbar}{2 m_0} \\left( \\phi^* \\vec{\\nabla} \\phi - \\phi \\vec{\\nabla} \\phi^* \\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-19\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAnother more general route for computing this probability gradient is to start with the \u003cstrong\u003econtinuity equation\u003c/strong\u003e in terms of the \u003cem\u003edensity\u003c/em\u003e \u003cspan class=\"math inline\"\u003e\\(\\rho\\)\u003c/span\u003e and \u003cem\u003ecurrent\u003c/em\u003e \u003cspan class=\"math inline\"\u003e\\(\\vec{J}\\)\u003c/span\u003e:\u003c/p\u003e\u003cp id=\"text-20\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial \\rho}{\\partial t} + \\vec{\\nabla} \\cdot \\vec{J} = 0\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-21\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial \\rho}{\\partial t} = - \\vec{\\nabla} \\cdot \\vec{J}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-22\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhere \u003cspan class=\"math inline\"\u003e\\(\\vec{\\nabla} \\cdot\\)\u003c/span\u003e is the \u003cem\u003edivergence\u003c/em\u003e of a vector quantity:\u003c/p\u003e\u003cp id=\"text-23\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{\\nabla} \\cdot \\vec{J} \\equiv \\frac{\\partial J_x}{\\partial x} + \\frac{\\partial J_y}{\\partial y} + \\frac{\\partial J_z}{\\partial z}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-24\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e(this is just the sum of the spatial derivatives along each spatial direction). We cover divergence in greater detail in \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e — it represents the amount of new “stuff” accumulating in a given region of space from the flow in from its neighbors. If the total amount of stuff is to remain constant, then this increment needs to be offset by a change in the amount of stuff in that region itself, which is \u003cspan class=\"math inline\"\u003e\\(\\frac{\\partial }{\\partial t}\\rho}\\)\u003c/span\u003e. This is what this equation captures.\u003c/p\u003e\u003cp id=\"text-25\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis continuity relationship can be expressed in a \u003ca href=\"four-vector\" target=\"_blank\"\u003efour-vector\u003c/a\u003e (space-time) derivative notation, in terms of a single four-vector charge / current variable \u003cspan class=\"math inline\"\u003e\\(J^{\\mu}\\)\u003c/span\u003e:\u003c/p\u003e\u003cp id=\"text-26\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\partial_{\\mu} J^{\\mu} = \\frac{\\partial J^0}{\\partial t} + \\frac{\\partial{J^1}}{\\partial{x}} + \\frac{\\partial{J^2}}{\\partial{y}} + \\frac{\\partial{J^3}}{\\partial{z}}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-27\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\partial_{\\mu} J^{\\mu} = \\frac{\\partial J^0}{\\partial t} + \\vec{\\nabla} \\cdot \\vec{J}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-28\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhere \u003cspan class=\"math inline\"\u003e\\(J^0 = c \\rho\\)\u003c/span\u003e (so, charge, like energy, is a \u003cem\u003etime-like\u003c/em\u003e quantity, whereas current is a space-like quantity). The continuity relationship in these terms is therefore:\u003c/p\u003e\u003cp id=\"text-29\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\partial_{\\mu} J^{\\mu} = 0\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-30\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis one expression succinctly captures the key idea that the time-like first element is trading-off against the three space-like elements to produce an overall conservation of current, very much in the same way that the basic wave equations involve a tradeoff between time and space derivatives.\u003c/p\u003e\u003cp id=\"text-31\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFrom here, we can go back to \u003ca href=\"conservation#eq_conserve-int\" target=\"_blank\"\u003eEq 1\u003c/a\u003e and establish the connection with the continuity equation, in the context of the KG wave function (\u003ca href=\"ref://Greiner00\" target=\"_blank\"\u003eGreiner, 2000\u003c/a\u003e; \u003ca href=\"ref://Gingrich04\" target=\"_blank\"\u003eGingrich, 2004\u003c/a\u003e), :\u003c/p\u003e\u003cp id=\"text-32\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\phi^* (\\partial_{\\mu} \\partial^{\\mu} + m_0^2) \\phi - \\phi (\\partial_{\\mu} \\partial^{\\mu} + m_0^2) \\phi^* = 0\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-33\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhere the KG wave equation can be written in four-vector notation as:\u003c/p\u003e\u003cp id=\"text-34\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n(\\partial_{\\mu} \\partial^{\\mu} + m_0^2) \\phi = 0\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-35\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ebecause two of these four-derivatives gives you:\u003c/p\u003e\u003cp id=\"text-36\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\partial_{\\mu} \\partial^{\\mu} = \\frac{\\partial^2}{\\partial t^2} - \\nabla^2\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-37\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThus, the starting expression amounts to multiplying the standard KG wave equation by the complex conjugate \u003cspan class=\"math inline\"\u003e\\(\\phi^*\\)\u003c/span\u003e, and subtracting the opposite configuration, which is the KG wave equation operating on the conjugate variable \u003cspan class=\"math inline\"\u003e\\(\\phi^*\\)\u003c/span\u003e, multiplied by the wave state \u003cspan class=\"math inline\"\u003e\\(\\phi\\)\u003c/span\u003e.\u003c/p\u003e\u003cp id=\"text-38\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIf we take the first half of this expression, it is:\u003c/p\u003e\u003cp id=\"text-39\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\phi^* (\\partial_{\\mu} \\partial^{\\mu} + m_0^2) \\phi\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-40\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\partial_{\\mu} (\\phi^* \\partial_{\\mu} \\phi) - (\\partial_{\\mu} \\phi^*)(\\partial^{\\mu} \\phi) + m_0^2 \\phi^* \\phi\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-41\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eand for the opposite configuration:\u003c/p\u003e\u003cp id=\"text-42\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\phi (\\partial_{\\mu} \\partial^{\\mu} + m_0^2) \\phi^*\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-43\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[ \n\\partial_{\\mu} (\\phi \\partial_{\\mu} \\phi^*) - (\\partial_{\\mu} \\phi)(\\partial^{\\mu} \\phi^*) + m_0^2 \\phi \\phi^*\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-44\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSo when you subtract them, the second and third terms in each expression are the same, and cancel out, leaving only the difference in the first terms:\u003c/p\u003e\u003cp id=\"text-45\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n(\\partial_{\\mu} \\phi^*)(\\partial^{\\mu} \\phi) - (\\partial_{\\mu} \\phi)(\\partial^{\\mu} \\phi^*) = 0\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-46\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\partial_{\\mu} (\\phi^* \\partial^{\\mu} \\phi - \\phi \\partial^{\\mu} \\phi^*) = 0\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-47\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eNow this is exactly what we were looking for, if we recognize that this is an expression where the four-derivative of something equals zero. That something must be the conserved four-current, \u003cspan class=\"math inline\"\u003e\\(J^{\\mu}\\)\u003c/span\u003e:\u003c/p\u003e\u003cp id=\"text-48\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\partial_{\\mu} J^{\\mu} = 0\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-49\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\partial_{\\mu} (\\phi^* \\partial^{\\mu} \\phi - \\phi \\partial^{\\mu} \\phi^*) = 0\n\\]\u003c/span\u003e\u003c/p\u003e\u003cdiv id=\"inline-container-51\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_charge-current-4vec\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_charge-current-4vec\"\u003e\u003cb\u003eEq 3:\u003c/b\u003e charge-current four vector\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-52\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[ \nJ^{\\mu} = \\phi^* \\partial^{\\mu} \\phi - \\phi \\partial^{\\mu} \\phi^*\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-53\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWe can then separate this into the charge (\u003cspan class=\"math inline\"\u003e\\(J^0\\)\u003c/span\u003e) and current components, as:\u003c/p\u003e\u003cp id=\"text-54\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n \\frac{J_0}{c} = \\rho \\equiv \\frac{i \\hbar e}{2m_0c^2} \\left( \\phi^* \\frac{\\partial \\phi}{\\partial t} - \\phi \\frac{\\partial \\phi^*}{\\partial t} \\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-55\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eand\u003c/p\u003e\u003cp id=\"text-56\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{J} \\equiv - \\frac{i \\hbar e}{2m_0} \\left( \\phi^* \\vec{\\nabla} \\phi - \\phi \\vec{\\nabla} \\phi^* \\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-57\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhere the extra conversion constants are inherited from the analogous expression for Schrödinger’s equation, and ensure that the KG equation reduces to it in the non-relativistic limit.\u003c/p\u003e\u003cp id=\"text-58\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eTo actually compute these in a simulation, we need to again break down the complex field value \u003cspan class=\"math inline\"\u003e\\(\\phi\\)\u003c/span\u003e into its two scalar components. We simplify \u003cspan class=\"math inline\"\u003e\\(\\hbar=1\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(c=1\\)\u003c/span\u003e, but preserve the mass term, and use \u003cspan class=\"math inline\"\u003e\\(a = \\varphi_a\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(b = \\varphi_b\\)\u003c/span\u003e for ease of calculation:\u003c/p\u003e\u003cp id=\"text-59\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\rho = \\frac{ie}{2m_0} \\left(\\phi^* \\frac{\\partial \\phi}{\\partial t} - \\phi \\frac{\\partial \\phi^*}{\\partial t} \\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-60\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\phi^* \\frac{\\partial \\phi}{\\partial t} - \\phi \\frac{\\partial \\phi^*}{\\partial t} = (a - ib) (\\dot a + i \\dot b) - (a + ib) (\\dot a - i \\dot b)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-61\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n= (a \\dot a + i a \\dot b - i b \\dot a + b \\dot b) - (a \\dot a - i a \\dot b + i b \\dot a + b \\dot b)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-62\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n= 2 i a \\dot b - 2 i b \\dot a\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-63\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\rho = \\frac{e}{m_0} (b \\dot a - a \\dot b) \n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-64\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\rho = \\frac{e}{m_0} (\\varphi_b \\dot \\varphi_a - \\varphi_a \\dot \\varphi_b)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-65\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eand for the spatial (current) terms, it boils down to the same kind of equation in the end:\u003c/p\u003e\u003cp id=\"text-66\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{J} = -\\frac{ie}{2m_0} \\left(\\phi^* \\vec{\\nabla} \\phi - \\phi \\vec{\\nabla} \\phi^* \\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-67\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\phi^* \\vec{\\nabla} \\phi - \\phi \\vec{\\nabla} \\phi^* = 2 i a \\vec{\\nabla} b - 2 i b \\vec{\\nabla} a\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-68\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{J} = \\frac{e}{m_0} (a \\vec{\\nabla} b - b \\vec{\\nabla} a)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-69\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{J} = \\frac{e}{m_0} (\\varphi_a \\vec{\\nabla} \\varphi_b - \\varphi_b \\vec{\\nabla} \\varphi_a)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-70\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThese are the expressions that are used in \u003ca href=\"dirac\" target=\"_blank\"\u003eDirac\u003c/a\u003e to couple with the electromagnetic field.\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"configuration-space\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"m274-450 248 248-42 42-320-320 320-320 42 42-248 248h526v60H274Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003ePrevious\u003c/p\u003e\u003c/a\u003e\u003cstretch id=\"content-buttons-go-167\" style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1ch;min-height:1em;font-weight:thin;text-align:start\"\u003e\u003c/stretch\u003e\u003ca id=\"content-buttons-go-168\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"contextual\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M686-450H160v-60h526L438-758l42-42 320 320-320 320-42-42 248-248Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Contextual","URL":"contextual","Title":"Contextual","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{"eq":["eq_commuting"],"figure":["figure_polarization"]},"Description":"The widely discussed distinction between \u003cstrong\u003econtextual\u003c/strong\u003e vs. “real” variables is a source of considerable confusion within the QM world (\u003ca href=\"ref://Shimony84\" target=\"_blank\"\u003eShimony, 1984\u003c/a\u003e; \u003ca href=\"ref://Gudder70\" target=\"_blank\"\u003eGudder, 1970\u003c/a\u003e; \u003ca href=\"ref://Khrennikov01\" target=\"_blank\"\u003eKhrennikov, 2001\u003c/a\u003e; \u003ca href=\"ref://Rovelli96\" target=\"_blank\"\u003eRovelli, 1996\u003c/a\u003e. A contextual variable is effectively something that cannot be discretely localized and quantified — its value depends in some necessary way on the surrounding \u003cem\u003econtext\u003c/em\u003e, which is usually taken to mean the state of the measurement apparatus. In the standard Copenhagen interpretation, \u003cem\u003eeverything\u003c/em\u003e could be described as being contextual, given that \u003cem\u003enothing\u003c/em\u003e is thought to exist in any localized, definite way prior to the measurement process.","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eContextual\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-7\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe widely discussed distinction between \u003cstrong\u003econtextual\u003c/strong\u003e vs. “real” variables is a source of considerable confusion within the QM world (\u003ca href=\"ref://Shimony84\" target=\"_blank\"\u003eShimony, 1984\u003c/a\u003e; \u003ca href=\"ref://Gudder70\" target=\"_blank\"\u003eGudder, 1970\u003c/a\u003e; \u003ca href=\"ref://Khrennikov01\" target=\"_blank\"\u003eKhrennikov, 2001\u003c/a\u003e; \u003ca href=\"ref://Rovelli96\" target=\"_blank\"\u003eRovelli, 1996\u003c/a\u003e. A contextual variable is effectively something that cannot be discretely localized and quantified — its value depends in some necessary way on the surrounding \u003cem\u003econtext\u003c/em\u003e, which is usually taken to mean the state of the measurement apparatus. In the standard Copenhagen interpretation, \u003cem\u003eeverything\u003c/em\u003e could be described as being contextual, given that \u003cem\u003enothing\u003c/em\u003e is thought to exist in any localized, definite way prior to the measurement process.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eHowever, in the pilot-wave framework, the positions of the particles (and \u003cem\u003eonly\u003c/em\u003e these variables) are given a privileged status as being \u003cem\u003ereal\u003c/em\u003e, non-contextual variables, and \u003cem\u003eeverything else\u003c/em\u003e about the quantum state is relegated to the usual \u003cem\u003econtextual\u003c/em\u003e status. Another way of stating this is that everything that must be computed from the wave function itself is contextual, and only the position values are excluded from this status.\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFor example, \u003ca href=\"ref://Norsen14\" target=\"_blank\"\u003eNorsen (2014)\u003c/a\u003e analyzed a pilot-wave model of \u003cem\u003espin\u003c/em\u003e, and showed very clearly that it is contextual in nature. There are no predefined, definite spin values for any particles (indeed this is mathematically impossible as discussed in a moment), and the interaction with a spin-detecting Stern-Gerlach magnetic field apparatus is responsible for \u003cem\u003ecreating\u003c/em\u003e a definite spin value along a given axis, where none existed previously. The same logic applies to the momentum and energy of the particles, which also depend on the wave function, as elaborated in \u003ca href=\"ref://Norsen17\" target=\"_blank\"\u003eNorsen (2017)\u003c/a\u003e.\u003c/p\u003e\u003cp id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIs there something fundamental within quantum theory that promotes position variables to this privileged status that they have in the pilot-wave framework? In general, the answer appears to be “no”, although there is ongoing philosophical debate on this topic (\u003ca href=\"ref://Schroeren22\" target=\"_blank\"\u003eSchroeren, 2022\u003c/a\u003e; \u003ca href=\"ref://Wallace20\" target=\"_blank\"\u003eWallace, 2020\u003c/a\u003e; \u003ca href=\"ref://North12\" target=\"_blank\"\u003eNorth, 2012\u003c/a\u003e; \u003ca href=\"ref://NeyAlbert13\" target=\"_blank\"\u003eNey \u0026 Albert, 2013\u003c/a\u003e). In the standard matrix mechanics approaches, the quantum state is chosen to be whatever is most convenient, and this could be a momentum basis instead of position, for example. However, if it turns out that the position variables somehow do provide a uniquely useful basis for “real” variables, as in the pilot-wave model, that would be an intriguing result.\u003c/p\u003e\u003cp id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThere is a further quantum property that is critical to appreciate in the context of contextuality, which is whether different quantum variables \u003cem\u003ecommute\u003c/em\u003e with each other or not. Mathematically, two variables \u003cspan class=\"math inline\"\u003e\\(A\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(B\\)\u003c/span\u003e commute if their order of application (multiplication) doesn’t affect the result:\u003c/p\u003e\u003cdiv id=\"inline-container-6\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_commuting\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_commuting\"\u003e\u003cb\u003eEq 1:\u003c/b\u003e Commuting variables\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-7\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nA B = B A\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-8\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eor:\u003c/p\u003e\u003cp id=\"text-9\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nAB + BA = 0\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-10\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn the mathematics of matrix mechanics, these variables are actually “operators”, that are like a measurement operation that extracts the value of a given variable. In any case, the key physical meaning of commuting variables is that \u003cem\u003ethe order in which you apply the measurements\u003c/em\u003e doesn’t matter. For example, if you measure the position of a particle in the X axis, that does not affect your measurement of the position in the Y axis, so these position operators commute. However, measuring the position of a particle specifically does \u003cem\u003enot\u003c/em\u003e commute with measuring its momentum, which is the key point of the Heisenberg uncertainty principle: the more accurately you measure position, the more that destroys your ability to measure momentum, and vice-versa.\u003c/p\u003e\u003cp id=\"text-11\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eCritically, the different components of \u003cem\u003espin\u003c/em\u003e do not commute with each other, which is why it is impossible for any quantum state to have a definite value for each of these different spin components (\u003ca href=\"ref://KochenSpecker90\" target=\"_blank\"\u003eKochen \u0026 Specker, 1990\u003c/a\u003e; \u003ca href=\"ref://Spekkens05\" target=\"_blank\"\u003eSpekkens, 2005\u003c/a\u003e). Thus, spin \u003cem\u003emust\u003c/em\u003e be a contextual value (as shown in \u003ca href=\"ref://Norsen14\" target=\"_blank\"\u003eNorsen, 2014\u003c/a\u003e), where the measurement process effectively rotates the spin axis onto one dimension, which at the same time makes the other axes indeterminate.\u003c/p\u003e\u003cdiv id=\"figure_polarization\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_polarization\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 1:\u003c/b\u003e \u003cp\u003eDemonstration that polarization actually rotates the “photons” in light — the polarized lens closest to the camera is oriented perpendicular to the polarization of the LCD screen, and thus blocks nearly all of that light. However, the other lens interposed between it and the screen rotates the light by roughly 45 degrees, so that it can then make it through the lens, as seen in their overlapping region.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cp id=\"text-14\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis contextual property of spin can also be easily demonstrated for polarization of light, as shown in \u003ca href=\"contextual#figure_polarization\" target=\"_blank\"\u003eFigure 1\u003c/a\u003e. The first pass through a polarizing sunglass lens effectively rotates polarized light onto that axis, such that it can then pass through a second lens, whereas it isn’t able to pass through that second lens directly.\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"conservation\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"m274-450 248 248-42 42-320-320 320-320 42 42-248 248h526v60H274Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003ePrevious\u003c/p\u003e\u003c/a\u003e\u003cstretch id=\"content-buttons-go-167\" style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1ch;min-height:1em;font-weight:thin;text-align:start\"\u003e\u003c/stretch\u003e\u003ca id=\"content-buttons-go-168\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"copenhagen\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M686-450H160v-60h526L438-758l42-42 320 320-320 320-42-42 248-248Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Copenhagen","URL":"copenhagen","Title":"Copenhagen","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{},"Description":"The \u003cstrong\u003eCopenhagen\u003c/strong\u003e interpretation of quantum mechanics (QM) was developed by Niels Bohr and Werner Heisenberg in the 1920’s, and it remains the dominant interpretation among working physicists to this day (\u003ca href=\"ref://Tegmark98\" target=\"_blank\"\u003eTegmark, 1998\u003c/a\u003e; \u003ca href=\"ref://SchlosshauerKoflerZeilinger13\" target=\"_blank\"\u003eSchlosshauer et al., 2013\u003c/a\u003e).","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eCopenhagen\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-8\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe \u003cstrong\u003eCopenhagen\u003c/strong\u003e interpretation of quantum mechanics (QM) was developed by Niels Bohr and Werner Heisenberg in the 1920’s, and it remains the dominant interpretation among working physicists to this day (\u003ca href=\"ref://Tegmark98\" target=\"_blank\"\u003eTegmark, 1998\u003c/a\u003e; \u003ca href=\"ref://SchlosshauerKoflerZeilinger13\" target=\"_blank\"\u003eSchlosshauer et al., 2013\u003c/a\u003e).\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eCentral to this interpretation is the notion that the physical world operates in two complementary modes: you are \u003cem\u003eeither\u003c/em\u003e making a \u003cem\u003emeasurement\u003c/em\u003e, which causes the wave function to \u003cem\u003ecollapse\u003c/em\u003e down to a single discrete particle-like point (via the \u003cstrong\u003eBorn rule\u003c/strong\u003e), \u003cem\u003eor\u003c/em\u003e physics is otherwise evolving according to the wave function, which critically preserves all the quantum uncertainty, and just rotates it around in a \u003cem\u003eunitary\u003c/em\u003e manner over time.\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eDuring this wave-function mode, the mathematical picture suggests that there is no definitive underlying state of the world: everything is in some kind of probabilistic superposition of possible states. Only once you measure something does it actually exist in any kind of definite way, leading to the mantra that “the world only exists when you measure it”.\u003c/p\u003e\u003cp id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis strong discretization of the laws of physics is at the root of many seeming paradoxes and puzzles in understanding the quantum world: what exactly defines a “measurement” at a fundamental level? How can the wave function, which could conceivably spread out over large macroscopic spaces over time, instantaneously collapse down to a single point within that entire space?\u003c/p\u003e\u003cp id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eDespite these conceptual difficulties, the mathematics of the framework allow straightforward calculations that match the outcomes of actual experiments, leading to a general attitude of “shut up and calculate”: don’t bother with unnecessary considerations of the actual underlying physical ontology, just do the math! This clearly puts this framework into the category of a \u003cem\u003ecalculational tool\u003c/em\u003e as discussed in \u003ca href=\"tools-vs-models\" target=\"_blank\"\u003etools vs models\u003c/a\u003e.\u003c/p\u003e\u003cp id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFrom a sociological perspective, the inability to properly appreciate the distinction between calculational tools and physical models has resulted in a century of the single most egregious form of “gaslighting” in any area of science. New students are trained that their own intuitions about the nature of the physical world are just “naive” biases inherited from the normal macroscopic realm, and that none of these things apply to this mysterious quantum world. You are just supposed to discard all such notions of what is physically plausible, and let the math tell you what is actually going on.\u003c/p\u003e\u003cp id=\"text-6\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe whole thing is just so impossibly preposterous as to be hilarious, except that generations of serious people have somehow convinced themselves to swallow these absurdities. And they make sure to enforce the dogma on everyone else too: it is truly a cult-like dynamic, right at the heart of the most foundational branch of science.\u003c/p\u003e\u003cp id=\"text-7\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe \u003ca href=\"pilot-wave\" target=\"_blank\"\u003epilot-wave\u003c/a\u003e framework, while still incomplete, provides such a striking contrast to the paradoxes present in the Copenhagen interpretation, and yet it remains a “fringe” theory. Sure, if you’re actually needing to run some calculations, go ahead and use the \u003ca href=\"hilbert-space\" target=\"_blank\"\u003eHilbert space\u003c/a\u003e formalism. But if you want to understand what might actually be going on in the underlying physics of Nature, the Copenhagen model is just obviously incoherent and nonsensical.\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"contextual\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"m274-450 248 248-42 42-320-320 320-320 42 42-248 248h526v60H274Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003ePrevious\u003c/p\u003e\u003c/a\u003e\u003cstretch id=\"content-buttons-go-167\" style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1ch;min-height:1em;font-weight:thin;text-align:start\"\u003e\u003c/stretch\u003e\u003ca id=\"content-buttons-go-168\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"dirac\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M686-450H160v-60h526L438-758l42-42 320 320-320 320-42-42 248-248Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Dirac","URL":"dirac","Title":"Dirac","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{"eq":["eq_dirac"]},"Description":"The \u003cstrong\u003eDirac\u003c/strong\u003e wave function builds on the considerable progress made in the \u003ca href=\"complex-kg\" target=\"_blank\"\u003ecomplex KG\u003c/a\u003e version of the \u003ca href=\"klein-gordon\" target=\"_blank\"\u003eKlein-Gordon\u003c/a\u003e (KG) wave function, which developed all of the tools needed to couple a matter wave that defines a \u003cstrong\u003ewave of charge\u003c/strong\u003e with the electromagnetic waves of \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e’s equations. This coupling of charge waves and EM waves has been pursued more recently in neoclassical self-coupled field theory (\u003ca href=\"ref://JaynesCummings63\" target=\"_blank\"\u003eJaynes \u0026 Cummings, 1963\u003c/a\u003e; \u003ca href=\"ref://CrispJaynes69\" target=\"_blank\"\u003eCrisp \u0026 Jaynes, 1969\u003c/a\u003e; \u003ca href=\"ref://BarutVanHuele85\" target=\"_blank\"\u003eBarut \u0026 Van Huele, 1985\u003c/a\u003e; \u003ca href=\"ref://BarutDowling90\" target=\"_blank\"\u003eBarut \u0026 Dowling, 1990\u003c/a\u003e; \u003ca href=\"ref://Crisp96\" target=\"_blank\"\u003eCrisp, 1996\u003c/a\u003e; \u003ca href=\"ref://FinsterSmollerYau99a\" target=\"_blank\"\u003eFinster et al., 1999\u003c/a\u003e; \u003ca href=\"ref://Radford03\" target=\"_blank\"\u003eRadford, 2003\u003c/a\u003e; \u003ca href=\"ref://MasielloDeumensOhrn05\" target=\"_blank\"\u003eMasiello et al., 2005\u003c/a\u003e).","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eDirac\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-9\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe \u003cstrong\u003eDirac\u003c/strong\u003e wave function builds on the considerable progress made in the \u003ca href=\"complex-kg\" target=\"_blank\"\u003ecomplex KG\u003c/a\u003e version of the \u003ca href=\"klein-gordon\" target=\"_blank\"\u003eKlein-Gordon\u003c/a\u003e (KG) wave function, which developed all of the tools needed to couple a matter wave that defines a \u003cstrong\u003ewave of charge\u003c/strong\u003e with the electromagnetic waves of \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e’s equations. This coupling of charge waves and EM waves has been pursued more recently in neoclassical self-coupled field theory (\u003ca href=\"ref://JaynesCummings63\" target=\"_blank\"\u003eJaynes \u0026 Cummings, 1963\u003c/a\u003e; \u003ca href=\"ref://CrispJaynes69\" target=\"_blank\"\u003eCrisp \u0026 Jaynes, 1969\u003c/a\u003e; \u003ca href=\"ref://BarutVanHuele85\" target=\"_blank\"\u003eBarut \u0026 Van Huele, 1985\u003c/a\u003e; \u003ca href=\"ref://BarutDowling90\" target=\"_blank\"\u003eBarut \u0026 Dowling, 1990\u003c/a\u003e; \u003ca href=\"ref://Crisp96\" target=\"_blank\"\u003eCrisp, 1996\u003c/a\u003e; \u003ca href=\"ref://FinsterSmollerYau99a\" target=\"_blank\"\u003eFinster et al., 1999\u003c/a\u003e; \u003ca href=\"ref://Radford03\" target=\"_blank\"\u003eRadford, 2003\u003c/a\u003e; \u003ca href=\"ref://MasielloDeumensOhrn05\" target=\"_blank\"\u003eMasiello et al., 2005\u003c/a\u003e).\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis final step in our long journey of progressively more complicated wave functions, is to come to terms with the full glory of the \u003ca href=\"electron\" target=\"_blank\"\u003eelectron\u003c/a\u003e. The standard model of physics has a set of parameters that are used to characterize the basic properties of the fundamental particles. One such property is the rest mass \u003cspan class=\"math inline\"\u003e\\(m_0\\)\u003c/span\u003e, which we introduced in \u003ca href=\"klein-gordon\" target=\"_blank\"\u003eKlein-Gordon\u003c/a\u003e to make the waves slow down and move at variable speeds.\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAnother such property is electrical charge, which we were able to extract from our wave equation once we used a complex-valued state variable, having two independent scalar values within it, in \u003ca href=\"complex-kg\" target=\"_blank\"\u003ecomplex KG\u003c/a\u003e. Furthermore, we found that this charge could come with two different signs, positive or negative. This turns out to be convenient, because electrons also come in a positive form, called a positron. The positron has the same mass as the electron, but just an opposite charge.\u003c/p\u003e\u003cp id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe third basic property of the electron is known as its \u003cstrong\u003espin\u003c/strong\u003e. It is known as a spin \u003cspan class=\"math inline\"\u003e\\(\\frac{1}{2}\\)\u003c/span\u003e particle, along with all of the other fundamental particles known (e.g., quarks). Unfortunately, our wave equations so far do not support this spin property, and so we’ll need to do a little bit more work. However, once we’re done, we’ll find that our equations capture all of the fundamental properties of the electron: we should have a 100% complete description of it!\u003c/p\u003e\u003cp id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eActually there is one last thing, which is that the electron is a member of the lepton family, whereas the other fundamental particles are quarks, and they have other fundamental properties in addition to those carried by leptons. But, this is presumably because quarks live in some other set of state variables separate from the lepton state variables we’re simulating here in our model. So, with that assumption, we might have captured everything about the electron.\u003c/p\u003e\u003cp id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSo what exactly does it mean for an electron to have spin \u003cspan class=\"math inline\"\u003e\\(\\frac{1}{2}\\)\u003c/span\u003e? The quantum mechanical concept of spin is perhaps one of the most difficult to grasp. Sometimes people try to think of a little point particle spinning about like a top on its axis, but this doesn’t actually fit the facts very well. In the end, the best strategy may be to just see what the equations we derive next actually do, and call that spin. Indeed, in the computer simulations, one can clearly see a spinning motion.\u003c/p\u003e\u003cp id=\"text-6\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis spin is very much like the first-order Schrödinger equation dynamics, where the two different elements of the complex variable rotate into each other. We also saw this kind of rotation earlier in the complex coupled KG equation, where the electromagnetic potential introduced a rotation among the complex variables. In the present case, we’re going to have four different variables, and they will all rotate amongst themselves to produce this mysterious spin.\u003c/p\u003e\u003cp id=\"text-7\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIncidentally, quantum physics holds that photons (which we think of as wave packets of the electromagnetic field that we’ve already characterized above) have a spin of 1. Furthermore, the charged complex KG wave equation is described as having a spin of 0. This latter case makes sense to me, in that the two components of the complex number do not rotate into each other, and thus they do not spin at all. However, the electromagnetic field case is a bit more confusing, because as we saw, the four components of this field do not interact with each other in the basic wave equations either!\u003c/p\u003e\u003cp id=\"text-8\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eTherefore, it would seem that it should have a spin of 0 as well. Countering this are two considerations. First, the observable variables of the electric and magnetic fields \u003cspan class=\"math inline\"\u003e\\(\\vec{E}\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(\\vec{B}\\)\u003c/span\u003e, which are derived from these non-interacting electrical potentials, do rotate around each other as the wave propagates. Second, when these potentials interact with our charge wave, the do so in a way that ends up coupling (and rotating) the two independent scalar values in the complex number, and thus they impart some spin on our otherwise spinless particle.\u003c/p\u003e\u003cp id=\"text-9\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOur first step is to introduce a new state variable \u003cspan class=\"math inline\"\u003e\\(\\psi\\)\u003c/span\u003e, to represent a field having four independent scalar values. Mathematically, this is defined as a vector of two complex numbers:\u003c/p\u003e\u003cp id=\"text-10\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\psi = \\begin{bmatrix} \\varphi_{1a} + i \\varphi_{1b} \\\\ \\varphi_{2a} + i \\varphi_{2b} \\end{bmatrix}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-11\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhere the first complex number is denoted with a subscript \u003cspan class=\"math inline\"\u003e\\(1\\)\u003c/span\u003e, and contains the two real-valued components \u003cspan class=\"math inline\"\u003e\\(\\varphi_{1a}\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(\\varphi_{1b}\\)\u003c/span\u003e, and the second has subscript \u003cspan class=\"math inline\"\u003e\\(2\\)\u003c/span\u003e, and contains \u003cspan class=\"math inline\"\u003e\\(\\varphi_{2a}\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(\\varphi_{2b}\\)\u003c/span\u003e. So, the spin is going to amount to these four variables rotating amongst themselves.\u003c/p\u003e\u003cp id=\"text-12\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eHow do we extend our basic complex-coupled KG equation to include this spin factor? Several authors in the literature have described a second-order version of the Dirac equation, which should look very familiar to you at this point, because it is essentially our current KG equation plus one additional spin term. One of the first references to such a thing comes from \u003ca href=\"ref://FeynmanGell-Mann58\" target=\"_blank\"\u003eFeynman \u0026 Gell-Mann (1958)\u003c/a\u003e, where they describe an equation that possesses all of the critical properties of the standard first-order Dirac equation, and note that it only requires four state variables instead of the eight required for the first-order equation. Indeed, Feynman states that he much prefers this form of the equation. This affection is presumably not widely shared, because there are relatively few other references to such an equation in the literature. Most of them come from a series of papers by Levere Hostler (e.g., \u003ca href=\"ref://Hostler82\" target=\"_blank\"\u003eHostler, 1982\u003c/a\u003e; \u003ca href=\"ref://Hostler83\" target=\"_blank\"\u003eHostler, 1983\u003c/a\u003e; \u003ca href=\"ref://Hostler85\" target=\"_blank\"\u003eHostler, 1985\u003c/a\u003e).\u003c/p\u003e\u003cp id=\"text-13\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe version of the equation described by \u003ca href=\"ref://FeynmanGell-Mann58\" target=\"_blank\"\u003eFeynman \u0026 Gell-Mann (1958)\u003c/a\u003e is:\u003c/p\u003e\u003cp id=\"text-14\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\left[ \\left(i {\\nabla}_\\mu - {A}_\\mu\\right)^2 + \\vec{\\sigma} \\cdot \\left(\\vec{B} + i \\vec{E} \\right) \\right] \\psi = m_0^2 \\psi\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-15\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhere \u003cspan class=\"math inline\"\u003e\\(\\vec{\\sigma}\\)\u003c/span\u003e are the standard Pauli matricies that we’ll describe in a moment. \u003ca href=\"ref://Hostler85\" target=\"_blank\"\u003eHostler (1985)\u003c/a\u003e describes a similar equation (which has the minus sign reversed in various places, but is otherwise the same):\u003c/p\u003e\u003cp id=\"text-16\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\left[ \\left(-i \\partial_\\mu - e {A}_\\mu\\right)^2 + m_0^2 + e i \\vec{\\sigma} \\cdot \\left(\\vec{E} + i \\vec{B}\\right) \\right] \\psi = 0\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-17\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIt should be clear that the first squared term is just the complex KG equation coupled to the EM field. Therefore, we can write this equation in our current notation as:\u003c/p\u003e\u003cp id=\"text-18\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\left[\\left(i \\hbar \\partial_\\mu - \\frac{e}{c}{A}_\\mu \\right)^2 + \\frac{e}{c} \\vec{\\sigma} \\cdot \\left(\\vec{B} + i \\vec{E} \\right) \\right] \\psi = m_0^2 c^2 \\psi\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-19\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eNow for the Pauli matricies \u003cspan class=\"math inline\"\u003e\\(\\vec{\\sigma}\\)\u003c/span\u003e. This is a vector of values \u003cspan class=\"math inline\"\u003e\\((\\sigma_x, \\sigma_y, \\sigma_z)\\)\u003c/span\u003e that enter into a dot product with the complex-valued vector composed of the magnetic and electric field values \u003cspan class=\"math inline\"\u003e\\(\\vec{B}\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(\\vec{E}\\)\u003c/span\u003e:\u003c/p\u003e\u003cp id=\"text-20\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{\\sigma} = \\left( \\begin{bmatrix} 0 \u0026 1\\\\\n1 \u0026 0 \\end{bmatrix}, \\begin{bmatrix} 0 \u0026 -i \\\\\ni \u0026 0 \\end{bmatrix}, \\begin{bmatrix} 1 \u0026 0 \\\\\n0 \u0026 -1 \\end{bmatrix} \\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-21\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn the end, the net result of the dot product with an arbitrary vector \u003cspan class=\"math inline\"\u003e\\(\\vec{p}\\)\u003c/span\u003e is:\u003c/p\u003e\u003cp id=\"text-22\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{\\sigma} \\cdot \\vec{p} = \\begin{bmatrix} p_z \u0026 p_x - i p_y \\\\\np_x + i p_y \u0026 -p_z \\end{bmatrix}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-23\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSo applied to our \u003cspan class=\"math inline\"\u003e\\(\\vec{B} + i\\vec{E}\\)\u003c/span\u003e vector, it is:\u003c/p\u003e\u003cp id=\"text-24\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{\\sigma} \\cdot \\left(\\vec{B} + i \\vec{E} \\right) =\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-25\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\begin{bmatrix} B_z + iE_z \u0026 B_x + iE_x - iB_y + E_y \\\\\nB_x + iE_x + i B_y -E_y \u0026 -B_z - iE_z \\end{bmatrix} =\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-26\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\begin{bmatrix} B_z + iE_z \u0026 B_x + E_y + i(E_x - B_y) \\\\\nB_x - E_y + i(E_x + B_y) \u0026 -B_z - iE_z \\end{bmatrix}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-27\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSo, now we’re getting some sense of how this works: different components of the electromagnetic field exert different forces on the different components of the \u003cspan class=\"math inline\"\u003e\\(\\psi\\)\u003c/span\u003e state, producing a rotational effect.\u003c/p\u003e\u003cp id=\"text-28\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis entire result then is multiplied by the two complex numbers in the \u003cspan class=\"math inline\"\u003e\\(\\psi\\)\u003c/span\u003e state:\u003c/p\u003e\u003cp id=\"text-29\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\begin{bmatrix} B_z + iE_z \u0026 B_x + E_y + i(E_x - B_y) \\\\\nB_x - E_y + i(E_x + B_y) \u0026 -B_z - iE_z \\end{bmatrix} \n\\times \\begin{bmatrix} \\varphi_{1a} + i \\varphi_{1b} \\\\\n\\varphi_{2a} + i \\varphi_{2b} \\end{bmatrix}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-30\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWhich produces this for the first complex number:\u003c/p\u003e\u003cp id=\"text-31\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\varphi_{1a} + i \\varphi_{1b} = (\\varphi_{1a} + i \\varphi_{1b})(B_z + iE_z) + (\\varphi_{2a} + i \\varphi_{2b})(B_x + E_y + i(E_x - B_y))\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-32\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhich decomposes into the two scalar variables as:\u003c/p\u003e\u003cp id=\"text-33\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\varphi_{1a} = \\varphi_{1a} B_z - \\varphi_{1b} E_z + \\varphi_{2a} (B_x + E_y) - \\varphi_{2b} (E_x - B_y)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-34\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\varphi_{1b} = \\varphi_{1b} B_z + \\varphi_{1a} E_z + \\varphi_{2b} (B_x + E_y) + \\varphi_{2a} (E_x - B_y)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-35\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAnd for the second complex number:\u003c/p\u003e\u003cp id=\"text-36\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\varphi_{2a} + i \\varphi_{2b} = (\\varphi_{2a} + i \\varphi_{2b})(-B_z - iE_z) + (\\varphi_{1a} + i \\varphi_{1b})(B_x - E_y + i(E_x + B_y))\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-37\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhich decomposes into:\u003c/p\u003e\u003cp id=\"text-38\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\varphi_{2a} = -\\varphi_{2a} B_z + \\varphi_{2b} E_z + \\varphi_{1a} (B_x - E_y) - \\varphi_{1b} (E_x + B_y)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-39\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\varphi_{2b} = -\\varphi_{2b} B_z - \\varphi_{2a} E_z + \\varphi_{1b} (B_x - E_y) + \\varphi_{1a} (E_x + B_y)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-40\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSo, with the minus sign flip that took place in the main equation, and the fact that all of the rest of the equation operates on each complex component of \u003cspan class=\"math inline\"\u003e\\(\\psi\\)\u003c/span\u003e separately, without any mixing across components, the final update equations for this second-order Dirac equation are just the basic complex KG equations plus these terms:\u003c/p\u003e\u003cdiv id=\"inline-container-42\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_dirac\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_dirac\"\u003e\u003cb\u003eEq 1:\u003c/b\u003e Dirac functions in real, second-order form\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-43\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\begin{array}{lcl}\n\\ddot{\\varphi}_{1a} \u0026 = \u0026 \\nabla^2 \\varphi_{1a} - m_0^2 \\varphi_{1a} + 2 e \\left(A_0 \\dot \\varphi_{1b} + \\vec{A} \\cdot \\vec{\\nabla} \\varphi_{1b} \\right) +\\\\\n\u0026 \u0026 e^2 \\varphi_{1a} \\left(A_0^2 - \\vec{A}^2 \\right) + e \\left( \\varphi_{1a} B_z - \\varphi_{1b} E_z + \\varphi_{2a} (B_x + E_y) - \\varphi_{2b} (E_x - B_y) \\right)\\\\\n\n\\ddot \\varphi_{1b} \u0026 = \u0026 \\nabla^2 \\varphi_{1b} - m_0^2 \\varphi_{1b} - 2 e \\left(A_0 \\dot \\varphi_{1a} + \\vec{A} \\cdot \\vec{\\nabla} \\varphi_{1a} \\right) +\\\\\n\u0026 \u0026 e^2 \\varphi_{1b} \\left(A_0^2 - \\vec{A}^2 \\right) + e \\left( \\varphi_{1b} B_z + \\varphi_{1a} E_z + \\varphi_{2b} (B_x + E_y) + \\varphi_{2a} (E_x - B_y) \\right)\\\\\n\n\\ddot \\varphi_{2a} \u0026 = \u0026 \\nabla^2 \\varphi_{2a} - m_0^2 \\varphi_{2a} + 2 e \\left(A_0 \\dot \\varphi_{2b} + \\vec{A} \\cdot \\vec{\\nabla} \\varphi_{2b} \\right) +\\\\\n\u0026 \u0026 e^2 \\varphi_{2a} \\left(A_0^2 - \\vec{A}^2 \\right) + e \\left( -\\varphi_{2a} B_z + \\varphi_{2b} E_z + \\varphi_{1a} (B_x - E_y) - \\varphi_{1b} (E_x + B_y) \\right)\\\\\n\n\\ddot \\varphi_{2b} \u0026 = \u0026 \\nabla^2 \\varphi_{2b} - m_0^2 \\varphi_{2b} - 2 e \\varphi_{2a} \\left(A_0 \\dot \\varphi_{2a} + \\vec{A} \\cdot \\vec{\\nabla} \\varphi_{2a} \\right) +\\\\\n\u0026 \u0026 e^2 \\varphi_{2b} \\left(A_0^2 - \\vec{A}^2 \\right) + e \\left( -\\varphi_{2b} B_z - \\varphi_{2a} E_z + \\varphi_{1b} (B_x - E_y) + \\varphi_{1a} (E_x + B_y) \\right)\\\\\n\\end{array}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-44\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAgain, it is fundamentally the wave equation, plus three additional terms that characterize the interaction with the electromagnetic field. Note that, as with the mixing across complex components \u003cspan class=\"math inline\"\u003e\\(\\varphi_a\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(\\varphi_b\\)\u003c/span\u003e that occurred in the previous version of the coupled KG equations, the mixing or spin across \u003cspan class=\"math inline\"\u003e\\(\\phi_1\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(\\phi_2\\)\u003c/span\u003e occurs via the electromagnetic field interaction. This time, the vector force fields are now required for the coupling, requiring that we compute them from the potentials, as described earlier (involving the \u003cspan class=\"math inline\"\u003e\\(\\vec{\\nabla}\\)\u003c/span\u003e first-order gradient and, for the first time, the \u003cspan class=\"math inline\"\u003e\\(\\vec{\\nabla} \\times\\)\u003c/span\u003e function, which is very similar in its discrete form to the \u003cspan class=\"math inline\"\u003e\\(\\vec{\\nabla}\\)\u003c/span\u003e function).\u003c/p\u003e\u003cp id=\"text-45\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eInterestingly, although we need to continue the broken symmetry from the previous coupled-complex KG equation, where we use the current values of \u003cspan class=\"math inline\"\u003e\\(\\dot \\varphi_{1a}\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(\\dot \\varphi_{2a}\\)\u003c/span\u003e to update the \u003cspan class=\"math inline\"\u003e\\(\\varphi_{1b}\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(\\varphi_{2b}\\)\u003c/span\u003e variables, we apparently do not need to perform a similar symmetry breaking for the new couplings in this Dirac equation.\u003c/p\u003e\u003cp id=\"text-46\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSo, to answer the question of “what is spin?”, we need only look at these equations. Spin, it seems, is this rotation of state values through the two complex variables in the \u003cspan class=\"math inline\"\u003e\\(\\psi\\)\u003c/span\u003e state: \u003cspan class=\"math inline\"\u003e\\(\\phi_1\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(\\phi_2\\)\u003c/span\u003e. As is evident, this spinning occurs via interactions with the electromagnetic field vectors oriented along the three different spatial directions. The fact that, in our CA model we actually fix these directions according to the underlying cubic grid may seem strange and arbitrary. However, this does not mean that stuff can only spin along these fixed directions, anymore than it means that waves can only propagate in certain directions. By having different continuous values along these dimensions, any “direction” of spin relative to the underlying grid can occur.\u003c/p\u003e\u003cp id=\"text-47\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAlthough somewhat complex, these equations should describe the entirety of the complexity of the electron’s interaction with the electromagnetic field, which is to say, with other electrons and positive electric charges in the nucleus. Therefore, as we know, a huge proportion of the known complexity of the universe stems from the consequences of these basic equations. So, perhaps they do not look so complex in comparison.\u003c/p\u003e\u003cp id=\"text-48\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOne final thing to note is that the charge and current density equations from the previous version of the coupled complex KG equation still hold for this Dirac version, because these additional terms do not enter into the covariant derivative, and are therefore canceled out in the subtraction, just like the mass term. The actual numerical calculation changes only to accommodate the \u003cspan class=\"math inline\"\u003e\\(\\psi\\)\u003c/span\u003e state value instead of the single complex \u003cspan class=\"math inline\"\u003e\\(\\phi\\)\u003c/span\u003e state. The charge and current equations are:\u003c/p\u003e\u003cp id=\"text-49\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\rho = \\frac{\\hbar e}{m_0 c^2} \\left((\\varphi_{1b} \\dot \\varphi_{1a} - \\varphi_{1a} \\dot \\varphi_{1b}) + (\\varphi_{2b} \\dot \\varphi_{2a} - \\varphi_{2a} \\dot \\varphi_{2b})\\right) - \\frac{e^2}{m_0 c^2} A_0 (\\varphi_{1a}^2 + \\varphi_{1b}^2 + \\varphi_{2a}^2 + \\varphi_{2b}^2)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-50\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{J} = \\frac{\\hbar e}{m_0 c^2} \\left((\\varphi_{1a} \\vec{\\nabla} \\varphi_{1b} - \\varphi_{1b} \\vec{\\nabla} \\varphi_{1a}) + (\\varphi_{2a} \\vec{\\nabla} \\varphi_{2b} - \\varphi_{2b} \\vec{\\nabla} \\varphi_{2a})\\right) - \\frac{e^2}{m_0 c^2} \\vec{A} (\\varphi_{1a}^2 + \\varphi_{1b}^2 + \\varphi_{2a}^2 + \\varphi_{2b}^2)\n\\]\u003c/span\u003e\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"copenhagen\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"m274-450 248 248-42 42-320-320 320-320 42 42-248 248h526v60H274Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003ePrevious\u003c/p\u003e\u003c/a\u003e\u003cstretch id=\"content-buttons-go-167\" style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1ch;min-height:1em;font-weight:thin;text-align:start\"\u003e\u003c/stretch\u003e\u003ca id=\"content-buttons-go-168\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"discrete-gradient\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M686-450H160v-60h526L438-758l42-42 320 320-320 320-42-42 248-248Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Discrete gradient","URL":"discrete-gradient","Title":"Discrete gradient","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{"eq":["eq_grad-approx","eq_grad-full"],"figure":["figure_gradient"]},"Description":"To compute the vector gradient in our discrete space-time cellular automaton, we need to introduce a new fundamental computation over the neighbors. The basic wave equation only requires a single neighborhood computation for the Laplacian: \u003cspan class=\"math inline\"\u003e\\(\\nabla^2\\)\u003c/span\u003e. This is one sense in which the model starts getting a bit more complex (it turns out that this computation will also be needed later for coupling with the electromagnetic field as well). First, in a single spatial dimension for state variable \u003cspan class=\"math inline\"\u003e\\(s\\)\u003c/span\u003e, the spatial gradient in one dimension can be approximated via a differenceas:","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eDiscrete gradient\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-10\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eTo compute the vector gradient in our discrete space-time cellular automaton, we need to introduce a new fundamental computation over the neighbors. The basic wave equation only requires a single neighborhood computation for the Laplacian: \u003cspan class=\"math inline\"\u003e\\(\\nabla^2\\)\u003c/span\u003e. This is one sense in which the model starts getting a bit more complex (it turns out that this computation will also be needed later for coupling with the electromagnetic field as well). First, in a single spatial dimension for state variable \u003cspan class=\"math inline\"\u003e\\(s\\)\u003c/span\u003e, the spatial gradient in one dimension can be approximated via a differenceas:\u003c/p\u003e\u003cdiv id=\"inline-container-2\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_grad-approx\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_grad-approx\"\u003e\u003cb\u003eEq 1:\u003c/b\u003e difference approximation for spatial gradient in one dimension\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial {s}}{\\partial {x}} \\approx \\frac{1}{2 \\epsilon} (s_{i+1} - s_{i-1})\n\\]\u003c/span\u003e\u003c/p\u003e\u003cdiv id=\"figure_gradient\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_gradient\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 1:\u003c/b\u003e \u003cp\u003eComputation of the spatial gradient using all 18 neighbors that have a non-zero projection along a given axis (in this case, looking at the x axis). The two face points (\u003cspan class=\"math inline\"\u003e\\(+,-\\)\u003c/span\u003e along the axis) have a full projection along the axis, and thus enter with a weight of 1. The 8 edge points each have a \u003cspan class=\"math inline\"\u003e\\(\\frac{1}{\\sqrt{2}}\\)\u003c/span\u003e projection of their overall distance along the axis, and thus contribute with that overall weighting. Similarly, the 8 corners have a \u003cspan class=\"math inline\"\u003e\\(\\frac{1}{\\sqrt{3}}\\)\u003c/span\u003e projection weighting. In computing the weighted average, the sum of all neighbor differences is divided by the sum of the weighting terms.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cp id=\"text-6\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn three dimensions, the computation can be made more accurate and robust by including more of the neighbors, just as we did for the computation of the Laplacian \u003cspan class=\"math inline\"\u003e\\(\\nabla^2\\)\u003c/span\u003e. The most relevant neighbors are the 18 that have some projection along an axis, as illustrated in \u003ca href=\"discrete-gradient#figure_gradient\" target=\"_blank\"\u003eFigure 1\u003c/a\u003e. These can be organized into 9 rays that project through the central point, so that the above approximation can be extended to:\u003c/p\u003e\u003cdiv id=\"inline-container-8\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_grad-full\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_grad-full\"\u003e\u003cb\u003eEq 2:\u003c/b\u003e 3D spatial gradient from 9 rays\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-9\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{1}{(2 + \\frac{8}{\\sqrt{2}} + \\frac{8}{\\sqrt{3}})} \\sum_{j \\in N_{9}} k_j (\\varphi_{j+} - \\varphi_{j-})\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-10\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWhere the neighborhood \u003cspan class=\"math inline\"\u003e\\(N_9\\)\u003c/span\u003e contains pairs of points \u003cspan class=\"math inline\"\u003e\\(j+\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(j-\\)\u003c/span\u003e that are opposite ends of the 9 rays through the central point, and the distance weighting factors \u003cspan class=\"math inline\"\u003e\\(k_j\\)\u003c/span\u003e are:\u003c/p\u003e\u003cul id=\"frame-11\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003cstrong\u003efaces:\u003c/strong\u003e \u003cspan class=\"math inline\"\u003e\\(k_j = \\pm 1 \\)\u003c/span\u003e\u003c/li\u003e\u003cli id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003cstrong\u003eedges:\u003c/strong\u003e \u003cspan class=\"math inline\"\u003e\\(k_j = \\pm \\frac{1}{\\sqrt{2}} \\)\u003c/span\u003e\u003c/li\u003e\u003cli id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003cstrong\u003ecorners:\u003c/strong\u003e \u003cspan class=\"math inline\"\u003e\\(k_j = \\pm \\frac{1}{\\sqrt{3}} \\)\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"dirac\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"m274-450 248 248-42 42-320-320 320-320 42 42-248 248h526v60H274Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003ePrevious\u003c/p\u003e\u003c/a\u003e\u003cstretch id=\"content-buttons-go-167\" style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1ch;min-height:1em;font-weight:thin;text-align:start\"\u003e\u003c/stretch\u003e\u003ca id=\"content-buttons-go-168\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"double-slit\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M686-450H160v-60h526L438-758l42-42 320 320-320 320-42-42 248-248Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Double slit","URL":"double-slit","Title":"Double slit","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{"figure":["figure_double-slit","figure_double-slit-elec","figure_double-slit-deb","figure_double-slit-kocsis"]},"Description":"The \u003cstrong\u003edouble-slit\u003c/strong\u003e experiment (also known as Young’s experiment) (\u003ca href=\"double-slit#figure_double-slit\" target=\"_blank\"\u003eFigure 1\u003c/a\u003e), is said to illustrate the full mystery of quantum mechanics, and nicely demonstrates some puzzling aspects of wave-particle duality. Interestingly, the double slit experiment was around long before quantum mechanics, as a way of generating interference patterns with waves, but it “just got weird” when the intensity of the light, or beam of electrons or other particles, is reduced to the point where there is only a \u003cem\u003esingle particle\u003c/em\u003e passing through the apparatus at a time.","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eDouble slit\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-11\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"figure_double-slit\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_double-slit\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 1:\u003c/b\u003e \u003cp\u003eThe double-slit experiment. Narrow openings in the slits cause the wave to spread through diffraction, and because of the different distances traveled in the path from the two different slits to a given point on the far screen P, the waves will experience either constructive or destructive interference, resulting in the wavy bands of light and dark as shown. The quantum paradox here is that this pattern obtains even when a \u003cem\u003esingle particle\u003c/em\u003e is emitted at a time. The particle only ever goes through \u003cem\u003eone\u003c/em\u003e slit or the other, but somehow the wave goes through both. Figure from wikimedia commons by Lacatosias.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"figure_double-slit-elec\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_double-slit-elec\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 2:\u003c/b\u003e \u003cp\u003eResults of a double-slit experiment using electrons, with increasing numbers of electrons recorded (11, 200, 6,000, 40,000, and 140,000). The interference pattern emerges over time, even though only single electrons are detected on each trial. Figure from by Dr. Tonomura via wikimedia commons.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cp id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe \u003cstrong\u003edouble-slit\u003c/strong\u003e experiment (also known as Young’s experiment) (\u003ca href=\"double-slit#figure_double-slit\" target=\"_blank\"\u003eFigure 1\u003c/a\u003e), is said to illustrate the full mystery of quantum mechanics, and nicely demonstrates some puzzling aspects of wave-particle duality. Interestingly, the double slit experiment was around long before quantum mechanics, as a way of generating interference patterns with waves, but it “just got weird” when the intensity of the light, or beam of electrons or other particles, is reduced to the point where there is only a \u003cem\u003esingle particle\u003c/em\u003e passing through the apparatus at a time.\u003c/p\u003e\u003cp id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSurprisingly, one still observes the interference effect in this case (\u003ca href=\"double-slit#figure_double-slit-elec\" target=\"_blank\"\u003eFigure 2\u003c/a\u003e). How can a single “hard little particle”, all by itself, produce this wave-like interference effect? There are \u003cem\u003emany\u003c/em\u003e other results that all add up to the strong conclusion that, somehow, elementary particles like electrons have \u003cem\u003eboth\u003c/em\u003e wave and particle properties.\u003c/p\u003e\u003cp id=\"text-6\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe \u003ca href=\"pilot-wave\" target=\"_blank\"\u003epilot-wave\u003c/a\u003e framework of de Broglie and Bohm provides the most natural, intuitive explanation of these effects: the wave goes through both slits, and the particle goes through one, but it is influenced by the wave.\u003c/p\u003e\u003cdiv id=\"figure_double-slit-deb\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_double-slit-deb\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 3:\u003c/b\u003e \u003cp\u003eTrajectories for particles in the double-slit experiment computed according to the de Broglie-Bohm pilot-wave model. The interference effects can be seen as relatively localized bumps in the trajectories, corresponding to steep gradients in the Schrödinger wave equation. Critically, the underlying trajectories are considered to exist at all points even if you don’t happen to observe them.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cp id=\"text-9\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003ca href=\"double-slit#figure_double-slit-deb\" target=\"_blank\"\u003eFigure 3\u003c/a\u003e shows what the underlying trajectories of particles under the pilot-wave framework look like in a double-slit experiment, and \u003ca href=\"double-slit#figure_double-slit-kocsis\" target=\"_blank\"\u003eFigure 4\u003c/a\u003e shows some recent data from an experiment where \u003cem\u003eweak measurements\u003c/em\u003e that minimally disturb the system allow one to infer particle trajectories, which look remarkably similar to those predicted by the pilot-wave model (\u003ca href=\"ref://KocsisBravermanRavetsEtAl11\" target=\"_blank\"\u003eKocsis et al., 2011\u003c/a\u003e).\u003c/p\u003e\u003cdiv id=\"figure_double-slit-kocsis\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_double-slit-kocsis\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 4:\u003c/b\u003e \u003cp\u003eReconstructed trajectories of photons in a double-slit experiment using a weak measurement technique that allows aggregate trajectory information to be reconstructed over many repeated samples that are post-sorted according to a weak additional modulation of the system — these are not individual particle trajectories. There is a striking correspondence to the predictions of the de Broglie-Bohm model. Figure from Kocsis et al, 2011.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"discrete-gradient\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"m274-450 248 248-42 42-320-320 320-320 42 42-248 248h526v60H274Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003ePrevious\u003c/p\u003e\u003c/a\u003e\u003cstretch id=\"content-buttons-go-167\" style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1ch;min-height:1em;font-weight:thin;text-align:start\"\u003e\u003c/stretch\u003e\u003ca id=\"content-buttons-go-168\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"duality\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M686-450H160v-60h526L438-758l42-42 320 320-320 320-42-42 248-248Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Duality","URL":"duality","Title":"Duality","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{},"Description":"From an \u003ca href=\"epistemic-vs-ontic#ontological\" target=\"_blank\"\u003eontological\u003c/a\u003e perspective (i.e., in terms of what we think is \u003cem\u003eactually real\u003c/em\u003e), there are two seemingly-conflicting entities at the heart of QM: \u003cstrong\u003ewaves\u003c/strong\u003e and \u003cstrong\u003eparticles\u003c/strong\u003e: the fundamental wave-particle \u003cstrong\u003eduality\u003c/strong\u003e. Much of the confusion and paradox in QM is tied up with this duality. The overall approach here follows the \u003ca href=\"pilot-wave\" target=\"_blank\"\u003epilot-wave\u003c/a\u003e framework in embracing the simultaneous and interacting reality of both particles and waves, and we spend a lot of time and effort understanding the seemingly “magical” properties of \u003ca href=\"waves\" target=\"_blank\"\u003ewaves\u003c/a\u003e in particular.","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eDuality\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-12\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFrom an \u003ca href=\"epistemic-vs-ontic#ontological\" target=\"_blank\"\u003eontological\u003c/a\u003e perspective (i.e., in terms of what we think is \u003cem\u003eactually real\u003c/em\u003e), there are two seemingly-conflicting entities at the heart of QM: \u003cstrong\u003ewaves\u003c/strong\u003e and \u003cstrong\u003eparticles\u003c/strong\u003e: the fundamental wave-particle \u003cstrong\u003eduality\u003c/strong\u003e. Much of the confusion and paradox in QM is tied up with this duality. The overall approach here follows the \u003ca href=\"pilot-wave\" target=\"_blank\"\u003epilot-wave\u003c/a\u003e framework in embracing the simultaneous and interacting reality of both particles and waves, and we spend a lot of time and effort understanding the seemingly “magical” properties of \u003ca href=\"waves\" target=\"_blank\"\u003ewaves\u003c/a\u003e in particular.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAt one point, I was hopeful that somehow one could do away with particles entirely, given that wave equations can account for so much of the known phenomena in physics, including the mind-bending space-time distortions of \u003ca href=\"special-relativity\" target=\"_blank\"\u003especial relativity\u003c/a\u003e, and the complex dynamics of electricity and magnetism (EM, see \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e). However, pure waves end up being just a bit too “squishy” and tend to ooze out all over the place.\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eParticles provide a kind of hard “skeleton” that is particularly important for maintaining the strict \u003ca href=\"conservation\" target=\"_blank\"\u003econservation\u003c/a\u003e laws that are so central in physics: Nature is above all a meticulous accountant, it seems. For example, there are strong conservation laws for charge, spin, and particle numbers of various sorts, along with the ubiquitous conservation of total energy. Although the \u003ca href=\"hamiltonian\" target=\"_blank\"\u003eHamiltonian\u003c/a\u003e framework shows how waves functions conserve total energy, the inevitable dispersion of the wave through space results in a very real practical problem, where these conserved quantities end up spread thinly across wide, discontinuous regions.\u003c/p\u003e\u003cp id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThere is no plausible mechanism within a purely wave-based model for how this distributed “mess” could be re-concentrated back into the discrete particles that are inevitably found in experiments. The conceptual problem is identical to the entirely paradoxical and unexplained “magic” behind the measurement process in the standard \u003ca href=\"copenhagen\" target=\"_blank\"\u003eCopenhagen\u003c/a\u003e interpretation.\u003c/p\u003e\u003cp id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFurthermore, the \u003ca href=\"pauli-exclusion-principle\" target=\"_blank\"\u003ePauli exclusion principle\u003c/a\u003e prevents there from being two of the same \u003cem\u003efermions\u003c/em\u003e (spin \u003csup\u003e1\u003c/sup\u003e⁄\u003csub\u003e2\u003c/sub\u003e particles like electrons and quarks) in the same quantum state, which in the pilot-wave framework means being in the same place at the same time with the same spin. This suggests from our computational, \u003ca href=\"cellular-automaton\" target=\"_blank\"\u003ecellular-automaton\u003c/a\u003e perspective that there is some kind of underlying constraint like “slots” in a lattice for holding at most one of each type of particle. This is both a welcome simplification for our models of these particles, and a tantalizing suggestion that this computational perspective might provide some unique insights into the underlying nature of the physical world.\u003c/p\u003e\u003cp id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis computational perspective also provides an interesting motivation for the need for waves. If you just have a simple discrete point-like particle sitting in some kind of lattice-like grid, it is very difficult to implement realistic force-field interactions among such particles, especially when using other discrete particles like “photons” to mediate these interactions.\u003c/p\u003e\u003cp id=\"text-6\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe purely particle picture of an electron constantly spewing baseball-like photons out in all directions to hit other electrons is very difficult to sustain. How does such a scheme ever achieve any kind of smooth field-like coverage of space using discrete point-like entities? How many balls per femtosecond does it have to spew? How do they manage to spread out uniformly over space and time, while properly conveying the dynamic interactions among the magnetic and electric aspects of the wave functions?\u003c/p\u003e\u003cp id=\"text-7\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eInstead, it is far more straightforward to use \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e’s EM wave equations to model the force field interactions among electrons. However, the ability of a discrete localized electron to “sense” such a force field as a distributed wave remains problematic: EM waves that influence electrons are widely distributed things, and small discrete samples at one point of a wave would not provide the proper net influence that the physical laws require. Thus, it works much better for the electron to also have its own wave field that is directly coupled with the EM wave field.\u003c/p\u003e\u003cp id=\"text-8\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn effect, the electron’s wave field acts like a kind of antenna that senses and responds to the EM forces, and then conveys the results to shape the unfolding trajectory of the discrete particle through space and time, as captured in the pilot-wave model. See \u003ca href=\"stochastic-particles\" target=\"_blank\"\u003estochastic particles\u003c/a\u003e for more details.\u003c/p\u003e\u003cp id=\"text-9\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOne approach is to implement a wave-particle model through coupled \u003ca href=\"dirac\" target=\"_blank\"\u003eDirac\u003c/a\u003e wave functions for the electron and \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e’s equations for EM, with the Dirac wave providing the guiding \u003ca href=\"pilot-wave\" target=\"_blank\"\u003epilot-wave\u003c/a\u003e for a discrete electron particle localized within a cubic lattice grid.\u003c/p\u003e\u003cp id=\"text-10\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe waves in this model are all implemented using the same cubic lattice grid that the discrete electron particles live on, where local neighborhood interactions among the lattice cells implement a highly spatially symmetric form of the \u003cem\u003eLaplacian\u003c/em\u003e spatial gradient function at the core of the wave function. In short, the entire model is essentially an elaborate form of \u003ca href=\"cellular-automaton\" target=\"_blank\"\u003ecellular automaton\u003c/a\u003e (CA), which has many appealing properties as the simplest-possible framework for a physical system, as advocated by a number of theorists over the years (John Von Neumann; Stanislaw Ulam; \u003ca href=\"ref://Zuse69\" target=\"_blank\"\u003eZuse, 1969\u003c/a\u003e; \u003ca href=\"ref://FredkinToffoli82\" target=\"_blank\"\u003eFredkin \u0026 Toffoli, 1982\u003c/a\u003e; \u003ca href=\"ref://Fredkin90\" target=\"_blank\"\u003eFredkin, 1990\u003c/a\u003e; \u003ca href=\"ref://tHooft05\" target=\"_blank\"\u003eHooft, 2005\u003c/a\u003e; \u003ca href=\"ref://tHooft16\" target=\"_blank\"\u003eHooft, 2016\u003c/a\u003e; \u003ca href=\"ref://Wolfram97\" target=\"_blank\"\u003eWolfram, 1997\u003c/a\u003e).\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"double-slit\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"m274-450 248 248-42 42-320-320 320-320 42 42-248 248h526v60H274Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003ePrevious\u003c/p\u003e\u003c/a\u003e\u003cstretch id=\"content-buttons-go-167\" style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1ch;min-height:1em;font-weight:thin;text-align:start\"\u003e\u003c/stretch\u003e\u003ca id=\"content-buttons-go-168\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"electron\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M686-450H160v-60h526L438-758l42-42 320 320-320 320-42-42 248-248Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Electron","URL":"electron","Title":"Electron","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{},"Description":"The \u003cstrong\u003eelectron\u003c/strong\u003e is the most basic version of a fermion (spin \u003csup\u003e1\u003c/sup\u003e⁄\u003csub\u003e2\u003c/sub\u003e), and is the focus of the modeling work here. We attempt to model it using the principles of \u003ca href=\"stochastic-particles\" target=\"_blank\"\u003estochastic particles\u003c/a\u003e, as it couples with the \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e EM field.","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eElectron\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-13\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe \u003cstrong\u003eelectron\u003c/strong\u003e is the most basic version of a fermion (spin \u003csup\u003e1\u003c/sup\u003e⁄\u003csub\u003e2\u003c/sub\u003e), and is the focus of the modeling work here. We attempt to model it using the principles of \u003ca href=\"stochastic-particles\" target=\"_blank\"\u003estochastic particles\u003c/a\u003e, as it couples with the \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e EM field.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThere is a quote somewhere about how if one could just understand this one thing: the electron coupled to the EM feld, then one would understand all the essential mysteries of quantum physics.\u003c/p\u003e\u003ch2 id=\"todo\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eTODO\u003c/h2\u003e\u003cul id=\"frame-3\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003ca href=\"ref://Holland05c\" target=\"_blank\"\u003eHolland, 2005\u003c/a\u003e – back-reaction\u003c/li\u003e\u003c/ul\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"duality\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"m274-450 248 248-42 42-320-320 320-320 42 42-248 248h526v60H274Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003ePrevious\u003c/p\u003e\u003c/a\u003e\u003cstretch id=\"content-buttons-go-167\" style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1ch;min-height:1em;font-weight:thin;text-align:start\"\u003e\u003c/stretch\u003e\u003ca id=\"content-buttons-go-168\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"epistemic-vs-ontic\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M686-450H160v-60h526L438-758l42-42 320 320-320 320-42-42 248-248Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Epistemic vs ontic","URL":"epistemic-vs-ontic","Title":"Epistemic vs ontic","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{},"Description":"The distinction between \u003cstrong\u003eepistemic\u003c/strong\u003e vs \u003cstrong\u003eontic\u003c/strong\u003e (also known as \u003cem\u003ealeatoric\u003c/em\u003e in other contexts) uncertainty is critical for understanding the difference between the standard interpretations of QM (e.g., the \u003ca href=\"copenhagen\" target=\"_blank\"\u003eCopenhagen\u003c/a\u003e interpretation and \u003ca href=\"hilbert-space\" target=\"_blank\"\u003eHilbert space\u003c/a\u003e approach) and the \u003ca href=\"pilot-wave\" target=\"_blank\"\u003epilot-wave\u003c/a\u003e approach.","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eEpistemic vs ontic\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-14\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe distinction between \u003cstrong\u003eepistemic\u003c/strong\u003e vs \u003cstrong\u003eontic\u003c/strong\u003e (also known as \u003cem\u003ealeatoric\u003c/em\u003e in other contexts) uncertainty is critical for understanding the difference between the standard interpretations of QM (e.g., the \u003ca href=\"copenhagen\" target=\"_blank\"\u003eCopenhagen\u003c/a\u003e interpretation and \u003ca href=\"hilbert-space\" target=\"_blank\"\u003eHilbert space\u003c/a\u003e approach) and the \u003ca href=\"pilot-wave\" target=\"_blank\"\u003epilot-wave\u003c/a\u003e approach.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eEpistemic uncertainty reflects our own \u003cem\u003elack of knowledge\u003c/em\u003e about the true underlying state of the system, but, critically, excludes any actual “true randomness” arising from the stochastic behavior of the system itself, that would obtain even if we had (counterfactually) perfect knowledge of the underlying state of the system. This latter type of uncertainty is the ontic (“ontologically real”) or aleatoric (derived from the latin word for dice) variety.\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIf the quantum wave function is largely (or even partially) reflecting epistemic uncertainty, then it seriously challenges the pilot-wave framework in a way that does not affect the purely probabilistic \u003ca href=\"copenhagen\" target=\"_blank\"\u003eCopenhagen\u003c/a\u003e approach. How would it make any sense for an \u003cem\u003eepistemic\u003c/em\u003e wave of uncertainty to be guiding the \u003cem\u003ereal\u003c/em\u003e physical positions of particles as they move about the world?\u003c/p\u003e\u003cp id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eBy contrast, the Copenhagen interpretation already takes a laissez-faire epistemic-level approach to the wave function in the first place: it is all just a big untouchable ball of mystery until you do a measurement anyway, so it might as well be epistemic or whatever! The Quantum Bayesianism (QBism) approach takes this to its logical extreme, with an entirely subjective epistemic treatment of the wave function (\u003ca href=\"ref://FuchsMerminSchack14\" target=\"_blank\"\u003eFuchs et al., 2014\u003c/a\u003e; \u003ca href=\"ref://Mermin18\" target=\"_blank\"\u003eMermin, 2018\u003c/a\u003e).\u003c/p\u003e\u003cp id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThere are increasingly strong theoretical and empirical attempts to show that a purely epistemic account contradicts quantum theory (\u003ca href=\"ref://PuseyBarrettRudolph12\" target=\"_blank\"\u003ePusey et al., 2012\u003c/a\u003e, \u003ca href=\"ref://RingbauerDuffusBranciardEtAl15\" target=\"_blank\"\u003eRingbauer et al., 2015\u003c/a\u003e), so there is good reason to believe in the central premise of \u003cstrong\u003ewave reality\u003c/strong\u003e (see also the \u003ca href=\"double-slit\" target=\"_blank\"\u003edouble-slit\u003c/a\u003e experiment).\u003c/p\u003e\u003cp id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eNevertheless, there is clear evidence from \u003cem\u003ewithin the pilot-wave approach itself\u003c/em\u003e that a not-insignificant portion of the pilot-wave actually does represent epistemic uncertainty, because many different possible initial starting states must be modeled to capture our very real uncertainty about the precise starting state of any actual experimental configuration.\u003c/p\u003e\u003cp id=\"text-6\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe Heisenberg \u003ca href=\"uncertainty-principle\" target=\"_blank\"\u003euncertainty principle\u003c/a\u003e dictates that there is a fundamental limit to which we can simultaneously determine all of the relevant degrees of freedom about a physical system, and in practice we almost certainly have well less certainty than this lower limit, because it is very difficult to make any kind of precise measurement of microscopic quantum-scale systems.\u003c/p\u003e\u003cp id=\"text-7\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe incorrect incorporation of epistemic uncertainty in the standard Schrödinger pilot-wave framework is also evident in the inevitable spreading out of the wave function over time. In the epistemic case, this spread represents a very sensible increase in uncertainty about where something might be located, given more time since the last time its position was known. But given that the pilot-wave model maintains exact locations of each particle over time, it really doesn’t seem to make sense for the wave function to spread out in this manner, at least for variables associated with particle positions.\u003c/p\u003e\u003cp id=\"text-8\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn summary, this quote from E. T. Jaynes (\u003ca href=\"ref://Jaynes90\" target=\"_blank\"\u003eJaynes, 1990\u003c/a\u003e) particularly apropos here:\u003c/p\u003e\u003cblockquote id=\"frame-9\" style=\"background:var(--surface-container-color);display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;margin:1em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e“But our present QM formalism is not purely epistemological; it is a peculiar mixture describing in part realities of Nature, in part incomplete human information about Nature — all scrambled up by Heisenberg and Bohr into an omelette that nobody has seen how to unscramble. Yet we think that the unscrambling is a prerequisite for any further advance in basic physical theory. For, if we cannot separate the subjective and objective aspects of the formalism, we cannot know what we are talking about; it is just that simple.”\u003c/p\u003e\u003c/blockquote\u003e\u003cp id=\"text-10\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFrom this perspective, one could make the following reasonable claim about the pilot-wave approach: it provides a very powerful \u003cem\u003edemonstration in principle\u003c/em\u003e that QM is compatible with a “realistic” underlying world where particles always have definite positions. Nevertheless the specific formulation in terms of the Schrödinger wave function operating in \u003ca href=\"configuration-space\" target=\"_blank\"\u003econfiguration space\u003c/a\u003e is very likely conflating epistemic and ontic uncertainty, and a more realistic wave function that only reflects whatever “real” aspect of the wave function remains after the epistemic part is subtracted away should be used instead.\u003c/p\u003e\u003cp id=\"text-11\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eA recent paper has attempted to disentangle the epistemic vs. ontic contributions to the wave function using a novel analytical technique, and concluded that different quantum behavior can be associated with each of these contributions (\u003ca href=\"ref://BudiyonoRohrlich17\" target=\"_blank\"\u003eBudiyono \u0026 Rohrlich, 2017\u003c/a\u003e). However, their approach assumes that the \u003ca href=\"uncertainty-principle\" target=\"_blank\"\u003euncertainty principle\u003c/a\u003e is purely epistemic, which is inconsistent with its fundamental basis in the basic properties of \u003ca href=\"waves\" target=\"_blank\"\u003ewaves\u003c/a\u003e. As usual, any analysis is only as good as its assumptions. As a consequence, they reject the pilot-wave approach because of its “incorrect” use of a \u003cem\u003epurely epistemic\u003c/em\u003e uncertainty wave (under their assumptions) to guide real particle trajectories.\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"electron\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"m274-450 248 248-42 42-320-320 320-320 42 42-248 248h526v60H274Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003ePrevious\u003c/p\u003e\u003c/a\u003e\u003cstretch id=\"content-buttons-go-167\" style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1ch;min-height:1em;font-weight:thin;text-align:start\"\u003e\u003c/stretch\u003e\u003ca id=\"content-buttons-go-168\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"four-vector\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M686-450H160v-60h526L438-758l42-42 320 320-320 320-42-42 248-248Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Four vector","URL":"four-vector","Title":"Four vector","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{"eq":["eq_wave","eq_cov","eq_con","eq_dot","eq_cov-deriv","eq_grad","eq_con-deriv","eq_maxwell"]},"Description":"The \u003cstrong\u003efour-vector\u003c/strong\u003e framework provides an especially compact way of representing the mathematics of wave functions and associated quantities, in terms of a single \u003cstrong\u003espace-time-vector\u003c/strong\u003e with the first element representing time, and the remaining three elements representing 3D space. This notation was initially developed by Minkowski for dealing with Einstein’s \u003ca href=\"special-relativity\" target=\"_blank\"\u003especial relativity\u003c/a\u003e theory, where space and time expand and contract together, and in some sense can be converted into each other.","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eFour vector\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-15\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe \u003cstrong\u003efour-vector\u003c/strong\u003e framework provides an especially compact way of representing the mathematics of wave functions and associated quantities, in terms of a single \u003cstrong\u003espace-time-vector\u003c/strong\u003e with the first element representing time, and the remaining three elements representing 3D space. This notation was initially developed by Minkowski for dealing with Einstein’s \u003ca href=\"special-relativity\" target=\"_blank\"\u003especial relativity\u003c/a\u003e theory, where space and time expand and contract together, and in some sense can be converted into each other.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe essence of the basic \u003ca href=\"waves#wave\" target=\"_blank\"\u003ewave\u003c/a\u003e function is indeed this dynamic interrelationship between the curvature of space (i.e., as the \u003cstrong\u003emomentum\u003c/strong\u003e factor in the \u003ca href=\"hamiltonian\" target=\"_blank\"\u003eHamiltonian\u003c/a\u003e) and the resulting acceleration in time (i.e., the \u003cstrong\u003eenergy\u003c/strong\u003e factor in the Hamiltonian), which is why special relativity can be seen as a consequence of the wave-like nature of physics:\u003c/p\u003e\u003cdiv id=\"inline-container-3\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_wave\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_wave\"\u003e\u003cb\u003eEq 1:\u003c/b\u003e the wave equation\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2 y}{\\partial t^2} = c^2 \\frac{\\partial^2 y}{\\partial x^2}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis interrelationship can be described as the \u003cstrong\u003ecovariance\u003c/strong\u003e or \u003cem\u003etrading-off\u003c/em\u003e of time against space, when you rearrange the terms and set them equal to zero, and see that any change in space has to be \u003cem\u003ecompensated for\u003c/em\u003e by a change in time, and vice-versa:\u003c/p\u003e\u003cp id=\"text-6\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2 y}{\\partial t^2} - c^2 \\frac{\\partial^2 y}{\\partial x^2} = 0\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-7\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe four-vector notation allows us to express this interrelationship in a particularly compact manner.\u003c/p\u003e\u003cp id=\"text-8\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFor the point \u003cspan class=\"math inline\"\u003e\\(a\\)\u003c/span\u003e in space-time, the \u003cstrong\u003ecovariant\u003c/strong\u003e form of the four-vector is defined as:\u003c/p\u003e\u003cdiv id=\"inline-container-10\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_cov\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_cov\"\u003e\u003cb\u003eEq 2:\u003c/b\u003e covariant four-vector\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-11\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n{a}_\\mu = (t,-x,-y,-z) = (a_t,-a_x,-a_y,-a_z) = (a_0,-a_1,-a_2,-a_3)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-12\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe little \u003cspan class=\"math inline\"\u003e\\(\\mu\\)\u003c/span\u003e (Greek “mu”) subscript goes from \u003cspan class=\"math inline\"\u003e\\(0..3\\)\u003c/span\u003e in counting out the different items in the vector, as shown. The time and space coordinates have different signs here in a way that directly matches their relationship in the wave equation, capturing their \u003cem\u003ecovariant\u003c/em\u003e nature.\u003c/p\u003e\u003cp id=\"text-13\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn the simplest form, we assume that \u003cspan class=\"math inline\"\u003e\\(c=1\\)\u003c/span\u003e, so you can just write a bare \u003cspan class=\"math inline\"\u003e\\(t\\)\u003c/span\u003e. The more general form requires converting units of time into units of space by multiplying by the speed of light, so you need to write \u003cspan class=\"math inline\"\u003e\\(ct\\)\u003c/span\u003e instead of just \u003cspan class=\"math inline\"\u003e\\(t\\)\u003c/span\u003e.\u003c/p\u003e\u003cp id=\"text-14\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe other form of the four-vector is the \u003cstrong\u003econtravariant\u003c/strong\u003e form, which is indicated by the use of \u003cstrong\u003esuperscripts\u003c/strong\u003e instead of subscripts:\u003c/p\u003e\u003cdiv id=\"inline-container-16\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_con\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_con\"\u003e\u003cb\u003eEq 3:\u003c/b\u003e contravariant four-vector\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-17\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n{a}^\\mu = (t,x,y,z) = (a^t,a^x,a^y,a^z) = (a^0,a^1,a^2,a^3)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-18\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe multiplication (\u003cstrong\u003edot product\u003c/strong\u003e) of two four-vectors is defined in terms of these covariant and contravariant forms:\u003c/p\u003e\u003cdiv id=\"inline-container-20\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_dot\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_dot\"\u003e\u003cb\u003eEq 4:\u003c/b\u003e four-vector dot product\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-21\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\na \\cdot b \\equiv {a}^\\mu {b}_\\mu = {a}_\\mu {b}^\\mu \\equiv a^0 b_0 + a^1 b_1 + a^2 b_2 + a^3 b_3 = a^t b^t - a^x b^x - a^y b^y - a^z b^z = \\sum^3_{\\mu = 0} {a}_\\mu {b}^\\mu\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-22\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAnd the derivative of a four-vector can also be defined. Just like four-vectors themselves, there are two versions, a covariant and a contravariant, where, potentially confusingly, the superscript / subscript relationship is \u003cem\u003eflipped\u003c/em\u003e for the derivatives!\u003c/p\u003e\u003cp id=\"text-23\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe \u003cem\u003ecovariant derivative\u003c/em\u003e doesn’t have any minus signs:\u003c/p\u003e\u003cdiv id=\"inline-container-25\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_cov-deriv\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_cov-deriv\"\u003e\u003cb\u003eEq 5:\u003c/b\u003e covariant derivative\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-26\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\partial_\\mu \\equiv \\frac{\\partial {}}{\\partial ^\\mu} \\equiv \\left(\\frac{\\partial {}}{\\partial {a^0}},\\frac{\\partial {}}{\\partial {a^1}},\\frac{\\partial {}}{\\partial {a^2}},\\frac{\\partial {}}{\\partial {a^3}}\\right) \\equiv \\left(\\frac{\\partial {}}{\\partial {t}}, \\vec{\\nabla} \\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-27\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhere the \u003cspan class=\"math inline\"\u003e\\(\\vec{\\nabla}\\)\u003c/span\u003e symbol represents the spatial \u003cem\u003egradient\u003c/em\u003e operator:\u003c/p\u003e\u003cdiv id=\"inline-container-29\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_grad\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_grad\"\u003e\u003cb\u003eEq 6:\u003c/b\u003e spatial gradient\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-30\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{\\nabla} \\equiv \\left(\\frac{\\partial {}}{\\partial {x}}, \\frac{\\partial {}}{\\partial {y}}, \\frac{\\partial {}}{\\partial {z}}\\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-31\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe \u003cem\u003econtravariant derivative\u003c/em\u003e is the same deal, except it now has the minus signs:\u003c/p\u003e\u003cdiv id=\"inline-container-33\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_con-deriv\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_con-deriv\"\u003e\u003cb\u003eEq 7:\u003c/b\u003e contravariant derivative\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-34\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\partial^\\mu \\equiv \\frac{\\partial {}}{\\partial _\\mu} \\left(\\frac{\\partial {}}{\\partial {a^0}},-\\frac{\\partial {}}{\\partial {a^1}},-\\frac{\\partial {}}{\\partial {a^2}},- \\frac{\\partial {}}{\\partial {a^3}}\\right) \\equiv \\left(\\frac{\\partial {}}{\\partial t}, -\\vec{\\nabla} \\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-35\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eNow, finally, for the payoff. If you take the second-order derivatives of a four-vector, you combine the vector multiplication rules with the derivative equations to get the following:\u003c/p\u003e\u003cp id=\"text-36\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\partial_\\mu \\partial^\\mu = \\frac{\\partial {}}{\\partial t} \\frac{\\partial {}}{\\partial t} - \\frac{\\partial {}}{\\partial {x}} \\frac{\\partial {}}{\\partial {x}} - \\frac{\\partial {}}{\\partial {y}} \\frac{\\partial {}}{\\partial {y}} - \\frac{\\partial {}}{\\partial {z}} \\frac{\\partial {}}{\\partial {z}}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-37\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n= \\frac{\\partial^2 {}}{\\partial t^2} - \\frac{\\partial^2 {}}{\\partial {x}^2} - \\frac{\\partial^2 {}}{\\partial {y}^2} - \\frac{\\partial^2 {}}{\\partial {z}^2}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-38\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n= \\frac{\\partial^2 {}}{\\partial t^2} - \\nabla^2\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-39\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSo we can now say that the basic wave equation is obtained by setting the second-order four-vector derivative to zero:\u003c/p\u003e\u003cp id=\"text-40\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\partial_\\mu \\partial^\\mu s = 0\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-41\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\left(\\frac{\\partial^2 {}}{\\partial t^2} - \\nabla^2 \\right) s = 0\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-42\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2 {s}}{\\partial t^2} - \\nabla^2 s = 0\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-43\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2 {s}}{\\partial t^2} = \\nabla^2 s\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-44\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAlthough this is equivalent to our basic wave equation, this way of computing the math, with time and space included in the same overall derivatives, will simplify calculations. For example, all of \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e’s equations for the electromagnetic field can be expressed in a single compact expression:\u003c/p\u003e\u003cdiv id=\"inline-container-46\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_maxwell\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_maxwell\"\u003e\u003cb\u003eEq 8:\u003c/b\u003e Maxwell's equation\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-47\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\partial_\\mu \\partial^\\mu A^\\mu = - k^\\mu J^\\mu\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-48\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eNotice that here the second-order derivative has a “source” term (instead of being \u003cspan class=\"math inline\"\u003e\\(=0\\)\u003c/span\u003e), which acts like a driving force on the waves: it represents the charge and currents that drive the electromagnetic field.\u003c/p\u003e\u003cp id=\"text-49\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFinally, we introduce just two more items of terminology. First, sometimes we’ll need to convert a contravariant four-vector into a covariant four-vector, and we can do this using something called the \u003cstrong\u003emetric tensor\u003c/strong\u003e, which has two equivalent forms (they differ for general relativity, but not for \u003ca href=\"special-relativity\" target=\"_blank\"\u003especial relativity\u003c/a\u003e):\u003c/p\u003e\u003cp id=\"text-50\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\ng_{\\mu\\nu} = g^{\\mu\\nu} = \\begin{bmatrix}\n1 \u0026 0 \u0026 0 \u0026 0 \\\\\n0 \u0026 -1 \u0026 0 \u0026 0 \\\\\n0 \u0026 0 \u0026 -1 \u0026 0 \\\\\n0 \u0026 0 \u0026 0 \u0026 -1\\\\\n\\end{bmatrix}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-51\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eTo convert from one form of four-vector to another, you just multiply (we arbitrarily choose \u003cspan class=\"math inline\"\u003e\\(g^{\\mu\\nu}\\)\u003c/span\u003e here):\u003c/p\u003e\u003cp id=\"text-52\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\na^\\mu = g^{\\mu\\nu} a_\\mu\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-53\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\na_\\mu = g^{\\mu\\nu} a^\\mu\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-54\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFinally, as if we needed an even simpler version of the wave equation (and one more symbol to memorize), the \u003cstrong\u003ed’Alembertian\u003c/strong\u003e \u003cspan class=\"math inline\"\u003e\\(\\sqcap\\)\u003c/span\u003e (note: \u003cspan class=\"math inline\"\u003e\\(\\sqcap\\)\u003c/span\u003e should actually just be a square box, but we don’t have that available for technical reasons):\u003c/p\u003e\u003cp id=\"text-55\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\sqcap \\equiv \\frac{\\partial^2 {}}{\\partial t^2} - \\nabla^2 = \\partial_\\mu \\partial^\\mu\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-56\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eallows you to write the wave equation in the simplest possible way, as:\u003c/p\u003e\u003cp id=\"text-57\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\sqcap s = 0\n\\]\u003c/span\u003e\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"epistemic-vs-ontic\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"m274-450 248 248-42 42-320-320 320-320 42 42-248 248h526v60H274Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003ePrevious\u003c/p\u003e\u003c/a\u003e\u003cstretch id=\"content-buttons-go-167\" style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1ch;min-height:1em;font-weight:thin;text-align:start\"\u003e\u003c/stretch\u003e\u003ca id=\"content-buttons-go-168\" 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style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Gravity","URL":"gravity","Title":"Gravity","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{},"Description":"The problem of reconciling quantum mechanics and gravitation as described by Einstein’s general theory of relativity is currently unsolved, and introduces many obstacles for the standard QM interpretations, which do not even admit to a well-specified source of gravitational mass / energy outside of the moments of wave function collapse.","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eGravity\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-16\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ch1 id=\"gravitation\" style=\"max-width:8in;margin:0.25em;font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eGravitation\u003c/h1\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe problem of reconciling quantum mechanics and gravitation as described by Einstein’s general theory of relativity is currently unsolved, and introduces many obstacles for the standard QM interpretations, which do not even admit to a well-specified source of gravitational mass / energy outside of the moments of wave function collapse.\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe Bohmian perspective in general provides a potentially fruitful avenue toward this reconciliation, because matter particles at least have a definite physical location at all times. However, given that these are point particles, they represent a gravitational singularity, which must be resolved, in the same way that the EM singularity must be resolved for electric charges. The WELD cellular automaton (CA) framework provides a natural “ultraviolet” cutoff point for any such singularities, in the form of the fixed underlying grid spacing (i.e., the \u003cem\u003ematrix\u003c/em\u003e).\u003c/p\u003e\u003cp id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eHowever, the size of this grid spacing is not directly constrained by the basic electrodynamic physics: we can “renormalize” the relevant constants to accommodate any such grid scale, so we need some additional form of physical data to provide relevant constraints. As the grid size gets smaller, the strength of gravitational effects increase, putting greater pressure on the need to incorporate gravitation as a central element of the framework.\u003c/p\u003e\u003cp id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe most natural way to incorporate a general relativity-like gravitational dynamic into the CA model is to have coupling constants between each cell that reflect the local curvature of space in each direction. As curvature increases, the effective distance between cells increases, reducing the strength of the wave transmission across them. As with the wave dynamics operating within the discrete CA framework, this would obscure the underlying flat grid space, except that, in the absence of any mass / energy generating these curvatures, “empty” space would be flat. Perhaps not coincidently, the universe is known to be flat. Furthermore, from the WELD CA perspective, time and space are both infinite, which is just slightly less implausible than having some kind of edge beyond which nothing exists.\u003c/p\u003e\u003cp id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis CA gravitational model avoids any kind of mathematical singularity in the context of black holes: there is just more energy and particles packed into a smaller space, which, due to Lorentz dynamics oscillate at slower and slower rates, and contract to smaller and smaller sizes, and drive more curvature, such that electromagnetic fields do not propagate out of the central region. Indeed, the ability to simulate such black hole dynamics represents a critical stress test for the overall framework.\u003c/p\u003e\u003cp id=\"text-6\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eBefore continuing to develop this gravitational model, we can consider whether there are available data to fix the approximate grid size.\u003c/p\u003e\u003ch2 id=\"constraints-from-cosmology\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eConstraints from Cosmology\u003c/h2\u003e\u003cp id=\"text-8\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe natural cosmology for a flat, infinite space populated by ever-growing non-singular black holes is to think of the big bang as resulting from the accumulation of matter / energy into a single huge black hole that grows up to a hypothesized point of a maximal density, at which point it explodes, spewing forth a new clean slate of matter to begin the process all over again. This is an appealing \u003ca href=\"https://en.wikipedia.org/wiki/Cyclic_model\" target=\"_blank\"\u003ecyclic model\u003c/a\u003e (wikipedia link) of cosmology, of which many forms have been developed.\u003c/p\u003e\u003cp id=\"text-9\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIf we can compute how dense matter must have been at the moment of the big bang, this might fix a grid scale capable of just representing this density.\u003c/p\u003e\u003cp id=\"text-10\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe most widely accepted (though still widely regarded as \u003cem\u003ead hoc\u003c/em\u003e) model of the big bang involves a period of \u003cem\u003einflation\u003c/em\u003e where a presumed singularity expands “magically” (no known physical force or process is a plausible candidate) for a very brief moment (from \u003cspan class=\"math inline\"\u003e\\(10^{-36}\\)\u003c/span\u003e sec to \u003cspan class=\"math inline\"\u003e\\(10^{-32}\\)\u003c/span\u003e sec after the big bang) at an exponential rate, such that the universe at the end of inflation is approximately the size of a grain of sand: \u003cspan class=\"math inline\"\u003e\\(0.88 x 10^-3\\)\u003c/span\u003e, according to this \u003ca href=\"https://physics.stackexchange.com/questions/32917/size-of-universe-after-inflation\" target=\"_blank\"\u003ephysics stackexchange\u003c/a\u003e.\u003c/p\u003e\u003cp id=\"text-11\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe physical consequence of such a process (i.e., the entire reason for its hypothesized existence) is to create the flat, relatively homogeneous, isotropic universe that we observe today. Without inflation, the huge energy density of the big bang would create massive amounts of “quantum foam” during expansion, which would make the universe very inhomogeneous and highly curved.\u003c/p\u003e\u003cp id=\"text-12\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn our alternative cosmological framework, this grain of sand, containing all of the mass / energy of the universe, would represent the maximum energy / particle density that a black hole could support. Because the flatness of space is a given in this framework, we would only need to explain why this initial black hole state would result in a relatively homogeneous and isotropic distribution of mass and energy. Perhaps because it spent a long time bouncing around in this grain of sand prior to blowing up?\u003c/p\u003e\u003cp id=\"text-13\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIt is highly uncertain (to me at least) what physics would look like within this primordial black hole. Is it just pure energy or do the fermion particles remain in some fashion? Presumably, due to the high energies present, we would need the “full” standard model including quarks etc to represent this state. I guess it might be some kind of \u003ca href=\"https://en.wikipedia.org/wiki/Quark-gluon_plasma\" target=\"_blank\"\u003equark-gluon plasma\u003c/a\u003e?\u003c/p\u003e\u003ch2 id=\"spin-coupling-to-gravity\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eSpin Coupling to Gravity\u003c/h2\u003e\u003cp id=\"text-15\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThere is a particularly relevant line of work from Nikodem Popławski, building on earlier work from Kopczyński (1972) in the Einstein-Cartan-Kibble-Sciama (ECKS) theory of gravity (Hehl et al, 1976). This form of gravity adds a coupling of the spin component of a fermion particle to the gravitational metric tensor, adding a \u003cem\u003etorsion\u003c/em\u003e component. The net effect of this torsion is a significant repulsive force when the density of fermions goes up (Popalawski 2010; 2012; 2016). This repulsive force would prevent a black hole from fully collapsing into a singularity, and instead it would “bounce” or oscillate as the mass / energy attractive gravitational force interacts with this repulsive force.\u003c/p\u003e\u003cp id=\"text-16\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ePopalawski (2012) gives the following estimates for the big bang parameters:\u003c/p\u003e\u003cul id=\"frame-17\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• critical temperature \u003cspan class=\"math inline\"\u003e\\(T_cr = .75 m_p\\)\u003c/span\u003e (what is m_p?)\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• critical radius \u003cspan class=\"math inline\"\u003e\\(≈ 5.9 × 10^{−4} m\\)\u003c/span\u003e (from which the density could be computed)\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• momentum? frequency? associated with the critical temperature: \u003cspan class=\"math inline\"\u003e\\(v(T_cr) ≈ 8.9 × 10^{34}\\)\u003c/span\u003e (total energy?)\u003c/p\u003e\u003c/ul\u003e\u003cp id=\"text-18\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ePopalawski (2010) defines the Cartan radius of an electron to be on the order of \u003cspan class=\"math inline\"\u003e\\(10^{-27} m\\)\u003c/span\u003e, which sets a minimum spatial scale for such a particle. The density of electrons all packed together at this radius would be \u003cspan class=\"math inline\"\u003e\\(ρ = m_e / r^3 = 10^{51} {kg} m^{−3}\\)\u003c/span\u003e.\u003c/p\u003e\u003ch2 id=\"pauli-exclusion-in-black-holes\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003ePauli Exclusion in Black Holes\u003c/h2\u003e\u003cp id=\"text-20\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe Pauli exclusion principle, i.e., the spin statistics theorem, says that two fermions cannot occupy the same quantum state, and this is understood to prevent a neutron star from further collapsing upon itself. Why doesn’t the same principle apply within a black hole? This is not well understood apparently, at least according to this \u003ca href=\"https://physics.stackexchange.com/questions/93988/does-black-hole-formation-contradict-the-pauli-exclusion-principle\" target=\"_blank\"\u003estackexchgange\u003c/a\u003e discussion, in large part because quantum mechanics and gravity have not been reconciled, and even the precise ways of working with singularities within black holes are unresolved (Geroch, 1967).\u003c/p\u003e\u003cp id=\"text-21\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn any case, it is unclear if the exclusion principle itself would provide the kind of critical density scaling parameters that we would like. Perhaps the above spin-torsion framework may provide a better answer? And what is the relationship between the gravitational spin dynamic and the exclusion principle anyway? Perhaps there is some deep connection?\u003c/p\u003e\u003ch2 id=\"planck-scale\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003ePlanck scale\u003c/h2\u003e\u003cp id=\"text-23\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAnother approach is to consider the Planck scale, where all the units involved in the gravitational equations take on a value of 1, which sets the Planck length on the order of \u003cspan class=\"math inline\"\u003e\\(10^{-35} m\\)\u003c/span\u003e. See \u003ca href=\"https://www.physicsforums.com/insights/hand-wavy-discussion-planck-length/\" target=\"_blank\"\u003ePhysics Forums\u003c/a\u003e and \u003ca href=\"https://math.ucr.edu/home/baez/lengths.html\" target=\"_blank\"\u003eJohn Baez Lengths\u003c/a\u003e more info. At this length scale, the gravitational force immediately surrounding an electron is as strong as its EM force. The smallest length that has been empirically measured, according to above link, is about \u003cspan class=\"math inline\"\u003e\\(10^{-22} m\\)\u003c/span\u003e, which is well below the Compton wavelength of an electron (order \u003cspan class=\"math inline\"\u003e\\(10^{-13}\\)\u003c/span\u003e).\u003c/p\u003e\u003ch2 id=\"the-extreme-coincidence-of-flat-space\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eThe Extreme Coincidence of Flat Space\u003c/h2\u003e\u003cp id=\"text-25\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eNothing in the math of general relativity requires that the overall spacetime metric of the universe be flat, and yet it is. In the standard inflationary big bang model of cosmology, a very precise set of parameters is required to end up with a flat result, making the entire construct of inflation even more implausible and ad hoc. The idea that flat spacetime just a coincidental consequence of these specific initial conditions seems to be missing a huge opportunity for understanding that the universe seems to telling us very clearly: \u003cstrong\u003espacetime is flat because space is fundamental\u003c/strong\u003e. Space is not just some arbitrary mathematical construct, but rather it is the substrate upon which physics operates, as captured in the cellular automaton (CA) framework. And flat spacetime is the natural consequence of the simplest cubic tiling of 3D space.\u003c/p\u003e\u003cp id=\"text-26\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eBecause we populate our CA space with waves, this underlying cubic structure disappears, and physics behaves in an invariant manner across reference frames, consistent with relativity. But the fact that it is so implausibly flat seems to be crying out for a deeper explanation. Also, the need for renormalization and to avoid all manner of singularities and infinities that arise in the limit of infintessimal, continuous space, strongly suggests that there is an “ultraviolet” cutoff: space is not continuous, but discrete, with a minimum underlying grid size.\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"four-vector\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"m274-450 248 248-42 42-320-320 320-320 42 42-248 248h526v60H274Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003ePrevious\u003c/p\u003e\u003c/a\u003e\u003cstretch id=\"content-buttons-go-167\" style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1ch;min-height:1em;font-weight:thin;text-align:start\"\u003e\u003c/stretch\u003e\u003ca id=\"content-buttons-go-168\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"hamiltonian\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M686-450H160v-60h526L438-758l42-42 320 320-320 320-42-42 248-248Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Hamiltonian","URL":"hamiltonian","Title":"Hamiltonian","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{"eq":["eq_total-energy","eq_momentum-op","eq_energy-op","eq_klein-gordon","eq_klein-gordon"]},"Description":"Using \u003ca href=\"special-relativity\" target=\"_blank\"\u003especial relativity\u003c/a\u003e, plus the notion of \u003cstrong\u003econservation of energy\u003c/strong\u003e — i.e., that the \u003cstrong\u003etotal energy\u003c/strong\u003e of the system is strictly conserved over time, we can derive the \u003ca href=\"klein-gordon\" target=\"_blank\"\u003eKlein-Gordon\u003c/a\u003e (KG) equation from first principles. In keeping with physicist’s penchant for assigning people’s names to concepts that would otherwise be very easy to understand if just spelled out, the total energy of the system is also called the \u003cstrong\u003eHamiltonian\u003c/strong\u003e (\u003cspan class=\"math inline\"\u003e\\(H\\)\u003c/span\u003e), and standard Newtonian physics can all be derived from the appropriate Hamiltonian (which is what W. R. Hamilton did).","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eHamiltonian\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-17\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eUsing \u003ca href=\"special-relativity\" target=\"_blank\"\u003especial relativity\u003c/a\u003e, plus the notion of \u003cstrong\u003econservation of energy\u003c/strong\u003e — i.e., that the \u003cstrong\u003etotal energy\u003c/strong\u003e of the system is strictly conserved over time, we can derive the \u003ca href=\"klein-gordon\" target=\"_blank\"\u003eKlein-Gordon\u003c/a\u003e (KG) equation from first principles. In keeping with physicist’s penchant for assigning people’s names to concepts that would otherwise be very easy to understand if just spelled out, the total energy of the system is also called the \u003cstrong\u003eHamiltonian\u003c/strong\u003e (\u003cspan class=\"math inline\"\u003e\\(H\\)\u003c/span\u003e), and standard Newtonian physics can all be derived from the appropriate Hamiltonian (which is what W. R. Hamilton did).\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis motif of using the total energy of the system to derive basic physical laws seems to work quite well in many cases, and is thus the primary way that such laws are derived for different definitions of the total energy. Essentially, the physical laws are latent in any given definition of total energy, and really amount to specifying the dynamics by which energy gets moved around in different ways, without ever gaining or losing any total energy.\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis \u003ca href=\"https://www.youtube.com/watch?v=Q10_srZ-pbs\" target=\"_blank\"\u003eVeritasium\u003c/a\u003e video is strongly recommended for the history of the development of this approach, which started with the \u003cstrong\u003eprinciple of least action\u003c/strong\u003e developed originally by Pierre Louis Maupertuis. The second video in this series of two shows how this same principle applies to the \u003ca href=\"https://www.youtube.com/watch?v=qJZ1Ez28C-A\" target=\"_blank\"\u003epath integral\u003c/a\u003e formulation of quantum mechanics, and compellingly demonstrates how quantum waves really must be spreading out through space in order to \u003cem\u003efind the right path for a particle to take\u003c/em\u003e.\u003c/p\u003e\u003cp id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn anthropomorphic terms, the quantum wave function is like a perceptual system for particles, telling them how to move in order to go in the most efficient way! And the Hamiltonian provides a mathematical framework for defining what “the most efficient way” actually means.\u003c/p\u003e\u003cp id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe form of the total energy that is used as a starting point determines the scope of the physics that the resulting equations of motion support. The KG equations are derived directly from the relativistic total energy (from Einstein’s \u003ca href=\"special-relativity\" target=\"_blank\"\u003especial relativity\u003c/a\u003e), so they therefore automatically produce all of that relativistic phenomena. However, \u003ca href=\"schrodinger\" target=\"_blank\"\u003eSchrodinger\u003c/a\u003e’s equation is derived from a \u003cem\u003eNewtonian\u003c/em\u003e total energy function, which means that it does \u003cem\u003enot\u003c/em\u003e handle relativistic phenomena. That derivation is shown in\u003c/p\u003e\u003cp id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe Hamiltonian can be extended to include spin and coupling to the EM field, to derive the \u003ca href=\"dirac\" target=\"_blank\"\u003eDirac\u003c/a\u003e equation (which is just a more complicated version of the KG equation). You will see that the total energy equation and the corresponding wave equation are very directly related mathematically, and thus this overall approach of using the total energy to derive the wave equation is a very powerful tool that is important to understand if you want to really understand what these wave equations are doing.\u003c/p\u003e\u003cp id=\"text-6\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eYou should be familiar with our computation of the total energy associated with a simple wave, which we calculated in \u003ca href=\"waves\" target=\"_blank\"\u003ewaves\u003c/a\u003e. There we saw that for each cell element in our wave matrix, the total energy was the sum of the \u003cstrong\u003ekinetic\u003c/strong\u003e and \u003cstrong\u003epotential\u003c/strong\u003e energy, where kinetic energy is a function of how fast the state value is moving, and potential energy is a function of how much stress or tension there was between the state and its neighbors (i.e., the curvature of the space).\u003c/p\u003e\u003cp id=\"text-7\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eNow, we’re going to try to formulate the total energy associated with \u003cem\u003ethe “particle” represented by the entire wave function\u003c/em\u003e, instead of thinking in terms of each individual cell within the wave state. We’ll see that we can compute the resulting total particle energy using local cell-level calculations, but the motivations and logic are different. There are still kinetic and potential contributions to this overall particle energy, but it is the overall velocity (actually momentum, which is just velocity times mass) of the particle, not the individual cell state, that we are concerned with in computing the kinetic energy.\u003c/p\u003e\u003cp id=\"text-8\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAs discussed in \u003ca href=\"special-relativity\" target=\"_blank\"\u003especial relativity\u003c/a\u003e, the relativistic total energy of a particle moving with momentum \u003cspan class=\"math inline\"\u003e\\(\\vec{p}\\)\u003c/span\u003e and having a rest mass \u003cspan class=\"math inline\"\u003e\\(m_0\\)\u003c/span\u003e is given by the following equation:\u003c/p\u003e\u003cdiv id=\"inline-container-10\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_total-energy\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_total-energy\"\u003e\u003cb\u003eEq 1:\u003c/b\u003e total relativistic energy\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-11\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nE^2 = \\vec{p}^2 c^2 + (m_0 c^2)^2\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-12\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nE = \\sqrt{\\vec{p}^2 c^2 + (m_0 c^2)^2}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-13\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe fact that these two components of energy, \u003cem\u003emomentum\u003c/em\u003e and \u003cem\u003erest mass\u003c/em\u003e, add together only when squared, ended up causing a remarkable amount of grief and confusion about the meaning of the KG and Dirac equations. Taking the square root of a sum is not a very friendly mathematical operation, and it automatically leads to the possibility of both positive and negative valued solutions. We are able to largely avoid these problems in our overall approach, however, in part by just using the squared energy instead of the raw energy. If the energy is conserved, so is its square!\u003c/p\u003e\u003cp id=\"text-14\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eTo do anything with this total energy equation, we need to be able to compute the momentum and the energy values from our wave states. We have already indicated that the momentum (velocity) of the wave is proportional to the amount of curvature in the wave state — more curve = faster velocity. This can be formalized with the appropriate constants in the following \u003cstrong\u003emomentum operator\u003c/strong\u003e (the little hat \u003cspan class=\"math inline\"\u003e\\(\\hat{}\\)\u003c/span\u003e indicates that this is an operator to be applied to a wave state) which operates over each cell in the entire wave state to produce the associated momentum of the particle:\u003c/p\u003e\u003cdiv id=\"inline-container-16\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_momentum-op\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_momentum-op\"\u003e\u003cb\u003eEq 2:\u003c/b\u003e momentum operator\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-17\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\hat{p} = -i \\hbar \\vec{\\nabla}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-18\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWe highlight this because we’ll keep using it again and again as we work our way up to the Dirac equation. Do not be alarmed by the presence of the \u003cspan class=\"math inline\"\u003e\\(-i\\)\u003c/span\u003e \u003cem\u003eimaginary number\u003c/em\u003e at the start of this equation — we’ll get rid of it soon enough. See \u003ca href=\"complex-numbers\" target=\"_blank\"\u003ecomplex numbers\u003c/a\u003e for more information if you want to brush up on your knowledge of these seemingly strange numbers at this point — you’ll need to really understand them in detail to understand the \u003ca href=\"schrodinger\" target=\"_blank\"\u003eSchrodinger\u003c/a\u003e equation. They really are very simple once you get past all the imaginary business and recognize their actual practical application.\u003c/p\u003e\u003cp id=\"text-19\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe total energy operator for the wave state is computed in terms of the overall rate of change across all the wave cells, consistent with the notion that energy is a function of the velocity (kinetic energy) of the cell states:\u003c/p\u003e\u003cdiv id=\"inline-container-21\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_energy-op\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_energy-op\"\u003e\u003cb\u003eEq 3:\u003c/b\u003e energy operator\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-22\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\hat{E} = i \\hbar \\frac{\\partial }{\\partial t}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-23\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eagain we see the imaginary number and \u003cspan class=\"math inline\"\u003e\\(\\hbar\\)\u003c/span\u003e constants here.\u003c/p\u003e\u003cp id=\"text-24\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eTo see how this all fits together, we now substitute in these operators in place of the momentum and energy terms in the above energy equation (squared version), and have them operate on the wave state values. The imaginary numbers turn into 1’s and -1’s in the process of the squaring, and thus very kindly disappear:\u003c/p\u003e\u003cp id=\"text-25\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nE^2 = \\vec{p}^2 c^2 + (m_0 c^2)^2\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-26\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n-\\hbar^2 \\frac{\\partial^2 {\\phi}}{\\partial t^2} = \\left (-\\hbar^2 \\nabla^2 c^2 + (m_0 c^2)^2 \\right) \\phi\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-27\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2 {\\phi}}{\\partial t^2} = \\left( c^2 \\nabla^2 - \\frac{(m_0 c^2)^2}{\\hbar^2} \\right) \\phi\n\\]\u003c/span\u003e\u003c/p\u003e\u003cdiv id=\"inline-container-29\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_klein-gordon\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_klein-gordon\"\u003e\u003cb\u003eEq 4:\u003c/b\u003e Klein-Gordon equation\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-30\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2 {\\phi}}{\\partial t^2} = c^2 \\left(\\nabla^2 - \\frac{m_0^2 c^2}{\\hbar^2}\\right) \\phi\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-31\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAnd amazingly, at the end, we recover exactly the \u003ca href=\"klein-gordon\" target=\"_blank\"\u003eKlein-Gordon\u003c/a\u003e equation!\u003c/p\u003e\u003cp id=\"text-32\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis means that the propagation of the particle defined by a KG wave function will conserve the total relativistic energy over time, simply by virtue of following the local wave dynamics. This means that a KG wave-particle will obey the physics of special relativity in every way. And in the case when the wave velocity is low compared to the speed of light, then the system will behave according to the laws of Newtonian physics, because special relativity reduces to standard Newtonian physics in this case.\u003c/p\u003e\u003cp id=\"text-33\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn short, \u003ca href=\"hamiltonian#eq_klein-gordon\" target=\"_blank\"\u003eEq 4\u003c/a\u003e contains such a huge scope of physics in such a simple package — it is truly the most \u003cem\u003eefficient\u003c/em\u003e representation of the \u003cem\u003eessence\u003c/em\u003e of physics. Nevertheless, it is not sufficient to describe any \u003cem\u003eactual\u003c/em\u003e physical particle, except perhaps the Higgs boson.\u003c/p\u003e\u003ch2 id=\"four-vector-version\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eFour-Vector Version\u003c/h2\u003e\u003cp id=\"text-35\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eBefore continuing, we explore the ultra-compact version of the above derivations that is possible using the four-vector space-time coordinate system introduced in \u003ca href=\"maxwell\" target=\"_blank\"\u003emaxwell\u003c/a\u003e. In this system, the \u003cem\u003efour-momentum\u003c/em\u003e operator is defined as:\u003c/p\u003e\u003cp id=\"text-36\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n{\\hat{p}^\\mu} = i \\hbar \\partial^\\mu\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-37\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n= i \\hbar \\left(\\frac{\\partial {}}{\\partial {ct}},-\\frac{\\partial {}}{\\partial {x}},-\\frac{\\partial {}}{\\partial {y}},- \\frac{\\partial {}}{\\partial {z}}\\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-38\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n= i \\hbar \\left(\\frac{\\partial {}}{\\partial {ct}}, -\\vec{\\nabla} \\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-39\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n= \\left(\\frac{\\hat{E}}{c}, \\hat{p}\\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-40\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAs indicated, it should be clear that the first (time) component of this is the energy operator given above, while the spatial components are the momentum operator (\u003ca href=\"hamiltonian#eq_momentum-op\" target=\"_blank\"\u003eEq 2\u003c/a\u003e):\u003c/p\u003e\u003cp id=\"text-41\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\hat{E} = i \\hbar \\frac{\\partial}{\\partial t}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-42\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\hat{p} = -i \\hbar \\vec{\\nabla}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-43\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIf you then square this momentum operator (using the standard covariant, contravariant four-vector definition of multiplying vectors), you get:\u003c/p\u003e\u003cp id=\"text-44\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n{\\hat{p}^\\mu} {\\hat{p}_\\mu} = \\frac{\\hat{E}^2}{c^2} - \\hat{p}^2\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-45\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eNow this is quite interesting, because it can be directly related to the relativistic energy-momentum equation:\u003c/p\u003e\u003cp id=\"text-46\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nE^2 = \\vec{p}^2 c^2 + (m_0 c^2)^2 \n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-47\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{E^2}{c^2} = \\vec{p}^2 + m_0^2 c^2\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-48\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{E^2}{c^2} - \\vec{p}^2 = m_0^2 c^2\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-49\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSo we can now put these two equations together, to get:\u003c/p\u003e\u003cp id=\"text-50\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n{\\hat{p}^\\mu} {\\hat{p}_\\mu} = m_0^2 c^2\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-51\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eNow, we also know that the wave equation arises from the second-order derivatives of a four-vector:\u003c/p\u003e\u003cp id=\"text-52\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\partial_\\mu \\partial^\\mu = \\frac{1}{c^2}\\frac{\\partial^2 {}}{\\partial t^2} - \\nabla^2\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-53\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eand this four-momentum operator is essentially just a first-order derivative with some additional constants (including the \u003cspan class=\"math inline\"\u003e\\(i\\)\u003c/span\u003e term, which, when squared produces a \u003cspan class=\"math inline\"\u003e\\(-1\\)\u003c/span\u003e), so when we apply these operators to our wave variable \u003cspan class=\"math inline\"\u003e\\(\\phi\\)\u003c/span\u003e, we get:\u003c/p\u003e\u003cp id=\"text-54\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n{\\hat{p}^\\mu} {\\hat{p}_\\mu} \\phi = m_0^2 c^2 \\phi\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-55\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\ni^2 \\hbar^2 \\partial_\\mu \\partial^\\mu \\phi = m_0^2 c^2 \\phi\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-56\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n- \\hbar^2 \\left(\\frac{1}{c^2}\\frac{\\partial^2}{\\partial t^2} - \\nabla^2\\right) \\phi = m_0^2 c^2 \\phi\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-57\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2 {\\phi}}{\\partial t^2} = c^2 \\left(\\nabla^2 - \\frac{m_0^2c^2}{\\hbar^2}\\right) \\phi\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-58\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWhich is right back to our KG wave equation.\u003c/p\u003e\u003cp id=\"text-59\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn summary, the core of the KG wave equation, and of special relativity, and much of quantum mechanics, can all be reduced to this one simple equation:\u003c/p\u003e\u003cdiv id=\"inline-container-61\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_klein-gordon\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_klein-gordon\"\u003e\u003cb\u003eEq 4:\u003c/b\u003e relativistic Hamiltonian\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-62\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\hat{p}^\\mu \\hat{p}_\\mu \\phi = m_0^2 c^2 \\phi\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-63\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis is in some sense the most fundamental equation of the universe, subsuming the more popular \u003cspan class=\"math inline\"\u003e\\(E = m c^2\\)\u003c/span\u003e and, as we’ll see, providing the basis for further extensions to the KG equation.\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"gravity\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"m274-450 248 248-42 42-320-320 320-320 42 42-248 248h526v60H274Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" 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style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Harmonic oscillator","URL":"harmonic-oscillator","Title":"Harmonic oscillator","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{"eq":["eq_force","eq_a","eq_","eq_"],"sim":["sim_sho"]},"Description":"The \u003cstrong\u003esimple harmonic oscillator\u003c/strong\u003e (SHO) captures the core oscillatory behavior of \u003ca href=\"waves\" target=\"_blank\"\u003ewaves\u003c/a\u003e, without any spatial dimensions to bother with. It can be seen as the 0-dimensional version of a wave, where the force that drives the oscillation comes not from neighbors, but from the position (height) of the wave itself. As such, it provides a potentially interesting role in the mechanics of \u003ca href=\"stochastic-particles\" target=\"_blank\"\u003estochastic particles\u003c/a\u003e because it can be entirely localized to one discrete grid cell within the \u003ca href=\"cellular-automaton\" target=\"_blank\"\u003ecellular automaton\u003c/a\u003e framework. Thus, a particle in this view can be considered to be a simple harmonic oscillator that periodically jumps between cells.","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eHarmonic oscillator\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-18\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe \u003cstrong\u003esimple harmonic oscillator\u003c/strong\u003e (SHO) captures the core oscillatory behavior of \u003ca href=\"waves\" target=\"_blank\"\u003ewaves\u003c/a\u003e, without any spatial dimensions to bother with. It can be seen as the 0-dimensional version of a wave, where the force that drives the oscillation comes not from neighbors, but from the position (height) of the wave itself. As such, it provides a potentially interesting role in the mechanics of \u003ca href=\"stochastic-particles\" target=\"_blank\"\u003estochastic particles\u003c/a\u003e because it can be entirely localized to one discrete grid cell within the \u003ca href=\"cellular-automaton\" target=\"_blank\"\u003ecellular automaton\u003c/a\u003e framework. Thus, a particle in this view can be considered to be a simple harmonic oscillator that periodically jumps between cells.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe basic equations from \u003ca href=\"waves\" target=\"_blank\"\u003ewaves\u003c/a\u003e for the SHO are:\u003c/p\u003e\u003cdiv id=\"inline-container-3\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_force\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_force\"\u003e\u003cb\u003eEq 1:\u003c/b\u003e restoring force\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nf = -c^2 y^t\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhere \u003cspan class=\"math inline\"\u003e\\(c\\)\u003c/span\u003e is the basic rate update constant, analogous to the speed of light in waves, which determines the effective strength of the restoring force, and thus the oscillation rate. The \u003cem\u003et\u003c/em\u003e suffix indicates the time step (only for variables that require integration over time). Everything else from this point onward is the same, in the basic Newtonian physics framework of acceleration, velocity, and position:\u003c/p\u003e\u003cdiv id=\"inline-container-7\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_a\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_a\"\u003e\u003cb\u003eEq 2:\u003c/b\u003e acceleration\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-8\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\na = \\frac{f}{m}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cdiv id=\"inline-container-10\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_\"\u003e\u003cb\u003eEq 3:\u003c/b\u003e new velocity\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-11\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nv^{t+1} = v^t + a\n\\]\u003c/span\u003e\u003c/p\u003e\u003cdiv id=\"inline-container-13\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_\"\u003e\u003cb\u003eEq 3:\u003c/b\u003e new state\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-14\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\ny^{t+1} = y^t + v^{t+1}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cdiv id=\"sim_sho\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"inline-container-16\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"sim_sho\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"sim_sho\"\u003e\u003cb\u003eSim 1:\u003c/b\u003e Simple harmonic oscillator\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003ccollapser id=\"collapser-1\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"frame-0\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.1em;font-weight:thin;text-align:start\"\u003e\u003cinput id=\"toggle\" style=\"color:var(--primary-color);display:flex;flex-direction:row;justify-content:center;align-items:center;font-weight:thin;text-align:center;border-radius:8px\" type=\"checkbox\" value=\"false\"\u003e\u003c/input\u003e\u003cp id=\"text-1\" style=\"font-weight:normal;line-height:1.5;text-align:start\"\u003eSimple harmonic oscillator\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"frame-1\" style=\"display:none;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ceditor id=\"editor-0\" style=\"background:var(--surface-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:10em;padding-top:0.5em;padding-right:0.5em;padding-bottom:0.5em;padding-left:0.5em;font-weight:thin;line-height:1.3;text-align:start;border-radius:16px\"\u003e\u003c/editor\u003e\u003c/div\u003e\u003c/collapser\u003e\u003cdiv id=\"frame-2\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cplot id=\"plot-0\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:512px;min-height:384px;width:512px;height:384px;font-weight:thin;text-align:start\"\u003e\u003c/plot\u003e\u003cp id=\"text-1\" style=\"min-width:80ch;width:80ch;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eTotal E: 0.51 \u003c/b\u003e\u003c/p\u003e\u003cp id=\"text-2\" style=\"min-width:40ch;width:40ch;font-weight:normal;line-height:1.5;text-align:start\"\u003emass: 0.5\u003c/p\u003e\u003cinput id=\"slider-3\" style=\"background:#E1E2EB;display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:20em;min-height:1em;width:20em;height:1em;padding-top:8px;padding-right:8px;padding-bottom:8px;padding-left:8px;font-weight:thin;text-align:start;border-radius:1e+09px\" type=\"range\" value=\"0.5\"\u003e\u003c/input\u003e\u003cp id=\"text-4\" style=\"min-width:40ch;width:40ch;font-weight:normal;line-height:1.5;text-align:start\"\u003ec: 0.2\u003c/p\u003e\u003cinput id=\"slider-5\" style=\"background:#E1E2EB;display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:20em;min-height:1em;width:20em;height:1em;padding-top:8px;padding-right:8px;padding-bottom:8px;padding-left:8px;font-weight:thin;text-align:start;border-radius:1e+09px\" type=\"range\" value=\"0.2\"\u003e\u003c/input\u003e\u003c/div\u003e\u003c/div\u003e\u003cp id=\"text-18\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe equivalent of the wavelength for the SHO is the \u003cem\u003eperiod\u003c/em\u003e, in time, for the cycle to repeat.\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"hamiltonian\"\u003e\u003csvg id=\"icon\" 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style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Hilbert space","URL":"hilbert-space","Title":"Hilbert space","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{},"Description":"By far the most widely-used calculational framework in standard QM is the algebraic \u003cstrong\u003ematrix mechanics\u003c/strong\u003e approach, pioneered by Heisbenberg, Dirac, Hilbert, von Neumann and others in the mid 1920s. It involves \u003cem\u003estate vector\u003c/em\u003e representations of the state of a system, encoded via complex-valued vectors representing \u003cem\u003eprobability amplitudes\u003c/em\u003e (i.e., a \u003cstrong\u003eHilbert space\u003c/strong\u003e). This state vector is a specific way of encoding the \u003ca href=\"configuration-space\" target=\"_blank\"\u003econfiguration space\u003c/a\u003e of the entire set of relevant variables, and is thus manifestly non-local, and represents the entire state a given point in time, in a way that is thus incompatible with the principles of relativity.","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eHilbert space\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-19\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eBy far the most widely-used calculational framework in standard QM is the algebraic \u003cstrong\u003ematrix mechanics\u003c/strong\u003e approach, pioneered by Heisbenberg, Dirac, Hilbert, von Neumann and others in the mid 1920s. It involves \u003cem\u003estate vector\u003c/em\u003e representations of the state of a system, encoded via complex-valued vectors representing \u003cem\u003eprobability amplitudes\u003c/em\u003e (i.e., a \u003cstrong\u003eHilbert space\u003c/strong\u003e). This state vector is a specific way of encoding the \u003ca href=\"configuration-space\" target=\"_blank\"\u003econfiguration space\u003c/a\u003e of the entire set of relevant variables, and is thus manifestly non-local, and represents the entire state a given point in time, in a way that is thus incompatible with the principles of relativity.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis state vector evolves under \u003cem\u003eunitary\u003c/em\u003e transformations (rotations in the complex vector space), which preserve the overall magnitudes of the vectors, even as they rotate around in the space. The unitary nature of the rotation transformations represents the behavior of the system when it is being governed by the \u003ca href=\"schrodinger\" target=\"_blank\"\u003eSchrodinger\u003c/a\u003e wave dynamics under the Copenhagen dualistic framework, which perfectly preserves the overall underlying probability space as long as nobody “looks at it the wrong way” (i.e., makes a measurement). Then, at the end, a “measurement” is made by collapsing the probability space down to a single discrete outcome (i.e., along an eigenvector of the resulting state).\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis matrix formalism is equivalent to a self-consistent form of probability theory, which can be derived from abstract axioms having nothing to do with quantum physics (\u003ca href=\"ref://Gleason75\" target=\"_blank\"\u003eGleason, 1975\u003c/a\u003e; \u003ca href=\"ref://Jaynes90\" target=\"_blank\"\u003eJaynes, 1990\u003c/a\u003e; \u003ca href=\"ref://CavesFuchsSchack02\" target=\"_blank\"\u003eCaves et al., 2002\u003c/a\u003e; \u003ca href=\"ref://FuchsMerminSchack14\" target=\"_blank\"\u003eFuchs et al., 2014\u003c/a\u003e; \u003ca href=\"ref://Mermin18\" target=\"_blank\"\u003eMermin, 2018\u003c/a\u003e). Indeed, this framework is so general that its only real physical commitment is that quantum physics obeys strict conservation laws: if you start with X amount of spin distributed however uncertainly across some state variables, then you have to end up with the same total uncertainty in spin distribution at the end, prior to the final measurement step, when everything collapses.\u003c/p\u003e\u003cp id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThus, the claim that standard QM is such a successful framework must be understood within this context: yes, it is accurate in capturing this basic fact of conservation, but it really isn’t going very far out on a limb here: nothing wagered, nothing lost; but also perhaps not so much gained.\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"harmonic-oscillator\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"m274-450 248 248-42 42-320-320 320-320 42 42-248 248h526v60H274Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003ePrevious\u003c/p\u003e\u003c/a\u003e\u003cstretch id=\"content-buttons-go-167\" style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1ch;min-height:1em;font-weight:thin;text-align:start\"\u003e\u003c/stretch\u003e\u003ca id=\"content-buttons-go-168\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"history\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M686-450H160v-60h526L438-758l42-42 320 320-320 320-42-42 248-248Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"History","URL":"history","Title":"History","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{"eq":["eq_energy","eq_borh","eq_momentum","eq_debh","eq_uncertainty"],"figure":["figure_double-slit-deb","figure_double-slit-kocsis"]},"Description":"The earliest seeds of quantum physics began right around the new century of 1900, with Planck’s solution to the \u003cem\u003eultraviolet catastrophe\u003c/em\u003e in the theory and data on black-body radiation. The classical mathematical treatment (according to the Rayleigh-Jeans law) predicts an infinite amount of energy should be released from an ideal perfectly-absorbent and perfectly re-radiating body (black body) in thermal equilibrium. This divergence arises as the frequency of absorbed and re-emitted light progressively gets higher, because higher-frequency light has more energy, and thus you end up with an infinite positive feedback loop from ever-higher frequencies.","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eHistory\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-20\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe earliest seeds of quantum physics began right around the new century of 1900, with Planck’s solution to the \u003cem\u003eultraviolet catastrophe\u003c/em\u003e in the theory and data on black-body radiation. The classical mathematical treatment (according to the Rayleigh-Jeans law) predicts an infinite amount of energy should be released from an ideal perfectly-absorbent and perfectly re-radiating body (black body) in thermal equilibrium. This divergence arises as the frequency of absorbed and re-emitted light progressively gets higher, because higher-frequency light has more energy, and thus you end up with an infinite positive feedback loop from ever-higher frequencies.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFurthermore the data from experiments consistently showed that these higher frequencies were in fact strongly diminished, and ultimately absent at higher frequencies, relative to the classical predictions. Planck found an empirically-derived equation that explained the data, by introducing a \u003cem\u003equantization\u003c/em\u003e in the process by which light is absorbed and emitted. Specifically, the absorption and emission of energy must occur in multiples of a parameter \u003cem\u003eh\u003c/em\u003e (Planck’s constant) times the frequency \u003cspan class=\"math inline\"\u003e\\(\\nu\\)\u003c/span\u003e:\u003c/p\u003e\u003cdiv id=\"inline-container-3\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_energy\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_energy\"\u003e\u003cb\u003eEq 1:\u003c/b\u003e Energy = h times frequency\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nE = h \\nu\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eEinstein took this empirical finding and turned it into a principle, by explaining the \u003cem\u003ephotoelectric effect\u003c/em\u003e, through the introduction of the idea of \u003cem\u003elight quanta\u003c/em\u003e (which were much later named \u003cem\u003ephotons\u003c/em\u003e; \u003ca href=\"ref://Klassen11\" target=\"_blank\"\u003eKlassen, 2011\u003c/a\u003e) that only come in these quantized units (\u003ca href=\"ref://Einstein05a\" target=\"_blank\"\u003eEinstein, 1905\u003c/a\u003e). Interestingly, this perspective introduced an important change in the nature of the physical model: instead of the absorption and emission \u003cem\u003eprocess\u003c/em\u003e being subject to quantization, Einstein hypothesized the quantization was due to the nature of light itself (see \u003ca href=\"semiclassical\" target=\"_blank\"\u003esemiclassical\u003c/a\u003e for more discussion).\u003c/p\u003e\u003cp id=\"text-6\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe light quanta idea was initially rejected by most physicists (\u003ca href=\"ref://Klassen11\" target=\"_blank\"\u003eKlassen, 2011\u003c/a\u003e), but found later resonance (so to speak) in the work of Niels Bohr trying to understand the nature of simple atoms like hydrogen, which has a single electron orbiting a nucleus. The dominant classical physical model of the atomic system in the early 1900’s was the Rutherford model of 1911, with electrons as tiny points of charge and mass, orbiting a nucleus, much like planets orbiting the sun. This model had important failings, which the full development of quantum mechanics resolved, thus cementing the demise of the classical worldview, and solidly establishing quantum mechanics.\u003c/p\u003e\u003cp id=\"text-7\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe major problem with the classical atom was that it is fundamentally unstable: the electron should emit electromagnetic radiation as it orbits around the nucleus, and thus lose energy. As it loses energy, the orbit must get tighter, and eventually the electron should just collapse into the nucleus, just like one of those quarters you roll around in a gravity well at a science museum. Furthermore, as its orbit gets tighter, it should emit higher frequency radiation, predicting a continuous and increasingly high frequency emission spectrum. Instead, it was known that atoms emit consistent, discrete frequencies of radiation.\u003c/p\u003e\u003cp id=\"text-8\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIt was yet another ultraviolet catastrophe! And the same solution came to the rescue. In 1913, Bohr postulated that electrons can only have orbits where the angular momentum (i.e., the effective period of the orbit) is restricted to an integer multiple \u003cspan class=\"math inline\"\u003e\\(n\\)\u003c/span\u003e of Planck’s constant:\u003c/p\u003e\u003cdiv id=\"inline-container-10\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_borh\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_borh\"\u003e\u003cb\u003eEq 2:\u003c/b\u003e Bohr wavelength = integer multiple of h\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-11\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nL = n \\hbar = n \\frac{h}{2\\pi}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-12\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAlthough the reason for this restriction was not clear, it immediately made sense of a great deal of data, including the Rydberg formula for hydrogen emission spectra. The justification for Bohr’s restriction on atomic orbits came in 1924, when Louis de Broglie proposed that electrons have a wave-like nature, and thus the only frequencies of electron wave vibration that are stable are standing waves. Standing waves must have an integer number of wavelengths, such that within the orbiting electron model, the electron orbits are constrained to have an integer number of such waves per orbit.\u003c/p\u003e\u003cp id=\"text-13\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eShortly thereafter, in 1926, Erwin Schrödinger developed his famous wave equation, which then gave a complete mathematical description of the behavior of bound electrons in atomic systems, which made sense of even more data than Bohr’s original model.\u003c/p\u003e\u003cp id=\"text-14\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe experimental confirmation of de Broglie’s matter wave hypothesis came in 1927 in an experiment by Davisson and Germer, who found that electrons moving through a crystal exhibit a diffraction pattern, consistent with a wave-like property. Calculations showed that the de Broglie wavelength predicted for the electrons fit the observed diffraction pattern quite well:\u003c/p\u003e\u003cdiv id=\"inline-container-16\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_momentum\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_momentum\"\u003e\u003cb\u003eEq 3:\u003c/b\u003e Momentum = h / de Broglie wavelength\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-17\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\np = \\frac{h}{\\lambda}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cdiv id=\"inline-container-19\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_debh\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_debh\"\u003e\u003cb\u003eEq 4:\u003c/b\u003e de Broglie wavelength = h / momentum\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-20\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\lambda = \\frac{h}{p}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-21\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis wavelength is about .165 nanometers for the electrons in the Davisson-Germer experiment (very tiny, but enough to produce a measurable diffraction pattern through the crystal).\u003c/p\u003e\u003cp id=\"text-22\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eBoth de Broglie and Schrödinger thought that these \u003ca href=\"matter-waves\" target=\"_blank\"\u003ematter waves\u003c/a\u003e were real physical things, like light waves. Furthermore, de Broglie suggested that the wave acted to \u003cem\u003eguide\u003c/em\u003e the point particle electron around, in his \u003ca href=\"pilot-wave\" target=\"_blank\"\u003epilot-wave\u003c/a\u003e theory. Schrödinger initially had an even more radical view, which abandoned the point electron entirely: he thought his wave equation described a wave of \u003cem\u003echarge density\u003c/em\u003e that \u003cem\u003eis\u003c/em\u003e the actual electron, without any need for a dual particle-like entity.\u003c/p\u003e\u003cp id=\"text-23\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eHowever, both of these attempts to provide a “physically realistic” perspective on the phenomena were quickly abandoned in the face of further evidence suggesting that the wave function fundamentally describes the \u003cem\u003eprobability\u003c/em\u003e that a particle might appear at a given point in space when measured. As such, the wave is somehow “non physical”, and yet exerts physically-measurable effects.\u003c/p\u003e\u003cp id=\"text-24\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis movement away from a physical model was furthered by Heisenberg and his \u003ca href=\"uncertainty-principle\" target=\"_blank\"\u003euncertainty principle\u003c/a\u003e, and the broader principle of \u003cem\u003ecomplementarity\u003c/em\u003e: there is a minimum, irreducible amount of ambiguity or uncertainty in the quantum world, and you can \u003cem\u003eeither\u003c/em\u003e push things more toward one perspective (e.g., the wave-like aspect) \u003cem\u003eor\u003c/em\u003e the other (the particle-like aspect), but not both at the same time. This philosophical stance was based initially on the simple fact that the uncertainty in the position (\u003cspan class=\"math inline\"\u003e\\(x\\)\u003c/span\u003e) or momentum (\u003cspan class=\"math inline\"\u003e\\(p\\)\u003c/span\u003e) of a particle has a minimum bound of a factor of Planck’s constant:\u003c/p\u003e\u003cdiv id=\"inline-container-26\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_uncertainty\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_uncertainty\"\u003e\u003cb\u003eEq 5:\u003c/b\u003e Uncertainty in position * momentum\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-27\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\sigma_x \\sigma_p \\geq \\frac{\\hbar}{2}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-28\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThus, any attempt to decrease the uncertainty in location \u003cspan class=\"math inline\"\u003e\\(\\sigma_x\\)\u003c/span\u003e necessarily increases the uncertainty in momentum \u003cspan class=\"math inline\"\u003e\\(\\sigma_p\\)\u003c/span\u003e. This can be seen as a natural consequence of \u003ca href=\"matter-waves\" target=\"_blank\"\u003ematter waves\u003c/a\u003e, and more generally, because a wave is a spatially distributed thing, it is very hard to pin down precisely.\u003c/p\u003e\u003cp id=\"text-29\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFrom this complementarity principle, Bohr and Heisenberg developed the \u003ca href=\"copenhagen\" target=\"_blank\"\u003eCopenhagen\u003c/a\u003e interpretation of QM in the late 1920’s, and this is still dominant today. Central to this interpretation is the notion that the physical world operates in two complementary modes: you are \u003cem\u003eeither\u003c/em\u003e making a \u003cem\u003emeasurement\u003c/em\u003e, which causes the wave function to \u003cem\u003ecollapse\u003c/em\u003e down to a single discrete particle-like point, \u003cem\u003eor\u003c/em\u003e physics is otherwise evolving according to the wave function, which critically preserves all the quantum uncertainty, and just rotates it around in a \u003cem\u003eunitary\u003c/em\u003e manner over time.\u003c/p\u003e\u003cp id=\"text-30\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eDuring this wave-function mode, the mathematical picture suggests that there is no definitive underlying state of the world: everything is in some kind of probabilistic superposition of possible states. Only once you measure something does it actually exist in any kind of definite way, leading to the mantra that “the world only exists when you measure it”.\u003c/p\u003e\u003cp id=\"text-31\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis strong discretization of the laws of physics is at the root of many seeming paradoxes and puzzles in understanding the quantum world: what exactly defines a “measurement” at a fundamental level? How can the wave function, which could conceivably spread out over large macroscopic spaces over time, instantaneously collapse down to a single point within that entire space? Despite these conceptual difficulties, the mathematics of the framework allow straightforward calculations that match the outcomes of actual experiments, leading to a general attitude of “shut up and calculate”: don’t bother with unnecessary considerations of the actual underlying physical ontology, just do the math!\u003c/p\u003e\u003cp id=\"text-32\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eHowever, in the 1950’s, David Bohm reinvented the original \u003cem\u003epilot-wave\u003c/em\u003e model of de Broglie, and showed that in fact it can fully explain the same phenomena as the standard Copenhagen QM framework. Critically, in this \u003cem\u003ede Broglie-Bohm pilot-wave\u003c/em\u003e framework, \u003cem\u003eparticles always have a definite well-defined location\u003c/em\u003e. There is no longer a complementary discretization of the world into measurement vs. wave-function evolution phases: the two are \u003cem\u003ealways\u003c/em\u003e operating hand-in-glove, all the time.\u003c/p\u003e\u003cdiv id=\"figure_double-slit-deb\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_double-slit-deb\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 1:\u003c/b\u003e \u003cp\u003eTrajectories for particles in the double-slit experiment computed according to the de Broglie-Bohm pilot-wave model. The interference effects can be seen as relatively localized bumps in the trajectories, corresponding to steep gradients in the Schrödinger wave equation. Critically, the underlying trajectories are considered to exist at all points even if you don’t happen to observe them.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cp id=\"text-35\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003ca href=\"history#figure_double-slit-deb\" target=\"_blank\"\u003eFigure 1\u003c/a\u003e shows what the underlying trajectories of particles under the pilot-wave framework look like in a double-slit experiment, and \u003ca href=\"history#figure_double-slit-kocsis\" target=\"_blank\"\u003eFigure 2\u003c/a\u003e shows some recent data from an experiment where \u003cem\u003eweak measurements\u003c/em\u003e that minimally disturb the system allow one to infer particle trajectories, which look remarkably similar to those predicted by the pilot-wave model (\u003ca href=\"ref://KocsisBravermanRavetsEtAl11\" target=\"_blank\"\u003eKocsis et al., 2011\u003c/a\u003e).\u003c/p\u003e\u003cdiv id=\"figure_double-slit-kocsis\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_double-slit-kocsis\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 2:\u003c/b\u003e \u003cp\u003eReconstructed trajectories of photons in a double-slit experiment using a weak measurement technique that allows aggregate trajectory information to be reconstructed over many repeated samples that are post-sorted according to a weak additional modulation of the system — these are not individual particle trajectories. There is a striking correspondence to the predictions of the de Broglie-Bohm model. Figure from Kocsis et al, 2011.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"hilbert-space\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"m274-450 248 248-42 42-320-320 320-320 42 42-248 248h526v60H274Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003ePrevious\u003c/p\u003e\u003c/a\u003e\u003cstretch id=\"content-buttons-go-167\" style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1ch;min-height:1em;font-weight:thin;text-align:start\"\u003e\u003c/stretch\u003e\u003ca id=\"content-buttons-go-168\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M686-450H160v-60h526L438-758l42-42 320 320-320 320-42-42 248-248Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Home","URL":"","Title":"","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{},"Description":"\u003cstrong\u003eWave reality\u003c/strong\u003e is dedicated to exploring the idea that the quantum wave function is \u003cem\u003ereal\u003c/em\u003e, and not just a description of our state of \u003ca href=\"epistemic-vs-ontic\" target=\"_blank\"\u003eepistemological\u003c/a\u003e ignorance. The reality of the wave function is strongly indicated by the classic \u003ca href=\"double-slit\" target=\"_blank\"\u003edouble-slit\u003c/a\u003e experiment results, where some kind of spatially-distributed wave-like interference phenomenon seems to be influencing the trajectories of discrete particles. In addition, there are increasingly strong theoretical and empirical attempts to show that a purely epistemic account contradicts quantum theory (\u003ca href=\"ref://PuseyBarrettRudolph12\" target=\"_blank\"\u003ePusey et al., 2012\u003c/a\u003e, \u003ca href=\"ref://RingbauerDuffusBranciardEtAl15\" target=\"_blank\"\u003eRingbauer et al., 2015\u003c/a\u003e).","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-21\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"image-0\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cstrong\u003eWave reality\u003c/strong\u003e is dedicated to exploring the idea that the quantum wave function is \u003cem\u003ereal\u003c/em\u003e, and not just a description of our state of \u003ca href=\"epistemic-vs-ontic\" target=\"_blank\"\u003eepistemological\u003c/a\u003e ignorance. The reality of the wave function is strongly indicated by the classic \u003ca href=\"double-slit\" target=\"_blank\"\u003edouble-slit\u003c/a\u003e experiment results, where some kind of spatially-distributed wave-like interference phenomenon seems to be influencing the trajectories of discrete particles. In addition, there are increasingly strong theoretical and empirical attempts to show that a purely epistemic account contradicts quantum theory (\u003ca href=\"ref://PuseyBarrettRudolph12\" target=\"_blank\"\u003ePusey et al., 2012\u003c/a\u003e, \u003ca href=\"ref://RingbauerDuffusBranciardEtAl15\" target=\"_blank\"\u003eRingbauer et al., 2015\u003c/a\u003e).\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe \u003ca href=\"pilot-wave\" target=\"_blank\"\u003epilot-wave\u003c/a\u003e framework of de Broglie and Bohm (\u003ca href=\"ref://Bohm52\" target=\"_blank\"\u003eBohm, 1952\u003c/a\u003e; \u003ca href=\"ref://Norsen22a\" target=\"_blank\"\u003eNorsen, 2022\u003c/a\u003e), which posits a real quantum wave function guiding a discrete particle around, very naturally and intuitively explains all of the otherwise strange phenomena in these and many other classic quantum physics experiments. A major conclusion from many converging angles discussed here is that this \u003cstrong\u003ewave-particle\u003c/strong\u003e \u003ca href=\"duality\" target=\"_blank\"\u003eduality\u003c/a\u003e is actually a critically important \u003cem\u003efeature\u003c/em\u003e of how physics must work to “solve” fundamental problems, rather than some kind of paradoxical bug that we just have to somehow learn to swallow.\u003c/p\u003e\u003cp id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis website contains a work-in-progress wiki-like collection of documentation in support of the development of a computational model of the phenomenology of quantum electrodynamics (\u003ca href=\"qed\" target=\"_blank\"\u003eQED\u003c/a\u003e), starting with the coupled \u003ca href=\"dirac\" target=\"_blank\"\u003eDirac\u003c/a\u003e – \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e wave functions, along with discrete \u003ca href=\"electron\" target=\"_blank\"\u003eelectron\u003c/a\u003e \u003ca href=\"stochastic-particles\" target=\"_blank\"\u003estochastic particles\u003c/a\u003e, consistent with the \u003ca href=\"pilot-wave\" target=\"_blank\"\u003epilot-wave\u003c/a\u003e framework. This computational model is based on the \u003ca href=\"cellular-automaton\" target=\"_blank\"\u003ecellular automaton\u003c/a\u003e framework, which is arguably the simplest way that physics could autonomously emerge in parallel, everywhere in the universe, all at once.\u003c/p\u003e\u003cp id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe primary goal of this project is to better understand the basic physics of electrons interacting with the electromagnetic field, and to try to sort through some of the notorious paradoxes and conceptual challenges that lie at the heart of quantum mechanics (QM). An easy-to-use GUI-based \u003ca href=\"waves-simulator\" target=\"_blank\"\u003ewaves simulator\u003c/a\u003e is integrated into this content, which allows one to interactively explore various physics models, providing a concrete and hands-on level of understanding. This provides a different and potentially valuable set of tools for someone trying to learn more about how quantum physics actually works, which may result in quicker and deeper understanding than staring at equations :)\u003c/p\u003e\u003cp id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis project is fully committed to \u003cstrong\u003eaccounting for all the empirical data\u003c/strong\u003e — there is no point in coming up with an elegant theory that is obviously false. We cannot impose our own aesthetic preferences onto Nature, and must humbly accept any irrefutable facts that have been reliably empirically established. However, there is a huge underexplored space of possible physical mechanisms that could account for all the data, while avoiding the kinds of obvious conceptual paradoxes that pervade the standard interpretations of quantum physics.\u003c/p\u003e\u003cp id=\"text-6\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFor example, Einstein famously rejected quantum mechanics because “God does not play dice with the universe” — that seems just a bit presumptuous. Indeed, a key element of the framework we develop here requires true randomness in the form of \u003ca href=\"stochastic-particles\" target=\"_blank\"\u003estochastic particles\u003c/a\u003e, and this one concession of nondeterminism can be traded in to retain several other more important (in my opinion) principles, such as a fundamental \u003cstrong\u003elocality\u003c/strong\u003e to physical mechanisms.\u003c/p\u003e\u003cp id=\"text-7\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eA critical conceptual foundation of this approach is to recognize the distinction between \u003cstrong\u003ecalculational tools\u003c/strong\u003e versus \u003cstrong\u003ephysical models\u003c/strong\u003e (\u003ca href=\"tools-vs-models\" target=\"_blank\"\u003etools vs models\u003c/a\u003e). The widespread failure to understand this distinction underlies many of the apparently intractable puzzles in quantum physics, that can be seen as arising from specific choices of calculational tools. There are typically many different ways to calculate a prediction of an experimental result, but presumably Nature is not strategically selecting different calculational tools based on different configurations of elements.\u003c/p\u003e\u003cp id=\"text-8\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eA fundamental assumption here is that Nature must be doing one consistent thing using one set of physical mechanisms, uniformly and consistently across space and time. That thing is what we seek to understand here.\u003c/p\u003e\u003ch2 id=\"standard-interpretations-and-quantum-non-locality\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eStandard interpretations and quantum non-locality\u003c/h2\u003e\u003cp id=\"text-10\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe \u003ca href=\"copenhagen\" target=\"_blank\"\u003eCopenhagen\u003c/a\u003e interpretation of QM, developed by Niels Bohr and Werner Heisenberg in the 1920’s (see \u003ca href=\"history\" target=\"_blank\"\u003ehistory\u003c/a\u003e), is the source of most of the apparent paradoxes and conundrums associated with quantum physics. At the heart of this is the interpretation of the wave function as a purely \u003ca href=\"epistemic-vs-ontic\" target=\"_blank\"\u003eepistemological\u003c/a\u003e, non-physical entity. If it is not actually real, then it isn’t subject to the kinds of constraints that we might otherwise expect it to have, like being local in space and time, which is \u003cem\u003erequired\u003c/em\u003e by the well-established phenomena of \u003ca href=\"special-relativity\" target=\"_blank\"\u003especial relativity\u003c/a\u003e.\u003c/p\u003e\u003cp id=\"text-11\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe central premise of the Copenhagen interpretation is that there is a special kind of process called \u003cem\u003emeasurement\u003c/em\u003e that somehow causes quantum wave functions to \u003cem\u003ecollapse\u003c/em\u003e down to a single point, with the wave function defining the \u003cem\u003eprobability\u003c/em\u003e of finding a discrete particle at any given point. Exactly how a potentially widely-distributed wave might somehow gather up all of its far-flung bits and shrink down into one randomly-chosen point, in an instantaneous, manifestly non-local process, is entirely beyond the explanatory scope of the theory.\u003c/p\u003e\u003cp id=\"text-12\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn the face of these kinds of obviously non-physical aspects of this framework, the standard answer is to “shut up and calculate”. This is the hallmark of a calculational tool, and as such, it seems prudent to consider this framework as such, and we will not spend any further effort here probing its fundamental strangeness.\u003c/p\u003e\u003cp id=\"text-13\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAnd yet, the Copenhagen interpretation remains the most popular interpretation according to informal surveys of working physicists (\u003ca href=\"ref://Tegmark98\" target=\"_blank\"\u003eTegmark, 1998\u003c/a\u003e; \u003ca href=\"ref://SchlosshauerKoflerZeilinger13\" target=\"_blank\"\u003eSchlosshauer et al., 2013\u003c/a\u003e). The next-most popular interpretation after Copenhagen according those surveys is the \u003cem\u003emany-worlds\u003c/em\u003e interpretation originated by \u003ca href=\"ref://Everett57\" target=\"_blank\"\u003eEverett (1957)\u003c/a\u003e, which postulates that the entire universe splits at each measurement event. This nominally avoids the need for wave function collapse, but at what cost? An infinite accumulation of new universes spawning everywhere? This is so completely physically implausible that it just defies belief that so many people could even contemplate such a theory, just because it simplifies the math.\u003c/p\u003e\u003cp id=\"text-14\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eRelatively few / none of the respondents in these surveys endorsed the \u003ca href=\"pilot-wave\" target=\"_blank\"\u003epilot-wave\u003c/a\u003e approach, despite the fact that it eliminates almost all of the strange paradoxes and counterintuitive ideas advanced in the more widely-accepted interpretations. A common objection to the pilot wave framework is that it is defined over \u003ca href=\"configuration-space\" target=\"_blank\"\u003econfiguration space\u003c/a\u003e, which grows exponentially in the number of particles. However, every other framework is \u003cem\u003ealso\u003c/em\u003e defined over configuration space, or it just pushes this exponential explosion into the forking of new universes in the case of the many-worlds framework.\u003c/p\u003e\u003cp id=\"text-15\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn effect, the other interpretations merely avoid confronting the difficulties of this exponential configuration space by denying its reality in one way or another, while still requiring it to calculate how the physics actually works. Indeed, the entire foundation of the Copenhagen interpretation is willful denial of reality: reality doesn’t exist until it is somehow “measured”. And there is no explanation for what actually constitutes a measurement, and how instantaneous collapse could possibly be made compatible with special relativity.\u003c/p\u003e\u003cp id=\"text-16\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFrom this perspective, the single most important unsolved problem in quantum physics is to eliminate the \u003ca href=\"configuration-space\" target=\"_blank\"\u003econfiguration space\u003c/a\u003e representation, and somehow derive a fully relativistic and local wave-particle \u003ca href=\"duality\" target=\"_blank\"\u003eduality\u003c/a\u003e where the nonlinear interactions that are otherwise captured in the calculational tool of configuration space naturally emerge. This new framework is likely to be fundamentally nondeterministic (see \u003ca href=\"stochastic-particles\" target=\"_blank\"\u003estochastic particles\u003c/a\u003e) and perhaps requires superliminal, but still finite, wave propagation speeds for the quantum waves. It is increasingly clear that the various existing “no-go” theorems such as Bell’s inequalities do not exclude this broader space of possible mechanisms.\u003c/p\u003e\u003ch2 id=\"developing-the-model\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eDeveloping the model\u003c/h2\u003e\u003cp id=\"text-18\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFollowing from the above background (including linked pages above), the following is a suggested sequence of how to proceed through this content:\u003c/p\u003e\u003cul id=\"frame-19\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003ca href=\"contextual\" target=\"_blank\"\u003eContextual\u003c/a\u003e variables, as contrasted with “real” variables, and their role in understanding the phenomenology of QM.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003ca href=\"semiclassical\" target=\"_blank\"\u003eSemiclassical\u003c/a\u003e models, that combine a classical treatment of the EM field according to \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e’s equations, with a quantum treatment of the \u003ca href=\"electron\" target=\"_blank\"\u003eelectron\u003c/a\u003e. This is the same approach used in this framework.\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003ca href=\"qed\" target=\"_blank\"\u003eQED\u003c/a\u003e provides an overview of \u003cem\u003equantum electrodynamics\u003c/em\u003e which provides a highly accurate \u003cem\u003edescription\u003c/em\u003e of the relevant phenomenology of interest. However, we argue that QED is a calculational tool, not a plausible physical model.\u003c/p\u003e\u003cp id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• The \u003ca href=\"zero-point\" target=\"_blank\"\u003eZero-point\u003c/a\u003e field and \u003cem\u003estochastic electrodynamics\u003c/em\u003e which provides an alternative formulation of QED, which the present framework shares some important similarities.\u003c/p\u003e\u003cp id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• The concept of a luminiferous \u003ca href=\"aether\" target=\"_blank\"\u003eaether\u003c/a\u003e, which was theoretically disproven by the famous Michelson-Morely experiment of 1887, but in fact is entirely consistent with a privileged reference frame as required by the CA framework, as long as that reference frame obeys the critical time / space distortion properties of \u003ca href=\"special-relativity\" target=\"_blank\"\u003especial relativity\u003c/a\u003e (which of course it must, to be consistent with well-established empirical data).\u003c/p\u003e\u003cp id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003ca href=\"waves\" target=\"_blank\"\u003eWaves\u003c/a\u003e provides the essential foundation for understanding the phenomenology and mathematical formulation of waves, within the CA framework.\u003c/p\u003e\u003cp id=\"text-6\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e’s equations for the electromagnetic (EM) force field, as a classical field, implemented through CA-based wave functions.\u003c/p\u003e\u003cp id=\"text-7\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003ca href=\"matter-waves\" target=\"_blank\"\u003eMatter waves\u003c/a\u003e discusses the general idea of a quantum wave equation that captures something about the properties of massive particles: but what is it actually representing? In the standard QM frameworks, it represents the \u003cem\u003eepistemic\u003c/em\u003e probability of finding a discrete particle at a given location at a given point in time. In the \u003ca href=\"pilot-wave\" target=\"_blank\"\u003epilot-wave\u003c/a\u003e framework adopted here, it represents a physically real wave permeating space, that guides the movement of a discrete particle.\u003c/p\u003e\u003cp id=\"text-8\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• The \u003ca href=\"klein-gordon\" target=\"_blank\"\u003eKlein-Gordon\u003c/a\u003e (KG) equation provides the simplest version of a relativistically-accurate quantum matter wave function, which can be seen as a kind of second-order version of the much more widely discussed \u003ca href=\"schrodinger\" target=\"_blank\"\u003eSchrodinger\u003c/a\u003e wave function.\u003c/p\u003e\u003cp id=\"text-9\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• In \u003ca href=\"complex-kg\" target=\"_blank\"\u003ecomplex KG\u003c/a\u003e, we apply the KG wave functions to a complex wave state, which enables a conserved quantity that acts like electrical charge to be computed. Although the \u003ca href=\"complex-numbers\" target=\"_blank\"\u003ecomplex numbers\u003c/a\u003e involved seem mysterious, we see that they are just a mathematical convenience, and we can eliminate them entirely in our second-order wave functions, which are computed using only real-valued numbers. This complex KG system can be directly coupled to \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e’s equations, getting very close to the ultimate goal of capturing all of the properties of an \u003ca href=\"electron\" target=\"_blank\"\u003eelectron\u003c/a\u003e.\u003c/p\u003e\u003cp id=\"text-10\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• The final step in the wave function development sequence is the second-order \u003ca href=\"dirac\" target=\"_blank\"\u003eDirac\u003c/a\u003e wave function, which builds on the complex KG system, and captures the phenomenon of \u003cem\u003espin\u003c/em\u003e. This then provides a physically complete description of the quantum dynamics of a particle like the electron. This is what drives all the amazing predictive accuracy of the \u003ca href=\"qed\" target=\"_blank\"\u003eQED\u003c/a\u003e framework within the standard model of physics, and is what we hypothesize exists as a real physical wave.\u003c/p\u003e\u003cp id=\"text-11\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Finally, the \u003ca href=\"electron\" target=\"_blank\"\u003eelectron\u003c/a\u003e is modeled as a discrete \u003ca href=\"stochastic-particles\" target=\"_blank\"\u003estochastic particle\u003c/a\u003e that moves with \u003cem\u003eintrinsic\u003c/em\u003e (\u003cem\u003eontic\u003c/em\u003e) noise under the influence of the Dirac wave function.\u003c/p\u003e\u003c/ul\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"history\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"m274-450 248 248-42 42-320-320 320-320 42 42-248 248h526v60H274Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003ePrevious\u003c/p\u003e\u003c/a\u003e\u003cstretch id=\"content-buttons-go-167\" style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1ch;min-height:1em;font-weight:thin;text-align:start\"\u003e\u003c/stretch\u003e\u003ca id=\"content-buttons-go-168\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"klein-gordon\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M686-450H160v-60h526L438-758l42-42 320 320-320 320-42-42 248-248Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Klein-Gordon","URL":"klein-gordon","Title":"Klein-Gordon","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{"eq":["eq_wave","eq_mass","eq_kg","eq_kg-4vec","eq_disc","eq_","eq_","eq_"],"figure":["figure_mass","figure_frequency"]},"Description":"The \u003cstrong\u003eKlein-Gordon\u003c/strong\u003e (KG) wave function is the simplest version of a \u003ca href=\"waves\" target=\"_blank\"\u003ewave\u003c/a\u003e equation that captures the known physics of particles such as an \u003ca href=\"electron\" target=\"_blank\"\u003eelectron\u003c/a\u003e. In fact, it explains a surprisingly wide range of physical phenomena, including Newtonian and relativistic equations of motion, the Lorentz transformations of \u003ca href=\"special-relativity\" target=\"_blank\"\u003especial relativity\u003c/a\u003e, and the quantum mechanical relationship between wave frequency and velocity (momentum), all with an incredibly simple equation.","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eKlein-Gordon\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-22\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe \u003cstrong\u003eKlein-Gordon\u003c/strong\u003e (KG) wave function is the simplest version of a \u003ca href=\"waves\" target=\"_blank\"\u003ewave\u003c/a\u003e equation that captures the known physics of particles such as an \u003ca href=\"electron\" target=\"_blank\"\u003eelectron\u003c/a\u003e. In fact, it explains a surprisingly wide range of physical phenomena, including Newtonian and relativistic equations of motion, the Lorentz transformations of \u003ca href=\"special-relativity\" target=\"_blank\"\u003especial relativity\u003c/a\u003e, and the quantum mechanical relationship between wave frequency and velocity (momentum), all with an incredibly simple equation.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe KG wave function can be derived directly from Einstein’s relativistic definition of total energy, via the \u003ca href=\"hamiltonian\" target=\"_blank\"\u003eHamiltonian\u003c/a\u003e strategy. After reading this chapter, which covers the phenomenology of the KG wave equation, it is recommended to read that derivation to obtain a deeper understanding of why and how the KG equations account for all of special relativity.\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe incredible scope of phenomena accounted for by the simple KG equation makes it tempting to think that particles are actually \u003ca href=\"matter-waves\" target=\"_blank\"\u003ematter waves\u003c/a\u003e, in the form of a spatially-localized \u003cem\u003ewave packet\u003c/em\u003e. However, despite all the amazing properties of the KG equation (and its more complicated iteration in the \u003ca href=\"dirac\" target=\"_blank\"\u003eDirac\u003c/a\u003e wave function), these matter waves have a fatal flaw: they inevitably diffuse away into amorphous blobs that fail to account for the precise localization of particles like electrons.\u003c/p\u003e\u003cp id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIf you were to imagine something like an electron to actually be a matter wave, then it becomes very difficult to understand how all of the widely distributed, far-flung bits could be somehow gathered up and accounted for, in order to satisfy strict conservation laws. Every time an electron is measured it has the exact same charge. And rest mass. This extremely difficult to imagine happening when you see what happens to the KG waves over time.\u003c/p\u003e\u003cp id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn fact, it is precisely as implausible as the standard \u003ca href=\"copenhagen\" target=\"_blank\"\u003eCopenhagen\u003c/a\u003e interpretation of QM, which requires the complete collapse of far-flung wave equations, which are thought to determine the probability of particle properties being measured in any given location.\u003c/p\u003e\u003cp id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eNevertheless, there is much to learn about physics and wave functions by understanding the almost miraculous properties of the KG wave function.\u003c/p\u003e\u003cp id=\"text-6\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eRecall that the wave equation can be written as a second-order differential equation:\u003c/p\u003e\u003cdiv id=\"inline-container-8\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_wave\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_wave\"\u003e\u003cb\u003eEq 1:\u003c/b\u003e standard wave equation\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-9\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2 {\\varphi}}{\\partial t^2} = c^2 \\nabla^2 \\varphi\n\\]\u003c/span\u003e\u003c/p\u003e\u003cdiv id=\"figure_mass\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_mass\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 1:\u003c/b\u003e \u003cp\u003eThe additional mass term \u003cspan class=\"math inline\"\u003e\\(-m_0^2 \\varphi\\)\u003c/span\u003e in the Klein-Gordon (KG) wave equation “drags down” the wave in proportion to the height of the waves (i.e., amplitude away from zero, either positive or negative). This fights against the curvature of the wave, computed by \u003cspan class=\"math inline\"\u003e\\(\\nabla^2\\)\u003c/span\u003e. Higher frequency waves have higher curvature, and thus move faster than lower frequency waves.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cp id=\"text-12\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWhat if we add a single new term to this equation, where we subtract away some \u003cem\u003emass\u003c/em\u003e (\u003cspan class=\"math inline\"\u003e\\(m_0\\)\u003c/span\u003e, a constant) from the Laplacian (\u003cspan class=\"math inline\"\u003e\\(\\nabla^2 \\varphi\\)\u003c/span\u003e) curvature driving force term (\u003ca href=\"klein-gordon#figure_mass\" target=\"_blank\"\u003eFigure 1\u003c/a\u003e):\u003c/p\u003e\u003cdiv id=\"inline-container-14\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_mass\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_mass\"\u003e\u003cb\u003eEq 2:\u003c/b\u003e mass subtraction\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-15\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2 {\\varphi}}{\\partial t^2} = c^2 \\left( \\nabla^2 \\varphi - \\frac{m_0^2}{\\hbar^2} \\varphi \\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-16\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eor, in somewhat simpler notation that we’ll use more frequently:\u003c/p\u003e\u003cdiv id=\"inline-container-18\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_kg\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_kg\"\u003e\u003cb\u003eEq 3:\u003c/b\u003e Klein-Gordon equation\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-19\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2 {\\varphi}}{\\partial t^2} = c^2 \\left(\\nabla^2 - \\frac{m_0^2}{\\hbar^2} \\right) \\varphi\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-20\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhere “hbar” \u003cspan class=\"math inline\"\u003e\\(\\hbar = \\frac{h}{2\\pi}\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(h\\)\u003c/span\u003e is Planck’s constant.\u003c/p\u003e\u003cp id=\"text-21\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe most natural interpretation of Planck’s constant here is as a scaling term on the impact of mass on the matter wave dynamics. For this reason, it is puzzling how \u003cem\u003eh\u003c/em\u003e could possibly show up in light waves, because they have no mass, and there is no role for this constant in \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e’s EM wave equations. Thus, one could view Einstein’s creation of the photon with energy \u003cspan class=\"math inline\"\u003e\\(E = h \\nu\\)\u003c/span\u003e as a calculational \u003ca href=\"tools-vs-models\" target=\"_blank\"\u003etool\u003c/a\u003e for representing the interaction between EM waves and matter in atomic systems, and it is the matter waves that impart the \u003cem\u003eh\u003c/em\u003e constant, not the EM “photon”. See \u003ca href=\"semiclassical\" target=\"_blank\"\u003esemiclassical\u003c/a\u003e for more discussion.\u003c/p\u003e\u003cp id=\"text-22\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn any case, this new equation (\u003ca href=\"klein-gordon#eq_kg\" target=\"_blank\"\u003eEq 3\u003c/a\u003e) is called the \u003cstrong\u003eKlein-Gordon (KG)\u003c/strong\u003e equation, named after Oskar Klein and Walter Gordon, who published the first papers on it (\u003ca href=\"ref://Klein26\" target=\"_blank\"\u003eKlein, 1926\u003c/a\u003e; \u003ca href=\"ref://Gordon27\" target=\"_blank\"\u003eGordon, 1927\u003c/a\u003e; see \u003ca href=\"ref://Kragh84\" target=\"_blank\"\u003eKragh, 1984\u003c/a\u003e for a detailed history of this equation, which was actually discovered by many individuals, including Schrödinger). This equation captures a surprising number of important phenomena, as we detail next.\u003c/p\u003e\u003cp id=\"text-23\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFirst, we’ll introduce some variations on how to write this equation, which are all obviously identical to the KG equation given above, but highlight different features of it, as we’ll see more later. Here’s one such variation:\u003c/p\u003e\u003cp id=\"text-24\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2 {\\varphi}}{\\partial t^2} - c^2 \\nabla^2\\varphi = -\\frac{c^2 m_0^2}{\\hbar^2} \\varphi\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-25\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eand another:\u003c/p\u003e\u003cp id=\"text-26\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\left(\\frac{\\partial^2 {}}{\\partial t^2} - c^2 \\nabla^2 + \\frac{c^2 m_0^2}{\\hbar^2}\\right) \\varphi = 0\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-27\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThese last two forms are useful for relating to the \u003ca href=\"four-vector\" target=\"_blank\"\u003efour-vector\u003c/a\u003e version of the wave equation, where we saw that:\u003c/p\u003e\u003cp id=\"text-28\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\partial_\\mu \\partial^\\mu = \\frac{\\partial^2}{\\partial t^2} - c^2 \\nabla^2\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-29\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eso that the equation can be written:\u003c/p\u003e\u003cdiv id=\"inline-container-31\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_kg-4vec\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_kg-4vec\"\u003e\u003cb\u003eEq 4:\u003c/b\u003e Klein-Gordon four-vector\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-32\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\partial_\\mu \\partial^\\mu \\varphi = - \\frac{c^2 m_0^2}{\\hbar^2} \\varphi\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-33\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eor:\u003c/p\u003e\u003cp id=\"text-34\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\left(\\partial_\\mu \\partial^\\mu + \\frac{c^2 m_0^2}{\\hbar^2}\\right) \\varphi = 0\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-35\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eTo actually implement this KG equation in our cellular automaton model, we make one modification to the acceleration term, to subtract off the mass:\u003c/p\u003e\u003cdiv id=\"inline-container-37\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_disc\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_disc\"\u003e\u003cb\u003eEq 5:\u003c/b\u003e discrete Klein-Gordon equation\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-38\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\ddot \\varphi_i^{t+1} = c^2 \\frac{3}{13}\\sum_{j \\in N_{26}} k_j (\\varphi_j - \\varphi_i) - \\frac{c^2 m_0^2}{\\hbar^2} \\varphi_i\n\\]\u003c/span\u003e\u003c/p\u003e\u003ch2 id=\"variable-speeds-momentum-from-frequency\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eVariable speeds: momentum from frequency\u003c/h2\u003e\u003cdiv id=\"figure_frequency\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_frequency\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 2:\u003c/b\u003e \u003cp\u003eRelationship between frequency and speed in the Klein-Gordon (KG) wave function, which derives from competition between the “mass drag” and the overall curvature of the wave. Higher frequency waves have more curvature and thus move faster.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cp id=\"text-42\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOne of the most important features of this KG equation is that waves now travel at \u003cem\u003evariable speeds\u003c/em\u003e, instead of always moving at exactly the same speed (the speed of light). This speed now depends on the relationship between the curvature (\u003cspan class=\"math inline\"\u003e\\(\\nabla^2 \\varphi\\)\u003c/span\u003e) and the squared-mass value \u003cspan class=\"math inline\"\u003e\\(\\frac{m_0^2}{\\hbar^2} \\varphi\\)\u003c/span\u003e. In essence, the mass “drags down” the wave propagation force conveyed by the local curvature, \u003cspan class=\"math inline\"\u003e\\(\\nabla^2 \\varphi\\)\u003c/span\u003e. Therefore, to get the wave to move faster, you need more curvature, which is to say, a higher frequency wave, because higher frequency waves have more waves per unit length, and this means overall greater “curvature” (\u003ca href=\"klein-gordon#figure_frequency\" target=\"_blank\"\u003eFigure 2\u003c/a\u003e).\u003c/p\u003e\u003cp id=\"text-43\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis relationship between frequency \u003cspan class=\"math inline\"\u003e\\(f\\)\u003c/span\u003e of a wave and the momentum (velocity * mass) of the particle that it describes is captured in one of the most basic equations of quantum physics:\u003c/p\u003e\u003cp id=\"text-44\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\np = \\frac{h}{c} f\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-45\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhere \u003cspan class=\"math inline\"\u003e\\(p\\)\u003c/span\u003e is the momentum, and \u003cspan class=\"math inline\"\u003e\\(h\\)\u003c/span\u003e is Planck’s constant. This can also be written in terms of the wavelength \u003cspan class=\"math inline\"\u003e\\(\\lambda\\)\u003c/span\u003e, which is the inverse of the frequency:\u003c/p\u003e\u003cp id=\"text-46\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nf = \\frac{c}{\\lambda}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-47\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\lambda = \\frac{c}{f}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-48\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\lambda f = c\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-49\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eso that momentum is inversely proportional to the length of the waves:\u003c/p\u003e\u003cp id=\"text-50\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\np = \\frac{h}{\\lambda}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-51\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAlthough it might be tempting to compute the velocity from the momentum expression given above (e.g., \u003cspan class=\"math inline\"\u003e\\(p = m v\\)\u003c/span\u003e so \u003cspan class=\"math inline\"\u003e\\(v = \\frac{p}{m}\\)\u003c/span\u003e), this is not accurate due to the effects of special relativity as we discuss in greater detail below. Instead, the appropriate equation that relates the momentum and the velocity is:\u003c/p\u003e\u003cdiv id=\"inline-container-53\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_\"\u003e\u003cb\u003eEq 6:\u003c/b\u003e relativistic momentum-velocity relationship\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-54\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\np = \\gamma m_0 v\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-55\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhere \u003cspan class=\"math inline\"\u003e\\(\\gamma\\)\u003c/span\u003e is the Lorentz factor as described in \u003ca href=\"special-relativity\" target=\"_blank\"\u003especial relativity\u003c/a\u003e. When we make all the necessary substitutions and do a bit of algebra, we end up with this expression for the velocity of the “particle” as a function of wavelength, rest mass, and the relevant constants:\u003c/p\u003e\u003cdiv id=\"inline-container-57\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_\"\u003e\u003cb\u003eEq 6:\u003c/b\u003e velocity as function of wavelength, relativistically correct\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-58\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nv = \\frac{h c}{\\sqrt{c^2 m_0^2 \\lambda^2 + h^2}}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-59\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSee \u003ca href=\"special-relativity#relativistic momentum and velocity\" target=\"_blank\"\u003erelativistic momentum and velocity\u003c/a\u003e in \u003ca href=\"special-relativity\" target=\"_blank\"\u003especial-relativity\u003c/a\u003e for all the algebraic steps in this derivation. In the exploration below we’ll confirm this equation experimentally. The complexity of this equation for velocity versus the much simpler expressions for momentum indicate why momentum and not velocity is the natural quantity to deal with in relativistic wave functions.\u003c/p\u003e\u003cp id=\"text-60\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAlso, as we will explore in greater detail later, the momentum can be computed directly from the wave function in terms of the first-order spatial derivative or gradient:\u003c/p\u003e\u003cdiv id=\"inline-container-62\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_\"\u003e\u003cb\u003eEq 6:\u003c/b\u003e essence of momentum operator\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-63\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{p} \\propto \\vec{\\nabla} \\varphi\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-64\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhere this spatial gradient is again going to be greater overall as the wave frequency increases, as suggested by the quantum mechanical relationships above.\u003c/p\u003e\u003cp id=\"text-65\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAgain, the main point for now is just that introducing the mass term makes the relative curvature or frequency of the wave matter in determining the overall velocity or momentum of the wave packet that describes a particle. Without this mass term, all waves travel at the speed of light.\u003c/p\u003e\u003ch2 id=\"newtonian-mechanics-f-ma\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eNewtonian mechanics: F = ma\u003c/h2\u003e\u003cp id=\"text-67\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWe can intuitively see that this very simple modification to the wave equation captures all of classical (Newtonian) mechanics for a “particle” characterized by a wave according to this equation. In the absence of any external forces, the wave will propagate along at a constant velocity (Newton’s first law of inertia), because the frequency of the wave does not decrease (and it would not change its overall direction of propagation). If a force is applied to this system, it will change the frequency of the oscillation of the wave, and thus result in a change in momentum, in accord with Newton’s second law:\u003c/p\u003e\u003cp id=\"text-68\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nF = \\frac{\\partial \\vec{p}}{\\partial t} = m \\frac{\\partial \\vec{v}}{\\partial t} = m \\vec{a}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-69\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAfter a few more developments, we can make this relationship much more formally accurate and precise, by considering the overall energy and momentum relationships computed by the KG wave equation. We will see that Schrödinger’s equation, which is the primary wave equation for basic quantum physics, captures classical Newtonian physics, and that the KG equation is a version of Schrödinger’s equation that also takes into account special relativity, which is important when particles are moving very fast (i.e., relatively close to the speed of light).\u003c/p\u003e\u003cp id=\"text-70\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAnticipating these results, and relying on intuition for now, we see that with one tiny addition, we now have an equation that can describe the motion of a massive particle through space (e.g., as a wave packet), in agreement with all the known physical laws (i.e., quantum physics and special relativity, which reduce in certain cases to the more familiar Newtonian mechanics).\u003c/p\u003e\u003ch2 id=\"what-is-the-mass\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eWhat is the mass?\u003c/h2\u003e\u003cp id=\"text-72\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe value \u003cspan class=\"math inline\"\u003e\\(m_0\\)\u003c/span\u003e in the KG equation is the \u003cstrong\u003erest mass\u003c/strong\u003e of the particle that it describes (the 0 subscript indicates “rest”). It is a fixed, constant value for a given type of particle, e.g., \u003cspan class=\"math inline\"\u003e\\(9.1x10^{-31}\\)\u003c/span\u003e kg for an \u003ca href=\"electron\" target=\"_blank\"\u003eelectron\u003c/a\u003e, which is an extremely tiny amount of mass.\u003c/p\u003e\u003cp id=\"text-73\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe value of \u003cspan class=\"math inline\"\u003e\\(m_0\\)\u003c/span\u003e also defines the \u003cstrong\u003eCompton wavelength\u003c/strong\u003e of a given particle, which is the wavelength of a particle at rest, due strictly to the rest mass. The formula for the Compton wavelength \u003cspan class=\"math inline\"\u003e\\(\\lambda_C\\)\u003c/span\u003e is:\u003c/p\u003e\u003cp id=\"text-74\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\lambda_C = \\frac{h}{m_0 c}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-75\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eInterestingly, consider what happens when you set the rest mass of our particle to zero: that extra termm in the KG equation drops out, and you recover the basic wave equation from before. Thus, it is immediately obvious that “particles” with a zero rest mass must move at the speed of light. This is a basic postulate of special relativity. Note also that it is impossible for a massive particle to travel at the speed of light, because it would have to have an infinitely high frequency, and this is not possible (even in a continuous spatial model).\u003c/p\u003e\u003cp id=\"text-76\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOne potential challenge with the KG equation is that this rest mass parameter must be “baked in” to the wave function equations. What if we are simulating different types of particles, beyond just electrons? For example, the leptons class includes the much more massive \u003cem\u003emuon\u003c/em\u003e and \u003cem\u003etau\u003c/em\u003e particles, which are otherwise identical to the electron. It seems rather inelegant to have to have different wave functions for each of these different types of particles, each with their own rest mass parameter.\u003c/p\u003e\u003cp id=\"text-77\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIndeed, the \u003ca href=\"stochastic-particles\" target=\"_blank\"\u003estochastic particles\u003c/a\u003e framework attempts to solve this problem by associating the rest mass with the internal dynamics of the particle: rest mass is just the energy associated with the internal gyrations of particles when they are not otherwise moving anywhere. This provides a much more elegant and natural explanation for the rest mass, and suggests that the \u003ca href=\"matter-waves\" target=\"_blank\"\u003ematter waves\u003c/a\u003e approach may not be correct.\u003c/p\u003e\u003ch2 id=\"smooth-continuous-motion\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eSmooth, continuous motion\u003c/h2\u003e\u003cp id=\"text-79\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAs a continuous-valued second-order wave equation, the KG equation still has all of the advantages discussed earlier about making the underlying grid disappear. Furthermore, it easily and naturally allows us to describe continuously variable rates of speed, in terms of continuous variation in the frequency or wavelength of wave oscillation. Particles described by such an equation can also travel in any direction, because of the essentially perfect spatial symmetry of wave propagation exhibited by this equation.\u003c/p\u003e\u003cp id=\"text-80\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn contrast, models with a discrete particle suffer from all manner of complexity in overcoming such problems. Thus, by describing a particle exclusively using this wave equation, we avoid many difficulties, and it remains to be seen if we encounter any other problems.\u003c/p\u003e\u003ch2 id=\"exploration-of-klein-gordon-waves\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eExploration of Klein-Gordon waves\u003c/h2\u003e\u003cp id=\"text-82\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eNow we explore the above properties of the Klein-Gordon wave equation, to get a concrete sense of how it works, before turning to a more extended discussion of how the KG wave equation explains the stretchy properties of matter implied by the Lorentz transformation and special relativity.\u003c/p\u003e\u003ch2 id=\"initial-conditions\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eInitial conditions\u003c/h2\u003e\u003cp id=\"text-84\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eCreating a moving wave packet that moves with a given velocity is a bit more complicated than for the simple wave equation, because we have more constraints to take into account.\u003c/p\u003e\u003ch2 id=\"schrödinger-s-equation-vs-klein-gordon\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eSchrödinger’s equation vs Klein-Gordon\u003c/h2\u003e\u003cp id=\"text-86\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe Klein-Gordon equation that we’ve been exploring is typically introduced as a strange and problematic alternative to the Schrödinger wave equation, which provides the cornerstone of standard quantum physics, even to this day. In \u003ca href=\"hamiltonian\" target=\"_blank\"\u003eHamiltonian\u003c/a\u003e (recommended as next reading), you can see how the KG equation can be derived from the relativistic total energy. By contrast, Schrödinger’s equation can be derived from \u003cem\u003eNewtonian\u003c/em\u003e total energy, and therefore it fails to account for the phenomena of \u003ca href=\"special-relativity\" target=\"_blank\"\u003especial relativity\u003c/a\u003e.\u003c/p\u003e\u003cp id=\"text-87\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe fact that Schrödinger’s equation remains the predominant tool used by practicing physicists can be attributed to its strict conservation properties, where the total probability value (computed as the complex conjugate of the wave values), is conserved as it propagates through space. In contrast, the KG equation does not have such a strict conservation behavior.\u003c/p\u003e\u003cp id=\"text-88\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFurthermore, the Schrödinger equation is a first-order wave equation, which has many advantages from an analytical perspective, even as it makes it very difficult for many people to understand, due to its reliance on \u003ca href=\"complex-numbers\" target=\"_blank\"\u003ecomplex numbers\u003c/a\u003e. In general, wave-like behavior can either be described by a second-order equation involving normal scalar variables (as we’ve been doing), or it can be described by a first-order equation involving complex numbers, exemplified by the Schrödinger equation.\u003c/p\u003e\u003cp id=\"text-89\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn the first-order version, you have two variables for every one variable in the second-order one — we’ll see later that this fact allows us to use only four variables to represent an electron using a second-order wave equation, whereas the standard first-order Dirac equation requires eight. The general intuition is that a first-order wave equation involves motion as rotation among its complex variables, in addition to motion through space, whereas the second-order equation just has motion through space.\u003c/p\u003e\u003cp id=\"text-90\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSee \u003ca href=\"schrodinger\" target=\"_blank\"\u003eSchrodinger\u003c/a\u003e for the full derivation, and exploration of its properties.\u003c/p\u003e\u003cp id=\"text-91\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFor the next step, move on to \u003ca href=\"complex-kg\" target=\"_blank\"\u003ecomplex KG\u003c/a\u003e for the application of the KG function to the two elements of a complex number, which opens up a major new set of phenomena in the interaction with the electromagnetic field.\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"m274-450 248 248-42 42-320-320 320-320 42 42-248 248h526v60H274Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003ePrevious\u003c/p\u003e\u003c/a\u003e\u003cstretch id=\"content-buttons-go-167\" style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1ch;min-height:1em;font-weight:thin;text-align:start\"\u003e\u003c/stretch\u003e\u003ca id=\"content-buttons-go-168\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"matter-waves\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M686-450H160v-60h526L438-758l42-42 320 320-320 320-42-42 248-248Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Matter waves","URL":"matter-waves","Title":"Matter waves","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{"figure":["figure_packet"]},"Description":"Historically, matter has typically been conceived of as something hard and solid: a particle of some form or another. However, it is also possible to produce many of the known properties of matter using a very simple extension of the simple second-order \u003ca href=\"waves\" target=\"_blank\"\u003ewave\u003c/a\u003e equation, in the form of the \u003ca href=\"klein-gordon\" target=\"_blank\"\u003eKlein-Gordon\u003c/a\u003e (KG) wave function.","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eMatter waves\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-23\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eHistorically, matter has typically been conceived of as something hard and solid: a particle of some form or another. However, it is also possible to produce many of the known properties of matter using a very simple extension of the simple second-order \u003ca href=\"waves\" target=\"_blank\"\u003ewave\u003c/a\u003e equation, in the form of the \u003ca href=\"klein-gordon\" target=\"_blank\"\u003eKlein-Gordon\u003c/a\u003e (KG) wave function.\u003c/p\u003e\u003cdiv id=\"figure_packet\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_packet\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 1:\u003c/b\u003e \u003cp\u003eA wave packet, which is a spatially localized wave disturbance that propagates through space as a coherent entity. This could serve as a model for a particle, except that the wave inevitably diffuses as it propagates over time. Mathematically, it can be constructed by multiplying a Gaussian function (normal bell-shaped distribution curve) times a sine wave.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cp id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe particle in this context would instead be something like a \u003cstrong\u003ewave packet\u003c/strong\u003e (\u003ca href=\"matter-waves#figure_packet\" target=\"_blank\"\u003eFigure 1\u003c/a\u003e). It can act like a particle in that it is somewhat spatially localized, and moves as a coherent entity. If you zoomed out very far, and blurred your eyes, you could imagine that a wave packet would look like a tiny point particle. Nevertheless, it fundamentally acts like a wave, in the sense that it is actually made of oscillations, and obeys a wave equation.\u003c/p\u003e\u003cp id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe simplicity and elegance of this matter wave in potentially explaining a wide range of physical phenomena, including Newtonian and relativistic equations of motion, the Lorentz transformations of \u003ca href=\"special-relativity\" target=\"_blank\"\u003especial relativity\u003c/a\u003e, and the quantum mechanical relationship between wave frequency and velocity (momentum), provides a compelling overall framework for thinking of matter in terms of waves.\u003c/p\u003e\u003cp id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAlthough the Klein-Gordon equation goes a long way, it has a few limitations, including the lack of strict conservation properties, and a failure to capture the quantum phenomenon of \u003cstrong\u003espin\u003c/strong\u003e. Both of these limitations are addressed by the \u003ca href=\"dirac\" target=\"_blank\"\u003eDirac\u003c/a\u003e wave function. In this context, the conserved value that we can compute from the wave function is actually the \u003cstrong\u003echarge\u003c/strong\u003e, not the mass, of the associated particle, and like electrons and their anti-particles the positron, this charge can be either negative or positive.\u003c/p\u003e\u003cp id=\"text-6\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe Dirac wave function shows how this charge, and associated current as the motion of the charge, couple with the electromagnetic waves described by \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e’s equations, to accurately capture the interaction between spin and the magnetic field.\u003c/p\u003e\u003cp id=\"text-7\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eHowever, the use of a wave equation as the basis for something like an \u003ca href=\"electron\" target=\"_blank\"\u003eelectron\u003c/a\u003e runs up against a number of seemingly insurmountable problems, the most significant being that these matter waves inevitably diffuse out and flatten over space and time, and do not exhibit any kind of emergent or spontaneous localization properties that would account for the apparent strong localization properties of elementary particles.\u003c/p\u003e\u003cp id=\"text-8\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe process by which a widely distributed matter wave function could somehow become localized, gathering up all of its far-flung bits in order to satisfy strict conservation laws, is precisely as implausible as the standard \u003ca href=\"copenhagen\" target=\"_blank\"\u003eCopenhagen\u003c/a\u003e interpretation of QM.\u003c/p\u003e\u003cp id=\"text-9\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIt is interesting to note that the Klein-Gordon equation has been almost completely neglected in the physics literature, which instead has been dominated by \u003ca href=\"schrodinger\" target=\"_blank\"\u003eSchrodinger\u003c/a\u003e’s equation and the \u003ca href=\"dirac\" target=\"_blank\"\u003eDirac\u003c/a\u003e equation. The enthusiasm for the KG equation here derives from its extreme elegance and simplicity from the perspective of a \u003ca href=\"cellular-automaton\" target=\"_blank\"\u003ecellular automaton\u003c/a\u003e framework, and its second-order nature both emerges naturally out of this framework, and solves a number of important problems (e.g., symmetric propagation in all directions, which does not occur in a first-order wave equation in the CA framework).\u003c/p\u003e\u003cp id=\"text-10\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn contrast, the rest of physics likes Schrödinger’s equation because it is more analytically tractable as a first-order equation. It is linear, and it also automatically produces a positive-valued conserved probability density, which fits perfectly with the standard probabilistic interpretation of quantum physics. The fact that it violates \u003ca href=\"special-relativity\" target=\"_blank\"\u003especial relativity\u003c/a\u003e is often overlooked, and anyway the Dirac equation solves that problem, while staying within a first-order framework (but at the cost of introducing 8 state variables interacting in a fairly complex way).\u003c/p\u003e\u003cp id=\"text-11\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThus, the overall difference is one of “mechanism” vs. “analysis,” where standard physics is strongly weighted toward analysis (as in \u003ca href=\"tools-vs-models\" target=\"_blank\"\u003etools vs models\u003c/a\u003e).\u003c/p\u003e\u003cp id=\"text-12\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eTODO: \u003ca href=\"ref://DemiralpRabitz97\" target=\"_blank\"\u003eDemiralp \u0026 Rabitz, 1997\u003c/a\u003e – dispersion-free wave packets!\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"klein-gordon\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"m274-450 248 248-42 42-320-320 320-320 42 42-248 248h526v60H274Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003ePrevious\u003c/p\u003e\u003c/a\u003e\u003cstretch id=\"content-buttons-go-167\" style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1ch;min-height:1em;font-weight:thin;text-align:start\"\u003e\u003c/stretch\u003e\u003ca id=\"content-buttons-go-168\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"maxwell\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M686-450H160v-60h526L438-758l42-42 320 320-320 320-42-42 248-248Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Maxwell","URL":"maxwell","Title":"Maxwell","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{"eq":["eq_scalar-potential","eq_scalar-pot-chg","eq_colomb","eq_curl","eq_e-vec","eq_b-vec","eq_divergence","eq_maxeq-i","eq_maxeq-ii","eq_maxeq-iii","eq_maxeq-iv","eq_lorenz","eq_b-source","eq_b-force","eq_e-force"],"figure":["figure_gradient"],"table":["table_maxwells"]},"Description":"The basic equations of electromagnetism were developed in the 1800’s, through the work of many people. James Clerk Maxwell put all this work together, with several important insights of his own, in a series of papers in 1861-2 that describe what are now known as Maxwell’s equations, which fully characterize the behavior of the electromagnetic fields, in relation to electric charges. Maxwell conceived of these as propagating through the luminiferous \u003ca href=\"aether\" target=\"_blank\"\u003eaether\u003c/a\u003e.","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eMaxwell\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-24\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe basic equations of electromagnetism were developed in the 1800’s, through the work of many people. James Clerk Maxwell put all this work together, with several important insights of his own, in a series of papers in 1861-2 that describe what are now known as Maxwell’s equations, which fully characterize the behavior of the electromagnetic fields, in relation to electric charges. Maxwell conceived of these as propagating through the luminiferous \u003ca href=\"aether\" target=\"_blank\"\u003eaether\u003c/a\u003e.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAlthough Maxwell’s equations describe the same kind of wave propagation we explored in \u003ca href=\"waves\" target=\"_blank\"\u003ewaves\u003c/a\u003e, they do so in a somewhat complex way, involving vector fields and various vector operators that can be difficult to understand. Therefore, we start with a formulation of the EM equations that will be immediately familiar from the previous chapter, using second-order wave equations. These second-order wave equations operate on the \u003cstrong\u003eelectromagnetic potentials\u003c/strong\u003e, instead of the electric and magnetic vector fields, which are described by the standard Maxwell’s equations.\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAlthough many people regard the vector fields as the primary physical reality underlying EM, there is solid physical evidence showing that the electromagnetic potentials are physically real, and exert measurable physical effects, for example the \u003cstrong\u003eAharonov-Bohm\u003c/strong\u003e effect (described later). Thus, in addition to being mathematically simpler, the potential-based formulation seems to be physically necessary. Nevertheless, there remain several important sources of controversy and confusion over this choice, which will be introduced as we go through.\u003c/p\u003e\u003cp id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThere are two electromagnetic potentials, the \u003cstrong\u003eelectrical scalar potential\u003c/strong\u003e, which is variously written as \u003cspan class=\"math inline\"\u003e\\(\\Phi\\)\u003c/span\u003e (capital Greek “Phi”) or \u003cspan class=\"math inline\"\u003e\\(V\\)\u003c/span\u003e or \u003cspan class=\"math inline\"\u003e\\(A_0\\)\u003c/span\u003e, and the \u003cstrong\u003emagnetic vector potential\u003c/strong\u003e, typically written as: \u003cspan class=\"math inline\"\u003e\\(\\vec{A} = (A_x, A_y, A_z)\\)\u003c/span\u003e. Most readers should be familiar with the notion of the electrical potential \u003cspan class=\"math inline\"\u003e\\(V\\)\u003c/span\u003e, in terms of the voltage of a battery or an electrical outlet — therefore, we’ll focus on it first. Then we’ll cover the more difficult vector potential, which underlies the magnetic field. After exploring each of these and obtaining a solid understanding of their behavior, we relate these potential wave equations back to the original Maxwell equations, and cover various important issues with our potential-based formulation of EM.\u003c/p\u003e\u003cp id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWhat should not get lost in all this discussion is the bare amazing fact that all of EM can be captured in the simple second-order wave equation (with appropriate source terms from electrical charge and current): this is the only equation we need to simulate the propagation of the EM fields over space and time. This wave equation naturally produces the \u003cstrong\u003einverse square law\u003c/strong\u003e of the electrical force, and it does so through strictly local wave propagation mechanisms, avoiding the apparent action-at-a-distance that the calculational tool of the usual Coulomb version of this force law, where you literally compute the force as a function of the distance between two charges.\u003c/p\u003e\u003cp id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFurthermore, the wave equation provides the framework for many aspects of \u003ca href=\"special-relativity\" target=\"_blank\"\u003especial relativity\u003c/a\u003e, such as the constant speed of light in a vacuum. The \u003ca href=\"four-vector\" target=\"_blank\"\u003efour-vector\u003c/a\u003e space-time notation establishes a deep connection between the way that space and time are interconnected in the wave equation, and in special relativity. This notation enables us to know immediately whether something is \u003cstrong\u003emanifestly covariant\u003c/strong\u003e, which means it is obviously compatible with the principles of special relativity (i.e., invariant with respect to the Lorentz transformation).\u003c/p\u003e\u003cp id=\"text-6\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThus, we see that a huge swath of fundamental physics falls right out of the basic wave equation, which in turn reflects the simplest form of cellular-automaton neighborhood interaction that does anything interesting.\u003c/p\u003e\u003ch2 id=\"the-electrical-scalar-potential\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eThe Electrical Scalar Potential\u003c/h2\u003e\u003cp id=\"text-8\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAn important issue with the electrical scalar potential is that it is always just a relative quantity — the potential is a measure of the voltage difference between two locations, and it varies depending on which two locations you happen to choose. Thus, people feel uncomfortable thinking of it as a physically real entity, because that would seem to imply that there is some kind of objective physical value for the potential at all locations in space, transforming it from a purely relative thing into something that must live in an absolute scale.\u003c/p\u003e\u003cp id=\"text-9\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eBut there doesn’t seem to be any objective way to fix this scale, because all that we can measure are these relative values, and the corresponding electric field, neither of which determine what the absolute potential values should be. This situation is known as \u003cstrong\u003egauge freedom\u003c/strong\u003e, and we’ll see that it becomes an increasingly important concept as we proceed further. We’ll see that it is actually not a problem in our simulation models, because the electric charges directly determine the resulting potential values. This is another example of an important discrepancy between a mathematical problem that really doesn’t seem to correspond to an actual physical problem. The gauge freedom is more apparent than real in this case, all things considered.\u003c/p\u003e\u003cp id=\"text-10\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn the absence of any charges (i.e., in empty space), the electrical scalar potential (we use the \u003cspan class=\"math inline\"\u003e\\(A_0\\)\u003c/span\u003e symbol, to be consistent with the \u003cspan class=\"math inline\"\u003e\\(\\vec{A}\\)\u003c/span\u003e vector potential) obeys the standard second-order wave equation:\u003c/p\u003e\u003cdiv id=\"inline-container-12\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_scalar-potential\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_scalar-potential\"\u003e\u003cb\u003eEq 1:\u003c/b\u003e electrical scalar potential, no charges\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-13\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2 {A_0}}{\\partial t^2} = c^2 \\nabla^2 A_0\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-14\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis means that you already understand exactly how this potential will behave, based on the explorations in \u003ca href=\"waves\" target=\"_blank\"\u003ewaves\u003c/a\u003e. It will propagate at the speed of light (\u003cem\u003ec\u003c/em\u003e), exhibit spreading over time due to the nature of the 3D Laplacian, etc.\u003c/p\u003e\u003cp id=\"text-15\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eTo include the effects of \u003cstrong\u003eelectrical charge\u003c/strong\u003e, we can extend the equation to include a simple additive factor that is proportional to the local charge density, written by convention as the Greek letter “rho” \u003cspan class=\"math inline\"\u003e\\(\\rho\\)\u003c/span\u003e:\u003c/p\u003e\u003cdiv id=\"inline-container-17\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_scalar-pot-chg\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_scalar-pot-chg\"\u003e\u003cb\u003eEq 2:\u003c/b\u003e electrical scalar potential, with charge\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-18\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2 {A_0}}{\\partial t^2} = c^2 \\nabla^2 A_0 + \\frac{1}{\\epsilon_0} \\rho\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-19\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewe’ll explain the \u003cspan class=\"math inline\"\u003e\\(\\epsilon_0\\)\u003c/span\u003e (Greek epsilon) constant, known as the \u003cstrong\u003epermittivity of free space\u003c/strong\u003e, in more detail later. For now, it is just a constant that determines the impact of charge on the electric field. Remembering that the left-hand-side of this wave equation represents the acceleration of the electrical potential, this equation just says that in addition to the local curvature of the field driving acceleration as in the standard wave equation, electrical charge will impart an additional acceleration. It should be clear how our discrete cellular-automaton based wave equation can be augmented to include this extra charge force — you literally just add this term into the acceleration, which then increments the velocity, which then increments the electrical potential state value.\u003c/p\u003e\u003cp id=\"text-20\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn the discrete space-time analog cellular automaton framework, the equations for the scalar potential are:\u003c/p\u003e\u003cp id=\"text-21\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\ddot A_0^{t+1} = \\frac{3}{13}\\sum_{j \\in N_{26}} k_j ({A_0}_j - {A_0}_i) + \\frac{1}{\\epsilon_0} \\rho\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-22\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhich is then integrated into a first-order term (velocity):\u003c/p\u003e\u003cp id=\"text-23\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\dot A_0^{t+1} = \\dot A_0^t + \\ddot A_0^{t+1}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-24\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eand finally the state variable is updated:\u003c/p\u003e\u003cp id=\"text-25\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nA_0^{t+1} = A_0^t + \\dot A_0^{t+1}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cdiv id=\"figure_gradient\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_gradient\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 1:\u003c/b\u003e \u003cp\u003eThe gradient, which is a vector consisting of the local slope along each of the different dimensions (two-dimensional case shown here). The electric field \u003cspan class=\"math inline\"\u003e\\(\\vec{E}\\)\u003c/span\u003e is the gradient of the electrical scalar potential field \u003cspan class=\"math inline\"\u003e\\(A_0\\)\u003c/span\u003e.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cp id=\"text-28\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis wave equation fully characterizes the behavior of the electrical field: \u003cem\u003ewe don’t need anything else to numerically simulate it, and account for all known behavior of the field itself.\u003c/em\u003e Thus, from a physical perspective, we can imagine that only this electrical potential field exists, and things like light waves are just wave propagation over this field.\u003c/p\u003e\u003cp id=\"text-29\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eHowever, to understand how this electrical field influences charged “particles” such as the electron, we do need to extract the electric field vector, which represents the force exerted by the electric field. We can think of this physically as reflecting the force impact of the electrical potential field, derived entirely from the potential, and not as a separate physical entity. Electrical forces ensue from the slope of change (i.e., the \u003cstrong\u003egradient\u003c/strong\u003e, as introduced in the waves chapter and pictured in (\u003ca href=\"maxwell#figure_gradient\" target=\"_blank\"\u003eFigure 1\u003c/a\u003e) of the electrical scalar potential across space, plus the rate of change of the magnetic vector field, which we’ll discuss later.\u003c/p\u003e\u003cp id=\"text-30\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eLoosely speaking, if you have more potential in one place than another, there is a pressure to flow “downhill” along this gradient to equalize the potential. Mathematically speaking, this can be expressed as:\u003c/p\u003e\u003cp id=\"text-31\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{E} = - \\vec{\\nabla} A_0 - \\frac{\\partial \\vec{A}}{\\partial t}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-32\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn words, the electrical field is the spatial gradient of the scalar potential (plus the temporal derivative of the magnetic vector potential \u003cspan class=\"math inline\"\u003e\\(\\vec{A}\\)\u003c/span\u003e). Recall the definition of the gradient operator \u003cspan class=\"math inline\"\u003e\\(\\vec{\\nabla}\\)\u003c/span\u003e from before: it computes the slope or amount of change in a scalar field along each of the three axes, yielding a vector of three values (\u003cspan class=\"math inline\"\u003e\\([E_x, E_y, E_z]\\)\u003c/span\u003e).\u003c/p\u003e\u003cp id=\"text-33\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eTo actually compute this vector quantity in our discrete 3D framework, we need a discrete gradient operator that is basically just the first-order version of the discrete Laplacian operator that we introduced in \u003ca href=\"waves\" target=\"_blank\"\u003ewaves\u003c/a\u003e. It is described in detail in this page \u003ca href=\"discrete-gradient\" target=\"_blank\"\u003ediscrete gradient\u003c/a\u003e.\u003c/p\u003e\u003ch3 id=\"1-r-potential-dropoff-1-r-2-force-field\" style=\"max-width:8in;margin:0.25em;font-size:22px;font-weight:normal;line-height:1.2727273;text-align:start\"\u003e1/r Potential Dropoff = 1/r^2 Force Field\u003c/h3\u003e\u003cp id=\"text-35\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOne of the first things people learn about the electric field is the \u003cstrong\u003eCoulomb force law\u003c/strong\u003e, which states that the electrical force between two charged entities is an \u003cstrong\u003einverse square law\u003c/strong\u003e as a function of the distance between the two:\u003c/p\u003e\u003cdiv id=\"inline-container-37\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_colomb\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_colomb\"\u003e\u003cb\u003eEq 3:\u003c/b\u003e Coulomb force law\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-38\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nF = \\frac{q_1 q_2}{r^2}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-39\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eHow does this derive from the second order wave equation and the charge acceleration? It turns out that the wave equation naturally produces a 1/r dropoff in the electrical potential, only in the 3D form of the wave equation (\u003ca href=\"ref://Whittaker03\" target=\"_blank\"\u003eWhittaker, 1903\u003c/a\u003e). We can see this in the simulation exploration in the next section. This 1/r dropoff in the potential is then translated into an inverse square function \u003cspan class=\"math inline\"\u003e\\(\\frac{1}{r^2}\\)\u003c/span\u003e in the process of computing the gradient force field from the scalar potential field.\u003c/p\u003e\u003cp id=\"text-40\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe fact that this critical inverse-square behavior emerges naturally from the wave equation is just another in a long series of amazing features of this equation.\u003c/p\u003e\u003ch3 id=\"exploration-of-the-electrical-potential\" style=\"max-width:8in;margin:0.25em;font-size:22px;font-weight:normal;line-height:1.2727273;text-align:start\"\u003eExploration of the Electrical Potential\u003c/h3\u003e\u003cp id=\"text-42\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eYou can now explore how charge drives the electrical potential, the 1/r falloff of the scalar potential, and how the electrical force field is computed from the gradient of the scalar potential field. Open the \u003ca href=\"WELDBook/Sims/EM/EM\" target=\"_blank\" title=\"wikilink\"\u003eEM\u003c/a\u003e simulation and follow the directions under the scalar electrical field section.\u003c/p\u003e\u003ch2 id=\"the-magnetic-vector-potential\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eThe Magnetic Vector Potential\u003c/h2\u003e\u003cp id=\"text-44\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eMagnetism is a bit more complex than the electrical field. Instead of a single scalar potential field, it requires a vector potential field, and each of the three components of this vector potential field propagates according to the basic second-order wave equation, with the driving source being the \u003cstrong\u003ecurrent\u003c/strong\u003e vector \u003cspan class=\"math inline\"\u003e\\(\\vec{J}\\)\u003c/span\u003e:\u003c/p\u003e\u003cp id=\"text-45\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2 \\vec{A}}{\\partial t^2} = c^2 \\nabla^2 \\vec{A} + \\mu_0 \\vec{J}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-46\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eInterestingly, the wave equation operates separately on each term. Therefore, the wave equation operating on the vector \u003cspan class=\"math inline\"\u003e\\(\\vec{A}\\)\u003c/span\u003e is the same thing as having three separate wave equations operating in parallel on each of the components of the vector:\u003c/p\u003e\u003cp id=\"text-47\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2 A_x}{\\partial t^2} = c^2 \\nabla^2 A_x + \\mu_0 J_x\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-48\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2 A_y}{\\partial t^2} = c^2 \\nabla^2 A_y + \\mu_0 J_y\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-49\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2 A_z}{\\partial t^2} = c^2 \\nabla^2 A_z + \\mu_0 J_z\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-50\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe computation of the vector potential terms \u003cspan class=\"math inline\"\u003e\\(\\vec{A}\\)\u003c/span\u003e similarly just follow the standard wave equations with an additional source term from the current vector \u003cspan class=\"math inline\"\u003e\\(\\vec{J}\\)\u003c/span\u003e. For the \u003cspan class=\"math inline\"\u003e\\(x\\)\u003c/span\u003e component of \u003cspan class=\"math inline\"\u003e\\(\\vec{A}\\)\u003c/span\u003e, the acceleration term is:\u003c/p\u003e\u003cp id=\"text-51\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\ddot A_x^{t+1} = \\frac{3}{13} \\sum_{j \\in N_{26}} k_j ({A_x}_j - {A_x}_i) + \\mu_0 J_x\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-52\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eand similar equations hold for the \u003cspan class=\"math inline\"\u003e\\(y\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(z\\)\u003c/span\u003e components.\u003c/p\u003e\u003cp id=\"text-53\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAs with the electric potential field, we can capture all of the known physics of the magnetic field propagation and how it is driven from the motion of charge (i.e., current) over time, using the above wave equation. Thus, we imagine that this vector potential is all that exists physically. In addition, as with the electric field, the force exerted by the magnetic field must be computed from this potential. This is where things get a little bit more complicated, because this relationship is not a simple gradient, but rather the \u003cstrong\u003ecurl\u003c/strong\u003e (\u003cspan class=\"math inline\"\u003e\\(\\vec{\\nabla} \\times\\)\u003c/span\u003e) of the magnetic vector potential: \u003cspan class=\"math inline\"\u003e\\(\\vec{B} = \\vec{\\nabla} \\times \\vec{A} \\)\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-54\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIntuitively, curl indicates the extent to which the arrows in a local region are spinning around. Formally, curl is defined as:\u003c/p\u003e\u003cdiv id=\"inline-container-56\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_curl\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_curl\"\u003e\u003cb\u003eEq 4:\u003c/b\u003e curl\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-57\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{\\nabla} \\times \\vec{B} \\equiv \\left( \\left\\[\\frac{\\partial {B_z}}{\\partial {y}} - \\frac{\\partial {B_y}}{\\partial {z}} \\right], \\left[\\frac{\\partial {B_x}}{\\partial {z}} - \\frac{\\partial {B_z}}{\\partial {x}} \\right], \\left[\\frac{\\partial {B_y}}{\\partial {x}} - \\frac{\\partial {B_x}}{\\partial {y}} \\right] \\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-58\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSo, the \u003cspan class=\"math inline\"\u003e\\(x\\)\u003c/span\u003e (first) component of the curl is the crossed spatial gradient of the other two dimensions (\u003cspan class=\"math inline\"\u003e\\(z\\)\u003c/span\u003e by \u003cspan class=\"math inline\"\u003e\\(y\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(y\\)\u003c/span\u003e by \u003cspan class=\"math inline\"\u003e\\(z\\)\u003c/span\u003e), and likewise for the remaining factors. Intuitively, each component measures how much the field is spinning around that dimension.\u003c/p\u003e\u003cp id=\"text-59\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eHere is how to compute the curl in the discrete space-time cellular-automaton framework: discrete curl. It is very similar to the way that the \u003ca href=\"discrete-gradient\" target=\"_blank\"\u003ediscrete gradient\u003c/a\u003e is computed, just with additional subtraction terms.\u003c/p\u003e\u003ch2 id=\"maxwell-s-equations\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eMaxwell’s Equations\u003c/h2\u003e\u003cp id=\"text-61\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eNow that you understand how the electromagnetic potential fields propagate over time, and are driven by charge and the motion of charge (current), in terms of the familiar second-order wave equation, we relate these equations to the four Maxwell’s equations that are covered in most standard electromagnetism courses, and frankly are much more complicated and difficult to understand than the potential formulation.\u003c/p\u003e\u003cp id=\"text-62\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eMaxwell’s equations are all in terms of the electric force vector field \u003cspan class=\"math inline\"\u003e\\(\\vec{E}\\)\u003c/span\u003e and magnetic force vector field \u003cspan class=\"math inline\"\u003e\\(\\vec{B}\\)\u003c/span\u003e:\u003c/p\u003e\u003cdiv id=\"inline-container-64\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_e-vec\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_e-vec\"\u003e\u003cb\u003eEq 5:\u003c/b\u003e electric force vector field\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-65\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{E} = (E_x, E_y, E_z)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cdiv id=\"inline-container-67\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_b-vec\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_b-vec\"\u003e\u003cb\u003eEq 6:\u003c/b\u003e magnetic force vector field\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-68\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{B} = (B_x, B_y, B_z)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-69\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eMaxwell’s four equations (in SI metric standard units) are:\u003c/p\u003e\u003cdiv id=\"table_maxwells\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eTable 1:\u003c/b\u003e Maxwell's equations\u003c/p\u003e\u003ctable id=\"table_maxwells\" style=\"display:grid;flex-direction:row;justify-content:center;align-items:start;columns:2;gap:6px;font-weight:thin;text-align:start\"\u003e\u003cth id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:bold;line-height:1.5;text-align:start\"\u003eName\u003c/th\u003e\u003cth id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:bold;line-height:1.5;text-align:start\"\u003eEquation\u003c/th\u003e\u003ctd id=\"text-2\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cstrong\u003ei. Gauss’s law:\u003c/strong\u003e\u003c/td\u003e\u003ctd id=\"text-3\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math inline\"\u003e\\(\\vec{\\nabla} \\cdot \\vec{E} = \\frac{1}{\\epsilon_0} \\rho \\)\u003c/span\u003e\u003c/td\u003e\u003ctd id=\"text-4\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cstrong\u003eii. Gauss’s law for magnetism:\u003c/strong\u003e\u003c/td\u003e\u003ctd id=\"text-5\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math inline\"\u003e\\(\\vec{\\nabla} \\cdot \\vec{B} = 0 \\)\u003c/span\u003e\u003c/td\u003e\u003ctd id=\"text-6\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cstrong\u003eiii. Faraday’s law of induction:\u003c/strong\u003e\u003c/td\u003e\u003ctd id=\"text-7\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math inline\"\u003e\\(\\vec{\\nabla} \\times \\vec{E} = - \\frac{\\partial \\vec{B}}{\\partial t} \\)\u003c/span\u003e\u003c/td\u003e\u003ctd id=\"text-8\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cstrong\u003eiv. Ampère’s law:\u003c/strong\u003e\u003c/td\u003e\u003ctd id=\"text-9\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math inline\"\u003e\\(\\vec{\\nabla} \\times \\vec{B} = \\mu_0 \\vec{J} + \\mu_0 \\epsilon_0 \\frac{\\partial \\vec{E}}{\\partial t} \\)\u003c/span\u003e\u003c/td\u003e\u003c/table\u003e\u003c/div\u003e\u003cp id=\"text-72\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhere again \u003cspan class=\"math inline\"\u003e\\(\\rho\\)\u003c/span\u003e is charge per unit volume (density) in a given location, and \u003cspan class=\"math inline\"\u003e\\(\\vec{J}\\)\u003c/span\u003e is the charge current density (where the charge is moving).\u003c/p\u003e\u003cp id=\"text-73\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe constants \u003cspan class=\"math inline\"\u003e\\(\\epsilon_0\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(\\mu_0\\)\u003c/span\u003e are the \u003cstrong\u003epermittivity of free space\u003c/strong\u003e and \u003cstrong\u003epermeability of free space\u003c/strong\u003e, respectively. They are related to the speed of light as follows:\u003c/p\u003e\u003cp id=\"text-74\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\mu_0 \\epsilon_0 = \\frac{1}{c^2}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-75\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nc = \\frac{1}{\\sqrt{\\mu_0 \\epsilon_0}}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-76\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe first, Gauss’s law, basically says that charge is the source of the electric field — this is equivalent to the \u003cspan class=\"math inline\"\u003e\\(\\rho\\)\u003c/span\u003e factor that we added to the standard wave equation for the electrical potential above. Instead of that being one part of the overall wave equation, it is pulled out separately in Maxwell’s equations, in terms of the \u003cstrong\u003edivergence operator\u003c/strong\u003e. Loosely speaking, the divergence operator \u003cspan class=\"math inline\"\u003e\\(\\vec{\\nabla} \\cdot\\)\u003c/span\u003e indicates how much “new stuff” is coming out of a given region of space, as compared to simply passing this stuff along from your neighbors. A non-zero divergence indicates that new stuff is being generated in that region, whereas a 0 divergence means that region of space is just passing its arrows along from its neighbors. More formally, the divergence operator is defined as:\u003c/p\u003e\u003cdiv id=\"inline-container-78\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_divergence\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_divergence\"\u003e\u003cb\u003eEq 7:\u003c/b\u003e divergence\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-79\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{\\nabla} \\cdot \\vec{E} \\equiv \\frac{\\partial {E_x}}{\\partial {x}} + \\frac{\\partial {E_y}}{\\partial {y}} + \\frac{\\partial {E_z}}{\\partial {z}}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-80\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhich is just summing up the spatial gradient along each of the three spatial dimensions. If you work through specific examples of vector fields under this operator, you can prove the above generalizations to yourself.\u003c/p\u003e\u003cp id=\"text-81\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eGiven what you now know about divergence, you should realize that the second law means that there are no sources for the magnetic field: its divergence is always 0. Therefore, each region of space is just passing the magnetic arrows along, without adding to them. If so, then how do they ever get created in the first place?\u003c/p\u003e\u003cp id=\"text-82\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis is the job of the fourth law (Ampère’s law, with Maxwell’s extension to it), which states that the source of the magnetic field is the charge current density \u003cspan class=\"math inline\"\u003e\\(\\vec{J}\\)\u003c/span\u003e, which we saw previously in the magnetic vector potential wave equation above.\u003c/p\u003e\u003cp id=\"text-83\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eHowever, Ampère’s law also includes the rate of change of the electrical field (\u003cspan class=\"math inline\"\u003e\\(\\frac{\\partial \\vec{E}}{\\partial t}\\)\u003c/span\u003e). Thus, moving charge and moving electrical fields create magnetism, but they do so not in a divergence-like way, but rather in terms of the \u003cem\u003ecurl\u003c/em\u003e, \u003cspan class=\"math inline\"\u003e\\(\\vec{\\nabla} \\times\\)\u003c/span\u003e (Figure~\\ref{fig.curl}). Intuitively, curl indicates the extent to which the arrows in a local region are spinning around.\u003c/p\u003e\u003cp id=\"text-84\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe third law (Faraday’s law of induction) states that the electric field can also exhibit curl, in proportion to the rate of change in the magnetic field. These last two laws have a certain symmetry to them, and indeed if you write Maxwell’s equations for the empty space where there is no charge at all (i.e., \u003cspan class=\"math inline\"\u003e\\(\\rho = 0\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(\\vec{J} = (0,0,0)\\)\u003c/span\u003e), you get a nicely symmetric set of equations:\u003c/p\u003e\u003cdiv id=\"inline-container-86\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_maxeq-i\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_maxeq-i\"\u003e\u003cb\u003eEq 8:\u003c/b\u003e i.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-87\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{\\nabla} \\cdot \\vec{E} = 0\n\\]\u003c/span\u003e\u003c/p\u003e\u003cdiv id=\"inline-container-89\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_maxeq-ii\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_maxeq-ii\"\u003e\u003cb\u003eEq 9:\u003c/b\u003e ii.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-90\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{\\nabla} \\cdot \\vec{B} = 0\n\\]\u003c/span\u003e\u003c/p\u003e\u003cdiv id=\"inline-container-92\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_maxeq-iii\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_maxeq-iii\"\u003e\u003cb\u003eEq 10:\u003c/b\u003e iii.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-93\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{\\nabla} \\times \\vec{E} = - \\frac{\\partial \\vec{B}}{\\partial t}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cdiv id=\"inline-container-95\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_maxeq-iv\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_maxeq-iv\"\u003e\u003cb\u003eEq 11:\u003c/b\u003e iv.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-96\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{\\nabla} \\times \\vec{B} = \\frac{1}{c^2} \\frac{\\partial \\vec{E}}{\\partial t}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-97\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis shows that the wave propagation dynamics in Maxwell’s equations are due to interactions between the \u003cspan class=\"math inline\"\u003e\\(\\vec{E}\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(\\vec{B}\\)\u003c/span\u003e fields, whereas interestingly in the potential-based formulation, we have four entirely separate second-order wave equations. It is not immediately obvious how this produces the same thing, but if we do the appropriate math, we can see that it all works out. See \u003ca href=\"maxwell#maxwell potential derivations\" target=\"_blank\"\u003emaxwell potential derivations\u003c/a\u003e for all the gory details.\u003c/p\u003e\u003ch2 id=\"the-lorenz-gauge-and-condition\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eThe Lorenz Gauge and Condition\u003c/h2\u003e\u003cp id=\"text-99\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThere is one important wrinkle in the connection between Maxwell’s equations and the simple wave equations operating on the potentials, which has to do with the \u003cstrong\u003eLorenz Condition\u003c/strong\u003e, which is why the wave equation version is known as the \u003cstrong\u003eLorenz gauge\u003c/strong\u003e. The wave equations only correspond to Maxwell’s standard four equations if this condition is met:\u003c/p\u003e\u003cdiv id=\"inline-container-101\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_lorenz\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_lorenz\"\u003e\u003cb\u003eEq 12:\u003c/b\u003e Lorenz condition\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-102\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{1}{c^2} \\frac{\\partial A_0}{\\partial t} + \\vec{\\nabla} \\cdot \\vec{A} = 0\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-103\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe question is, what does this expression mean, and what can we do to ensure that it is properly satisfied? This condition is known as the \u003cstrong\u003econtinuity equation\u003c/strong\u003e, and it corresponds to a situation where the total quantity of something (in this case the total scalar and vector potential) is conserved over time. This makes sense intuitively, because the first term is the rate of change in the scalar potential, and the second term is the divergence or source of the vector potential field (recall that we saw this in the first of Maxwell’s equations, where the divergence of the electric force is equal to the charge density) — this says that any source of the vector potential must then translate into corresponding changes in the scalar potential. Thus, the sources are balanced out with temporal changes, producing a net balance of zero — no change. It is important to appreciate that although there are no sources for the magnetic vector force \u003cspan class=\"math inline\"\u003e\\(\\vec{B}\\)\u003c/span\u003e, there \u003cem\u003eare\u003c/em\u003e sources for the magnetic vector \u003cem\u003epotential\u003c/em\u003e \u003cspan class=\"math inline\"\u003e\\(\\vec{A}\\)\u003c/span\u003e, namely the current density:\u003c/p\u003e\u003cdiv id=\"inline-container-105\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_b-source\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_b-source\"\u003e\u003cb\u003eEq 13:\u003c/b\u003e source of magnetic potential\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-106\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{\\nabla} \\cdot \\vec{A} = \\mu_0 \\vec{J}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-107\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThus, to satisfy the Lorenz condition, we also need to ensure that wherever there are current sources, the electric scalar potential experiences a corresponding negative rate of change:\u003c/p\u003e\u003cp id=\"text-108\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{1}{c^2} \\frac{\\partial A_0}{\\partial t} = -\\mu_0 \\vec{J}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-109\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWe will keep this condition in mind as we consider how the electron wave function generates both the electric and magnetic potential fields. Outside of the region where there are currents, this constraint is not relevant, and the simple second-order wave propagation can take place.\u003c/p\u003e\u003ch3 id=\"lorentz-invariance-of-the-wave-equation\" style=\"max-width:8in;margin:0.25em;font-size:22px;font-weight:normal;line-height:1.2727273;text-align:start\"\u003eLorentz Invariance of the Wave Equation\u003c/h3\u003e\u003cp id=\"text-111\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe \u003ca href=\"four-vector\" target=\"_blank\"\u003efour-vector\u003c/a\u003e notation provides a powerful and compact way of understanding the relationship between \u003ca href=\"special-relativity\" target=\"_blank\"\u003especial relativity\u003c/a\u003e and wave equations. Now is a good time to read that page, so the following analysis of the EM wave functions will make sense.\u003c/p\u003e\u003cp id=\"text-112\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSpecifically, the four-vector framework allows us to write all of EM using a single equation, in terms of the \u003cstrong\u003efour potential:\u003c/strong\u003e\u003c/p\u003e\u003cp id=\"text-113\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n{A}^\\mu = (A_0, \\vec{A}) = (A_0, A_x, A_y, A_z)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-114\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn this four-vector notation, the wave equation arises as the double-application of the four-derivative operator:\u003c/p\u003e\u003cp id=\"text-115\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\partial_\\mu \\partial^\\mu = \\frac{\\partial^2 {}}{\\partial t^2} - \\nabla^2\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-116\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eTherefore, we can write:\u003c/p\u003e\u003cp id=\"text-117\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\partial_\\mu \\partial^\\mu {A}^\\mu = {k}^\\mu {J}^\\mu\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-118\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhere \u003cspan class=\"math inline\"\u003e\\({J}^\\mu = (J_0, J_x, J_y, J_z)\\)\u003c/span\u003e, and \u003cspan class=\"math inline\"\u003e\\(J_0 = \\rho\\)\u003c/span\u003e, and \u003cspan class=\"math inline\"\u003e\\({k}^\\mu = \\left( \\frac{1}{\\epsilon_0}, \\mu_0, \\mu_0, \\mu_0 \\right)\\)\u003c/span\u003e.\u003c/p\u003e\u003cp id=\"text-119\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis extreme level of simplicity accurately expresses the fundamental point that the electromagnetic force can be described using only the basic wave equation, plus the source driving terms. The charge density \u003cspan class=\"math inline\"\u003e\\(\\rho\\)\u003c/span\u003e and current density \u003cspan class=\"math inline\"\u003e\\(\\vec{J}\\)\u003c/span\u003e provide an external driving force on the electromagnetic field equations (and are thus the sources of the fields). Interestingly, this potential formalism just requires four variables, which is intriguingly convenient for the four-vector space-time framework.\u003c/p\u003e\u003cp id=\"text-120\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOne immediate payoff from the above mathematical detour into space-time coordinates and special relativity is that it provides a preliminary indication that the wave equation obeys the fundamental constraints of special relativity. Indeed, our model holds that the wave equation embodies special relativity. Specifically, the wave equation has the covariant form where time = space, and, equivalently, time and space are represented with opposite signs:\u003c/p\u003e\u003cp id=\"text-121\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2 {s}}{\\partial t^2} = \\nabla^2 s\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-122\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2 {s}}{\\partial t^2} - \\nabla^2 s = 0\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-123\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAs explored in \u003ca href=\"special-relativity\" target=\"_blank\"\u003especial relativity\u003c/a\u003e, this covariant form means that \u003cstrong\u003ethe wave equation works the same in any reference frame\u003c/strong\u003e. In other words, it is \u003cem\u003einvariant\u003c/em\u003e with respect to Lorentz transformations, which are the conversion operations that take you from one reference frame to another. Therefore, if reality happens to be a wave equation operating within one specific reference frame (the reference frame of our grid of cells), this wave equation will automatically appear the same to all observers regardless of their relative velocities with respect to this underlying grid.\u003c/p\u003e\u003cp id=\"text-124\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn other words, just because our CA model happened to produce the wave equation (based on more “bottom up” considerations of simplicity of underlying mechanisms) we also get special relativity for free in the bargain!\u003c/p\u003e\u003ch3 id=\"the-lorenz-condition-in-four-vector-notation\" style=\"max-width:8in;margin:0.25em;font-size:22px;font-weight:normal;line-height:1.2727273;text-align:start\"\u003eThe Lorenz Condition in Four-Vector Notation\u003c/h3\u003e\u003cp id=\"text-126\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFinally, it is useful to express the Lorenz condition in four-vector terminology:\u003c/p\u003e\u003cp id=\"text-127\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\partial_\\mu {A}^\\mu = 0\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-128\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ebecause:\u003c/p\u003e\u003cp id=\"text-129\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\partial_\\mu {A}^\\mu = 0\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-130\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial {A^0}}{\\partial t} + \\vec{\\nabla} \\cdot \\vec{A} = 0\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-131\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial {A^0}}{\\partial t} = -\\vec{\\nabla} \\cdot \\vec{A}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-132\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAs noted above, this represents a continuity equation, and when this continuity equation holds, the total amount of the four-vector quantity \u003cspan class=\"math inline\"\u003e\\(A^\\mu\\)\u003c/span\u003e is conserved over time: it can move around to different locations, but the total amount of it integrated across all of space never changes over time. The four-vector notation provides a single unified quantity that is conserved. Therefore, the Lorenz condition is effectively just saying that the system must conserve the potential values, which is true of the wave equations, except where there are source terms, so that is where we need to focus on the Lorenz condition.\u003c/p\u003e\u003ch2 id=\"maxwell-potential-derivations\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eMaxwell Potential Derivations\u003c/h2\u003e\u003cp id=\"text-134\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn this subsection, we derive the second-order wave equations operating on the electrical scalar potential and the magnetic vector potential.\u003c/p\u003e\u003cp id=\"text-135\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThese are the relationships between the potentials and the force vector fields:\u003c/p\u003e\u003cdiv id=\"inline-container-137\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_b-force\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_b-force\"\u003e\u003cb\u003eEq 14:\u003c/b\u003e magnetic force from vector potential\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-138\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{B} = \\vec{\\nabla} \\times \\vec{A}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cdiv id=\"inline-container-140\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_e-force\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_e-force\"\u003e\u003cb\u003eEq 15:\u003c/b\u003e electric force from scalar potential\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-141\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{E} = - \\vec{\\nabla} A_0\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-142\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003ca href=\"maxwell#eq_maxeq-ii\" target=\"_blank\"\u003eEq 9\u003c/a\u003e is automatically satisfied by the fact that the magnetic field vector is the curl of the magnetic vector potential, because the divergence of the curl is always 0. Similarly, \u003ca href=\"maxwell#eq_maxeq-i\" target=\"_blank\"\u003eEq 8\u003c/a\u003e is just the basic source equation that is easily incorporated into the second-order wave equation as we saw in the main chapter. Thus, the key challenge to explain is how the \u003ca href=\"maxwell#eq_maxeq-iii\" target=\"_blank\"\u003eEq 10\u003c/a\u003e and \u003ca href=\"maxwell#eq_maxeq-iv\" target=\"_blank\"\u003eEq 11\u003c/a\u003e equations correspond to simple wave equations on the potentials.\u003c/p\u003e\u003cp id=\"text-143\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWe start by inserting the definition for \u003cspan class=\"math inline\"\u003e\\(\\vec{B}\\)\u003c/span\u003e into \u003ca href=\"maxwell#eq_maxeq-iii\" target=\"_blank\"\u003eEq 10\u003c/a\u003e, to get:\u003c/p\u003e\u003cp id=\"text-144\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{\\nabla} \\times \\vec{E} = - \\frac{\\partial \\vec{B}}{\\partial t}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-145\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{\\nabla} \\times \\vec{E} = - \\frac{\\partial}{\\partial t}\\left(\\vec{\\nabla} \\times \\vec{A}\\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-146\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{\\nabla} \\times \\vec{E} = - \\vec{\\nabla} \\times \\frac{\\partial \\vec{A}}{\\partial t}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-147\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{\\nabla} \\times \\vec{E} + -\\vec{\\nabla} \\times \\frac{\\partial \\vec{A}}{\\partial t} = 0\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-148\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{\\nabla} \\times \\left(\\vec{E} + \\frac{\\partial \\vec{A}}{\\partial t} \\right) = 0\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-149\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eBecause this quantity in the parentheses has zero curl according to this equation, it means that it can be written in terms of a gradient of a scalar potential (gradients of scalar fields can’t have curl; for example, a ball rolling down a surface can only roll, not spin):\u003c/p\u003e\u003cp id=\"text-150\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{E} + \\frac{\\partial \\vec{A}}{\\partial t} = - \\vec{\\nabla}{A_0}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-151\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eTherefore, we can just re-arrange these terms to get a definition of the electric vector field in terms of the scalar field:\u003c/p\u003e\u003cp id=\"text-152\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{E} = - \\vec{\\nabla} A_0 - \\frac{\\partial \\vec{A}}{\\partial t}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-153\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis definition agrees with the simple gradient equation given earlier, but it also adds the first temporal derivative of the vector potential as a contributor to the electrical field. To remove this extra term, we need to remove one extra degree of freedom from our system, by making the following definition:\u003c/p\u003e\u003cp id=\"text-154\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial A_0}{\\partial t} = - c^2 \\vec{\\nabla} \\cdot \\vec{A}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-155\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis is known as the \u003cstrong\u003eLorenz gauge\u003c/strong\u003e or condition (not Lorentz, as some incorrectly state), which is also covered above.\u003c/p\u003e\u003cp id=\"text-156\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eNow we use this within \u003ca href=\"maxwell#eq_maxeq-i\" target=\"_blank\"\u003eEq 8\u003c/a\u003e and \u003ca href=\"maxwell#eq_maxeq-iv\" target=\"_blank\"\u003eEq 11\u003c/a\u003e. For \u003ca href=\"maxwell#eq_maxeq-i\" target=\"_blank\"\u003eEq 8\u003c/a\u003e, we get:\u003c/p\u003e\u003cp id=\"text-157\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{\\nabla} \\cdot \\vec{E} = \\frac{1}{\\epsilon_0} \\rho\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-158\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{\\nabla} \\cdot \\left( \\vec{\\nabla} A_0 + \\frac{\\partial \\vec{A}}{\\partial t} \\right) = - \\frac{1}{\\epsilon_0} \\rho\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-159\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\nabla^2 A_0 + \\vec{\\nabla} \\cdot \\frac{\\partial \\vec{A}}{\\partial t} = - \\frac{1}{\\epsilon_0} \\rho\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-160\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eand for \u003ca href=\"maxwell#eq_maxeq-iv\" target=\"_blank\"\u003eEq 11\u003c/a\u003e, we get:\u003c/p\u003e\u003cp id=\"text-161\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{\\nabla} \\times \\vec{B} = \\mu_0 \\vec{J} + \\mu_0 \\epsilon_0 \\frac{\\partial \\vec{E}}{\\partial t}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-162\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{\\nabla} \\times \\left( \\vec{\\nabla} \\times \\vec{A} \\right) = \\mu_0 \\vec{J} + \\mu_0 \\epsilon_0 \\frac{\\partial}{\\partial t} \\left( -\\vec{\\nabla} A_0 - \\frac{\\partial \\vec{A}}{\\partial t} \\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-163\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\left( \\nabla^2 \\vec{A} - \\frac{1}{c^2} \\frac{\\partial^2 \\vec{A}}{\\partial t^2} \\right) - \\vec{\\nabla} \\left( \\vec{\\nabla} \\cdot \\vec{A} + \\frac{1}{c^2} \\frac{\\partial {A_0}}{\\partial t} \\right) = -\\mu_0\\vec{J}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-164\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\sqcap \\vec{A} - \\vec{\\nabla} \\left( \\vec{\\nabla} \\cdot \\vec{A} + \\frac{1}{c^2} \\frac{\\partial {A_0}}{\\partial t} \\right) = -\\mu_0 \\vec{J}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-165\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSo, perhaps you can see that now we are getting somewhat closer to a wave equation. We now have the \u003cspan class=\"math inline\"\u003e\\(\\nabla^2\\)\u003c/span\u003e terms showing up in both equations, and in the latter we have a \u003cspan class=\"math inline\"\u003e\\(\\frac{\\partial^2 {}}{\\partial t^2}\\)\u003c/span\u003e term, such that we get the classic wave equation signature, as indicated by the last line where we substituted in the d’Alembertian operator (note: \u003cspan class=\"math inline\"\u003e\\(\\sqcap\\)\u003c/span\u003e should actually just be a square box, but we don’t have that available for technical reasons):\u003c/p\u003e\u003cp id=\"text-166\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\sqcap = \\frac{\\partial^2 {}}{\\partial t^2} - \\nabla^2\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-167\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhich encapsulates the wave equation dynamics of second-order time minus second-order space differentials.\u003c/p\u003e\u003cp id=\"text-168\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eBut these equations are still quite messy, and certainly are not purely wave equations. Furthermore, there is still some extra degrees of freedom in these potentials in terms of their implications for the observable \u003cspan class=\"math inline\"\u003e\\(\\vec{E}\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(\\vec{B}\\)\u003c/span\u003e fields. For example, you can add any kind of constant numerical offset to the entire electrical potential \u003cspan class=\"math inline\"\u003e\\(A_0\\)\u003c/span\u003e, and this will not change the behavior of the system, because the observable electrical force is defined only in terms of the gradient or slope of this potential field, not its absolute magnitude.\u003c/p\u003e\u003cp id=\"text-169\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn more formal parlance, it is said that one can choose different \u003cem\u003egauges\u003c/em\u003e for these potentials, and this choice will affect the form of the equations. In the Lorenz gauge mentioned earlier, this extra degree of freedom is removed by defining:\u003c/p\u003e\u003cp id=\"text-170\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial {A_0}}{\\partial t} = - c^2 \\vec{\\nabla} \\cdot \\vec{A}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-171\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFor later convenience, this also means that:\u003c/p\u003e\u003cp id=\"text-172\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{\\nabla} \\cdot \\vec{A} = - \\frac{1}{c^2}\\frac{\\partial {A_0}}{\\partial t}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-173\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWhen you take this latter form and plug it into the above two Maxwell equations, you end up canceling some of the nasty bits out, and you get a very nice form of standard wave equations. For \u003ca href=\"maxwell#eq_maxeq-i\" target=\"_blank\"\u003eEq 8\u003c/a\u003e, we get:\u003c/p\u003e\u003cp id=\"text-174\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\nabla^2 A_0 + \\vec{\\nabla} \\cdot \\frac{\\partial {\\vec{A}}}{\\partial t}} = - \\frac{1}{\\epsilon_0} \\rho\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-175\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\nabla^2 A_0 + \\frac{\\partial {}}{\\partial t}\\left(\\vec{\\nabla} \\cdot \\vec{A}\\right) = - \\frac{1}{\\epsilon_0} \\rho\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-176\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\nabla^2 A_0 - \\frac{\\partial {}}{\\partial t}\\left(\\frac{1}{c^2}\\frac{\\partial {A_0}}{\\partial t}\\right) = - \\frac{1}{\\epsilon_0} \\rho\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-177\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\nabla^2 A_0 - \\frac{1}{c^2} \\frac{\\partial^2 {A_0}}{\\partial t^2} = - \\frac{1}{\\epsilon_0} \\rho\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-178\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2 {A_0}}{\\partial t^2} = c^2 \\nabla^2 A_0 + \\frac{1}{\\epsilon_0} \\rho\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-179\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e(where the boxes indicate the location of the substitution). The result is clearly a basic wave equation with an additional “driving” term of \u003cspan class=\"math inline\"\u003e\\(\\frac{1}{\\epsilon_0} \\rho\\)\u003c/span\u003e.\u003c/p\u003e\u003cp id=\"text-180\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFor \u003ca href=\"maxwell#eq_maxeq-iv\" target=\"_blank\"\u003eEq 11\u003c/a\u003e, you get:\u003c/p\u003e\u003cp id=\"text-181\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\left( \\nabla^2 \\vec{A} - \\frac{1}{c^2} \\frac{\\partial^2 \\vec{A}}{\\partial t^2} \\right) - \\vec{\\nabla} \\left( \\vec{\\nabla} \\cdot \\vec{A} + \\frac{1}{c^2} \\frac{\\partial {A_0}}{\\partial t} \\right) = -\\mu_0\\vec{J}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-182\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\left( \\nabla^2 \\vec{A} - \\frac{1}{c^2} \\frac{\\partial^2 \\vec{A}}{\\partial t^2} \\right) - \\vec{\\nabla} \\left( \\vec{\\nabla} \\cdot \\vec{A} - \\frac{1}{c^2}c^2 \\vec{\\nabla} \\cdot \\vec{A} \\right) = -\\mu_0\\vec{J}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-183\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\left( \\nabla^2 \\vec{A} - \\frac{1}{c^2} \\frac{\\partial^2 \\vec{A}}{\\partial t^2} \\right) - \\vec{\\nabla} \\left( \\vec{\\nabla} \\cdot \\vec{A} - \\vec{\\nabla} \\cdot \\vec{A} \\right) = -\\mu_0\\vec{J}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-184\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\nabla^2 \\vec{A} - \\frac{1}{c^2} \\frac{\\partial^2 \\vec{A}}{\\partial t^2} = -\\mu_0\\vec{J}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-185\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2 \\vec{A}}{\\partial t^2} = c^2 \\nabla^2 \\vec{A} + \\mu_0\\vec{J}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-186\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAgain, somewhat miraculously, a wave equation emerges, again with a driving term.\u003c/p\u003e\u003ch2 id=\"todo\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eTODO\u003c/h2\u003e\u003cul id=\"frame-188\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003ca href=\"ref://Franklin07\" target=\"_blank\"\u003eFranklin, 2007\u003c/a\u003e shows that energy is all in the charge, and none in the field.\u003c/li\u003e\u003c/ul\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"matter-waves\"\u003e\u003csvg id=\"icon\" 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style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Non locality","URL":"non-locality","Title":"Non locality","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{},"Description":"One of the primary challenges of adopting a \u003ca href=\"cellular-automaton\" target=\"_blank\"\u003ecellular automaton\u003c/a\u003e (CA) framework for physics is reconciling the local interactions among neighboring cells, which so naturally produces a relativistic speed-of-light limit (one time step update per lattice cell), with the now irrefutable evidence for some kind of non-locality in quantum physics among \u003cem\u003eentangled\u003c/em\u003e particles. Recent work within the \u003ca href=\"pilot-wave\" target=\"_blank\"\u003epilot-wave\u003c/a\u003e framework has helped to significantly clarify the nature of these non-local interactions, and the broader conflict that they actually pose for all of QM, despite many attempts to downplay these issues from within the standard QM frameworks (\u003ca href=\"ref://DurrGoldsteinNorsenEtAl14\" target=\"_blank\"\u003eDürr et al., 2014\u003c/a\u003e).","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eNon locality\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-25\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOne of the primary challenges of adopting a \u003ca href=\"cellular-automaton\" target=\"_blank\"\u003ecellular automaton\u003c/a\u003e (CA) framework for physics is reconciling the local interactions among neighboring cells, which so naturally produces a relativistic speed-of-light limit (one time step update per lattice cell), with the now irrefutable evidence for some kind of non-locality in quantum physics among \u003cem\u003eentangled\u003c/em\u003e particles. Recent work within the \u003ca href=\"pilot-wave\" target=\"_blank\"\u003epilot-wave\u003c/a\u003e framework has helped to significantly clarify the nature of these non-local interactions, and the broader conflict that they actually pose for all of QM, despite many attempts to downplay these issues from within the standard QM frameworks (\u003ca href=\"ref://DurrGoldsteinNorsenEtAl14\" target=\"_blank\"\u003eDürr et al., 2014\u003c/a\u003e).\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAt the heart of all of this is a fundamental, pervasive confusion of what this non-locality means, where it comes from within the existing QM formalisms, and how it might actually work in a more physically plausible, localist manner.\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFirst, there is a widespread confusion that just because you cannot use quantum non-locality to actually communicate \u003cem\u003earbitrary\u003c/em\u003e information at superliminal (faster than light) speeds (\u003ca href=\"ref://BallentineJarrett87\" target=\"_blank\"\u003eBallentine \u0026 Jarrett, 1987\u003c/a\u003e), that it does not somehow violate the absolute constraints of \u003ca href=\"special-relativity\" target=\"_blank\"\u003especial relativity\u003c/a\u003e. Despite considerable wishful thinking to the contrary, there are many solid arguments indicating that this is not true. The observed non-local correlations across entangled space-like separated particles (i.e., more than the speed-of-light away) \u003cem\u003erequire\u003c/em\u003e some form of superliminal physics, or its equivalent in the form of a manifest instantaneous non-locality (\u003ca href=\"ref://Norsen11\" target=\"_blank\"\u003eNorsen, 2011\u003c/a\u003e; \u003ca href=\"ref://Maudlin11\" target=\"_blank\"\u003eMaudlin, 2011\u003c/a\u003e; \u003ca href=\"ref://Shimony93\" target=\"_blank\"\u003eShimony, 1993\u003c/a\u003e).\u003c/p\u003e\u003cp id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe inability to use this superliminal transmission for communication is due to the fundamental lack of external control we have over these correlations, which must be analyzed “offline” and in aggregate, in comparison with the results from the other observers, and does not contradict the relativistic violation (\u003ca href=\"ref://Norsen11\" target=\"_blank\"\u003eNorsen, 2011\u003c/a\u003e). Furthermore, the standard \u003ca href=\"copenhagen\" target=\"_blank\"\u003eCopenhagen\u003c/a\u003e interpretation covers up this glaring conflict with relativity by adopting a purely \u003ca href=\"epistemic-vs-ontic\" target=\"_blank\"\u003eepistemological\u003c/a\u003e, non-physical interpretation of the quantum wave-driven phenomena: if it isn’t “real” then it isn’t a “real violation” somehow?\u003c/p\u003e\u003cp id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn the current framework, the CA models typically require a superliminal update rate to achieve numerical stability, while still propagating light waves at the speed of light. Thus, there is no fundamental requirement that everything in such a framework operate at the the speed-of-light update rate, so it is entirely possible that the quantum wave function is actually communicated significantly faster.\u003c/p\u003e\u003cp id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOn balance, the adjustment of an arbitrary update rate seems much less consequential than abandoning the foundational requirement of local mechanisms, which are essential for the overall simplicity of the underlying physical system. Non-locality is fundamentally about complexity: what is the scope of communication if not local? How is that determined? What could possibly support the bandwidth necessary for extensive non-local communication? It really just doesn’t make any sense at all from a mechanistic perspective.\u003c/p\u003e\u003cp id=\"text-6\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn this context, a recent paper showed that superliminal \u003cem\u003ecausal\u003c/em\u003e mechanisms should lead to a violation of the no-signalling constraint (\u003ca href=\"ref://BancalPironioAcinEtAl12\" target=\"_blank\"\u003eBancal et al., 2012\u003c/a\u003e). However, the analysis there did not take into account the role of true randomness (nondeterminism, see \u003ca href=\"stochastic-particles\" target=\"_blank\"\u003estochastic particles\u003c/a\u003e), which is essential (\u003ca href=\"ref://Popescu14\" target=\"_blank\"\u003ePopescu, 2014\u003c/a\u003e; \u003ca href=\"ref://AharonovRohrlich08\" target=\"_blank\"\u003eAharonov \u0026 Rohrlich, 2008\u003c/a\u003e). The non-local correlations in quantum physics are much weaker than theoretically possible, reflecting this randomness. Furthermore, other assumptions may yet produce the observed results in a more naturalistic manner (\u003ca href=\"ref://ReidDrummond26\" target=\"_blank\"\u003eReid \u0026 Drummond, 2026\u003c/a\u003e).\u003c/p\u003e\u003cp id=\"text-7\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSecond, the existing QM formalisms are \u003cem\u003eall\u003c/em\u003e based on the use of a \u003ca href=\"configuration-space\" target=\"_blank\"\u003econfiguration space\u003c/a\u003e representation of the physical world, which is manifestly non-local, and thus builds in the non-locality from the start. There is an obvious double-standard where the \u003ca href=\"pilot-wave\" target=\"_blank\"\u003epilot-wave\u003c/a\u003e framework is dismissed for its reliance on this configuration space representation, while all other standard approaches likewise use precisely the same representation (e.g., in the \u003ca href=\"copenhagen\" target=\"_blank\"\u003eCopenhagen\u003c/a\u003e interpretation and the \u003ca href=\"hilbert-space\" target=\"_blank\"\u003eHilbert space\u003c/a\u003e matrix mechanics).\u003c/p\u003e\u003cp id=\"text-8\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOne further, critical point here is that the use of configuration space in existing frameworks has been essential because the underlying equations are \u003cem\u003elinear\u003c/em\u003e, so configuration space is the only way to capture nonlinear interactions. Linear systems are tremendously advantageous for mathematical analysis, but the computational CA-based approach has no problem incorporating nonlinear dynamics. Therefore, that represents a specific “competitive advantage” of the present approach, in trying to find the right kind of nonlinear dynamics that gives rise to such a simple emergent behavior as captured in the linear configuration-space models.\u003c/p\u003e\u003ch2 id=\"physical-models-are-local\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003ePhysical models are local\u003c/h2\u003e\u003cp id=\"text-10\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThere is a clear pattern in the examples from \u003ca href=\"tools-vs-models\" target=\"_blank\"\u003etools vs models\u003c/a\u003e of the plausible physical models vs. calculational tools. All of the physical models are based on \u003cem\u003elocal\u003c/em\u003e propagation of signals according to simple laws, whereas the calculational tools tend to employ non-local equations. This difference is directly tied to the fundamental tradeoffs at work: the calculational tools need non-locality to enable simple single-step calculations, whereas the physical models use local dynamics to enable iterative, autonomous calculations to work in the general case. Indeed, it is difficult to imagine how an autonomous model could be strongly non-local: the amount of computation and communication required per step would become prohibitive.\u003c/p\u003e\u003cp id=\"text-11\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWe have also seen that standard QM calculational tools including \u003ca href=\"configuration-space\" target=\"_blank\"\u003econfiguration space\u003c/a\u003e, \u003ca href=\"hilbert-space\" target=\"_blank\"\u003eHilbert space\u003c/a\u003e matrix mechanics state vectors, and Fourier space \u003ca href=\"qed\" target=\"_blank\"\u003equantum field theory\u003c/a\u003e are all fully non-local state representations, and thus cannot help but to produce non-local results. Again, this is analogous to using Newton’s gravitational law or the Coulomb equations for EM: it is baked right into the model. Nevertheless, there are strong empirical results suggesting that at least some of these non-local effects are real (\u003ca href=\"ref://AspectDalibardRoger82\" target=\"_blank\"\u003eAspect et al., 1982\u003c/a\u003e; \u003ca href=\"ref://AspectGrangierRoger82\" target=\"_blank\"\u003eAspect et al., 1982\u003c/a\u003e; \u003ca href=\"ref://TittelBrendelGisinEtAl98\" target=\"_blank\"\u003eTittel et al., 1998\u003c/a\u003e). Furthermore, in many cases they make good physical sense, in reflecting the strict conservation of some property such as spin.\u003c/p\u003e\u003cp id=\"text-12\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThus, the challenge here is to try to better understand how the underlying physical processes of quantum wave field interactions, unfolding over time through strictly local propagation mechanisms, can end up producing non-local effects consistent with the empirical data. An important available degree of freedom here is that while the speed of light is strictly obeyed by the Maxwell wave equations, it is unclear if such a constraint actually applies to the quantum wave fields. Furthermore, our initial implementations of coupled Dirac — Maxwell equations in the CA framework demonstrate that a faster rate of updating, with smaller incremental update steps, is needed for numerical stability, relative to the simple CA one-cell-per-unit-time speed of light value.\u003c/p\u003e\u003cp id=\"text-13\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThus, we are already necessarily in a “superluminal” space. For example, it seems logical that if the quantum wave fields updated at twice the speed of light (in the EM field), then they would always be able to “keep in touch” with each other, even for particles moving apart at near light speed, and this could mediate observed non-local effects. Again, we need to be flexible here and explore whatever mechanisms might actually work to capture the established empirical data. In my estimation, requiring 2x light speed quantum wave function propagation is far more reasonable than the completely non-local interaction assumed by the standard QM calculational models.\u003c/p\u003e\u003ch3 id=\"entanglement-and-non-locality\" style=\"max-width:8in;margin:0.25em;font-size:22px;font-weight:normal;line-height:1.2727273;text-align:start\"\u003eEntanglement and non-locality\u003c/h3\u003e\u003cp id=\"text-15\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe phenomenology of quantum non-locality is fascinating and confusing, and provides some insights into the relevant physical properties of the quantum realm. The primary line of investigation traces back to a paper that Einstein wrote with Podolsky and Rosen in 1935, known as the EPR paper, about the strange implications of quantum \u003cem\u003eentanglement\u003c/em\u003e. In the standard formalisms, entanglement occurs whenever the aggregate quantum state of a system is not a simple product of its constituents: i.e., there is some kind of interdependency between the elements. This is closely related to the issue of \u003ca href=\"contextual\" target=\"_blank\"\u003econtextual\u003c/a\u003e effects, and is particularly clear in the case of quantum \u003cem\u003espin\u003c/em\u003e, which is represented by state variables that do \u003cem\u003enot commute\u003c/em\u003e with each other, meaning that their states are irrevocably intertwined with each other, and it is impossible to simultaneously specify all of them.\u003c/p\u003e\u003cp id=\"text-16\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFurthermore, there is a conservation law associated with spin, so that the total spin of a system must remain conserved over time. Thus, if a spin zero particle splits into two spin \u003csup\u003e1\u003c/sup\u003e⁄\u003csub\u003e2\u003c/sub\u003e particles, these two particles must maintain opposite spin states (+\u003csup\u003e1\u003c/sup\u003e⁄\u003csub\u003e2\u003c/sub\u003e and -\u003csup\u003e1\u003c/sup\u003e⁄\u003csub\u003e2\u003c/sub\u003e) to conserve overall spin, and this represents a strong form of entanglement. Thus, if you were to measure the spin state of one particle, you should be able to predict that the other’s spin state is the opposite. The extra challenge here is that, because spin is necessarily contextual, the measurement process actually \u003cem\u003ecreates\u003c/em\u003e a specific spin state in a particle.\u003c/p\u003e\u003cp id=\"text-17\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eTherefore, logically, it seems as though the measurement process operating on one particle must somehow “inform” a measurement process operating on the other particle, so that it produces the opposite result. In practice, these two measurements could be (and have been) performed on particles moving away from each other at or close to the speed of light, with sufficient space-like separation that it would be impossible for any actual light-speed communication between the measuring devices.\u003c/p\u003e\u003cp id=\"text-18\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eHowever, there is a \u003cem\u003eno-signaling\u003c/em\u003e proof, based on the standard QM formalism, that shows that it would be impossible for the measurement process in one location to actually communicate information to the other process (\u003ca href=\"ref://BallentineJarrett87\" target=\"_blank\"\u003eBallentine \u0026 Jarrett, 1987\u003c/a\u003e). Specifically, if “Alice” is conducting measurements in one location on particle A, and “Bob” is doing the same on B, there is no way for Alice to send some kind of message to Bob. In other words, there is no way for Bob to know, \u003cem\u003ejust by looking at the outcomes of his own measurement device\u003c/em\u003e, what Alice is doing.\u003c/p\u003e\u003cp id=\"text-19\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIntuitively, this makes sense because neither knows the initial state of the particles, nor the state of the other’s measuring device, so they just record a bunch of seemingly-random spin measurements that would be indistinguishable from any other such experiment. It is only when Alice and Bob get together later and compare their results, that they can then discover the presence of \u003cem\u003ecorrelations\u003c/em\u003e in the outcomes of their different measurements. It is these correlations that the famous “Bell’s inequalities” (\u003ca href=\"ref://Bell64\" target=\"_blank\"\u003eBell, 1964\u003c/a\u003e) are based on, which form the basis for the various empirical tests of quantum non-locality. Critically, these correlations are “preordained” in the laws of QM, and thus do not represent an \u003cem\u003eadditional\u003c/em\u003e degree of freedom that could be used to send new information. That is all that the no-signaling proof shows.\u003c/p\u003e\u003cp id=\"text-20\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe above argument serves to satisfy many people that somehow standard QM formalisms are not violating the speed-of-light constraints of special relativity. But this really does not square with the original intuition that somehow the two “measurement contexts” of Alice and Bob must be doing \u003cem\u003esomething\u003c/em\u003e physical to establish these correlations, especially given the strong constraint that spin measurements are necessarily \u003ca href=\"contextual\" target=\"_blank\"\u003econtextual\u003c/a\u003e (\u003ca href=\"ref://Norsen11\" target=\"_blank\"\u003eNorsen, 2011\u003c/a\u003e; \u003ca href=\"ref://Maudlin11\" target=\"_blank\"\u003eMaudlin, 2011\u003c/a\u003e; \u003ca href=\"ref://Shimony93\" target=\"_blank\"\u003eShimony, 1993\u003c/a\u003e). Furthermore, the pilot-wave framework unambiguously shows that entanglement phenomena directly require non-local interactions between the two particles (\u003ca href=\"ref://Norsen14\" target=\"_blank\"\u003eNorsen, 2014\u003c/a\u003e; \u003ca href=\"ref://NorsenMarianOriols15\" target=\"_blank\"\u003eNorsen et al., 2015\u003c/a\u003e).\u003c/p\u003e\u003cp id=\"text-21\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSpecifically, by replacing the standard \u003ca href=\"configuration-space\" target=\"_blank\"\u003econfiguration space\u003c/a\u003e formalism with separate wave functions for each particle, Norsen and colleagues can isolate the direct particle-particle interactions necessary to replicate the predictions that are otherwise obtained by the full configuration-space model. When the quantum state is not at all entangled, then no such particle interactions are necessary. However, with any amount of entanglement, these interactions are necessary, and, especially in the case of spin, would require some kind of effective non-local communication to replicate the observed results.\u003c/p\u003e\u003cp id=\"text-22\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThus, consistent with the original concerns of Einstein and colleagues, it really does seem as though quantum physics requires “spooky action-at-a-distance” in a way that is incompatible with simple local speed-of-light dynamics. The no-signaling proof does not actually eliminate this problem. Most people, adopting the standard QM formalisms that are inherently non-local, are not particularly bothered by this, and have already swallowed the “red pill” of physical ignorance anyway. See this \u003ca href=\"https://www.youtube.com/watch?v=NIk_0AW5hFU\" target=\"_blank\"\u003eVeritasium video\u003c/a\u003e for a nice thorough exploration.\u003c/p\u003e\u003cp id=\"text-23\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eHowever, in our current attempt to provide a complete physical picture of how the quantum world operates, we must somehow account for these results, within a CA-like framework employing only local interactions among cells. Therefore, it seems as though some kind of superluminal quantum wave dynamics are likely to be required. Furthermore, it is critical to appreciate that the only localized elements of the quantum state are the positions of the particles, and everything else (momentum, energy, spin) is directly tied up with the wave interactions between the \u003ca href=\"dirac\" target=\"_blank\"\u003eDirac\u003c/a\u003e and \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e fields, which are widely physically distributed over space, and continuously mutually interacting with the fields generated by other particles.\u003c/p\u003e\u003cp id=\"text-24\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIt does not seem implausible that the necessary quantum correlations could emerge from such a system, and by simulating and analyzing this system in detail, and exploring different rates of wave function updating and ways in which the quantum fields from different particles interact, we should be able to better understand the otherwise mysterious nature of quantum non-locality.\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"maxwell\"\u003e\u003csvg id=\"icon\" 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style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003ePhenomenology\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-27\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis page provides an accounting of all the major phenomenology of (quantum) physics that strongly constrains theorizing, along with how different frameworks account for these phenomena.\u003c/p\u003e\u003ch2 id=\"space-time\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eSpace-time\u003c/h2\u003e\u003ch3 id=\"core-phenomena\" style=\"max-width:8in;margin:0.25em;font-size:22px;font-weight:normal;line-height:1.2727273;text-align:start\"\u003eCore phenomena\u003c/h3\u003e\u003cul id=\"frame-3\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Physics appears to operate within 3 spatial dimensions, evolving over time.\u003c/li\u003e\u003cli id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• With the notable exception of non-locality effects in QM, many phenomena are compatible with strictly local interactions within spacetime — the light-speed cone of causality etc.\u003c/li\u003e\u003cli id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Cosmologically, space seems remarkably, suspiciously flat.\u003c/li\u003e\u003cli id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Philosophically, it seems overall more elegant to imagine that space is infinite in extent and time — this avoids the inevitable conundrums of “edges” in spacetime — something outside the bubble of the big-bang or whatever.\u003c/li\u003e\u003c/ul\u003e\u003ch3 id=\"implications\" style=\"max-width:8in;margin:0.25em;font-size:22px;font-weight:normal;line-height:1.2727273;text-align:start\"\u003eImplications\u003c/h3\u003e\u003cul id=\"frame-5\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• From the mechanistic, autonomous perspective, the \u003ca href=\"cellular-automaton\" target=\"_blank\"\u003ecellular automaton\u003c/a\u003e framework (CA) fits exceptionally well with these basic facts of spacetime. Space is primary, and physical rules are defined as update equations operating on cubic cells of primary \u003cem\u003estuff.\u003c/em\u003e The speed of light emerges naturally from this framework. It can easily accommodate all of classical EM — we have a nice implementation of \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e’s equations within this framework.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• But QM \u003ca href=\"non-locality\" target=\"_blank\"\u003enon-locality\u003c/a\u003e flies in the face of this framework. Now that we have to take this seriously, we need to see if all the “pros” of the CA framework can be somehow augmented with whatever it takes to accommodate nonlocality, or whether it needs to be discarded entirely as a nice idea that simply doesn’t “work”.\u003c/p\u003e\u003c/ul\u003e\u003ch2 id=\"waves\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eWaves\u003c/h2\u003e\u003ch3 id=\"core-phenomena-1\" style=\"max-width:8in;margin:0.25em;font-size:22px;font-weight:normal;line-height:1.2727273;text-align:start\"\u003eCore phenomena\u003c/h3\u003e\u003cul id=\"frame-8\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• EM is well described by \u003ca href=\"waves\" target=\"_blank\"\u003ewaves\u003c/a\u003e, and the surprising success of the \u003ca href=\"semiclassical\" target=\"_blank\"\u003esemiclassical\u003c/a\u003e approach suggests that perhaps the \u003ca href=\"photon\" target=\"_blank\"\u003ephoton\u003c/a\u003e is unnecessary.\u003c/li\u003e\u003cli id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003ca href=\"dirac\" target=\"_blank\"\u003eDirac\u003c/a\u003e wave equation — relativistic generalization of Schroedinger equation — can be implemented nicely within CA framework (in its second order form).\u003c/li\u003e\u003cli id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003ca href=\"double-slit\" target=\"_blank\"\u003edouble-slit\u003c/a\u003e interference effects, etc.\u003c/li\u003e\u003c/ul\u003e\u003ch3 id=\"implications-1\" style=\"max-width:8in;margin:0.25em;font-size:22px;font-weight:normal;line-height:1.2727273;text-align:start\"\u003eImplications\u003c/h3\u003e\u003cul id=\"frame-10\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003ca href=\"ref://Bialynicki-Birula94\" target=\"_blank\"\u003eBialynicki-Birula, 1994\u003c/a\u003e and \u003ca href=\"ref://Meyer96\" target=\"_blank\"\u003eMeyer, 1996\u003c/a\u003e models: uniqueness of conservative (unitary) local transfer functions, require bidirectional flow — need to investigate those, esp BB.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Another approach: lattice particles with wave-functions on the links can potentially regularize wave dynamics — digitizes wave functions at every step.\u003c/p\u003e\u003c/ul\u003e\u003ch2 id=\"particles\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eParticles\u003c/h2\u003e\u003ch3 id=\"core-phenomena-2\" style=\"max-width:8in;margin:0.25em;font-size:22px;font-weight:normal;line-height:1.2727273;text-align:start\"\u003eCore phenomena\u003c/h3\u003e\u003cul id=\"frame-13\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Strictly quantized, conserved charge.\u003c/li\u003e\u003cli id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Conserved fermion number.\u003c/li\u003e\u003cli id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Scattering, scattering, scattering! how do waves ever scatter!? Cross sections appear pointlike?\u003c/li\u003e\u003cli id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Fundamental frequency / momentum relationship: \u003cspan class=\"math inline\"\u003e\\(\\lambda = \\frac{h}{p}\\)\u003c/span\u003e, \u003cspan class=\"math inline\"\u003e\\(p = \\frac{h}{\\lambda}\\)\u003c/span\u003e – this is the absolute essence of QM, reflected in wave equations, but what about discrete \u003ca href=\"stochastic-particles\" target=\"_blank\"\u003estochastic particles\u003c/a\u003e?\u003c/li\u003e\u003cli id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Photoelectric effect? maybe more about atoms than EM? maybe not though? \u003ca href=\"semiclassical\" target=\"_blank\"\u003eSemiclassical\u003c/a\u003e models can get pretty far — what are the remaining phenomena that we need to deal with? photon-based entanglement maybe the most strong?\u003c/li\u003e\u003cli id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Particles transform into other particles! what are the rules that govern creation and destruction of particles?? this basic fact greatly complicates any particle model — would have been much simpler to think of particles as hard little permanent things — but they are only partially that. Other times they are rather fluid..\u003c/li\u003e\u003cli id=\"text-6\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Furthermore, many particles are strictly temporary and unstable — what makes them unstable?\u003c/li\u003e\u003cli id=\"text-7\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Virtual particles — short-lived but “real” — have measurable effects.. what is the deal!?\u003c/li\u003e\u003c/ul\u003e\u003ch3 id=\"implications-2\" style=\"max-width:8in;margin:0.25em;font-size:22px;font-weight:normal;line-height:1.2727273;text-align:start\"\u003eImplications\u003c/h3\u003e\u003cul id=\"frame-15\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• all really challenge pure wave-based approach, \u003cem\u003eexcept\u003c/em\u003e the fluidity of particles and and virtual particles — but emergent particles within a pure wave framework doesn’t seem likely to handle all the strong particle phenomena.\u003c/li\u003e\u003cli id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• as we know from QM, once you fully acknowledge the dual wave-particle reality, all the conundrums emerge. the pure-wave dodge isn’t going to work. we need to confront this head-on.\u003c/li\u003e\u003cli id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• classic copenhagen / Born rule approach is non-starter as mechanistic theory: doesn’t define when collapse happens, depends on vaugely-defined “measurement” events, etc.\u003c/li\u003e\u003cli id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Bohm is very appealing: particles always somewhere, have well-defined trajectories, no collapse. all the nonlocality is in the wave function — particles themselves are fully in 3D spacetime.\n\n\u003c/li\u003e\u003cul id=\"frame-4\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:8ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e◦ doesn’t have a good relativistic formulation? actually it does — H. Nikolic has shown this.\u003c/li\u003e\u003cli id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e◦ doesn’t deal with fluidity of particles — also work by Nikolic attempts to merge QFT particle creation / destruction with Bohmian world view.\u003c/li\u003e\u003c/ul\u003e\u003cli id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Quantum Field approach (QED): fourier space, particles are modes of vibration — naturally non-local, deals with particle transformations, virtual particles..\u003c/li\u003e\u003c/ul\u003e\u003ch2 id=\"zero-point-energy-and-coherent-background-field-effects\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eZero point energy and coherent background field effects\u003c/h2\u003e\u003cul id=\"frame-17\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• There is a non-zero background energy level in a 2nd quantized system — this plays a critical role in the stochastic electrodynamics (SED) and stochastic optics models (Marshall \u0026 Santos, 1988; Marshall \u0026 Santos, 1997; de la Pena \u0026 Cetto, 1996) — the ZPF = Zero Point Field.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• In the oil drop simulations of QM, the wave field emerges from a coordinated shaking of the fluid.\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• I think the ZPF / ZPE seems to emerge because of the synchronization of all the points in a lattice?? if we have a big lattice system, all phenomena may emerge as disturbances on top of a coordinated synchronized vibrating field. Disturbances themselves can propagate at the “speed of light” in such a system, but perhaps some nonlocal things could happen in terms of coordinated changes across the entire field???\u003c/p\u003e\u003c/ul\u003e\u003ch2 id=\"quantum-harmonic-oscillator-qho-and-second-quantization\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eQuantum Harmonic Oscillator (QHO) and second quantization\u003c/h2\u003e\u003cp id=\"text-19\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eTrying to make sense of the foundations of QFT — key pages: \u003ca href=\"https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator\" target=\"_blank\"\u003ehttps://en.wikipedia.org/wiki/Quantum_harmonic_oscillator\u003c/a\u003e \u003ca href=\"https://en.wikipedia.org/wiki/Phonon\" target=\"_blank\"\u003ehttps://en.wikipedia.org/wiki/Phonon\u003c/a\u003e \u003ca href=\"https://en.wikipedia.org/wiki/Normal_mode\" target=\"_blank\"\u003ehttps://en.wikipedia.org/wiki/Normal_mode\u003c/a\u003e \u003ca href=\"https://en.wikipedia.org/wiki/Zero-point_energy\" target=\"_blank\"\u003ehttps://en.wikipedia.org/wiki/Zero-point_energy\u003c/a\u003e\u003c/p\u003e\u003cul id=\"frame-20\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• The basic QHO and phonon theory is derived from classical wave equations, where the acceleration is proportional to distances from neighbors, as derived here: \u003ca href=\"https://grey.colorado.edu/WELD/index.php/WELDBook/Waves\" target=\"_blank\"\u003ehttps://grey.colorado.edu/WELD/index.php/WELDBook/Waves\u003c/a\u003e\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• The “mass” and “spring constant” coupling relationship of the particles in the lattice model determine the speed of propagation, equivalent to the speed-of-light squared (c^2).\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• As near as I can tell, second quantization of this lattice field amounts to the following moves:\u003c/p\u003e\u003cp id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Representing everything in terms of the normal modes of oscillation. The normal modes are an orthogonal basis set of sine waves across the entire state, in multiples of the fundamental frequency. In other words, Fourier space.\u003c/p\u003e\u003cul id=\"frame-4\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:8ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e◦ This also clarifies the situation a bit: \u003ca href=\"https://physics.stackexchange.com/questions/122570/which-is-more-fundamental-fields-or-particles\" target=\"_blank\"\u003ehttps://physics.stackexchange.com/questions/122570/which-is-more-fundamental-fields-or-particles\u003c/a\u003e — the original state-based formulation in terms of wave functions for particles is a problem because particles are indistinquishable from each other and thus it has 2x the amount of info and needs to subtract that out — representing in terms of Fourier space doesn’t have that problem.\u003c/li\u003e\u003cli id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e◦ It seems implicit that the states one is considering are all “contextualized” or “conditioned” on a particular situation — they are not true \u003ca href=\"autonomous\" target=\"_blank\" title=\"wikilink\"\u003eautonomous\u003c/a\u003e states — in other words, they are like the violin string — about a particular constrained object like an electron in an atom — they cannot apply more generally because then the particles extend across the entire universe in the Fourier representation.\n\n\u003c/li\u003e\u003cul id=\"frame-2\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:12ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Indeed, \u003ca href=\"https://physics.stackexchange.com/questions/248754/what-do-the-wave-functions-associated-to-the-fock-states-of-each-mode-of-a-bound\" target=\"_blank\"\u003ehttps://physics.stackexchange.com/questions/248754/what-do-the-wave-functions-associated-to-the-fock-states-of-each-mode-of-a-bound\u003c/a\u003e states that “Each fock state has an associated wave function” — i.e., the QFT bookkeeping is not the full picture — there is an additional wave function associated with each such state! This might be relevant: \u003ca href=\"http://iopscience.iop.org/article/10.1088/1751-8121/aa70ba/meta\" target=\"_blank\"\u003ehttp://iopscience.iop.org/article/10.\u003csup\u003e1088\u003c/sup\u003e⁄\u003csub\u003e1751\u003c/sub\u003e-8121/aa70ba/meta\u003c/a\u003e\u003c/li\u003e\u003c/ul\u003e\u003cli id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e◦ Critically, \u003cem\u003ethere is no fundamental quantization\u003c/em\u003e to the field. In the continuous Fourier space, there isn’t any natural cutoff to the frequencies, or preferred fundamental frequency. This is truly just a bookkeeping device for tracking whatever frequencies or energies might be relevant for a given “contextual” situation — you apply the QFT to describing a \u003cem\u003especific\u003c/em\u003e state that does have relevant quantization dynamics due to atomic confinement etc..\u003c/li\u003e\u003cli id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e◦ In the discrete, lattice-like QHO model, there \u003cem\u003eis\u003c/em\u003e a cutoff, highest frequency, and Planck’s constant could possibly be related to the distance between lattice sites here. Need to look at \u003ca href=\"https://en.wikipedia.org/wiki/Natural_units\" target=\"_blank\"\u003ehttps://en.wikipedia.org/wiki/Natural_units\u003c/a\u003e — e.g., the Planck scale could be defined much higher if we ignore G, which is what drives things so tiny in space and time, and so huge in mass scale. E = hv proportionality doesn’t imply any specific limit on frequency or wavelength.. Paper about discretization cutoff with various relevant discussion: \u003ca href=\"https://arxiv.org/abs/1210.1847\" target=\"_blank\"\u003ehttps://arxiv.org/abs/1210.1847\u003c/a\u003e\u003c/li\u003e\u003cli id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e◦ There is still \u003cem\u003esome\u003c/em\u003e kind of quantization because particles are represented as integer-valued multiples of these normal modes? The amplitudes are not continuous, but quantized. This is perhaps the key discrepancy from the classical system?\u003c/li\u003e\u003cli id=\"text-6\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e◦ Due to the Fourier representation of the system, a particle corresponds to one of these discretized modes being created across the entire universe — i.e., there is no spatial localization. what!?\u003c/li\u003e\u003cli id=\"text-7\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e◦ The zero point energy corresponds to a non-zero fluctuation in this field — various ways of understanding this: for real particles, they have to have a wave function, and a wave function always implies some kind of oscillation (e.g., rotation among the real / complex components) — this oscillation persists even when there is no overt motion of the particle (i.e., the momentum factor is zero — momentum = curvature of wave, so this would be a perfectly “flat” wave..)\u003c/li\u003e\u003c/ul\u003e\u003c/ul\u003e\u003cul id=\"frame-21\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Bottom line: as usual, the question is what is the physics and what is the math here? seems like quantization and resonance are key — is this just discretization?\u003c/li\u003e\u003cli id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• do neighbors really interact, or not? the fourier expansion seems to get rid of them..\u003c/li\u003e\u003cli id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• how is a localized particle ever represented!? phase interference effects of course.\u003c/li\u003e\u003c/ul\u003e\u003ch1 id=\"nonlocality-of-entanglement\" style=\"max-width:8in;margin:0.25em;font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eNonlocality of entanglement\u003c/h1\u003e\u003cp id=\"text-23\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSo many attempts to explain this phenomenon really fail to confront the nonlocality head-on, and invoke things like the no-communication theorem or some kind of weird predeterminism or even the many-worlds view, all of which fail to truly confront the nonlocal nature of the phenomenon. For example, this article by a nobel-prize winning physicist: \u003ca href=\"https://www.quantamagazine.org/entanglement-made-simple-20160428/\" target=\"_blank\"\u003ehttps://www.quantamagazine.org/entanglement-made-simple-20160428/\u003c/a\u003e denies the mystery of nonlocality by appealing to a hidden variable account, without even so much as realizing it:\u003c/p\u003e\u003cp id=\"text-24\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e“Nor is it paradoxical to find that distant events are correlated. After all, if I put each member of a pair of gloves in boxes, and mail them to opposite sides of the earth, I should not be surprised that by looking inside one box I can determine the handedness of the glove in the other. Similarly, in all known cases the correlations between an EPR pair must be imprinted when its members are close together, though of course they can survive subsequent separation, as though they had memories. Again, the peculiarity of EPR is not correlation as such, but its possible embodiment in complementary forms.”\u003c/p\u003e\u003cp id=\"text-25\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThat is exactly a hidden variable account: entanglement creates complementary states in two particles (like the two hands of a pair of gloves), and those two gloves fly off in different directions, to be measured later — absolutely reasonable and sensible that they should have opposite handedness. Unfortunately, this is completely wrong from a QM perspective as several commenters point out, and it completely elides the entire absurdity of the nonlocality!\u003c/p\u003e\u003cp id=\"text-26\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe logic really isn’t that complicated: the wave function for an entangled system fundamentally expresses the necessary unitary nature of the conservation of some kind of property (spin, polarization, etc). If you start out with X amount of spin (e.g., 0), then whatever spin is measured at A must be compensated for by an offsetting spin at B so that the net spin is still 0. This, together with the critical idea that the state is truly indeterminate until measured, necessitates that the measurement process on one part of the wavefunction must have immediate, nonlocal implications for the measurement on the other part of the wavefunction. If both end up measuring “spin up” then there is a clear violation of the conservation of spin. There really is no way to have this happen “physically” without \u003cem\u003ewhatever\u003c/em\u003e is happening during measurement at location A somehow be “communicated” back to location B to properly influence the measurement outcome there. Whether “Alice” can send a message to “Bob” using this mysterious nonlocal communication channel (the no-communication theorem) is quite beside the point: \u003cem\u003esomething\u003c/em\u003e is truly nonlocal about this process!\u003c/p\u003e\u003cp id=\"text-27\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOne major problem with this conundrum is that there isn’t a “physical” model for \u003cem\u003eany\u003c/em\u003e of the processes involved. We don’t even know if the wavefunction itself is “real”, and we certainly don’t know what a “measurement” is. So that makes it a bit strange to start talking about the physical nature of the nonlocal communication process when a wavefunction collapses during measurement. It is all very abstract math, and at that level, it makes for nice clean calculations\u003c/p\u003e\u003cp id=\"text-28\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe main “physical” attempt to understand measurement is through decoherence: as a quantum state interacts with the measurement apparatus, the wavefunction does not actually collapse, it is just so strongly “hammered” by the measurement process that only one of the relevant eigenstates survives. But I haven’t seen any explanations of how this kind of decoherence process would ensure that the other half of the wavefunction should always collapse the other way. For example, why couldn’t a \u003cem\u003elocal\u003c/em\u003e conservation of spin operate within each half of the overall wavefunction, when taking into account the full state of the measurement system together with the incoming thing being measured (photon, electron, or whatever)?\u003c/p\u003e\u003cp id=\"text-29\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis entanglement / nonlocality really cuts to the heart of core issues in QM. One major question is whether the wavefunction is something “physical” or just “epistemological” — something that describes our state of knowledge, but not something that actually “exists” in nature. If it is purely epistemological, then somehow the nonlocality problem seems to evaporate: the collapse of a non-physical wavefunction presumably cannot itself be a physical process, so, somehow, thinking about a physical signal going across space doesn’t make any sense either. But in this case, we are still left with the fundamental mystery: changing the measurement settings at A \u003cem\u003esomehow\u003c/em\u003e really does affect outcomes measured at B. Calling the wavefunction epistemological seems to just avoid any attempt to answer this problem. Indeed, this is really the problem with the entire Copenhagen interpretation: it avoids all the hard questions and essentially amounts to the classic “shut up and calculate” approach — just stick to the abstract math and don’t worry about the physics.\u003c/p\u003e\u003cp id=\"text-30\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe deBroglie-Bohm framework ascribes physical reality to the wavefunction, and explicitly makes it nonlocal to account for the relevant data. This seems like the only framework that isn’t just ducking the problem using one sleight-of-hand or another.\u003c/p\u003e\u003cp id=\"text-31\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSome other relevant points:\u003c/p\u003e\u003cul id=\"frame-32\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Are we putting too much “reality” into the outcomes of measurements? There is this assumption that measurement extracts some “true” property of a thing. But measurement is known to be “contextual”, and really it makes more sense to think of it as “creating its own reality” — e.g., the above figure of polarization just rotating light around — intermediate filtering can “rescue” light that would otherwise have been blocked. There is evidence that nonlocality and contextuality are manifestations of the same thing: \u003ca href=\"https://phys.org/news/2016-03-features-quantum-mechanics.html\" target=\"_blank\"\u003ehttps://phys.org/news/2016-03-features-quantum-mechanics.html\u003c/a\u003e — \u003ca href=\"https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.090401\" target=\"_blank\"\u003ehttps://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.090401\u003c/a\u003e . The problem is that contextuality makes perfect sense, while nonlocality makes no sense, but it doesn’t seem like we can just always have contextuality and somehow get rid of nonlocality? Anyway, understanding this more clearly seems like an important direction.\u003c/li\u003e\u003c/ul\u003e\u003cul id=\"frame-33\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• if wavefunction collapse is an illusion, and measurement is always “contextual”, then what are the implications for the conservation logic behind nonlocal entanglement? There isn’t any reality to the thing being measured in the first place — it is just a “construction”. So why does measurement in one place have to have any implication for measurement elsewhere? Except that apparently it does.. This seems to get back to the contextuality vs. nonlocality tradeoff issue — where contextuality is absent, nonlocality holds..\u003c/li\u003e\u003c/ul\u003e\u003cul id=\"frame-34\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Standard Hilbert-space formulations of QM reinforce the notion that the choice of a basis set is essentially arbitrary. But the Schrödinger equation, and more strongly the Dirac equation, and QED and all the field theoretic frameworks of the standard model, are manifestly \u003cem\u003enot\u003c/em\u003e formulated about arbitrary probability distributions floating around. They are about a specific Hamiltonian describing particles with specific masses interacting with EM fields in specific ways, etc. The frequency (spatial gradient) of the wave function describes the momentum of the particle, while second-order temporal derivative describes its energy. There is manifestly here a very specific ontology and a strongly preferred basis. Mathematically perhaps things can be rotated and transformed in various ways, but it really seems like there are strongly relevant properties being represented here, that constitute the reality of something like an electron. And these are represented by wave functions. The whole apparatus of the standard model is just one big wave function after another. So it really seems like the wave function is what is real, if anything is. This is the initial premise of the WELD approach.\u003c/li\u003e\u003c/ul\u003e\u003cul id=\"frame-35\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• If the Dirac wave function is the only “true” reality of the electron, and it never collapses, then all measurement is definitely a “contextual” fiction — the wave always remains a distributed thing across space and time. But how then do we explain that if we find something in one place, we don’t also find it somewhere else? The wavefunction has to get squeezed into a small space — in general in our WELD models this kind of squeezing of the wavefunction has been difficult. This is the mystery of the Born rule collapse function: it magically digitizes these crazy waves. Again we circle back to these fundamental mysteries. The pure wave picture seems to be incompatible with this Born magic, unless we can find some better ways of keeping the waves localized. Second quantization seems like the key thing missing from the simple classical-esque pure wave framework. Need to learn more about that.\u003c/li\u003e\u003c/ul\u003e\u003cul id=\"frame-36\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• It is worth pointing out that the wave fields are fully conservative — they are indeed derived on the basis of conserving energy, and total charge. Thus, we don’t explicitly need nonlocality per se to obtain conservation — pure wave propagation without ever collapsing would automatically conserve everything properly. It is really only when collapse itself is invoked that nonlocal logic arises: if we don’t reify the existence of “spin up” as a precise discrete thing at the moment of collapse, then we just have ongoing continuous wave propagation, with some kind of macroscopic illusion of having learned something “definitive” about the state of these waves.\u003c/li\u003e\u003c/ul\u003e\u003cul id=\"frame-37\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• There is something deep and mysterious about the positive and negative charge states of the complex KG and Dirac equations, which seems somehow related to the need for bidirectional flow in simpler CA models from Bialynicki-Birula (1993) and David Meyer (1996?). Is there something here that could explain nonlocality: these two channels are somehow coupled and paired, and when something affects one, it immediately affects the other!? Relatedly, why do we need spin? complex KG seems fine. but coupling with EM requires spin. Which then does introduce a coupling between the otherwise separate fields of complex KG. maybe this coupling is essential to allow for nonlocal conservative processes of some sort?\u003c/li\u003e\u003c/ul\u003e\u003cul id=\"frame-38\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• todo: examine this more carefully: \u003ca href=\"https://web.archive.org/web/20151117174141/http://www.mth.kcl.ac.uk/~streater/EPR.html\" target=\"_blank\"\u003ehttps://web.archive.org/web/20151117174141/http://www.mth.kcl.ac.uk/~streater/EPR.html\u003c/a\u003e — makes strong claims of locality but not sure what the error might be..\u003c/li\u003e\u003c/ul\u003e\u003cul id=\"frame-39\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003ca href=\"http://www.mat.univie.ac.at/~neum/physfaq/therm/thermalMain.html\" target=\"_blank\"\u003ehttp://www.mat.univie.ac.at/~neum/physfaq/therm/thermalMain.html\u003c/a\u003e: “Therefore in the EPR sense, sources and beams are much more real than particles. The former, not the latter, are the real players in solid foundations. That’s why an inappropriate focus on the particle aspect of quantum mechanics creates the appearance of mystery. It is a historical accident that one continues to use the name particle in the many microscopic situations where it is grossly inappropriate if one thinks of it with the classical meaning of a tiny bullet moving through space. Restrict the use of the particle concept to where it is appropriate, or don’t think of particles as \u003cem\u003eobjects\u003c/em\u003e - in both cases all mystery is gone, and the foundations become fully rational.”\u003c/li\u003e\u003c/ul\u003e\u003ch1 id=\"lattice-field-model\" style=\"max-width:8in;margin:0.25em;font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eLattice Field Model\u003c/h1\u003e\u003cp id=\"text-41\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe lattice field model (\u003ca href=\"https://en.wikipedia.org/wiki/Lattice_field_theory\" target=\"_blank\"\u003ehttps://en.wikipedia.org/wiki/Lattice_field_theory\u003c/a\u003e) of QED / QCD has many potentially appealing aspects:\u003c/p\u003e\u003cul id=\"frame-42\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• QED as typically formulated in Fock space etc is founded on a \u003cem\u003eparticle-identity\u003c/em\u003e framework — it is keeping track (indexed) by an infinite list of particles — these particles are either created or destroyed by the appropriate operators.\u003c/li\u003e\u003c/ul\u003e\u003cul id=\"frame-43\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Instead, in the lattice model, you just have potential (“virtual”) particles at each location in the lattice, and these are either on or off. This is a powerful inversion of the traditional logic:\n\n\u003c/li\u003e\u003cul id=\"frame-1\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:8ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e◦ The fundamental notion of QM, that something doesn’t exist until it is observed, is taken to a kind of extreme: particles do not exist as identifiable, stable entities — there is just a conservation of relevant factors (charge, energy, momentum, etc) that governs the activation of these “virtual” particles at different locations.\u003c/li\u003e\u003cli id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e◦ In effect, each discretized location is space is continuously performing a measurement, deciding whether a virtual particle should be actualized momentarily at that location — instead of quantum measurement being a rare, ill-defined process, it is a constant, continuously occurring process. Critically, unlike wave-function collapse versions of this general idea (\u003ca href=\"https://en.wikipedia.org/wiki/Objective_collapse_theory\" target=\"_blank\"\u003ehttps://en.wikipedia.org/wiki/Objective_collapse_theory\u003c/a\u003e), the measurement process here is just the actualization of a particle — the wave function continues to propagate as usual, and decoherence (\u003ca href=\"https://en.wikipedia.org/wiki/Quantum_decoherence\" target=\"_blank\"\u003ehttps://en.wikipedia.org/wiki/Quantum_decoherence\u003c/a\u003e) can explain the classical measurement issues.\u003c/li\u003e\u003cli id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e◦ This potentially puts “virtual” and “real” particles on the same footing — presumably this is a known feature of lattice methods — virtual particles in QED / QFT in general have had a somewhat mysterious ontological status — clearly playing an essential role in the math, but often neglected in various philosophical / interpretational discussions.\u003c/li\u003e\u003cli id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e◦ Also overall consistent with the notion of particles as a digital rectification of an underlying analog system — each lattice location computes such a rectification.\u003c/li\u003e\u003c/ul\u003e\u003c/ul\u003e\u003cul id=\"frame-44\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Lattice model is fundamentally like a cellular automaton and thus consistent with overall approach.\u003c/li\u003e\u003c/ul\u003e\u003cul id=\"frame-45\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Major issues: nonlocality as ever — does conservation require some kind of nonlocal tracking of particles? when a particle is realized in one location, how do you ensure that it doesn’t also show up somewhere else?? tracking particles by identity like this is fundamentally at odds with all the advantages listed above — particles shouldn’t have an identity as such — the wave function should not be about an individual particle — it should be about all particles of a given type.. but can this work in practice? and does it just reduce to the same kind of sloppy fluid dynamics as the pure wave approach?\n\n\u003c/li\u003e\u003cul id=\"frame-1\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:8ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e◦ key idea: yes, nonlocality is fundamentally required for particle discretization in the context of particle number conservation (baryon number). otherwise, you end up with goofy local handoff of particle bits from one lattice cell to another — logic is very hacky. Nikolic shows that Bohm model in relativistic case requires faster-than-speed-of-light motion — this is not compatible with some kind of local particle passing algorithm. So the crux of the problem is how to heuristically discretize the continuous wave field in a way that does not actually require tracking particles as separate entities (which is implausible due to fluidity of particle creation and destruction etc — fundamental constraint — number is only kinda conserved and all particles are identical so you can’t get into business of enumerating them EVER — hard constraint).\u003c/li\u003e\u003cli id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e◦ one idea: integrate over a given spatial extent at every step and put a particle at the local max if probability density is above some threshold. kinda crazy but might just work.. how big is the area?? interesting..\u003c/li\u003e\u003cli id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e◦ having many update cycles of the wave equation within each particle update event or something like that would be far preferable to true nonlocality — a very fast clock can look like nonlocality, and again the barrier of nonlocality is so high in terms of the central feature of \u003ca href=\"autonomous\" target=\"_blank\" title=\"wikilink\"\u003eautonomous\u003c/a\u003e models (how do you do any computation that requires infinite sums!?) — we try to avoid it at all costs..\u003c/li\u003e\u003c/ul\u003e\u003c/ul\u003e\u003ch2 id=\"links\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eLinks\u003c/h2\u003e\u003cul id=\"frame-47\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003ca href=\"http://www.physik.uni-bielefeld.de/~laine/lattice/cover.html\" target=\"_blank\"\u003ehttp://www.physik.uni-bielefeld.de/~laine/lattice/cover.html\u003c/a\u003e — lecture notes on lattice models (hand written)\u003c/li\u003e\u003cli id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003ca href=\"http://www.thphys.uni-heidelberg.de/~sexty/lattice/\" target=\"_blank\"\u003ehttp://www.thphys.uni-heidelberg.de/~sexty/lattice/\u003c/a\u003e — class on lattice — also hand-written notes\u003c/li\u003e\u003cli id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Smit book: \u003ca href=\"https://books.google.com/books?id=KIHHW9NtbuAC\u0026printsec=frontcover#v=onepage\u0026q\u0026f=false\" target=\"_blank\"\u003ehttps://books.google.com/books?id=KIHHW9NtbuAC\u0026printsec=frontcover#v=onepage\u0026q\u0026f=false\u003c/a\u003e\u003c/li\u003e\u003cli id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003ca href=\"http://www.qubit.it/research/publications/1601.04842.pdf\" target=\"_blank\"\u003ehttp://www.qubit.it/research/publications/1601.04842.pdf\u003c/a\u003e — recent CA models paper\u003c/li\u003e\u003cli id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003ca href=\"https://arxiv.org/abs/quant-ph/0603164\" target=\"_blank\"\u003ehttps://arxiv.org/abs/quant-ph/0603164\u003c/a\u003e — stochastic schrodinger — why are people still using the schrodinger equation!? crazy.\u003c/li\u003e\u003c/ul\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" 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style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"photon\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M686-450H160v-60h526L438-758l42-42 320 320-320 320-42-42 248-248Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Photon","URL":"photon","Title":"Photon","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{},"Description":"TODO: fourier representation, fock space, \u003ca href=\"ref://Lamb95\" target=\"_blank\"\u003eLamb, 1995\u003c/a\u003e anti-photon etc from surely joking paper.","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003ePhoton\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-28\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eTODO: fourier representation, fock space, \u003ca href=\"ref://Lamb95\" target=\"_blank\"\u003eLamb, 1995\u003c/a\u003e anti-photon etc from surely joking paper.\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" 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style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003ePrevious\u003c/p\u003e\u003c/a\u003e\u003cstretch id=\"content-buttons-go-167\" style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1ch;min-height:1em;font-weight:thin;text-align:start\"\u003e\u003c/stretch\u003e\u003ca id=\"content-buttons-go-168\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"pilot-wave\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M686-450H160v-60h526L438-758l42-42 320 320-320 320-42-42 248-248Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Pilot wave","URL":"pilot-wave","Title":"Pilot wave","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{},"Description":"When David Bohm reinvented the then-neglected pilot-wave framework of de Broglie, he naturally applied it within the prevalent framework of the time, namely the Schrödinger wave equation operating within non-local, high-dimensional \u003ca href=\"configuration-space\" target=\"_blank\"\u003econfiguration space\u003c/a\u003e. He was able to show that you can use the gradient of the Schrödinger wave to guide the motion of discrete particles through 3D space, without ever requiring a final wave collapse event.","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003ePilot wave\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-29\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWhen David Bohm reinvented the then-neglected pilot-wave framework of de Broglie, he naturally applied it within the prevalent framework of the time, namely the Schrödinger wave equation operating within non-local, high-dimensional \u003ca href=\"configuration-space\" target=\"_blank\"\u003econfiguration space\u003c/a\u003e. He was able to show that you can use the gradient of the Schrödinger wave to guide the motion of discrete particles through 3D space, without ever requiring a final wave collapse event.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eInstead, the final positions of these particles represent the predicted outcome of an experiment, and, critically, you have to perform many different “runs” of the experiment with the particles starting in different locations that would be consistent with the actual experimental uncertainty in starting state, as captured in the initial configuration of the Schrödinger wave function. Furthermore, you can simply include the “measurement apparatus” as one of the elements in your configuration space, to get specific predictions about the position of a needle or other readout device.\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOn the one hand, this is a startling result from the Copenhagen perspective: all that crazy stuff about reality not existing until you measure it, and the non-physical nature of wave function collapse can just be dispensed with entirely. However, many scientists, including Einstein, rejected Bohm’s new pilot-wave theory \u003cem\u003especifically because of its use of the high-dimensional configuration space\u003c/em\u003e (\u003ca href=\"ref://NorsenMarianOriols15\" target=\"_blank\"\u003eNorsen et al., 2015\u003c/a\u003e). Why did they not similarly complain about the exclusive use of this same objectionable device in standard QM frameworks? Or in standard classical \u003ca href=\"hamiltonian\" target=\"_blank\"\u003eHamiltonian\u003c/a\u003e applications?\u003c/p\u003e\u003cp id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe answer is evidently that, unlike these other applications, the pilot-wave framework strongly requires that the wave function is somehow a \u003cem\u003ereal\u003c/em\u003e thing! It must actually influence the real trajectories of particles as they move through space, and thus it must actually be something real itself. And if it is real, then the requirement that it be this strange high-dimensional, non-local beast is just as (if not more) unsatisfying as all the bizarre aspects of the standard framework.\u003c/p\u003e\u003cp id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eBut wait a second. If this very same objectionable configuration space is being used in the standard framework, and it is somehow determining the probabilities for where things end up in actual real experiments, then \u003cem\u003ewhy isn’t it just as real for the standard framework as well\u003c/em\u003e? This seems like a serious double standard if there ever was one.\u003c/p\u003e\u003cp id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIt is really just a verbal legerdemain that avoids the obvious conclusion that if there is an actual interference effect being observed, there must be an actual \u003cem\u003ecause\u003c/em\u003e of that interference effect, and that cause cannot just be wished away with a bunch of vague platitudes about complementarity principles etc.\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"photon\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"m274-450 248 248-42 42-320-320 320-320 42 42-248 248h526v60H274Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003ePrevious\u003c/p\u003e\u003c/a\u003e\u003cstretch id=\"content-buttons-go-167\" style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1ch;min-height:1em;font-weight:thin;text-align:start\"\u003e\u003c/stretch\u003e\u003ca id=\"content-buttons-go-168\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"qed\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M686-450H160v-60h526L438-758l42-42 320 320-320 320-42-42 248-248Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"QED","URL":"qed","Title":"QED","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{"figure":["figure_fourier"]},"Description":"The most successful quantum model in terms of generating precise empirical predictions is \u003cstrong\u003equantum electrodynamics (QED)\u003c/strong\u003e, which introduces significantly different conceptual and mathematical frameworks compared to the relatively simple \u003ca href=\"hilbert-space\" target=\"_blank\"\u003eHilbert space\u003c/a\u003e matrix mechanics and \u003ca href=\"schrodinger\" target=\"_blank\"\u003eSchrodinger\u003c/a\u003e wave functions. There are two key conceptual elements to QED: the path integral and quantum field theory, which enable it to accurately represent the detailed interactions between an electron and the EM field.","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eQED\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-30\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe most successful quantum model in terms of generating precise empirical predictions is \u003cstrong\u003equantum electrodynamics (QED)\u003c/strong\u003e, which introduces significantly different conceptual and mathematical frameworks compared to the relatively simple \u003ca href=\"hilbert-space\" target=\"_blank\"\u003eHilbert space\u003c/a\u003e matrix mechanics and \u003ca href=\"schrodinger\" target=\"_blank\"\u003eSchrodinger\u003c/a\u003e wave functions. There are two key conceptual elements to QED: the path integral and quantum field theory, which enable it to accurately represent the detailed interactions between an electron and the EM field.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe \u003cem\u003epath integral\u003c/em\u003e method developed by Richard Feynmann (originated by Dirac) works by iteratively enumerating all the possible events that might take place in an interaction between electron and photon, and computing the total probability of different outcomes by integrating across all these paths. This method is intuitively illustrated by the famous \u003cem\u003eFeynmann diagrams\u003c/em\u003e showing the different possible scenarios in each path. Critically, at this level, the key elements of the theory are the coupled \u003ca href=\"dirac\" target=\"_blank\"\u003eDirac\u003c/a\u003e equation for the electron and \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e’s equations for the EM field.\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eA central problem with this path integral framework is that the path integral sums diverge into infinity. A kind of mathematical “hack” (according to Feymann and others) called \u003cem\u003erenormalization\u003c/em\u003e was finally able to resolve this divergence. Despite its seemingly arbitrary “pragmatic” basis, renormalization proved successful and allowed the framework to be extended to other domains, involving the weak and strong forces. Clearly, a mathematical framework based on divergent infinite integrals is not a particularly promising basis for a plausible physical model, and is thus clearly a \u003ca href=\"tools-vs-models\" target=\"_blank\"\u003ecalculational tool\u003c/a\u003e.\u003c/p\u003e\u003cp id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAt higher energy levels, the possible paths include the spontaneous creation of new particles out of raw energy (i.e., \u003cem\u003epair production\u003c/em\u003e). Dealing with these kinds of events in an efficient way mathematically required an entirely new framework called \u003cstrong\u003equantum field theory (QFT)\u003c/strong\u003e, which does away with the configuration-space representation based on a specific number of particles, and instead considers an entire spatially-extended field where particles can more easily be created and destroyed as “excitations of the field”.\u003c/p\u003e\u003cdiv id=\"figure_fourier\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_fourier\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 1:\u003c/b\u003e \u003cp\u003eThe Fourier transform, which is the basis of the photon model in QED. A fourier transform converts a function from normal physical space into an orthogonal basis space of sine waves parameterized according to their amplitude, phase and frequency. No position parameter is retained in Fourier space, as the sine waves are infinite in extent. The QED model of the photon is, implausibly, one of these Fourier sine waves, similarly infinite in extent, without any physical localization.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cp id=\"text-6\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe mathematics of this field are based on a \u003cem\u003eFourier space\u003c/em\u003e representation (\u003ca href=\"qed#figure_fourier\" target=\"_blank\"\u003eFigure 1\u003c/a\u003e), where each “particle” is associated with a specific Fourier component, which is a sine wave having a specific frequency and phase. Adding or removing a particle amounts to just adding or removing a corresponding Fourier component.\u003c/p\u003e\u003cp id=\"text-7\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFourier space is the \u003cem\u003emomentum space\u003c/em\u003e representation complementary to \u003cem\u003eposition space\u003c/em\u003e: it is defined in terms of frequency and phase coordinates, where the frequency is proportional to the momentum of a particle as we saw in the discussion of the original motivation for matter waves by de Broglie. This Fourier representation is particularly convenient for keeping track of total energy, which is proportional to momentum and frequency, and allows one to more easily represent different types of particles splitting up some total amount of energy in different ways.\u003c/p\u003e\u003cp id=\"text-8\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAs captured in the Heisenberg \u003ca href=\"uncertainty-principle\" target=\"_blank\"\u003euncertainty principle\u003c/a\u003e, momentum and position are “conjugate” (complementary) variables, so when you represent everything in precisely in Fourier space, the position information is completely lost. \u003cem\u003eThus, “particles” in Fourier space have no positions: they spread across the entire space.\u003c/em\u003e\u003c/p\u003e\u003cp id=\"text-9\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ePosition information only arises in Fourier space in terms of the constructive and destructive interference effects of different phases and frequencies. Thus, representing a particle with some specificity of spatial position requires many different sine waves “working together” to add up in one part of space and cancel out in other parts of space.\u003c/p\u003e\u003cp id=\"text-10\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eRepresenting something with a fully precise spatial position requires an \u003cem\u003einfinite\u003c/em\u003e number of such sine waves, in the same way that representing a fully precise momentum (i.e, frequency) requires a continuum infinity of particle positions oscillating according to a specific precise frequency. This is again the Heisenberg uncertainty principle, and it is a basic property of \u003ca href=\"waves\" target=\"_blank\"\u003ewaves\u003c/a\u003e.\u003c/p\u003e\u003cp id=\"text-11\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eUnder the primary postulate of the \u003ca href=\"pilot-wave\" target=\"_blank\"\u003epilot-wave\u003c/a\u003e framework, that particles have a specific position at all times, it is clear that this QFT Fourier space representation is exactly the wrong one for describing a physically accurate model. But, as usual, people have a difficult time recognizing that this QFT framework is just another \u003ca href=\"tools-vs-models\" target=\"_blank\"\u003ecalculational tool\u003c/a\u003e that is convenient for computing certain kinds of problems, and they end up thinking of the QFT particle representation as a physical model.\u003c/p\u003e\u003cp id=\"text-12\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSpecifically, QFT provides the only mathematically tractable model of a photon as some kind of discrete particle-like entity, but for all of the reasons explained above, such a photon is nothing like any other kind of particle we would recognize in the real world. It has infinite spatial extent, and is defined only in terms of its frequency and phase.\u003c/p\u003e\u003cp id=\"text-13\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn summary, QED and QFT are amazing calculational tools that have been used to make some of the most accurate predictions in all of physics. But they have a number of obvious problems as physical models for how nature actually operates, especially relative to the assumptions of the \u003ca href=\"pilot-wave\" target=\"_blank\"\u003epilot-wave\u003c/a\u003e framework. Nevertheless, the coupled \u003ca href=\"dirac\" target=\"_blank\"\u003eDirac\u003c/a\u003e and \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e fields at the heart of this theory are clearly the key basic ingredients, and it is possible that a more computationally-based approach, with the strong constraint of discrete electron particles in specific spatial locations, could provide the basis for an accurate physical model.\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"pilot-wave\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"m274-450 248 248-42 42-320-320 320-320 42 42-248 248h526v60H274Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003ePrevious\u003c/p\u003e\u003c/a\u003e\u003cstretch id=\"content-buttons-go-167\" style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1ch;min-height:1em;font-weight:thin;text-align:start\"\u003e\u003c/stretch\u003e\u003ca id=\"content-buttons-go-168\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"references\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M686-450H160v-60h526L438-758l42-42 320 320-320 320-42-42 248-248Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"References","URL":"references","Title":"References","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{},"Description":"Aharonov, Y., Davidovich, L., \u0026 Zagury, N. 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Quantum random walks. \u003ci\u003ePhysical Review A, 48\u003c/i\u003e, 1687–1690. \u003ca href=\"https://link.aps.org/doi/10.1103/PhysRevA.48.1687\"\u003ehttps://link.aps.org/doi/10.1103/PhysRevA.48.1687\u003c/a\u003e\u003ca href=\"http://doi.org/10.1103/PhysRevA.48.1687\"\u003e http://doi.org/10.1103/PhysRevA.48.1687\u003c/a\u003e","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eReferences\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-31\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"AharonovDavidovichZagury93\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAharonov, Y., Davidovich, L., \u0026 Zagury, N. 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On the partial differential equations of mathematical physics. \u003ci\u003eMathematische Annalen, 57\u003c/i\u003e, 333–355. \u003ca href=\"https://doi.org/10.1007/BF01444290\"\u003ehttps://doi.org/10.1007/BF01444290\u003c/a\u003e\u003ca href=\"http://doi.org/10.1007/BF01444290\"\u003e http://doi.org/10.1007/BF01444290\u003c/a\u003e\u003c/p\u003e\u003cp id=\"Wolfram97\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWolfram, S. (1997). \u003ci\u003eA new kind of science. \u003c/i\u003e\u003c/p\u003e\u003cp id=\"Zuse69\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eZuse, K. (1969). Rechnender Raum (Calculating Space) \u003ci\u003eSchriften Zur Dataverarbeitung, 1\u003c/i\u003e, \u003c/p\u003e\u003cp id=\"Zuse70\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eZuse, K. (1970). Calculating Space (Rechnender Raum) \u003ci\u003eMassachusetts Institute of Technology Technical Translation AZT-70-164-GEMIT., \u003ca href=\"https://www.worldscientific.com/doi/abs/10.1142/9789814374309_0036\"\u003ehttps://www.worldscientific.com/doi/abs/10.1142/9789814374309_0036\u003c/a\u003e\u003c/i\u003e\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"qed\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"m274-450 248 248-42 42-320-320 320-320 42 42-248 248h526v60H274Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003ePrevious\u003c/p\u003e\u003c/a\u003e\u003cstretch id=\"content-buttons-go-167\" style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1ch;min-height:1em;font-weight:thin;text-align:start\"\u003e\u003c/stretch\u003e\u003ca id=\"content-buttons-go-168\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"schrodinger\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M686-450H160v-60h526L438-758l42-42 320 320-320 320-42-42 248-248Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Schrodinger","URL":"schrodinger","Title":"Schrodinger","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{"eq":["eq_momentum","eq_energy","eq_gradient","eq_kinetic","eq_kv","eq_schrodinger","eq_schrodinger","eq_KG"]},"Description":"Compared to the \u003ca href=\"hilbert-space\" target=\"_blank\"\u003eHilbert space\u003c/a\u003e approach, the explicit use of the \u003cstrong\u003eSchrödinger wave equation\u003c/strong\u003e represents an increased level of commitment to the details involved in the dynamics of the wave updating, its frequency and phase characteristics, and how it spreads out over time. Schrödinger’s wave equation captures basic non-relativistic Newtonian physics in a simple linear, first-order framework, and can be derived from a \u003ca href=\"hamiltonian\" target=\"_blank\"\u003eHamiltonian\u003c/a\u003e representing the total energy of the system, which is strictly conserved over time. It captures the fundamental relationships between momentum and wave frequency at the heart of quantum physics, as discussed in \u003ca href=\"klein-gordon\" target=\"_blank\"\u003eKlein-Gordon\u003c/a\u003e.","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eSchrodinger\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-32\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eCompared to the \u003ca href=\"hilbert-space\" target=\"_blank\"\u003eHilbert space\u003c/a\u003e approach, the explicit use of the \u003cstrong\u003eSchrödinger wave equation\u003c/strong\u003e represents an increased level of commitment to the details involved in the dynamics of the wave updating, its frequency and phase characteristics, and how it spreads out over time. Schrödinger’s wave equation captures basic non-relativistic Newtonian physics in a simple linear, first-order framework, and can be derived from a \u003ca href=\"hamiltonian\" target=\"_blank\"\u003eHamiltonian\u003c/a\u003e representing the total energy of the system, which is strictly conserved over time. It captures the fundamental relationships between momentum and wave frequency at the heart of quantum physics, as discussed in \u003ca href=\"klein-gordon\" target=\"_blank\"\u003eKlein-Gordon\u003c/a\u003e.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eHowever, it has a rather simplistic treatment for how forces affect charged particles in terms of overall scalar potentials, and says nothing in detail about how electric charge generates the EM wave field (or photons for that matter), or the detailed way in which different particles might interact with each other. Indeed, because the Schrödinger wave equation is linear, it is incapable of capturing particle interactions, because the waves simply superpose (additively combine) past each other, without impacting each other at all.\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThus, in order to capture relevant interactions, the Schrödinger wave equation is applied to a multi-dimensional \u003ca href=\"configuration-space\" target=\"_blank\"\u003econfiguration space\u003c/a\u003e representation that is essentially equivalent to the state space representation in matrix mechanics. For example, if there are two interacting particles, then they each get their own set of 3D dimensional coordinates within this configuration space, and the entire wave function evolves over time so as to conserve the overall energy / probability represented in the configuration space.\u003c/p\u003e\u003cp id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAs noted above, this configuration space is entirely \u003ca href=\"non-locality\" target=\"_blank\"\u003enon-local\u003c/a\u003e by its very construction, representing at each instant of time the entire configuration of the system, regardless of how far apart any of the particles might be. Interestingly, exactly such a configuration space model is used in \u003cem\u003eclassical\u003c/em\u003e applications of the Hamiltonian framework, and yet somehow its use there is widely recognized as just being a calculational tool.\u003c/p\u003e\u003cp id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn summary, the high-dimensional non-local configuration space is very different from anything anyone would recognize as actual 3D physical space. Nevertheless, one of the most striking and challenging results from these standard QM models is that the non-local effects that they predict actually do appear to be empirically validated. Thus, a significant challenge remains to understand the underlying physical nature of these effects, and how they can occur without violating everything else we have come to regard as strict physical laws, specifically the speed-of-light constraints of special relativity (\u003ca href=\"ref://DurrGoldsteinNorsenEtAl14\" target=\"_blank\"\u003eDürr et al., 2014\u003c/a\u003e).\u003c/p\u003e\u003cp id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAlso, while the dimensionality of configuration space increases linearly in the number of particles involved, the underlying computational complexity of the space grows exponentially, and quickly becomes computationally intractable for even relatively moderately-sized such spaces. This is precisely what makes quantum computers so attractive. Nevertheless, it remains unclear how Nature might get around such prohibitive exponential scaling problems, in whatever computation it is performing.\u003c/p\u003e\u003ch2 id=\"schrodinger-s-equation\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eSchrodinger’s equation\u003c/h2\u003e\u003cp id=\"text-7\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eUsing the total energy (\u003ca href=\"hamiltonian\" target=\"_blank\"\u003eHamiltonian\u003c/a\u003e) approach, we can derive Schrödinger’s equation, using the same energy and momentum operators that we used in the derivation of the \u003ca href=\"klein-gordon\" target=\"_blank\"\u003eKlein-Gordon\u003c/a\u003e equation (strongly recommend reading that page first, for the introductory treatment of this approach). To remind, these operators are:\u003c/p\u003e\u003cdiv id=\"inline-container-9\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_momentum\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_momentum\"\u003e\u003cb\u003eEq 1:\u003c/b\u003e momentum operator\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-10\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\hat{p} = -i \\hbar \\vec{\\nabla}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cdiv id=\"inline-container-12\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_energy\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_energy\"\u003e\u003cb\u003eEq 2:\u003c/b\u003e energy operator\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-13\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\hat{E} = i \\hbar \\frac{\\partial }{\\partial t}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cdiv id=\"inline-container-15\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_gradient\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_gradient\"\u003e\u003cb\u003eEq 3:\u003c/b\u003e gradient operator\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-16\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{\\nabla} = \\left(\\frac{\\partial {}}{\\partial {x}}, \\frac{\\partial {}}{\\partial {y}}, \\frac{\\partial}{\\partial {z}}\\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-17\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eNext, we need to define the total energy Hamiltonian. Instead of the relativistic total energy, we use the classical Newtonian expression for the kinetic energy of a particle, in terms of its velocity \u003cspan class=\"math inline\"\u003e\\(\\vec{v}\\)\u003c/span\u003e, just as we did in the simple wave energy calculation in \u003ca href=\"waves\" target=\"_blank\"\u003ewaves\u003c/a\u003e:\u003c/p\u003e\u003cdiv id=\"inline-container-19\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_kinetic\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_kinetic\"\u003e\u003cb\u003eEq 4:\u003c/b\u003e kinetic energy of particle\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-20\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nK = \\frac{1}{2} m_0 \\vec{v}^2 = \\frac{1}{2 m_0} \\vec{p}^2\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-21\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe second form uses the Newtonian relationship of momentum to velocity (just \u003cspan class=\"math inline\"\u003e\\(\\vec{p} = m_0 \\vec{v}\\)\u003c/span\u003e) — because we have a momentum operator, we need to use this momentum form.\u003c/p\u003e\u003cp id=\"text-22\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWe also include a potential energy term that is a function of any kind of electrical or other force potential that the particle experiences. We won’t deal much with such forces at this point, so we just call this potential energy \u003cspan class=\"math inline\"\u003e\\(V\\)\u003c/span\u003e for now, and focus on the kinetic energy. The total energy or Hamiltonian in abstract terms is just the kinetic energy \u003cspan class=\"math inline\"\u003e\\(K\\)\u003c/span\u003e plus this potential energy:\u003c/p\u003e\u003cdiv id=\"inline-container-24\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_kv\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_kv\"\u003e\u003cb\u003eEq 5:\u003c/b\u003e kinetic and potential energy\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-25\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nE = K + V\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-26\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nE = \\frac{1}{2 m_0} \\vec{p}^2 + V\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-27\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWe can now just apply our momentum and energy operators to these expressions, and the result is in fact:\u003c/p\u003e\u003cdiv id=\"inline-container-29\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_schrodinger\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_schrodinger\"\u003e\u003cb\u003eEq 6:\u003c/b\u003e Schrödinger's equation\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-30\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\ni \\hbar \\frac{\\partial {\\phi}}{\\partial t} = -\\frac{\\hbar^2}{2 m_0} \\nabla^2 \\phi + V \\phi\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-31\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe net result is that we can conclude that Schrödinger’s equation provides an accurate description of the flow of energy and momentum over time of a “particle” described by a wave, such that it obeys classical Newtonian physical laws. Note that in comparison with the KG equation, there is no speed-of-light factor \u003cspan class=\"math inline\"\u003e\\(c\\)\u003c/span\u003e in this equation, consistent with its non-relativistic nature.\u003c/p\u003e\u003cp id=\"text-32\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOmitting various constants (factors of \u003cspan class=\"math inline\"\u003e\\(h\\)\u003c/span\u003e) and any external force potential, Schrödinger’s equation is:\u003c/p\u003e\u003cdiv id=\"inline-container-34\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_schrodinger\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_schrodinger\"\u003e\u003cb\u003eEq 6:\u003c/b\u003e Schrödinger's equation, essence\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-35\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\ni \\frac{\\partial {\\phi}}{\\partial t} = - \\frac{1}{2m_0} \\nabla^2 \\phi\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-36\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhere \u003cspan class=\"math inline\"\u003e\\(m_0\\)\u003c/span\u003e is again the rest mass of the particle in question. This is clearly very similar to the basic second-order KG wave equation:\u003c/p\u003e\u003cdiv id=\"inline-container-38\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_KG\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_KG\"\u003e\u003cb\u003eEq 8:\u003c/b\u003e Klein-Gordon equation\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-39\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2 {\\varphi}}{\\partial t^2} = c^2 \\nabla^2 \\varphi - \\frac{m_0^2}{\\hbar^2} \\varphi\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-40\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eexcept that the temporal derivative is first-order, and mass enters in a different way. Nevertheless, the driving force is still the overall curvature of the wave, computed by \u003cspan class=\"math inline\"\u003e\\(\\nabla^2 \\varphi\\)\u003c/span\u003e. As we noted above, the multiplication by the \u003cspan class=\"math inline\"\u003e\\(i\\)\u003c/span\u003e term causes things to rotate — this rotation is key for making the first-order equation behave like a wave.\u003c/p\u003e\u003cp id=\"text-41\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eTo see this effect more explicitly, we can write out Schrödinger’s equation in terms of the two underlying scalar values:\u003c/p\u003e\u003cp id=\"text-42\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\ni \\frac{\\partial ({\\varphi_a + i \\varphi_b})}{\\partial t} = - \\frac{1}{2m_0} \\nabla^2 (\\varphi_a + i \\varphi_b)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-43\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n-\\frac{\\partial {\\varphi_b}}{\\partial t} + \\frac{\\partial {i \\varphi_a}}{\\partial t} = -\\frac{1}{2m_0} \\nabla^2 \\varphi_a - i \\nabla^2 \\varphi_b\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-44\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhere \u003cspan class=\"math inline\"\u003e\\(\\varphi_a\\)\u003c/span\u003e indicates a scalar state variable that is the \u003cspan class=\"math inline\"\u003e\\(a\\)\u003c/span\u003e component of \u003cspan class=\"math inline\"\u003e\\(\\phi\\)\u003c/span\u003e, and \u003cspan class=\"math inline\"\u003e\\(\\varphi_b\\)\u003c/span\u003e is the \u003cspan class=\"math inline\"\u003e\\(b\\)\u003c/span\u003e component of \u003cspan class=\"math inline\"\u003e\\(\\phi\\)\u003c/span\u003e. Note that the derivatives operate separately on each of the two variables. At this point, we now can just separate all the terms that involve an \u003cspan class=\"math inline\"\u003e\\(i\\)\u003c/span\u003e from those that do not, to get update equations for each of the two variables. For the real-valued components (without the \u003cspan class=\"math inline\"\u003e\\(i\\)\u003c/span\u003e):\u003c/p\u003e\u003cp id=\"text-45\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n-\\frac{\\partial {\\varphi_b}}{\\partial t} = - \\frac{1}{2m_0} \\nabla^2 \\varphi_a\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-46\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial {\\varphi_b}}{\\partial t} = \\frac{1}{2m_0} \\nabla^2 \\varphi_a\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-47\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eand for the imaginary components (dropping the \u003cspan class=\"math inline\"\u003e\\(i\\)\u003c/span\u003e now, because we no longer need it to keep the variables separated):\u003c/p\u003e\u003cp id=\"text-48\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial {\\varphi_a}}{\\partial t} = - \\frac{1}{2m_0} \\nabla^2 \\varphi_b\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-49\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn a discrete-space and time \u003ca href=\"cellular-automaton\" target=\"_blank\"\u003ecellular automaton\u003c/a\u003e implementation, these equations would be written:\u003c/p\u003e\u003cp id=\"text-50\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\dot {\\varphi_a}_i^{t+1} = - \\frac{3}{26 m_0} \\sum_{j \\in N_{26}} k_j ({\\varphi_b}_j^t - {\\varphi_b}_i^t)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-51\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n{\\varphi_a}_i^{t+1} = {\\varphi_a}_i^t + \\dot {\\varphi_a}_i^{t+1}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-52\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eand:\u003c/p\u003e\u003cp id=\"text-53\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\dot {\\varphi_b}_i^{t+1} = \\frac{3}{26 m_0}\\sum_{j \\in N_{26}} k_j ({\\varphi_a}_j^t - {\\varphi_a}_i^t)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-54\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n{\\varphi_b}_i^{t+1} = {\\varphi_b}_i^t + \\dot {\\varphi_b}_i^{t+1}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-55\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSo, in the end, Schrödinger’s equation really just boils down to two very simple differential equations. Interestingly, these equations are \u003cem\u003ecoupled\u003c/em\u003e, in the sense that it is the curvature of \u003cspan class=\"math inline\"\u003e\\(\\varphi_a\\)\u003c/span\u003e that drives the change in \u003cspan class=\"math inline\"\u003e\\(\\varphi_b\\)\u003c/span\u003e, and vice-versa. This is the rotational aspect of the equation mentioned earlier, which is caused by the presence of the \u003cspan class=\"math inline\"\u003e\\(i\\)\u003c/span\u003e in the equation.\u003c/p\u003e\u003cp id=\"text-56\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWhen you actually implement Schrödinger’s equation on a computer using the update rules given above, the resulting system is numerically unstable. In other words, the resulting numbers quickly blow up to infinity. This is not due to any kind of numerical roundoff error from limited precision floating point numbers on the computer, but rather due to the way that changes in state values reverberate back and forth across the two scalar values. However, it is possible to overcome it relatively simply by just alternating the update: on one time step you compute one value, and on the next you update the other. This is what is done for illustrative purposes in the computer explorations.\u003c/p\u003e\u003cp id=\"text-57\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe basic phenomenology of Schrödinger’s equation is that wave packets propagate through space, with a speed that is proportional to \u003cspan class=\"math inline\"\u003e\\(\\nabla^2 \\phi\\)\u003c/span\u003e, which in turn is proportional to the frequency of the wave. In other words, it describes exactly the same behavior as the KG equation, where particle speed is proportional to frequency.\u003c/p\u003e\u003cp id=\"text-58\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOne critical property of Schrödinger’s equation (which the scalar \u003ca href=\"klein-gordon\" target=\"_blank\"\u003eKlein-Gordon\u003c/a\u003e equation does not have) is that it preserves the overall magnitude of the \u003cspan class=\"math inline\"\u003e\\(\\phi\\)\u003c/span\u003e state values across all of space, for all time. This is to say, if you compute the sum of \u003cspan class=\"math inline\"\u003e\\(\\phi \\phi^*\\)\u003c/span\u003e for each point in space, this sum will remain the same across time under the Schrödinger equation. This conserved value is interpreted as a probability in standard quantum mechanics.\u003c/p\u003e\u003cp id=\"text-59\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFor example, we can initialize the state with a localized wave packet (see \u003ca href=\"matter-waves\" target=\"_blank\"\u003ematter waves\u003c/a\u003e) to represent the initial probability for the location and velocity of a particle (velocity being a function of the frequency of the wave packet). If we then apply the Schrödinger equation repeatedly, we can interpret the resulting \u003cspan class=\"math inline\"\u003e\\(\\phi \\phi^*\\)\u003c/span\u003e values as the probability of the particle having moved to the corresponding location.\u003c/p\u003e\u003cp id=\"text-60\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn other words, the wave packet defines a kind of “cloud of probability” for finding a discrete particle within its midst. However, these probabilities have different meanings in different scenarios, and it is notoriously difficult to come up with a intuitively sensible interpretation of what these probability clouds mean (see \u003ca href=\"copenhagen\" target=\"_blank\"\u003eCopenhagen\u003c/a\u003e for discussion).\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"references\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"m274-450 248 248-42 42-320-320 320-320 42 42-248 248h526v60H274Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003ePrevious\u003c/p\u003e\u003c/a\u003e\u003cstretch id=\"content-buttons-go-167\" style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1ch;min-height:1em;font-weight:thin;text-align:start\"\u003e\u003c/stretch\u003e\u003ca id=\"content-buttons-go-168\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"semiclassical\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M686-450H160v-60h526L438-758l42-42 320 320-320 320-42-42 248-248Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Semiclassical","URL":"semiclassical","Title":"Semiclassical","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{},"Description":"\u003cstrong\u003eSemiclassical\u003c/strong\u003e models of electrodynamics feature a classical electromagnetic field evolving according to \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e’s differential equations, interacting with an atomic system that has quantum mechanical properties (\u003ca href=\"ref://JaynesCummings63\" target=\"_blank\"\u003eJaynes \u0026 Cummings, 1963\u003c/a\u003e; \u003ca href=\"ref://Jaynes73\" target=\"_blank\"\u003eJaynes, 1973\u003c/a\u003e; \u003ca href=\"ref://Mandel76\" target=\"_blank\"\u003eMandel, 1976\u003c/a\u003e; \u003ca href=\"ref://Grandy91\" target=\"_blank\"\u003eGrandy, 1991\u003c/a\u003e; \u003ca href=\"ref://MarshallSantos97\" target=\"_blank\"\u003eMarshall \u0026 Santos, 1997\u003c/a\u003e; \u003ca href=\"ref://GerryKnight05\" target=\"_blank\"\u003eGerry \u0026 Knight, 2005\u003c/a\u003e). This contrasts with the standard QM model of electrodynamics (\u003ca href=\"qed\" target=\"_blank\"\u003eQED\u003c/a\u003e), which treats the electromagnetic field in terms of discrete \u003cem\u003ephoton\u003c/em\u003e particles, instead of the classical differential equations.","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eSemiclassical\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-33\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cstrong\u003eSemiclassical\u003c/strong\u003e models of electrodynamics feature a classical electromagnetic field evolving according to \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e’s differential equations, interacting with an atomic system that has quantum mechanical properties (\u003ca href=\"ref://JaynesCummings63\" target=\"_blank\"\u003eJaynes \u0026 Cummings, 1963\u003c/a\u003e; \u003ca href=\"ref://Jaynes73\" target=\"_blank\"\u003eJaynes, 1973\u003c/a\u003e; \u003ca href=\"ref://Mandel76\" target=\"_blank\"\u003eMandel, 1976\u003c/a\u003e; \u003ca href=\"ref://Grandy91\" target=\"_blank\"\u003eGrandy, 1991\u003c/a\u003e; \u003ca href=\"ref://MarshallSantos97\" target=\"_blank\"\u003eMarshall \u0026 Santos, 1997\u003c/a\u003e; \u003ca href=\"ref://GerryKnight05\" target=\"_blank\"\u003eGerry \u0026 Knight, 2005\u003c/a\u003e). This contrasts with the standard QM model of electrodynamics (\u003ca href=\"qed\" target=\"_blank\"\u003eQED\u003c/a\u003e), which treats the electromagnetic field in terms of discrete \u003cem\u003ephoton\u003c/em\u003e particles, instead of the classical differential equations.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe basic intuition behind these semiclassical models is that electrons are locked into bound states in the atomic system, and a minimum resonant frequency is required to wedge them out of these states. Any wave that is below this minimum frequency just doesn’t resonate properly with the wave field of the electron, and passes right through.\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThese bound electrons have discrete, quantized energy levels because they obey wave equations, and essentially these waves must vibrate like drums or guitar strings, with an integral number of wavelengths fitting within the overall space available in an atom. The frequency dependence and quantized nature of the atomic system would hold if it interacted with \u003cem\u003eanything\u003c/em\u003e — it would be impossible for the EM field to behave other than in this discretized manner in its interactions with atoms.\u003c/p\u003e\u003cp id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAnother clue that there may be something fundamentally misplaced in the photon model is the presence of Planck’s constant \u003cem\u003eh\u003c/em\u003e, which arises directly from adding mass to the wave equations, where the waves travel at speeds less than the speed of light (i.e., the \u003ca href=\"klein-gordon\" target=\"_blank\"\u003eKlein-Gordon\u003c/a\u003e and \u003ca href=\"dirac\" target=\"_blank\"\u003eDirac\u003c/a\u003e equations). Because light (electromagnetic radiation) has no mass, there is no reason for there to be such a constant associated with it, and the classical EM equations have no place for this constant.\u003c/p\u003e\u003cp id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAlthough the photoelectric effect has a fairly compelling semiclassical explanation, there are other phenomena that are harder to explain within this framework. For example, it is possible to have a system that emits a single “photon” of EM energy at a time, and this photon can then be detected later. Advocates of the photon model argue that it is only detected in one specific location, which seems like evidence for a localized little particle, and not a more broadly distributed wave.\u003c/p\u003e\u003cp id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eHowever, we must appreciate that the source of the EM field with sufficient energy to excite an atom is typically the spontaneous emission of photons from other atomic systems. This means that these photons were “created” by a kind of mirror image of the very same discrete process involved in detecting the photons. This should impart a temporal, spatial, and energy-level discreteness to the EM radiation in the first place.\u003c/p\u003e\u003cp id=\"text-6\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThere are other statistical properties of photon emission (e.g., anticorrelations; \u003ca href=\"ref://GrangierRogerAspect86\" target=\"_blank\"\u003eGrangier et al., 1986\u003c/a\u003e; and antibunching; \u003ca href=\"ref://HongOuMandel87\" target=\"_blank\"\u003eHong et al., 1987\u003c/a\u003e) that have been proposed to be inconsistent with the semiclassical approach. Nevertheless, semiclassical accounts of these phenomena have been provided, by leveraging an additional stochastic process associated with the hypothesized \u003ca href=\"zero-point\" target=\"_blank\"\u003ezero point\u003c/a\u003e field (\u003ca href=\"ref://MarshallSantos07\" target=\"_blank\"\u003eMarshall \u0026 Santos, 2007\u003c/a\u003e; \u003ca href=\"ref://MarshallSantos97\" target=\"_blank\"\u003eMarshall \u0026 Santos, 1997\u003c/a\u003e), but this work has failed to overturn the status quo belief in photons, perhaps in part because of various important outstanding issues associated with this zero point field construct.\u003c/p\u003e\u003cp id=\"text-7\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOverall, this semiclassical physical model requires much more complex calculations and conceptual frameworks than the simple ideas and math associated with the photon model, so from the \u003ca href=\"tools-vs-models\" target=\"_blank\"\u003etools vs models\u003c/a\u003e perspective, there isn’t much reason for people to adopt this more complex model.\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"schrodinger\"\u003e\u003csvg id=\"icon\" 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style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003ePrevious\u003c/p\u003e\u003c/a\u003e\u003cstretch id=\"content-buttons-go-167\" style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1ch;min-height:1em;font-weight:thin;text-align:start\"\u003e\u003c/stretch\u003e\u003ca id=\"content-buttons-go-168\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"special-relativity\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M686-450H160v-60h526L438-758l42-42 320 320-320 320-42-42 248-248Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Special relativity","URL":"special-relativity","Title":"Special relativity","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{"eq":["eq_factor","eq_contract","eq_dilate","eq_esq-p","eq_esq-simp","eq_p-start","eq_flip-v","eq_flip-v","eq_sq","eq_sep","eq_cons","eq_cons2","eq_elim","eq_get-v","eq_move-v","eq_flip-v"],"figure":["figure_contract","figure_lorentz","figure_lorentz2","figure_lorentz-factor","figure_lorentz-coord","figure_lorentz-coord-c"],"table":["table_examples"]},"Description":"\u003cstrong\u003eSpecial relativity\u003c/strong\u003e is the first version of Einstein’s famous relativity theories, published in 1904. Interestingly, most of the mathematical principles and phenomena were already well established prior to this publication. What the theory added was a comprehensive new framework for understanding these otherwise disparate phenomena.","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eSpecial relativity\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-34\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cstrong\u003eSpecial relativity\u003c/strong\u003e is the first version of Einstein’s famous relativity theories, published in 1904. Interestingly, most of the mathematical principles and phenomena were already well established prior to this publication. What the theory added was a comprehensive new framework for understanding these otherwise disparate phenomena.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eApparently Einstein was motivated to come up with special relativity in part by thinking about what it would be like to catch up with a beam of light. Turns out you can’t: light always moves at the same speed away from you, no matter how fast you are going. Thus, the \u003ca href=\"waves\" target=\"_blank\"\u003ewaves\u003c/a\u003e described by the basic wave equation are always speeding along at the same speed, and are relativistic in a fairly straightforward but also somewhat uninteresting way: they just cruise around at the same speed of light, and not much else can be said for them.\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn contrast, things that have a non-zero rest mass are subject to three major effects as their velocities increase (\u003ca href=\"special-relativity#figure_contract\" target=\"_blank\"\u003eFigure 1\u003c/a\u003e, \u003ca href=\"special-relativity#figure_lorentz\" target=\"_blank\"\u003eFigure 2\u003c/a\u003e; all of these are as measured relative to a static observer watching such a thing whiz by — in our model, we can conveniently use the underlying grid reference frame):\u003c/p\u003e\u003cul id=\"frame-3\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• They shrink along the direction of travel (Lorentz contraction).\u003c/li\u003e\u003cli id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Time slows down (time dilation).\u003c/li\u003e\u003cli id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Apparent (relativistic) mass increases.\u003c/li\u003e\u003c/ul\u003e\u003cdiv id=\"figure_contract\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_contract\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 1:\u003c/b\u003e \u003cp\u003eIntuitive explanation of Lorentz contraction of space that occurs as something moves faster, in terms of the relationship between wave frequency and speed in the Klein-Gordon wave equation. Because faster movement is associated with higher frequency and shorter wavelength, the system contracts in the direction of motion as it speeds up. Thus, any measurements made in the faster system will have their basic constituents, including yard sticks and everything else, shrunk in this way. This is one of the main effects of special relativity.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"figure_lorentz\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_lorentz\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 2:\u003c/b\u003e \u003cp\u003eThe Lorentz Transformation, a central property of special relativity, which causes length in the direction of motion to shrink and time to expand (dilate) as a function of relative speed, in just such a way as to preserve the observed speed of light regardless of how fast one is going. The matter wave equation exhibits exactly this behavior, which completely masks any fixed matrix in which such waves might be implemented. In this example, a speeding light ray is observed at a given time t by an observer in a train speeding along at 86.6% of the speed of light \u003cspan class=\"math inline\"\u003e\\((c \\approx 3.0x10^8)\\)\u003c/span\u003e, and by “us” sitting in a stopped train (on a siding presumably). All the measurements in black are what we observe in this stopped reference frame, while those in red are what the speeding train guy observes. If we wait 100 nanoseconds (ns) (\u003cspan class=\"math inline\"\u003e\\(1x10^{-7}\\)\u003c/span\u003e seconds — 100 times slower than the clock rate on a 1Ghz computer chip), then this light ray will have moved 30 meters. However, from our stopped perspective, the speeding train will be partially keeping up with the light ray, so that it will appear to have traveled only 4m relative to the moving train. Thus, in this stopped reference frame, where 100ns have passed for this light to appear to have traveled 4m, we might naively assume that someone on the speeding train would measure the speed of light as only \u003cspan class=\"math inline\"\u003e\\(4x10^{7} m/s\\)\u003c/span\u003e — oops! But the Lorentz transformations of length and time exactly compensate. The length of the train in the direction of motion shrinks in half, so that people on the train measure the 4m in the stopped reference frame as 8m in the moving reference frame — twice as long. Furthermore, time moves more slowly for the speeding train, such that the 100ns in our reference frame is measured as only 50ns in the speeding train reference frame (at a static reference point in the speeding train, which is the very back of the coal tender in this example). The measurement of time is very strange in special relativity, because what is observed as occurring at the same time (simultaneity) across different reference frames depends on both time “and location”. Thus, when the light ray is measured at 8m ahead of the back of the coal tender, this registers as only 26.8ns of elapsed time! If you divide this 8m by that amount of time, it comes out to exactly the same speed of light as in the stopped frame. The time transformation equation is: \u003cspan class=\"math inline\"\u003e\\(t’ = \\gamma (t-vx/c^2)\\)\u003c/span\u003e and the position transformation is: \u003cspan class=\"math inline\"\u003e\\(x’ = \\gamma (x-vt)\\)\u003c/span\u003e, where t’ and x’ are as measured on the speeding train and t, x are on the stopped one, and \u003cspan class=\"math inline\"\u003e\\(\\gamma = 1/\\sqrt{1-v^2/c^2}\\)\u003c/span\u003e.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cp id=\"text-8\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAlmost miraculously, all of these effects can be derived directly from the \u003ca href=\"klein-gordon\" target=\"_blank\"\u003eKlein-Gordon\u003c/a\u003e equation. Intuitively, the contraction of space along the direction of travel occurs because the length of a given thing is measured in terms of the wavelengths of the underlying particles that compose it. As these things move with greater velocity, their frequency increases and their wavelengths decrease, and this results in a shorter overall length (as viewed from an outside observer).\u003c/p\u003e\u003cp id=\"text-9\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSimilarly, the increase in apparent mass comes from the increased difficulty in further accelerating the system. Mass is measured in terms of the inertia of a system: its resistance to further acceleration. It happens that the amount of force required to increase the frequency of a given wave packet further is proportional to its existing frequency, so that it becomes harder and harder to accelerate something as it moves faster and faster. This is manifest as an increase in the apparent mass of the particle. The increase in apparent mass of the particle also has the effect of causing the apparent time to slow down: time is fundamentally measured as a rate of change, which depends on accelerations, which are affected by the increase in apparent mass.\u003c/p\u003e\u003cp id=\"text-10\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe very stretchiness and malleability of space and time postulated by special relativity is a very strong argument in favor of waves over particles. Waves are inherently stretchy and malleable (as just invoked above to explain the key features of special relativity), whereas particles as hard discrete points just don’t seem to fit with this picture. What would it mean for a point particle to contract along its direction of travel?\u003c/p\u003e\u003cp id=\"text-11\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOne of the main puzzles that people often have with special relativity is understanding how light can appear to move at the same fixed speed, when an observer is screaming along at nearly the speed of light. You would think that you would catch up to a photon (as in Einstein’s motivating thought experiment), and even if you couldn’t see it actually standing still, it sure should seem to move more slowly relative to a speeding observer than to someone sitting still.\u003c/p\u003e\u003cp id=\"text-12\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe trick is, as an observer speeds up, the time dilation and space contraction effects conspire to alter the perception of the relative motion of other things, as illustrated in \u003ca href=\"special-relativity#figure_lorentz\" target=\"_blank\"\u003eFigure 2\u003c/a\u003e. The wavelength and velocity of the moving light wave never actually change (they are always the same relative to the underlying reference frame of the grid), but relative to the moving observer, they appear to be much faster. This is because the observer’s rulers have shrunk, and their clocks have slowed down, so that the same light wave appears to be moving much faster to them.\u003c/p\u003e\u003cp id=\"text-13\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn contrast, a static observer watching the moving observer chasing after the light beam will perceive the two as having a smaller relative velocity difference. These changes occur in exactly the right way to make the light wave appear to travel at the speed of light to the moving observer (which in fact it is).\u003c/p\u003e\u003ch2 id=\"details\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eDetails\u003c/h2\u003e\u003cdiv id=\"figure_lorentz2\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_lorentz2\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 3:\u003c/b\u003e \u003cp\u003eIllustration of Lorentz transformation between a static reference frame F (which is the same reference frame as the page, for example), and a frame F’ moving at velocity \u003cspan class=\"math inline\"\u003e\\(v\\)\u003c/span\u003e relative to F. Both frames contain a rigid rod, which when F’ was at rest relative to F were the same length \u003cspan class=\"math inline\"\u003e\\(l\\)\u003c/span\u003e, as indicated by the ruler marks. These rods now appear to be of different lengths in the two frames — from the perspective of F, the rod in F’ appears to have shrunk to a shorter length \u003cspan class=\"math inline\"\u003e\\(l’\\)\u003c/span\u003e. Interestingly, from the perspective of an observer moving along with the F’ reference frame, its rod appears to be of length \u003cspan class=\"math inline\"\u003e\\(l\\)\u003c/span\u003e (same as when it was stationary), and the other rod in F appears to have shrunk.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"figure_lorentz-factor\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_lorentz-factor\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 4:\u003c/b\u003e \u003cp\u003eThe amount of shrinkage as a function of velocity \u003cspan class=\"math inline\"\u003e\\(v\\)\u003c/span\u003e is determined by the Lorentz factor \u003cspan class=\"math inline\"\u003e\\(\\gamma\\)\u003c/span\u003e, which is plotted here (in natural units where the speed of light \u003cspan class=\"math inline\"\u003e\\(c = 1\\)\u003c/span\u003e). Not much happens until you get very close to the speed of light (e.g., above 90% or .9).\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"figure_lorentz-coord\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_lorentz-coord\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 5:\u003c/b\u003e \u003cp\u003eLorentz coordinate transformations, plotted in one spatial coordinate (x) and time (vertical axis). In the resting frame F, two points (think of them as marbles) just sit motionless, and thus form vertical trajectories through increasing time. When these are transformed into a reference frame F’ (aligned at point (t=0,x=0) with F) moving to the left at \u003cspan class=\"math inline\"\u003e\\(v = -.866\\)\u003c/span\u003e (Lorentz factor \u003cspan class=\"math inline\"\u003e\\(\\gamma = 2\\)\u003c/span\u003e), several interesting features of the Lorentz transformation are evident. If we follow the marble that was originally located at (0,x) as it sits in frame F for 3 time steps, we see that it appears to move to the right in frame F’, in the opposite direction of F’ motion. Furthermore, because of the Lorentz factor, the 3 seconds in frame F amount to 6 seconds in F’, and the distance it should travel due to the relative motion, computed in frame F (\u003cspan class=\"math inline\"\u003e\\(-vt = .866 \\times 3 = 2.598\\)\u003c/span\u003e) corresponds to 5.196 in frame F’. The second marble reveals a critical and somewhat counter-intuitive effect, where two events that are \u003cem\u003esimultaneous\u003c/em\u003e in frame F (i.e., t=0 for both of these), occur at “different times” in frame F’. Specifically the second marble sitting at rest at x=2 at t=0 is not \u0026amp;quot;encountered\u0026amp;quot; by the moving frame F’ until 3.464 time seconds later (in F’ time units), due to it being offset in space from the first marble. It takes the frame F’ \u003cspan class=\"math inline\"\u003e\\((-vx = .866 * 2 = 1.732)\\)\u003c/span\u003e time units to get to this second point (in the units of the F frame), and when this gets subject to the time dilation effect, you end up with the 3.464. From this starting point for the second marble, the same time and space increments as for the first marble occur for the subsequent point 3 time units later.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"figure_lorentz-coord-c\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_lorentz-coord-c\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 6:\u003c/b\u003e \u003cp\u003eAn alternative situation to the previous Figure, where the points in frame F are now moving at the speed of light (indicated by their slopes being 1; they are now photons instead of marbles). The Lorentz conversions preserve these slopes, so that the speeds are still 1 in the F’ frame. Thus, the speed of light is always the same to all observers.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cp id=\"text-23\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eHere we provide a more thorough treatment of \u003cstrong\u003especial relativity\u003c/strong\u003e, including the math needed to transform between different reference frames. We never actually have to use this in our numerical simulations, because all of our wave equations are automatically consistent with special relativity (i.e., they are \u003cstrong\u003emanifestly covariant\u003c/strong\u003e). But it is important to understand how all this works in any case.\u003c/p\u003e\u003cp id=\"text-24\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe mathematics of special relativity involves the computation of relative distance and time measurements in different reference frames that are moving relative to each other (\u003ca href=\"special-relativity#figure_lorentz2\" target=\"_blank\"\u003eFigure 3\u003c/a\u003e). As illustrated, a rigid rod of the same length \u003cspan class=\"math inline\"\u003e\\(l\\)\u003c/span\u003e can appear shortened if it is moving along at a high velocity \u003cspan class=\"math inline\"\u003e\\(v\\)\u003c/span\u003e relative to a static reference frame F (i.e., not moving relative to this page of paper). We denote the moving rod’s reference frame F’.\u003c/p\u003e\u003cp id=\"text-25\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe factor for transforming between reference frames is the \u003cstrong\u003eLorentz factor\u003c/strong\u003e gamma:\u003c/p\u003e\u003cdiv id=\"inline-container-27\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_factor\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_factor\"\u003e\u003cb\u003eEq 1:\u003c/b\u003e Lorentz factor\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-28\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\gamma = \\frac{1}{\\sqrt{1 - \\frac{v^2}{c^2}}}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-29\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe shape of this function is shown in \u003ca href=\"special-relativity#figure_lorentz-factor\" target=\"_blank\"\u003eFigure 4\u003c/a\u003e. Because the velocity enters into this function as a squared term, the function has a parabolic shape, such that not much happens until the velocity gets very close to the speed of light.\u003c/p\u003e\u003cp id=\"text-30\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFor the situation illustrated in , the moving rod in frame F’ appears shortened in the static frame F by a factor of \u003cspan class=\"math inline\"\u003e\\(\\frac{1}{\\gamma}\\)\u003c/span\u003e:\u003c/p\u003e\u003cdiv id=\"inline-container-32\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_contract\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_contract\"\u003e\u003cb\u003eEq 2:\u003c/b\u003e Lorentz contraction\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-33\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nl' = \\frac{l}{\\gamma}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-34\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSimilarly, the duration of a given fixed interval \u003cspan class=\"math inline\"\u003e\\(t\\)\u003c/span\u003e (e.g., a second) in the moving reference frame F’ appears longer from the perspective of F by the same factor:\u003c/p\u003e\u003cdiv id=\"inline-container-36\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_dilate\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_dilate\"\u003e\u003cb\u003eEq 3:\u003c/b\u003e Time dilation\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-37\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nt' = \\gamma t\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-38\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIt is important to keep in mind that in these equations, the \u003cspan class=\"math inline\"\u003e\\(l'\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(t'\\)\u003c/span\u003e refer to what something in the moving F’ reference frame looks like to someone in the static frame, F, relative to these same quantities as measured in the static frame (\u003cspan class=\"math inline\"\u003e\\(l\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(t\\)\u003c/span\u003e). But this assumes that we have previously established in a common reference frame that the rod in F’ actually has the same length as the one in F (and the second similarly has the same duration).\u003c/p\u003e\u003cp id=\"text-39\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAn alternative (and more conventional, but sometimes more confusing) way of using the prime notation is to directly convert between coordinate systems of the two reference frames, where the primed and unprimed cases both refer to \u003cem\u003ethe exact same event\u003c/em\u003e from the two different perspectives. In this case \u003cspan class=\"math inline\"\u003e\\(l'\\)\u003c/span\u003e would refer to how an observer \u003cem\u003ein F’\u003c/em\u003e would measure the rod length, whereas \u003cspan class=\"math inline\"\u003e\\(l\\)\u003c/span\u003e refers to what someone in F would measure \u003cem\u003efor the very same rod that is moving in F’\u003c/em\u003e, not for the “standard length” of the rod when both reference frames where static.\u003c/p\u003e\u003cp id=\"text-40\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn this case, things are exactly flipped, and we would say that \u003cspan class=\"math inline\"\u003e\\(l'\\)\u003c/span\u003e is the original rod length (say 1 meter), because it is in the F’ reference frame that the rod is not moving, whereas in the F reference frame, the rod has shrunk to a shorter apparent length \u003cspan class=\"math inline\"\u003e\\(l\\)\u003c/span\u003e (say .5 meters, if \u003cspan class=\"math inline\"\u003e\\(v=.866\\)\u003c/span\u003e). Similarly, the second measured in F’ (\u003cspan class=\"math inline\"\u003e\\(t'\\)\u003c/span\u003e) corresponds to a longer time interval \u003cspan class=\"math inline\"\u003e\\(t\\)\u003c/span\u003e in F (e.g., 2 seconds). It is definitely complicated to keep track of all this moving back and forth between reference frames!\u003c/p\u003e\u003cp id=\"text-41\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn this way of doing things, in frame F, we designate an event as occurring at space-time location \u003cspan class=\"math inline\"\u003e\\((t,x,y,z)\\)\u003c/span\u003e (this is a \u003ca href=\"four-vector\" target=\"_blank\"\u003efour-vector\u003c/a\u003e or \u003cstrong\u003espace-time coordinate\u003c/strong\u003e in \u003cem\u003eMinkowski\u003c/em\u003e space. In frame F’, this same event has coordinates \u003cspan class=\"math inline\"\u003e\\((t',x',y',z')\\)\u003c/span\u003e, where the two coordinate systems are aligned such that the origin (0,0,0,0) is the same in both. As before, we specify that the relative velocity \u003cspan class=\"math inline\"\u003e\\(v\\)\u003c/span\u003e between the two frames is entirely along the \u003cspan class=\"math inline\"\u003e\\(x\\)\u003c/span\u003e axis, for simplicity. We can compute these F’ coordinates directly from our F coordinates, using Lorentz transformations (again in natural units where \u003cspan class=\"math inline\"\u003e\\(c=1\\)\u003c/span\u003e, and \u003cspan class=\"math inline\"\u003e\\(v\\)\u003c/span\u003e goes between 0 and 1):\u003c/p\u003e\u003cul id=\"frame-42\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003cspan class=\"math inline\"\u003e\\(t' = \\gamma (t - vx) \\)\u003c/span\u003e\u003c/li\u003e\u003cli id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003cspan class=\"math inline\"\u003e\\(x' = \\gamma (x - vt) \\)\u003c/span\u003e\u003c/li\u003e\u003cli id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003cspan class=\"math inline\"\u003e\\(y' = y \\)\u003c/span\u003e\u003c/li\u003e\u003cli id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003cspan class=\"math inline\"\u003e\\(z' = z \\)\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cp id=\"text-43\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003ca href=\"special-relativity#figure_lorentz-coord\" target=\"_blank\"\u003eFigure 5\u003c/a\u003e shows where these formulas come from, and demonstrates their concrete application. The \u003cspan class=\"math inline\"\u003e\\(-vx\\)\u003c/span\u003e term in the transformation of time, and the \u003cspan class=\"math inline\"\u003e\\(-vt\\)\u003c/span\u003e term in the transformation of the \u003cspan class=\"math inline\"\u003e\\(x\\)\u003c/span\u003e coordinate, are critical and possibly counter-intuitive factors in these equations. As the figure explains, the \u003cspan class=\"math inline\"\u003e\\(-vt\\)\u003c/span\u003e is more sensible, as it simply reflects the relative motion of the reference frame over time. The \u003cspan class=\"math inline\"\u003e\\(-vx\\)\u003c/span\u003e term is somewhat less intuitive, but it reflects the fact that two events separated by some spatial distance will not be experienced as simultaneous in a moving reference frame! Instead, the spatial separation turns into a temporal separation due to the relative motion of the frame.\u003c/p\u003e\u003cp id=\"text-44\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOne can use these equations to compute the transformed velocity of an object moving at velocity \u003cspan class=\"math inline\"\u003e\\(w\\)\u003c/span\u003e along the x axis in frame F as it would appear to an observer in frame F’. We can compute this by taking velocity as distance over time, and noting that in time \u003cspan class=\"math inline\"\u003e\\(t\\)\u003c/span\u003e in frame F, the object will travel a distance \u003cspan class=\"math inline\"\u003e\\(x = wt\\)\u003c/span\u003e. Thus, we need to compute the transformed velocity as distance over time in the F’ frame, as:\u003c/p\u003e\u003cp id=\"text-45\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nw' = \\frac{x'}{t'}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-46\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n= \\frac{\\gamma (x - v t)}{\\gamma (t - v x)}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-47\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n= \\frac{\\gamma (wt - v t)}{\\gamma (t - v w t)}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-48\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n= \\frac{\\gamma t (w - v)}{\\gamma t (1 - v w)}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-49\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n= \\frac{w - v}{1 - wv}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-50\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e(the intermediate steps are included to make everything as clear as possible, tracking the substitution of \u003cspan class=\"math inline\"\u003e\\(x\\)\u003c/span\u003e with \u003cspan class=\"math inline\"\u003e\\(wt\\)\u003c/span\u003e).\u003c/p\u003e\u003cp id=\"text-51\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eInterestingly, if both velocities \u003cspan class=\"math inline\"\u003e\\(w\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\(v\\)\u003c/span\u003e are small (relative to the speed of light, \u003cspan class=\"math inline\"\u003e\\(c=1\\)\u003c/span\u003e), then the denominator is close to 1, and you get the more intuitive result that the relative velocities just add (or subtract, as the case may be). This is known as the \u003cstrong\u003eGalilean transformation\u003c/strong\u003e, which holds for non-relativistic speeds, and is intuitive to most people. However, as either of these speeds increase, the denominator starts to get smaller, and the relative velocities do not add linearly. Here are several illustrative examples of how this equation works:\u003c/p\u003e\u003cdiv id=\"table_examples\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eTable 1:\u003c/b\u003e Examples of relative velocities\u003c/p\u003e\u003ctable id=\"table_examples\" style=\"display:grid;flex-direction:row;justify-content:center;align-items:start;columns:4;gap:6px;font-weight:thin;text-align:start\"\u003e\u003cth id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:bold;line-height:1.5;text-align:start\"\u003e\u003c/th\u003e\u003cth id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:bold;line-height:1.5;text-align:start\"\u003e\u003c/th\u003e\u003cth id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:bold;line-height:1.5;text-align:start\"\u003e\u003c/th\u003e\u003cth id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:bold;line-height:1.5;text-align:start\"\u003e\u003c/th\u003e\u003ctd id=\"text-4\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003ew = 0\u003c/td\u003e\u003ctd id=\"text-5\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003ew’ = -v\u003c/td\u003e\u003ctd id=\"text-6\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003c/td\u003e\u003ctd id=\"text-7\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003eonly relative motion\u003c/td\u003e\u003ctd id=\"text-8\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003ew = v\u003c/td\u003e\u003ctd id=\"text-9\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003ew’ = 0\u003c/td\u003e\u003ctd id=\"text-10\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003c/td\u003e\u003ctd id=\"text-11\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003esame speed\u003c/td\u003e\u003ctd id=\"text-12\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003ew = .2\u003c/td\u003e\u003ctd id=\"text-13\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003ev = .1\u003c/td\u003e\u003ctd id=\"text-14\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003ew’ = .102\u003c/td\u003e\u003ctd id=\"text-15\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003eslightly faster than .1\u003c/td\u003e\u003ctd id=\"text-16\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003ew = -.1\u003c/td\u003e\u003ctd id=\"text-17\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003ev = .1\u003c/td\u003e\u003ctd id=\"text-18\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003ew’ = .199\u003c/td\u003e\u003ctd id=\"text-19\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003eslightly slower than .2\u003c/td\u003e\u003ctd id=\"text-20\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003ew = .9\u003c/td\u003e\u003ctd id=\"text-21\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003ev = .1\u003c/td\u003e\u003ctd id=\"text-22\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003ew’ = .87\u003c/td\u003e\u003ctd id=\"text-23\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003emuch faster than .8\u003c/td\u003e\u003ctd id=\"text-24\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003ew = -.9\u003c/td\u003e\u003ctd id=\"text-25\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003ev = .1\u003c/td\u003e\u003ctd id=\"text-26\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003ew’ = .99\u003c/td\u003e\u003ctd id=\"text-27\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003emuch slower than 1.1\u003c/td\u003e\u003ctd id=\"text-28\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003ew = 1.0\u003c/td\u003e\u003ctd id=\"text-29\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003ev = .1\u003c/td\u003e\u003ctd id=\"text-30\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003ew’ = 1.0\u003c/td\u003e\u003ctd id=\"text-31\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003espeed of light is same\u003c/td\u003e\u003ctd id=\"text-32\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003ew = -1.0\u003c/td\u003e\u003ctd id=\"text-33\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003ev = .1\u003c/td\u003e\u003ctd id=\"text-34\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003ew’ = 1.0\u003c/td\u003e\u003ctd id=\"text-35\" style=\"max-width:8in;margin:0em;font-weight:normal;line-height:1.5;text-align:start\"\u003espeed of light is same\u003c/td\u003e\u003c/table\u003e\u003c/div\u003e\u003cp id=\"text-54\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOne other important point to be made about special relativity is that there is an invariant calculation that can be made on the coordinates measured in any given reference frame, which will yield the same result as in any other reference frame. This is a kind of distance metric between two points \u003cspan class=\"math inline\"\u003e\\((t_1, x_1, y_1, z_1)\\)\u003c/span\u003e and \u003cspan class=\"math inline\"\u003e\\((t_2, x_2, y_2, z_2)\\)\u003c/span\u003e (both of which must be measured in the same reference frame):\u003c/p\u003e\u003cp id=\"text-55\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nds^2 = (t_1 - t_2)^2 - (x_1 - x_2)^2 - (y_1 - y_2)^2 - (z_1 - z_2)^2\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-56\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eYou can try this out on the coordinates in for the same points in the different reference frames: you’ll get the same results for either (with small differences due to round-off errors). Interestingly, when you try it for the second case with the photons moving at the speed of light, you see that this value is always 0. Thus, in effect, the distance in space is equal to the distance in time. If we just have motion in one spatial dimension (e.g., \u003cspan class=\"math inline\"\u003e\\(x\\)\u003c/span\u003e), this is clear:\u003c/p\u003e\u003cp id=\"text-57\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n(t_1 - t_2)^2 - (x_1 - x_2)^2 = 0\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-58\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n(t_1 - t_2)^2 = (x_1 - x_2)^2\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-59\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis is exactly the same time = space relationship for the basic wave equation. Perhaps now the fundamental importance of this for relativity is clearer.\u003c/p\u003e\u003cp id=\"text-60\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis distance metric also tells you in what way two events are separated. If the distance metric is positive, then the two points are separated by time, and if it is negative, they are separated by space.\u003c/p\u003e\u003cp id=\"text-61\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eInterestingly, the Lorentz transformations can be derived directly from this distance metric, and this distance metric can in turn be derived from the principle that the speed of light is a constant in any reference frame. Therefore, all of these effects are really just different manifestations of the same basic set of constraints, which are captured directly and automatically in the wave equations.\u003c/p\u003e\u003cp id=\"text-62\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOne additional property of special relativity has to do with the relationship between energy and momentum. If an object is at rest, its only energy is that associated with its rest mass, according to the famous equation:\u003c/p\u003e\u003cp id=\"text-63\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nE_0 = m_0 c^2\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-64\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAs the object moves faster, it gains energy in proportion to the Lorentz factor \u003cspan class=\"math inline\"\u003e\\(\\gamma\\)\u003c/span\u003e:\u003c/p\u003e\u003cp id=\"text-65\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nE = \\gamma m_0 c^2\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-66\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFurthermore, the relativistic momentum of this object is just the Lorentz factor times the standard Newtonian definition of momentum for motion at velocity \u003cspan class=\"math inline\"\u003e\\(\\vec{v}\\)\u003c/span\u003e:\u003c/p\u003e\u003cp id=\"text-67\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{p} = \\gamma m_0 \\vec{v}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-68\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThese two variables can be related in the relativistic energy-momentum equation:\u003c/p\u003e\u003cdiv id=\"inline-container-70\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_esq-p\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_esq-p\"\u003e\u003cb\u003eEq 4:\u003c/b\u003e relativistic energy-momentum\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-71\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nE^2 = \\vec{p}^2 c^2 + (m_0 c^2)^2\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-72\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nE = \\sqrt{\\vec{p}^2 c^2 + (m_0 c^2)^2}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-73\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhich is different way of expressing the energy of the system. It is very sensible, in that the two main contributors to energy are the momentum (i.e., kinetic energy) and the rest energy. If you set \u003cspan class=\"math inline\"\u003e\\(c=1\\)\u003c/span\u003e, it is even simpler:\u003c/p\u003e\u003cp id=\"text-74\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nE^2 = \\vec{p}^2+ m_0^2\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-75\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nE = \\sqrt{\\vec{p}^2+ m_0^2}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-76\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eYou can think of the rest energy and momentum as two legs of a right triangle, such that the total energy is the hypotenuse, according to the pythagorean theorem.\u003c/p\u003e\u003cp id=\"text-77\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn the limit of a slow velocity (relative to the speed of light), this expression approaches the Newtonian expression for kinetic energy (plus the rest mass energy):\u003c/p\u003e\u003cp id=\"text-78\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nE \\approx \\frac{1}{2}m_0 \\vec{v}^2 + m_0 c^2\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-79\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eInterestingly, the \u003ca href=\"klein-gordon\" target=\"_blank\"\u003eKlein-Gordon\u003c/a\u003e equation can be derived directly from the full \u003cspan class=\"math inline\"\u003e\\(E^2\\)\u003c/span\u003e equation!\u003c/p\u003e\u003ch2 id=\"simplification-of-the-energy-factor\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eSimplification of the energy factor\u003c/h2\u003e\u003cp id=\"text-81\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe relativistic energy factor as normally expressed in \u003ca href=\"special-relativity#eq_esq-p\" target=\"_blank\"\u003eEq 4\u003c/a\u003e actually just contains 3 essential variables, when you apply the Lorentz factor to the momentum expression. Thus, it can be simplified considerably to combine all these factors together, which reveals a rather interesting expression:\u003c/p\u003e\u003cdiv id=\"inline-container-83\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_esq-simp\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_esq-simp\"\u003e\u003cb\u003eEq 5:\u003c/b\u003e simplified relativistic energy-momentum\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-84\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nE^2 = \\frac{c^6 m_0^2}{c^2 - v^2}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-85\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe \u003cspan class=\"math inline\"\u003e\\(c^6\\)\u003c/span\u003e in the numerator is particularly striking, in that one rarely sees such a large power in a fundamental equation such as this. If the “basic” power for a squared equation such as this is \u003cspan class=\"math inline\"\u003e\\(c^2\\)\u003c/span\u003e, then it seems that there are 3 such elements coming together in this energy expression. Does that perhaps have something to do with the 3 spatial dimensions? Interestingly, in \u003ca href=\"stochastic-particles\" target=\"_blank\"\u003estochastic particles\u003c/a\u003e, we create a system of 3 simple harmonic oscillators to encode the momentum of a discrete particle, one for each spatial dimension, so perhaps this is in fact represented in the total energy of the particle.\u003c/p\u003e\u003cp id=\"text-86\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe denominator is also another way of understanding the Lorentz factor, where as the velocity \u003cspan class=\"math inline\"\u003e\\(v\\)\u003c/span\u003e increases toward \u003cspan class=\"math inline\"\u003e\\(c\\)\u003c/span\u003e, this denominator gets smaller, and thus the \u003cspan class=\"math inline\"\u003e\\(E^2\\)\u003c/span\u003e factor grows toward infinity.\u003c/p\u003e\u003ch3 id=\"algebraic-steps\" style=\"max-width:8in;margin:0.25em;font-size:22px;font-weight:normal;line-height:1.2727273;text-align:start\"\u003eAlgebraic steps\u003c/h3\u003e\u003cp id=\"text-88\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFor concreteness, here are the steps taken to arrive at this simplification:\u003c/p\u003e\u003cp id=\"text-89\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nE^2 = \\frac{c^2 m_0^2 v^2}{1 - \\frac{v^2}{c^2}} + (c^4 m_0^2)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-90\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nE^2 = \\frac{c^2 m_0^2 v^2}{\\frac{c^2 - v^2}{c^2}} + (c^4 m_0^2)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-91\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nE^2 = \\frac{c^4 m_0^2 v^2}{c^2 - v^2} + (c^4 m_0^2)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-92\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nE^2 = \\frac{c^4 m_0^2 v^2}{c^2 - v^2} + \\frac{(c^4 m_0^2)(c^2 - v^2)}{c^2 - v^2}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-93\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nE^2 = \\frac{c^4 m_0^2 (v^2 + c^2 - v^2)}{c^2 - v^2}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-94\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nE^2 = \\frac{c^6 m_0^2}{c^2 - v^2}\n\\]\u003c/span\u003e\u003c/p\u003e\u003ch2 id=\"relativistic-momentum-and-velocity\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eRelativistic momentum and velocity\u003c/h2\u003e\u003cp id=\"text-96\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe following derivation shows all the mathematical steps needed to go from the relativistic relationship between momentum and velocity, to an expression for the velocity as a function of the wavelength \u003cspan class=\"math inline\"\u003e\\(\\lambda\\)\u003c/span\u003e, rest mass \u003cspan class=\"math inline\"\u003e\\(m_0\\)\u003c/span\u003e, and the other standard constants (c, h). The \u003cspan class=\"math inline\"\u003e\\(\\gamma\\)\u003c/span\u003e is the Lorentz factor used throughout special relativity.\u003c/p\u003e\u003cdiv id=\"inline-container-98\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_p-start\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_p-start\"\u003e\u003cb\u003eEq 6:\u003c/b\u003e momentum\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-99\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\np = \\gamma m_0 v\n\\]\u003c/span\u003e\u003c/p\u003e\u003cdiv id=\"inline-container-101\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_flip-v\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_flip-v\"\u003e\u003cb\u003eEq 7:\u003c/b\u003e flip to solve for v\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-102\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nv = \\frac{1}{\\gamma m_0} p\n\\]\u003c/span\u003e\u003c/p\u003e\u003cdiv id=\"inline-container-104\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_flip-v\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_flip-v\"\u003e\u003cb\u003eEq 7:\u003c/b\u003e substitute definition of p in terms of wavelength\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-105\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nv = \\frac{\\sqrt{1 - \\frac{v^2}{c^2}}} {m_0} \\frac{h}{\\lambda}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cdiv id=\"inline-container-107\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_sq\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_sq\"\u003e\u003cb\u003eEq 9:\u003c/b\u003e square everything\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-108\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nv^2 = \\frac{(1 - \\frac{v^2}{c^2})h^2} {m_0^2 \\lambda^2}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cdiv id=\"inline-container-110\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_sep\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_sep\"\u003e\u003cb\u003eEq 10:\u003c/b\u003e separate terms\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-111\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nv^2 = \\frac{h^2} {m_0^2 \\lambda^2} - \\frac{v^2 h^2} {c^2 m_0^2 \\lambda^2}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cdiv id=\"inline-container-113\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_cons\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_cons\"\u003e\u003cb\u003eEq 11:\u003c/b\u003e consolidate v's on lhs\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-114\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nv^2 + \\frac{v^2 h^2} {c^2 m_0^2 \\lambda^2} = \\frac{h^2} {m_0^2 \\lambda^2}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cdiv id=\"inline-container-116\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_cons2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_cons2\"\u003e\u003cb\u003eEq 12:\u003c/b\u003e more consolidation\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-117\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{v^2 c^2 m_0^2 \\lambda^2 + v^2 h^2} {c^2 m_0^2 \\lambda^2} = \\frac{h^2} {m_0^2 \\lambda^2}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cdiv id=\"inline-container-119\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_elim\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_elim\"\u003e\u003cb\u003eEq 13:\u003c/b\u003e eliminate redundant denominators\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-120\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{v^2 c^2 m_0^2 \\lambda^2 + v^2 h^2} {c^2} = h^2\n\\]\u003c/span\u003e\u003c/p\u003e\u003cdiv id=\"inline-container-122\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_get-v\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_get-v\"\u003e\u003cb\u003eEq 14:\u003c/b\u003e pull out v term uniquely\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-123\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{v^2 (c^2 m_0^2 \\lambda^2 + h^2)} {c^2} = h^2\n\\]\u003c/span\u003e\u003c/p\u003e\u003cdiv id=\"inline-container-125\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_move-v\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_move-v\"\u003e\u003cb\u003eEq 15:\u003c/b\u003e multiply by inverse of v factor to move to other side\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-126\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nv^2 = \\frac{h^2 c^2}{c^2 m_0^2 \\lambda^2 + h^2}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cdiv id=\"inline-container-128\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_flip-v\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_flip-v\"\u003e\u003cb\u003eEq 7:\u003c/b\u003e final expression: take square root\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-129\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nv = \\frac{h c}{\\sqrt{c^2 m_0^2 \\lambda^2 + h^2}}\n\\]\u003c/span\u003e\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"semiclassical\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"m274-450 248 248-42 42-320-320 320-320 42 42-248 248h526v60H274Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003ePrevious\u003c/p\u003e\u003c/a\u003e\u003cstretch id=\"content-buttons-go-167\" style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1ch;min-height:1em;font-weight:thin;text-align:start\"\u003e\u003c/stretch\u003e\u003ca id=\"content-buttons-go-168\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"spin\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M686-450H160v-60h526L438-758l42-42 320 320-320 320-42-42 248-248Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Spin","URL":"spin","Title":"Spin","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{},"Description":"\u003cstrong\u003eSpin\u003c/strong\u003e is a uniquely quantum property that lies at the heart of the underlying mechanisms implemented in the current model. There are two known values of spin, 1 and +/- \u003csup\u003e1\u003c/sup\u003e⁄\u003csub\u003e2\u003c/sub\u003e:","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eSpin\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-35\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cstrong\u003eSpin\u003c/strong\u003e is a uniquely quantum property that lies at the heart of the underlying mechanisms implemented in the current model. There are two known values of spin, 1 and +/- \u003csup\u003e1\u003c/sup\u003e⁄\u003csub\u003e2\u003c/sub\u003e:\u003c/p\u003e\u003cul id=\"frame-1\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Bosons are nominally the particles that carry force fields, and they have a spin of 1, with the \u003ca href=\"photon\" target=\"_blank\"\u003ephoton\u003c/a\u003e being the canonical example. A spin of 1 is effectively like a spin of 0: in a very abstract sense, they complete a full revolution every “step”. It is unclear whether a boson is actually a thing or not under the current model, as the \u003ca href=\"photon\" target=\"_blank\"\u003ephoton\u003c/a\u003e may not really exist, and the weak and strong charge carriers may be something else entirely.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Fermions are massive particles like \u003ca href=\"electron\" target=\"_blank\"\u003eelectrons\u003c/a\u003e and quarks, with a spin of +/- \u003csup\u003e1\u003c/sup\u003e⁄\u003csub\u003e2\u003c/sub\u003e. Interestingly, the mysterious neutrino can be thought of as a particle that \u003cem\u003eonly\u003c/em\u003e has spin and nothing else (no charge) – it also has a spin of \u003csup\u003e1\u003c/sup\u003e⁄\u003csub\u003e2\u003c/sub\u003e, but the “matter” neutrino is \u003cem\u003ealways\u003c/em\u003e spinning in the left-hand direction, while the “antimatter” anti-neutrino spins in the left-hand direction. Although they are thought to have mass, it is possible that neutrinos have these fixed spin directions because they move at the speed of light in a kind of fixed helical pattern in relation to their spin, and they don’t have a proper mass-like momentum property after all.\u003c/p\u003e\u003c/ul\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe \u003ca href=\"dirac\" target=\"_blank\"\u003eDirac\u003c/a\u003e equation introduced “Dirac spinors” as a mathematical formulation of spin for describing the spin \u003csup\u003e1\u003c/sup\u003e⁄\u003csub\u003e2\u003c/sub\u003e property of fermions.\u003c/p\u003e\u003cp id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn general, the spin of something like an electron is thought to be a strongly \u003ca href=\"contextual\" target=\"_blank\"\u003econtextual\u003c/a\u003e property, meaning that it is easily shaped through interactions with the environment, and is not something that is otherwise strongly constrained by the intrinsic properties of the particle (except for neutrinos).\u003c/p\u003e\u003ch2 id=\"spin-as-particle-momentum-and-mass-mechanism\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eSpin as particle momentum and mass mechanism\u003c/h2\u003e\u003cp id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eUnder the current \u003ca href=\"stochastic-particles\" target=\"_blank\"\u003estochastic particles\u003c/a\u003e plans, the phenomenon of spin reflects an internal dynamic process within discrete particles, operating over internal state values (similar to the spinors), that is responsible for the following particle properties:\u003c/p\u003e\u003cul id=\"frame-6\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Maintaining the particle’s momentum value across time and space.\u003c/li\u003e\u003cli id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Causing a massive particle to have a resting mass / energy.\u003c/li\u003e\u003cli id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• And, of course, its actual spin direction.\u003c/li\u003e\u003c/ul\u003e\u003cp id=\"text-7\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe neutrino is a particle that \u003cem\u003eonly\u003c/em\u003e has spin, indicating that this spin mechanism can dissociate from other particle properties. Thus, modeling this most mysterious of particles is a first-order goal, along with that of the \u003ca href=\"electron\" target=\"_blank\"\u003eelectron\u003c/a\u003e.\u003c/p\u003e\u003cp id=\"text-8\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe neutrino interacts only via the \u003ca href=\"weak\" target=\"_blank\"\u003eweak\u003c/a\u003e force, which is strongly localized to around \u003cspan class=\"math inline\"\u003e\\(10^−16\\)\u003c/span\u003em — on the scale of the charge radius of a proton (\u003cspan class=\"math inline\"\u003e\\(10^-15\\)\u003c/span\u003em). This length scale is short because the bosons that carry the weak force are \u003cem\u003emassive\u003c/em\u003e (indeed they are much heavier than the mass of a proton).\u003c/p\u003e\u003cp id=\"text-9\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe weak force is thus the obvious force for understanding dynamics within fermion particles. In particular, the leptons (electron, muon, tau) are all different \u003cem\u003eflavors\u003c/em\u003e or \u003cem\u003egenerations\u003c/em\u003e of the same particle, and the muon and tau decay back to the stable electron through the weak force, while emitting neutrinos (and energy in the form of photons??). Thus, it is clear that the weak force is somehow critically involved in these spin dynamics, which in turn are involved in the increased masses of these higher generation particle variants.\u003c/p\u003e\u003cp id=\"text-10\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThere are 3 different generations (flavors) of quark, analogous to those of the leptons, with up and down being the first generation.\u003c/p\u003e\u003cp id=\"text-11\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThus, it makes sense to think of these generations as essentially excited states of some kind of internal spin-like dynamic, which are unstable, and decay back to the stable ground states, while shedding extra spin in the form of neutrinos, which are emitted with zero-sum spin, in the form of a neutrino and an anti-neutrino. So it isn’t like there is extra \u003cem\u003enet\u003c/em\u003e spin, but somehow the internal spin dynamics are excited, as if there is just more of something spin-like in there.\u003c/p\u003e\u003cp id=\"text-12\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe property of \u003cstrong\u003eweak isospin\u003c/strong\u003e is effectively like the charge value that drives the weak force, coming in positive and negative \u003csup\u003e1\u003c/sup\u003e⁄\u003csub\u003e2\u003c/sub\u003e increments.\u003c/p\u003e\u003cp id=\"text-13\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe charged \u003cstrong\u003eW\u003c/strong\u003e boson is unstable and decays into an electron and a neutrino, so it can be thought of as a kind of anomolous excitation of this weak-spin system, with the extra spin component of the neutrino added to that of the electron.\u003c/p\u003e\u003cp id=\"text-14\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe neutral \u003cstrong\u003eZ\u003c/strong\u003e boson is characterized by a weak charge value that is also a function of the momentum of the relevant particles, therefore supporting the connection between spin, momentum, and the weak force.\u003c/p\u003e\u003cp id=\"text-15\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe masses of these weak bosons is all tied up with the Higgs mechanism, so that also needs to be understood.\u003c/p\u003e\u003cp id=\"text-16\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eBasically, the whole electroweak / spin soup needs to be probed to understand how an elementary particle like the \u003ca href=\"electron\" target=\"_blank\"\u003eelectron\u003c/a\u003e actually works. It seems clear given the mutation rules that an electron is not really \u003cem\u003eelementary\u003c/em\u003e anymore: the elementary components are \u003cem\u003espin\u003c/em\u003e and \u003cem\u003echarge\u003c/em\u003e and some kinds of interesting dynamics that make these all produce the relevant particle-like behavior.\u003c/p\u003e\u003cp id=\"text-17\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis all seems like an interesting case where adding more complexity to the model may indeed make it easier to solve, because it gives us a lot of constraints on understanding the connections among charge and spin dynamics.\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"special-relativity\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"m274-450 248 248-42 42-320-320 320-320 42 42-248 248h526v60H274Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003ePrevious\u003c/p\u003e\u003c/a\u003e\u003cstretch id=\"content-buttons-go-167\" style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1ch;min-height:1em;font-weight:thin;text-align:start\"\u003e\u003c/stretch\u003e\u003ca id=\"content-buttons-go-168\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"stochastic-particles\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M686-450H160v-60h526L438-758l42-42 320 320-320 320-42-42 248-248Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Stochastic particles","URL":"stochastic-particles","Title":"Stochastic particles","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{"eq":["eq_kg","eq_kg-current","eq_kg-charge"],"figure":["figure_pf-origin"]},"Description":"Wave equations provide such a natural explanation of so many physical phenomena, that it is very tempting to imagine that the entirety of physics can be produced exclusively through various coupled wave functions, as in the coupled \u003ca href=\"dirac\" target=\"_blank\"\u003eDirac\u003c/a\u003e – \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e system. However, having implemented such a system, it is evident that the Achilles heel of waves is a fundamental problem: they inevitably just spread out and diffuse over time.","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eStochastic particles\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-36\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWave equations provide such a natural explanation of so many physical phenomena, that it is very tempting to imagine that the entirety of physics can be produced exclusively through various coupled wave functions, as in the coupled \u003ca href=\"dirac\" target=\"_blank\"\u003eDirac\u003c/a\u003e – \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e system. However, having implemented such a system, it is evident that the Achilles heel of waves is a fundamental problem: they inevitably just spread out and diffuse over time.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe obvious solution to this problem is to have some kind of discrete particle that cannot spread out, and remains localized in a specific point in space at any given point in time. In this context, the fundamental wave-particle duality in quantum physics suddenly starts to look like an essential \u003cem\u003efeature\u003c/em\u003e of the system, instead of some kind of paradoxical bug that nature threw at us for no good reason.\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe basic \u003ca href=\"waves\" target=\"_blank\"\u003ewave\u003c/a\u003e equation represents a fundamental interaction between \u003cstrong\u003espace\u003c/strong\u003e (the spatial gradient, i.e., the \u003cem\u003emomentum\u003c/em\u003e operator in the \u003ca href=\"hamiltonian\" target=\"_blank\"\u003eHamiltonian\u003c/a\u003e) and \u003cstrong\u003etime\u003c/strong\u003e (acceleration within a single grid element, i.e., the Hamiltonian \u003cem\u003eenergy\u003c/em\u003e operator). In this context, consistent with \u003ca href=\"special-relativity\" target=\"_blank\"\u003especial relativity\u003c/a\u003e, a massive particle lives strictly in the time component. This can be seen in the \u003ca href=\"klein-gordon\" target=\"_blank\"\u003eKlein-Gordon\u003c/a\u003e equation:\u003c/p\u003e\u003cdiv id=\"inline-container-4\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_kg\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_kg\"\u003e\u003cb\u003eEq 1:\u003c/b\u003e Klein-Gordon equation\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2 {\\varphi}}{\\partial t^2} = c^2 \\nabla^2 \\varphi - \\frac{c^2 m_0^2}{\\hbar^2} \\varphi\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-6\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhere the mass-dependent factor enters without any spatial gradient operator, and directly subtracts away from the energy factor (acceleration) on the left-hand side. This lack of spatial extent is consistent with a discrete particle entity, that somehow consumes energy from the wave field in proportion to its rest mass. This is the qualitative picture for how our discrete particles emerge, with the energy consumption propelling an internal oscillation within the discrete cell in the cubic lattice of the \u003ca href=\"cellular-automaton\" target=\"_blank\"\u003ecellular automaton\u003c/a\u003e (CA) framework.\u003c/p\u003e\u003cp id=\"text-7\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAn obvious problem with this notion of something being contained entirely within a discrete cell is that it becomes challenging to imagine how it might ever move to another such cell. Such a move would have to happen in a discrete jump, creating a major discontinuity in the overall wave state, and potentially making the particle trajectory seemingly discontinuous and anisotrophic.\u003c/p\u003e\u003cp id=\"text-8\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe only way to overcome those difficulties is to use \u003cstrong\u003estochastic\u003c/strong\u003e discrete jumps, such that, on longer time averages, the timing and spatial distribution of such jumps smooths out into a continuous, isotrophic distribution. Thus, contrary to Einstein’s oft-cited objection that “God does not play dice with the universe”, in fact it seems that an essential form of randomness is \u003cem\u003enecessary\u003c/em\u003e for discrete particles to move in a physically plausible manner.\u003c/p\u003e\u003cp id=\"text-9\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFurthermore, although the continuum limit is mathematically approachable through the tools of calculus, it is problematic from a physics perspective due to the nearly-infinite field strengths (and thus energies) that would be present in the immediate vicinity of a charged particle. This \u003cem\u003eultraviolet catastrophe\u003c/em\u003e is a recurring theme throughout the \u003ca href=\"history\" target=\"_blank\"\u003ehistory\u003c/a\u003e of quantum physics, and it is nicely resolved through the use of the discrete cubic lattice of the CA framework.\u003c/p\u003e\u003cp id=\"text-10\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe seminal work in analyzing stochastic discrete particle motion within the context of quantum physics was done by \u003ca href=\"ref://Nelson66\" target=\"_blank\"\u003eNelson (1966)\u003c/a\u003e, who showed that a form of stochastic discrete particle motion actually results in the \u003ca href=\"schrodinger\" target=\"_blank\"\u003eSchrodinger\u003c/a\u003e wave function in the continuous time-average limit. Interestingly, this work builds on the original work by Einstein on Brownian random-walk motion, back in 1905. Subsequent work has developed these ideas in multiple ways (\u003ca href=\"ref://Cufaro-PetroniVigier83\" target=\"_blank\"\u003eCufaro-Petroni \u0026 Vigier, 1983\u003c/a\u003e; \u003ca href=\"ref://CufaroPetroniVigier79\" target=\"_blank\"\u003eCufaro Petroni \u0026 Vigier, 1979\u003c/a\u003e; \u003ca href=\"ref://Ord96\" target=\"_blank\"\u003eOrd, 1996\u003c/a\u003e; \u003ca href=\"ref://Sciarretta18\" target=\"_blank\"\u003eSciarretta (2018)\u003c/a\u003e; \u003ca href=\"ref://Sciarretta21\" target=\"_blank\"\u003eSciarretta (2021)\u003c/a\u003e and others reviewed therein).\u003c/p\u003e\u003cp id=\"text-11\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eCritically none of this existing work involves an integrated wave-particle \u003ca href=\"duality\" target=\"_blank\"\u003eduality\u003c/a\u003e; it focuses exclusively on the time-average distributions of discrete particle motion. Thus, unlike the \u003ca href=\"pilot-wave\" target=\"_blank\"\u003epilot-wave\u003c/a\u003e framework, the quantum wave function in a purely stochastic particle model is entirely \u003ca href=\"epistemic-vs-ontic\" target=\"_blank\"\u003eepistemic\u003c/a\u003e: it just describes the expected value of a discrete particle’s random walk trajectories over time. There is no physical reality to such a wave. By contrast, our goal here is to derive an integration of discrete particle motion with quantum wave functions.\u003c/p\u003e\u003cdiv id=\"figure_pf-origin\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_pf-origin\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 1:\u003c/b\u003e \u003cp\u003eStochastic origin of quantum momentum / frequency relationship. The momentum on the left is 0.5c while on the right is 0. The distribution of position is on the vertical axis, while time is on the horizontal axis, with each point centered at the origin in the center (i.e., the temporal autocorrelation function). The variance on the left is half of that on the right.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cp id=\"text-14\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOne key intuition for why discrete particle motion naturally exhibits quantum wave-like behavior is that a slow drift rate produces a wide cloud of space where particle could be, corresponding to a long wavelength in the probability cloud that the Schrödinger wave function describes. However, when the particle has high momentum (velocity), it moves more deterministically in a given direction, resulting in a narrower range of variance around the particle’s mean trajectory, resulting in a narrower effective wavelength (\u003ca href=\"stochastic-particles#figure_pf-origin\" target=\"_blank\"\u003eFigure 1\u003c/a\u003e).\u003c/p\u003e\u003cp id=\"text-15\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWhile this pure particle-based approach is appealing in its simplicity, it does not appear to provide an explanation for phenomena such as the \u003ca href=\"double-slit\" target=\"_blank\"\u003edouble-slit\u003c/a\u003e experiment, where somehow a particle can interfere \u003cem\u003ewith itself\u003c/em\u003e, but only if the other slit is open. Furthermore, it cannot be the case that these interference effects only arise in the rare cases when a discrete particle happens to wander so aimlessly as to go through both slits somehow.\u003c/p\u003e\u003cp id=\"text-16\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFurthermore, discrete particles require some other continuous state values embedded in the same discrete lattice to drive the \u003cem\u003eprobabilities\u003c/em\u003e underlying their stochastic behavior. The nature of such values is often unaddressed in existing frameworks. The \u003ca href=\"pilot-wave\" target=\"_blank\"\u003epilot-wave\u003c/a\u003e framework of de Broglie and Bohm provides a natural mechanism for these continuous momentum values, where the continuous-valued wave function guides the motion of a discrete particle in some way.\u003c/p\u003e\u003cp id=\"text-17\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOverall, the discrete particle framework does a great job of keeping the accounting tight, in comparison to the fundamentally sloppy \u003ca href=\"matter-waves\" target=\"_blank\"\u003ematter waves\u003c/a\u003e that end up diffusing over time and space. The discrete particles can be strictly conserved, and always tightly localized, while the surrounding wave functions diffuse outward from this central island of stability.\u003c/p\u003e\u003cp id=\"text-18\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eHowever, a system based purely on discrete particles has a very hard time managing interactions among the particles via forces: the odds of any other discrete particle “hitting” another are very small, whereas these big sloppy waves provide a nice continuous, broadly diffusing and saturating medium for force transmission. In effect, the surrounding wave field is essential for the particle to be able to properly “sense” the force field effects (from electromagnetic, weak and strong forces). Thus, the wave field functions like an \u003cem\u003eantenna\u003c/em\u003e or “whiskers” in sensing forces over a broader space, beyond its own singular cell.\u003c/p\u003e\u003cp id=\"text-19\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFrom a “design” perspective, this \u003cem\u003ehybrid\u003c/em\u003e framework of discrete particles and continuous wave fields represents a “best of both worlds” solution, relative to something based exclusively on discrete particles, or exclusively on continuous wave functions. This then provides a compelling reason for the otherwise central puzzle of the quantum world: \u003cem\u003ewhy\u003c/em\u003e is there this wave-particle duality? Why is nature so strangely complicated in this way?\u003c/p\u003e\u003cp id=\"text-20\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eRelative to the standard QM interpretations, the pilot-wave nature of this approach avoids all the impossible conundrums of instantaneous wave collapse at the point of measurement, as the discrete particles are there at every point in time and space, even when you’re not looking. The primary challenge for such a framework is dealing with the apparent \u003ca href=\"non-locality\" target=\"_blank\"\u003enon locality\u003c/a\u003e of quantum entanglement.\u003c/p\u003e\u003ch2 id=\"remaining-degrees-of-freedom\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eRemaining degrees of freedom\u003c/h2\u003e\u003cp id=\"text-22\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWithin this broad framework of discrete particles interacting via continuous waves, there are many degrees of freedom in specifying the details, in a way that ends up being physically accurate. Here are some of the primary questions:\u003c/p\u003e\u003cul id=\"frame-23\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• An original inspiration for the stochastic particle approach was Feynman’s path integral approach to \u003ca href=\"qed\" target=\"_blank\"\u003eQED\u003c/a\u003e, which is based on the idea that every possible pathway is somehow being sampled with an associated probability, via \u003cem\u003evirtual particles\u003c/em\u003e that are only transiently extant. This “quantum foam” provides a different kind of picture relative to a lone “real” discrete particle bumping around in space. Instead, at every moment, particles are constantly popping into and out of existence, most with tiny probabilities, and this is what shapes the behavior of the “real” particles that are observed.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe key question here is whether the continuous wave function plus stochastic behavior in a real particle ends up capturing everything that is otherwise captured by the virtual particles? From a complexity management perspective, it makes sense in any case to start with the much simpler frameork of the known \u003ca href=\"dirac\" target=\"_blank\"\u003eDirac\u003c/a\u003e wave function operating on a real \u003ca href=\"electron\" target=\"_blank\"\u003eelectron\u003c/a\u003e, and see how far that goes.\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• How does the particle influence the wave fields? This is the \u003cem\u003eback reaction\u003c/em\u003e question that has long bedeviled attempts to understand the detailed “mechanistic” physics taking place up close around a charged particle (\u003ca href=\"ref://FordOConnell91\" target=\"_blank\"\u003eFord \u0026 O’Connell, 1991\u003c/a\u003e; \u003ca href=\"ref://FordOConnell93\" target=\"_blank\"\u003eFord \u0026 O’Connell, 1993\u003c/a\u003e). Specifically, very near the particle itself, the electromagnetic potential would be very strong, and thus correspond to a very high energy density, which should then end up producing virtual particles etc. This is known as the \u003cem\u003eultraviolet divergence\u003c/em\u003e problem. It can be “solved” by having some kind of cutoff at short length scales, but that seems fairly arbitrary. Also, as the particle moves discretely to neighboring grid points, that creates ripples in the force field.\u003c/p\u003e\u003cp id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOne solution, consistent with the \u003ca href=\"pilot-wave\" target=\"_blank\"\u003epilot-wave\u003c/a\u003e approach, is that the discrete particle itself doesn’t actually generate any back reaction, and is instead purely a “surfer” on the waves. This would imply that the \u003ca href=\"dirac\" target=\"_blank\"\u003eDirac\u003c/a\u003e field itself would generate the corresponding \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e EM field, which also solves many conceptual difficulties because it is smoothly distributed over space. In short, the same arguments that favor the Dirac field as a kind of antenna for the force field also apply in the reverse direction, for the generation of force fields from charged matter.\u003c/p\u003e\u003cp id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eHowever, the Dirac field itself is likely representing both epistemological vs ontic contributions to overall uncertainty. So in principle, the \u003cem\u003eactual\u003c/em\u003e current location of the particle should be driving updating of the Dirac field. But there is little in the way of existing guidance for how this might happen. Mathematically, this should be just like the state right after a particle has been localized to one point, but then the momentum is gone. Considerable work needs to be done investigating useful continuous momentum wave functions that can be driven by the localization of a discrete particle.\u003c/p\u003e\u003cp id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eA further wrinkle is the possibility that particles leave some kind of \u003cem\u003etrace\u003c/em\u003e on the wave field that then influences subsequent particles, as a way of explaining the apparent \u003ca href=\"non-locality\" target=\"_blank\"\u003enon-locality\u003c/a\u003e phenomena (\u003ca href=\"ref://Sciarretta18\" target=\"_blank\"\u003eSciarretta, 2018\u003c/a\u003e; \u003ca href=\"ref://Sciarretta21\" target=\"_blank\"\u003eSciarretta, 2021\u003c/a\u003e).\u003c/p\u003e\u003cp id=\"text-6\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Hidden particle states. The \u003ca href=\"ref://Ord96\" target=\"_blank\"\u003eOrd (1996)\u003c/a\u003e model involves 2 state variables for each particle, that correspond qualitatively to the \u003ca href=\"spin\" target=\"_blank\"\u003espin\u003c/a\u003e degrees of freedom in a fermion particle. The \u003ca href=\"dirac\" target=\"_blank\"\u003eDirac\u003c/a\u003e equation in its second order formulation likewise has 4 wave state variables that mutually interact to produce spin, via the \u003cem\u003espinor\u003c/em\u003e dynamics. These also produce the phase dynamics of quantum wave functions, and are important for the conservation properties of these functions. Thus, the possibility of these internal state degrees of freedom, beyond just a single binary state value, must be properly explored, especially as it might interact with the Dirac spinor states.\u003c/p\u003e\u003cp id=\"text-7\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOne intuition is that the resting energy / mass of a particle is associated with this constant cycling through the \u003ca href=\"spin\" target=\"_blank\"\u003espin\u003c/a\u003e states, and that somehow this cycling dynamic within a single cell is capable of maintaining some kind of momentum value, as it couples with the Dirac-like wave state that it is generating. This may be related to the phenomenon of \u003cem\u003ezitterbewegung\u003c/em\u003e (\u003ca href=\"ref://Hestenes08\" target=\"_blank\"\u003eHestenes, 2008\u003c/a\u003e; \u003ca href=\"ref://Hestenes90]\" target=\"_blank\"\u003eHestenes90]\u003c/a\u003e; \u003ca href=\"ref://Sidharth09\" target=\"_blank\"\u003eSidharth, 2009\u003c/a\u003e; \u003ca href=\"ref://RomanRosoPlaja03\" target=\"_blank\"\u003eRoman et al., 2003\u003c/a\u003e; \u003ca href=\"ref://BarutBracken81\" target=\"_blank\"\u003eBarut \u0026 Bracken, 1981\u003c/a\u003e; \u003ca href=\"ref://WangZhang01\" target=\"_blank\"\u003eWang \u0026 Zhang, 2001\u003c/a\u003e). Thus, the actual momentum represents a \u003cem\u003espatial imbalance\u003c/em\u003e in this constant internal spin motion within the particle state.\u003c/p\u003e\u003cp id=\"text-8\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOne tempting idea in this space is that you could somehow imagine a spatially distributed collection of “sub states” for each particle, kind of like a very compact localized wave function. However, once you stray outside of a single discrete cell, it becomes logistically very difficult to keep track of these distributed states, and preventing them from leaking out all over the place.\u003c/p\u003e\u003cp id=\"text-9\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe initial specific model here would involve at least two values for each cubic 3D axis (6 total), plus two for the “staying still” component, and the relative balance between the values along each axis determines the net momentum along that axis. These values are then updated in response to the gradient along each such axis, computed across the 26 point cubic neighborhood as usual (\u003ca href=\"discrete-gradient\" target=\"_blank\"\u003ediscrete gradient\u003c/a\u003e). The same technique as used in the \u003ca href=\"klein-gordon\" target=\"_blank\"\u003eKlein-Gordon\u003c/a\u003e and \u003ca href=\"dirac\" target=\"_blank\"\u003eDirac\u003c/a\u003e wave functions, where the mass term drags against the wave-based oscillation frequency, could be used to obtain the fundamental quantum frequency relationship.\u003c/p\u003e\u003cp id=\"text-10\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Multiparticle states are also an essential constraint on the system. The wave state variables must be shared across all particles of a given type (otherwise you’re back in \u003ca href=\"configuration-space\" target=\"_blank\"\u003econfiguration space\u003c/a\u003e), but if they are holding important state for an individual particle, then how does that work when another particle’s wave function gets close by? For example, if it is spinning the other way? Does that interfere with the spin of the other guy? This is also a good reason to keep the wave function less widely distributed, as should happen when the particle is the generator.\u003c/p\u003e\u003cp id=\"text-11\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• The particle zoo: muon and tau vs \u003ca href=\"electron\" target=\"_blank\"\u003eelectrons\u003c/a\u003e, antiparticles, neutrinos are “pure spin” without any charge. All of this suggests that charge and spin are two separate factors that could potentially dissociate, but yet stick together. Neutrinos are always left-handed, and anti-neutrinos are right-handed: this is a key constraint on the nature of spin and mass. Charge gives rise to (lots of) mass, but spin does not (much)? Maybe just kinetic energy in spin? The spin model should definitely accommodate neutrinos. Also, they have muon and tau flavors. So overall, this all sounds like a factorizable dynamic system.\u003c/p\u003e\u003cp id=\"text-12\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe muon decays after an average lifetime of 2.2 millionths of a second into an electron, a neutrino, and an antineutrino. Ok, so the muon is somehow a regular electron plus these two additional neutrinos worth of spin? And that adds up to 200x the rest mass of the electron, and yet the neutrinos themselves are nearly massless! And all of this is tied up with the \u003ca href=\"weak\" target=\"_blank\"\u003eweak\u003c/a\u003e force, which must then be considered. Perhaps the short-range nature of this force plays a critical role in the localized stochastic dynamics. Also, the random nature of weak-mediated decay processes is broadly consistent with the true randomness required here.\u003c/p\u003e\u003c/ul\u003e\u003cp id=\"text-24\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOverall, this is a relatively large space to explore, and there are many potential tradeoffs in terms of the risks of taking on too much versus, paradoxically, too little, where it may be the case that adding additional complexity (e.g., in the form of the quantum foam of virtual particles and multiparticle states more generally) solves problems that would otherwise plague simpler, reduced, single-particle frameworks.\u003c/p\u003e\u003ch2 id=\"photons\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003ePhotons\u003c/h2\u003e\u003cp id=\"text-26\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eUnlike fermion particles such as electrons, the EM field is not amenable to a discrete particle-like framework: photons have many problematic issues as discrete particles of the EM field. Therefore, it makes more sense to retain a “classical” \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e EM field, interacting with the discretized fermion cells, as in the \u003ca href=\"semiclassical\" target=\"_blank\"\u003esemiclassical\u003c/a\u003e approach developed by a number of researchers (see \u003ca href=\"ref://Struyve20\" target=\"_blank\"\u003eStruyve, 2020\u003c/a\u003e; \u003ca href=\"ref://Santos15\" target=\"_blank\"\u003eSantos, 2015\u003c/a\u003e).\u003c/p\u003e\u003ch2 id=\"virtual-particles-the-discrete-lattice-and-probability-waves\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eVirtual particles, the discrete lattice, and probability waves\u003c/h2\u003e\u003cp id=\"text-28\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eVirtual particles are an essential feature of QED / quantum field theory, and yet their “ontological” status is clearly somewhat confusing: they aren’t the “real” particles that we observe, and yet their fleeting existence is necessary for the theory to work, so in some sense they must be just as real as the “real” particles.\u003c/p\u003e\u003cp id=\"text-29\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe discrete particle lattice framework provides a potential resolution to this conundrum. If any given “real” particle can potentially occupy any given cell in a discrete lattice, then there must effectively be a “slot” reserved for such a particle in each cell. These empty slots could provide an appealing basis for virtual particles, and the propagation and interactions of particles in the matrix.\u003c/p\u003e\u003cp id=\"text-30\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn particular, a simple schema is that the probability waves associated with the standard interpretation of QM reflect a rippling propagation of probability factors across virtual particle slots in the matrix, with a real particle having a special status as being the current “true” location. Each possible jump to a neighboring cell involves a full transition matrix dependent upon the total energy (mass + kinetic) of the source: if the source is sufficiently energetic, it has some probability of activating a different combination of real particles as it makes the leap, accounting for the splitting tracks observed in particle accelerator experiments. Perhaps some of the “trace” in the matrix represents residual bits of this probability field propagating out and being left behind as real particles move around.\u003c/p\u003e\u003cp id=\"text-31\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIt is essential that these probability computations are all propagated in terms of \u003cem\u003eamplitudes\u003c/em\u003e, not the probability values themselves, which are obtained by the product with the complex conjugate (“squaring”).\u003c/p\u003e\u003ch2 id=\"todo\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eTODO\u003c/h2\u003e\u003cul id=\"frame-33\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• zitterbewegung and helical spin in electron: \u003ca href=\"ref://Hestenes08\" target=\"_blank\"\u003eHestenes, 2008\u003c/a\u003e; \u003ca href=\"ref://Hestenes90]\" target=\"_blank\"\u003eHestenes90]\u003c/a\u003e; \u003ca href=\"ref://Sidharth09\" target=\"_blank\"\u003eSidharth, 2009\u003c/a\u003e; \u003ca href=\"ref://RomanRosoPlaja03\" target=\"_blank\"\u003eRoman et al., 2003\u003c/a\u003e; \u003ca href=\"ref://BarutBracken81\" target=\"_blank\"\u003eBarut \u0026 Bracken, 1981\u003c/a\u003e; \u003ca href=\"ref://WangZhang01\" target=\"_blank\"\u003eWang \u0026 Zhang, 2001\u003c/a\u003e\u003c/li\u003e\u003c/ul\u003e\u003ch2 id=\"stochastic-particle-equations\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eStochastic particle equations\u003c/h2\u003e\u003cp id=\"text-35\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe basic behavior of a stochastic particle independent of any waves is described by \u003ca href=\"ref://Sciarretta18\" target=\"_blank\"\u003eSciarretta (2018)\u003c/a\u003e (in the 1D case; \u003ca href=\"ref://Sciarretta21\" target=\"_blank\"\u003eSciarretta, 2021\u003c/a\u003e extends to the 3D case with spin). In this non-relativistic model, the particle has an associated real-valued 3-component normalized (range -1..1) momentum vector \u003cspan class=\"math inline\"\u003e\\(\\vec{\\nu}\\)\u003c/span\u003e that drives a \u003cem\u003estable\u003c/em\u003e trajectory over time, despite the stochastic nature of each movement step. Any forces accumulate in this momentum vector, and it propagates with the particle. The motion of the particle is defined per dimension \u003cspan class=\"math inline\"\u003e\\(\\mu\\)\u003c/span\u003e (per the \u003ca href=\"four-vector\" target=\"_blank\"\u003efour-vector\u003c/a\u003e notation), in reference to an energy-like factor:\u003c/p\u003e\u003cp id=\"text-36\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003etodo: figure out if e should in fact be across all dims – seems like it should. just run an empirical sim.\u003c/p\u003e\u003cp id=\"text-37\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\ne_{\\mu} = \\frac{1 + \\nu_{\\mu}^2}{2} \n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-38\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe probability of moving in a positive or negative direction, or staying put, along a given dimension is given by:\u003c/p\u003e\u003cp id=\"text-39\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nP(+1) = \\frac{e_{\\mu} + \\nu_{\\mu}}{2}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-40\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nP(-1) = \\frac{e_{\\mu} - \\nu_{\\mu}}{2}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-41\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nP(0) = 1 - e_{\\mu}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-42\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe energy-like factor \u003cspan class=\"math inline\"\u003e\\(e\\)\u003c/span\u003e is .5 when the momentum \u003cspan class=\"math inline\"\u003e\\(\\nu=0\\)\u003c/span\u003e, and thus there is a .25 probability of going in either direction, and a .5 probability of staying in the same location. As momentum increases in say the positive direction, the probability of moving in the positive direction increases in a proportional manner, while the opposite direction decreases. The squaring of the energy factor ensures a division among the staying and move directions works out.\u003c/p\u003e\u003cp id=\"text-43\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIt would be useful to derive a relativistic version of these individual motion equations, where the effective mass, and thus momentum, increases with velocity. This is what the wave function naturally does.\u003c/p\u003e\u003cp id=\"text-44\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eKey points:\u003c/p\u003e\u003cul id=\"frame-45\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• key point about simple \u003ca href=\"harmonic-oscillator\" target=\"_blank\"\u003eharmonic oscillator\u003c/a\u003e (SHO): the height position \u003cem\u003eitself\u003c/em\u003e provides the acceleration force pulling back – this is even simpler than a wave! So there is a 4 vector of SHO that represent the momentum of the particle, and are coupled to the wave variables! The time-like one is the “heartbeat” of the particle, representing the rest mass, while the 3 spatial ones represent the velocity direction.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• each velocity axis (X,Y,Z) has a phase relative to the central time-like beat, and this phase represents the -1..+1 velocity value. This phase relationship is now demonstrated in \u003ca href=\"complex-kg\" target=\"_blank\"\u003ecomplex KG\u003c/a\u003e. The extreme nutrino-level particle would have the extreme case. There would have to be some kind of coordination across the 3 pairs, so the total norm could not exceed 1.\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• the resting state is all four oscillating in sync with 0 phase, and then forces act by boosting or lagging an axis wave relative to the central one. this could be pretty natural. the amount of boost needs to be dependent on existing phase to capture relativistic effects.\u003c/p\u003e\u003cp id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• the wave oscillation frequency is determined by the relativistic E^2 energy which goes up as momentum increases. This captures the key momentum / frequency relationship with QM. Never quite gets to CSq with massive particles.\u003c/p\u003e\u003cp id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• probability of jumping is then proportional to these values. can just implement that. it might emerge more naturally from some kind of phase offset dynamic, but that can be a later stage.\u003c/p\u003e\u003cp id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• coupling to surrounding wave is directly via drive from the central time-like oscillator, which provides the driving input to the field in its neighborhood.\u003c/p\u003e\u003cp id=\"text-6\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• An acceleration kick just bumps the phase? doesn’t capture the conservation dynamics among the SHOs.\u003c/p\u003e\u003c/ul\u003e\u003cp id=\"text-46\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWeyl wave couples spin with direction as a helical thing.\u003c/p\u003e\u003cp id=\"text-47\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThere is a literature on coupling of a stochastic particle with a “heat bath”, somewhat like the \u003ca href=\"zero-point\" target=\"_blank\"\u003ezero-point\u003c/a\u003e field, and trying to understand the aggregate behavior of such a system. \u003ca href=\"ref://DunkelHanggi05a\" target=\"_blank\"\u003eDunkel \u0026 Hänggi (2005)\u003c/a\u003e, \u003ca href=\"ref://DunkelHanggi05\" target=\"_blank\"\u003eDunkel \u0026 Hänggi, 2005\u003c/a\u003e provide a relatively accessible treatment, building on foundational work (\u003ca href=\"ref://Dudley65\" target=\"_blank\"\u003eDudley, 1965\u003c/a\u003e, \u003ca href=\"ref://Dudley73\" target=\"_blank\"\u003eDudley, 1973\u003c/a\u003e, \u003ca href=\"ref://GuerraRuggiero78\" target=\"_blank\"\u003eGuerra \u0026 Ruggiero, 1978\u003c/a\u003e, \u003ca href=\"ref://Nakagomi88\" target=\"_blank\"\u003eNakagomi, 1988\u003c/a\u003e). This all builds on Langevin equations, which are stochastic equations of motion, with connections to Ornstein-Ullenbach and Fokker-Planck etc. The specific restriction to heat bath dynamics vs. some kind of other intrinsic stochastic process is perhaps overly restrictive, but they nevertheless have a four-vector representation that seems to involve a conservation of energy between the time and momentum factors, which is really the essential calculus for the SHO model.\u003c/p\u003e\u003cp id=\"text-48\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOne key “no-go” finding from \u003ca href=\"ref://Dudley65\" target=\"_blank\"\u003eDudley (1965)\u003c/a\u003e is that a purely Markovian position-based system doesn’t capture particle motion – you \u003cem\u003eneed\u003c/em\u003e an additional momentum / velocity vector as part of the state. This is definitely key.\u003c/p\u003e\u003cul id=\"frame-49\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• compton wavelength as function of rest mass – does this fall out?\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003cspan class=\"math inline\"\u003e\\(\\nu\\)\u003c/span\u003e is already 0-c normalized – v/c\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• wave function trades energy against momentum – momentum is \u003cspan class=\"math inline\"\u003e\\(\\gamma m0 \\nu\\)\u003c/span\u003e, \u003cspan class=\"math inline\"\u003e\\(E^2 = p^2 + m0^2\\)\u003c/span\u003e so the 1 in above eq is like m0^2 – not clear where the \u003csup\u003e1\u003c/sup\u003e⁄\u003csub\u003e2\u003c/sub\u003e comes from but whatever.\u003c/p\u003e\u003cp id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• key idea that m0 is the internal motion of the particle rotating through spin, so need to just have that always going on as an “anchor”, and then there are these extra \u003cspan class=\"math inline\"\u003e\\(\\nu_{\\mu}\\)\u003c/span\u003e factors where the x^2 + y^2 + z^2 hypotenuse of the momentum-velocity is v^2 relative to c^2 – i.e., need to constrain total length to c^2.\u003c/p\u003e\u003cp id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• probabilistically when it stays still it rotates the internal spin. if it never stays still it never rotates the spin. the nutrino very rarely rotates the spin, but does sometimes. spin coupling couples the two helicies of the Weyl. need to go back to that. are there 2 neutrinos trapped inside one electron??\u003c/p\u003e\u003cp id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• todo:\u003c/p\u003e\u003c/ul\u003e\u003col id=\"frame-50\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e1. Ord GN, Schrödinger’s Equation and Classical Brownian Motion, Fortschr. Phys. 46, 6–8, 889–\n896 (1998).\u003c/li\u003e\u003cli id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e2. Janaswamy R, Transitional probabilities for the 4-state random walk on a lattice, J. Phys. A: Math.\nTheor. 41, 1–11 (2008).\u003c/li\u003e\u003cli id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e3. Badiali JP, Entropy, time-irreversibility and the Schrödinger equation in a primarily discrete\nspacetime, J. Phys. A: Math. Gen. 38(13), 2835–2848 (2005).\u003c/li\u003e\u003cli id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e4. Snyder HS, Quantized space-time, Phys. Rev. 71(1), 38-41 (1947).\u003c/li\u003e\u003cli id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e5. Finkelstein D, Saller H, and Tang Z, Quantum spacetime, in: P. Pronin, et al. (Eds.), Gravity,\nParticles and Space Time, World Scientific, Singapore (1996).\u003c/li\u003e\u003cli id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e6. Sidharth BG, The Thermodynamic Universe: Exploring the Limits of Physics, World Scientific,\nSingapore (2008).\u003c/li\u003e\u003c/ol\u003e\u003ch2 id=\"stochastic-kg\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eStochastic KG\u003c/h2\u003e\u003cp id=\"text-52\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis is a first-pass attempt at an integrated wave-particle system based on the complex KG system.\u003c/p\u003e\u003cul id=\"frame-53\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• The mass-drag term only exists at one discrete point, otherwise it is just standard wave equation.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• The probability of a motion jump to the next cell is computed using the standard current density equation:\u003c/p\u003e\u003c/ul\u003e\u003cdiv id=\"inline-container-55\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_kg-current\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_kg-current\"\u003e\u003cb\u003eEq 2:\u003c/b\u003e Klein-Gordon current density\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-56\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{J} \\equiv - \\frac{i \\hbar e}{2m_0} \\left( \\phi^* \\vec{\\nabla} \\phi - \\phi \\vec{\\nabla} \\phi^* \\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-57\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn terms of the underlying scalar state variables (and again for natural units), this is:\u003c/p\u003e\u003cp id=\"text-58\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\vec{J} = \\frac{e}{m_0} (\\varphi_a \\vec{\\nabla} \\varphi_b - \\varphi_b \\vec{\\nabla} \\varphi_a)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cul id=\"frame-59\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• The probability of staying in the same location is from the charge density:\u003c/li\u003e\u003c/ul\u003e\u003cdiv id=\"inline-container-61\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_kg-charge\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_kg-charge\"\u003e\u003cb\u003eEq 3:\u003c/b\u003e Klein-Gordon charge density\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-62\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\rho \\equiv \\frac{i \\hbar e}{2m_0c^2} \\left( \\phi^* \\frac{\\partial \\phi}{\\partial t} - \\phi \\frac{\\partial \\phi^*}{\\partial t} \\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-63\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\rho_i = \\frac{e}{m_0} ({\\varphi_b}_i \\dot {\\varphi_a}_i - {\\varphi_a}_i \\dot {\\varphi_b}_i)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cul id=\"frame-64\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• everything is normalized by the sum, and then the stochastic choice is made on these normalized probabilities.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• if staying put, then the energy equivalent of the mass-dependent factor:\u003c/p\u003e\u003c/ul\u003e\u003cp id=\"text-65\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{c^2 m_0^2}{\\hbar^2} \\varphi\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-66\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eis converted into an acceleration of the complex state variables. How exactly??\u003c/p\u003e\u003cul id=\"frame-67\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• if moving, this same quantity is used to drive the motion in the new cell that is chosen as a weighted function of the projection of the charge density vector onto the laplacian neighbor vectors.\u003c/li\u003e\u003c/ul\u003e\u003cp id=\"text-68\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eagain, the force drives the rotation of the complex state values but how exactly?\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"spin\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"m274-450 248 248-42 42-320-320 320-320 42 42-248 248h526v60H274Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003ePrevious\u003c/p\u003e\u003c/a\u003e\u003cstretch id=\"content-buttons-go-167\" style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1ch;min-height:1em;font-weight:thin;text-align:start\"\u003e\u003c/stretch\u003e\u003ca id=\"content-buttons-go-168\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"tools-vs-models\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M686-450H160v-60h526L438-758l42-42 320 320-320 320-42-42 248-248Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Tools vs models","URL":"tools-vs-models","Title":"Tools vs models","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{"eq":["eq_newton"]},"Description":"The differences between the standard Copenhagen interpretation vs. the \u003ca href=\"pilot-wave\" target=\"_blank\"\u003epilot-wave\u003c/a\u003e model reviewed in \u003ca href=\"history\" target=\"_blank\"\u003ehistory\u003c/a\u003e nicely exemplify the broader distinctions between \u003cstrong\u003ecalculational tools\u003c/strong\u003e vs. \u003cstrong\u003ephysical models\u003c/strong\u003e. A calculational tool is a mathematical framework that makes it easy to compute relevant results, but the underlying processes implied by the nature of the calculations have no direct mapping onto underlying “real” physical properties of the system.","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eTools vs models\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-37\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe differences between the standard Copenhagen interpretation vs. the \u003ca href=\"pilot-wave\" target=\"_blank\"\u003epilot-wave\u003c/a\u003e model reviewed in \u003ca href=\"history\" target=\"_blank\"\u003ehistory\u003c/a\u003e nicely exemplify the broader distinctions between \u003cstrong\u003ecalculational tools\u003c/strong\u003e vs. \u003cstrong\u003ephysical models\u003c/strong\u003e. A calculational tool is a mathematical framework that makes it easy to compute relevant results, but the underlying processes implied by the nature of the calculations have no direct mapping onto underlying “real” physical properties of the system.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFor example, Newton’s theory of gravitation (still widely used in practice) is a calculational tool that enables gravitational effects to be conveniently computed in terms of the respective masses (\u003cspan class=\"math inline\"\u003e\\(m_1\\)\u003c/span\u003e, \u003cspan class=\"math inline\"\u003e\\(m_2\\)\u003c/span\u003e) and distance \u003cem\u003er\u003c/em\u003e between the centers of mass of two bodies:\u003c/p\u003e\u003cdiv id=\"inline-container-3\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_newton\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_newton\"\u003e\u003cb\u003eEq 1:\u003c/b\u003e Newton's gravitation law\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nF = G \\frac{m_1 m_2}{r^2}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eBut this is not a plausible physical model because the math requires the use of the distances between relevant objects (typically many millions of miles in the usual astronomical applications), and their respective aggregate masses. Not only is this a (highly) non-local computation (“action at a distance”), Nature would presumably have to run this computation for all other bodies within some relevant radius, which could get a bit hairy.\u003c/p\u003e\u003cp id=\"text-6\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eBy contrast, a plausible physical model should compute gravitational forces directly from the collective effects of each individual atom within all the different celestial bodies in the universe, and propagate these forces via local mechanisms at the speed of light.\u003c/p\u003e\u003cp id=\"text-7\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIndeed, Einstein’s \u003cem\u003egeneral relativity\u003c/em\u003e provides exactly this kind of satisfying physical model, and it would be hard to find a serious physicist who did not recognize and appreciate these distinctions between the Newtonian and Einstein versions of gravitation. In particular, nobody would argue that nature actually implements the Newtonian model, while it is very easy to see how general relativity could happen “naturally”.\u003c/p\u003e\u003cp id=\"text-8\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn the case of Copenhagen vs. pilot-wave frameworks, both produce the same end results, and more generally, it is almost always the case that there are multiple different ways of framing a problem that end up producing the same results. From a purely pragmatic, “shut up and calculate” perspective, one should just pick the one that is simplest or otherwise most effective to use for a given situation, and be done with it. If all you care about is getting the right answer, why should you care about the particular “aesthetic” details of a given model? For this reason, the pilot-wave model is generally ignored by most working physicists, because it is more complicated than the standard framework.\u003c/p\u003e\u003cp id=\"text-9\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eHowever, problems arise when people mistake their calculational tools for physical models. Many (most?) people somehow regard the Copenhagen measurement and wave collapse processes as a real physical model, and thus get confused by how strikingly non-physical its core mechanisms are. Going further, the \u003cem\u003emany worlds\u003c/em\u003e interpretation (MWI, \u003ca href=\"ref://Everett57\" target=\"_blank\"\u003eEverett, 1957\u003c/a\u003e) attempts to avoid the measurement problem entirely by forking off \u003cem\u003ean entire new Universe\u003c/em\u003e at each possible point of measurement!\u003c/p\u003e\u003cp id=\"text-10\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAccording to surveys, a relatively large number of working physicists endorse this perspective (\u003ca href=\"ref://Tegmark10\" target=\"_blank\"\u003eTegmark, 2010\u003c/a\u003e)! While this idea seems absurd at face value from a physical plausibility perspective, it is entirely compatible with the mathematics of the standard framework, which ultimately requires a massively high-dimensional \u003ca href=\"configuration-space\" target=\"_blank\"\u003econfiguration space\u003c/a\u003e to describe the entire universe as a single quantum state, which in some sense contains all possible physical realities. MWI simply argues that this incomprehensibly large space always continues to update purely according to the unitary wave function, and never actually collapses. However, the framework is somewhat less clear about exactly corresponds to our subjective experience associated with the measurement process, where things really do seem to be in definite states.\u003c/p\u003e\u003cp id=\"text-11\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eInstead of regarding the standard Copenhagen framework as a plausible physical model, we can instead think of it as a purely calculational tool, in the same category as Newton’s gravitational framework, and thus not something that we need to try to interpret literally as how the physics actually works. The difficulty here is that we don’t yet have a plausible physical model playing the role of general relativity in the gravitational space (and indeed the conflict between quantum mechanics and general relativity is itself a major unresolved problem).\u003c/p\u003e\u003cp id=\"text-12\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThus, it is difficult for people to avoid thinking of the standard framework as a physical model, because they don’t have anything else available, and the implied physical properties of the standard framework (or MWI) are so radically non-physical that it is very difficult to even imagine that a physically-plausible model could exist at all. In effect, everyone’s conceptual understanding of quantum mechanics is so deeply shaped by the standard model that it is very difficult to break free of it and consider alternatives that would be more physically plausible.\u003c/p\u003e\u003cp id=\"text-13\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn this context, a somewhat smaller subset of physicists have continued to pursue the de Broglie-Bohm \u003ca href=\"pilot-wave\" target=\"_blank\"\u003epilot-wave\u003c/a\u003e model as perhaps the most promising way forward in developing a physically plausible model of the quantum world. Specifically, the pilot-wave framework shows that it is \u003cem\u003epossible\u003c/em\u003e to have a “truly unitary” physical model that does not have to (arbitrarily) switch between the complementary and seemingly incompatible modes of unitary wave propagation vs. measurement-induced collapse of the wave function (or somehow magically avoiding the latter in the MWI framework) (\u003ca href=\"ref://NorsenMarianOriols15\" target=\"_blank\"\u003eNorsen et al., 2015\u003c/a\u003e).\u003c/p\u003e\u003cp id=\"text-14\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis critical proof-of-concept that a dramatically different, and clearly more physically-plausible, conceptual interpretation is possible, opens the door for recognizing that the standard framework is just a calculational tool that works very well as such, but it need not strongly constrain an alternative pursuit of a physically plausible model.\u003c/p\u003e\u003cp id=\"text-15\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe pilot-wave model nevertheless does have a number of important remaining issues to be resolved, reviewed below, and several of these issues have likely impeded the broader engagement with this framework. We will see that one of the most pressing issues is that the pilot-wave model is also based on the same high-dimensional configuration space as the standard models, and it is clear that this mathematical construct is at the root of many of the remaining problems that need to be resolved in order to achieve a plausible physical model. Fortunately, some exciting recent progress has been made in moving beyond the configuration space framework, providing a critical window into what needs to be done to move past this mathematical barrier (\u003ca href=\"ref://NorsenMarianOriols15\" target=\"_blank\"\u003eNorsen et al., 2015\u003c/a\u003e; \u003ca href=\"ref://Norsen22\" target=\"_blank\"\u003eNorsen, 2022\u003c/a\u003e).\u003c/p\u003e\u003ch2 id=\"other-examples-of-calculational-tools\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eOther examples of calculational tools\u003c/h2\u003e\u003cp id=\"text-17\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eBefore continuing, it is useful to review some of the other clear examples of calculational tools vs. physical models in different domains of physics, which provide further elaboration of the importance of this distinction.\u003c/p\u003e\u003cp id=\"text-18\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFor example, there is a similar case as the gravitational example above, for Coulomb’s law for the strength of the electric field as a function of distances between charged particles, compared to the Lorenz gauge formulation of Maxwell’s EM equations.\u003c/p\u003e\u003cp id=\"text-19\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eCoulomb’s law is of the same form as Newton’s gravitational formula (and likewise represents a useful calculational tool), while the Lorenz gauge formulation of Maxwell’s equations describes purely local, light-speed wave propagation dynamics. Even within Maxwell’s wave equations, there is a Coulomb gauge version that implies immediate action-at-a-distance for the electrical potential, which is clearly incompatible with special relativity. It turns out that some non-localities in this framework actually enable the observed EM fields to propagate at the speed of light, but one can still get into trouble using this gauge incorrectly \u003ca href=\"ref://BrillGoodman67\" target=\"_blank\"\u003eBrill \u0026 Goodman, 1967\u003c/a\u003e; \u003ca href=\"ref://Jackson02\" target=\"_blank\"\u003eJackson, 2002\u003c/a\u003e; \u003ca href=\"ref://Onoochin01\" target=\"_blank\"\u003eOnoochin, 2001\u003c/a\u003e).\u003c/p\u003e\u003cp id=\"text-20\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn these and many other cases, people still use Newton’s gravitational equation instead of Einstein’s equations, and prefer the Coulomb gauge over the Lorenz gauge, because it makes the calculations simpler for relevant practical applications. But rarely do you find people being confused over which best describes the actual underlying physical processes involved in these domains. However, until Einstein came up with his gravitational framework, people did fret about the action-at-a-distance property of Newton’s laws, demonstrating that it is difficult to appreciate the calculational status of a given framework until a more plausible alternative is at hand.\u003c/p\u003e\u003cp id=\"text-21\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAs explored in \u003ca href=\"hilbert-space\" target=\"_blank\"\u003eHilbert space\u003c/a\u003e, there is a fairly deep connection between these Newtonian and Coulomb formulations, and the problematic configuration space representation in standard QM models. The configuration space represents the entire configuration of relevant physical parameters at a given moment in time, and is thus manifestly non-local, in the same way that these other classical laws are. The mathematical utility of having all the relevant variables represented in one set of equations at the same point in time is clear; it should also be clear that this is really a mathematical convenience, and not something that we must accept as a property of the physical world.\u003c/p\u003e\u003cp id=\"text-22\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn short, here’s a useful mantra: \u003cstrong\u003eDon’t confuse the math for the physics!\u003c/strong\u003e Math can represent anything, and there are many ways of solving the same problem. Nature presumably operates in only one specific way, which may not be the most convenient for solving our specific problems of interest.\u003c/p\u003e\u003ch2 id=\"the-seduction-of-elegant-models\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eThe seduction of elegant models\u003c/h2\u003e\u003cp id=\"text-24\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAn important corollary of the “don’t confuse the math for the physics” mantra is: \u003cstrong\u003edon’t be seduced by elegant mathematical models\u003c/strong\u003e. For example, there is an overarching framework based on \u003cem\u003esymmetry groups\u003c/em\u003e for organizing the elements of the standard model. Many people have tried to push this framework in various ways (e.g., “supersymmetry”) to make novel predictions, all of which have failed to date. This perhaps suggests that this elegant framework of symmetry is not really what is driving the structure of fundamental physics: it works as far as it goes, but it does not appear to be a truly \u003cem\u003egenerative\u003c/em\u003e principle that allows one to get deeper insight than the hard-won, empirically-based standard model.\u003c/p\u003e\u003cp id=\"text-25\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAn earlier iteration of the present attempt here was based on the idea of using only interacting wave equations, in large part because of the amazing ability of very simple and elegant wave equations to capture so many critical properties of particles, as especially evident in the \u003ca href=\"klein-gordon\" target=\"_blank\"\u003eKlein-Gordon\u003c/a\u003e equations, which are then inherited by the \u003ca href=\"dirac\" target=\"_blank\"\u003eDirac\u003c/a\u003e equations that accurately captures many properties of electrons. Thus, I was seduced by the elegance of these wave functions, without properly recognizing the fundamental limitations of waves for capturing the conservation and discrete localization properties of particles.\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"stochastic-particles\"\u003e\u003csvg id=\"icon\" 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style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eUncertainty principle\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-38\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe \u003cstrong\u003euncertainty principle\u003c/strong\u003e articulated by Werner Heisenberg is a basic property of any wave, under the \u003ca href=\"hamiltonian\" target=\"_blank\"\u003eHamiltonian\u003c/a\u003e interpretation of the wave properties in terms of \u003cem\u003emomentum\u003c/em\u003e and \u003cem\u003eposition\u003c/em\u003e operators.\u003c/p\u003e\u003cdiv id=\"figure_frequency\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_frequency\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 1:\u003c/b\u003e \u003cp\u003eRelationship between frequency and speed in the Klein-Gordon (KG) wave function, which derives from competition between the “mass drag” and the overall curvature of the wave. Higher frequency waves have more curvature and thus move faster.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cp id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eSpecifically the property of a wave function that corresponds to \u003cem\u003emomentum\u003c/em\u003e is proportional to the \u003cem\u003efrequency\u003c/em\u003e of the wave (\u003ca href=\"uncertainty-principle#figure_frequency\" target=\"_blank\"\u003eFigure 1\u003c/a\u003e), while more precisely encoding the position depends on having most of the wave amplitude concentrated into a narrow region of space.\u003c/p\u003e\u003cp id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThese two properties are \u003cstrong\u003ecomplementary\u003c/strong\u003e (\u003cstrong\u003ecanonically conjugate\u003c/strong\u003e), because frequency is only precisely well defined for an infinite sine wave, and becomes increasingly imprecise as the wave amplitude is concentrated into a narrow range of locations. But the converse is true for position, as noted above. So as you improve the quality of the momentum encoding in a wave, it necessarily degrades the position information, and vice-versa.\u003c/p\u003e\u003cp id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis can be expressed in the following inequality:\u003c/p\u003e\u003cdiv id=\"inline-container-7\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_uncertainty\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_uncertainty\"\u003e\u003cb\u003eEq 1:\u003c/b\u003e Uncertainty inequality\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-8\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\sigma_x \\sigma_p \\geq \\frac{\\hbar}{2}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-9\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWhere \u003cspan class=\"math inline\"\u003e\\(\\sigma_x\\)\u003c/span\u003e is the uncertainty (standard deviation) of position, and \u003cspan class=\"math inline\"\u003e\\(\\sigma_p\\)\u003c/span\u003e is the uncertainty in momentum, and the best-case scenario is that the are equal to the reduced Plank constant \u003cspan class=\"math inline\"\u003e\\(\\hbar\\)\u003c/span\u003e over 2. As one goes down, the other must go up to maintain this relationship.\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"tools-vs-models\"\u003e\u003csvg id=\"icon\" 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style=\"display:flex;flex-direction:row;justify-content:center;align-items:center;gap:0.5em;padding-top:8px;padding-right:16px;padding-bottom:8px;padding-left:16px;font-weight:thin;text-align:center;border-radius:8px\"\u003e\u003cp id=\"text\" style=\"font-weight:normal;line-height:1.5;text-align:center\"\u003eWave\u003c/p\u003e\u003c/select\u003e\u003cp id=\"label-Config • Size\" style=\"font-weight:normal;line-height:1.5;text-align:start\"\u003eConfig • Size\u003c/p\u003e\u003cform id=\"value-Config • Size-math32.Vector3i\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"label-X\" style=\"font-weight:normal;line-height:1.5;text-align:start\"\u003eX\u003c/p\u003e\u003cinput id=\"value-X-int32\" style=\"background:var(--surface-container-color);display:flex;flex-direction:row;justify-content:start;align-items:center;min-width:14ch;min-height:1.1em;max-width:22ch;padding-top:8px;padding-right:12px;padding-bottom:8px;padding-left:12px;font-weight:thin;line-height:1.4;text-align:start;border-radius:4px\" type=\"number\" value=\"80\"\u003e\u003c/input\u003e\u003cp id=\"label-Y\" style=\"font-weight:normal;line-height:1.5;text-align:start\"\u003eY\u003c/p\u003e\u003cinput id=\"value-Y-int32\" style=\"background:var(--surface-container-color);display:flex;flex-direction:row;justify-content:start;align-items:center;min-width:14ch;min-height:1.1em;max-width:22ch;padding-top:8px;padding-right:12px;padding-bottom:8px;padding-left:12px;font-weight:thin;line-height:1.4;text-align:start;border-radius:4px\" type=\"number\" value=\"1\"\u003e\u003c/input\u003e\u003cp id=\"label-Z\" style=\"font-weight:normal;line-height:1.5;text-align:start\"\u003eZ\u003c/p\u003e\u003cinput 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style=\"background:var(--surface-container-color);display:flex;flex-direction:row;justify-content:start;align-items:center;min-width:14ch;min-height:1.1em;max-width:22ch;padding-top:8px;padding-right:12px;padding-bottom:8px;padding-left:12px;font-weight:thin;line-height:1.4;text-align:start;border-radius:4px\" type=\"number\" value=\"8\"\u003e\u003c/input\u003e\u003cp id=\"label-Config • Velocity\" style=\"font-weight:normal;line-height:1.5;text-align:start\"\u003eConfig • Velocity\u003c/p\u003e\u003cform id=\"value-Config • Velocity-math32.Vector3\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"label-X\" style=\"font-weight:normal;line-height:1.5;text-align:start\"\u003eX\u003c/p\u003e\u003cinput id=\"value-X-float32\" style=\"background:var(--surface-container-color);display:flex;flex-direction:row;justify-content:start;align-items:center;min-width:14ch;min-height:1.1em;max-width:22ch;padding-top:8px;padding-right:12px;padding-bottom:8px;padding-left:12px;font-weight:thin;line-height:1.4;text-align:start;border-radius:4px\" type=\"number\" value=\"0.2\"\u003e\u003c/input\u003e\u003cp id=\"label-Y\" style=\"font-weight:normal;line-height:1.5;text-align:start\"\u003eY\u003c/p\u003e\u003cinput id=\"value-Y-float32\" style=\"background:var(--surface-container-color);display:flex;flex-direction:row;justify-content:start;align-items:center;min-width:14ch;min-height:1.1em;max-width:22ch;padding-top:8px;padding-right:12px;padding-bottom:8px;padding-left:12px;font-weight:thin;line-height:1.4;text-align:start;border-radius:4px\" type=\"number\" value=\"0\"\u003e\u003c/input\u003e\u003cp id=\"label-Z\" style=\"font-weight:normal;line-height:1.5;text-align:start\"\u003eZ\u003c/p\u003e\u003cinput id=\"value-Z-float32\" style=\"background:var(--surface-container-color);display:flex;flex-direction:row;justify-content:start;align-items:center;min-width:14ch;min-height:1.1em;max-width:22ch;padding-top:8px;padding-right:12px;padding-bottom:8px;padding-left:12px;font-weight:thin;line-height:1.4;text-align:start;border-radius:4px\" type=\"number\" value=\"0\"\u003e\u003c/input\u003e\u003c/form\u003e\u003cp id=\"label-Config • MaxSteps\" style=\"font-weight:normal;line-height:1.5;text-align:start\"\u003eConfig • MaxSteps\u003c/p\u003e\u003cinput id=\"value-Config • MaxSteps-int\" style=\"background:var(--surface-container-color);display:flex;flex-direction:row;justify-content:start;align-items:center;min-width:14ch;min-height:1.1em;max-width:22ch;padding-top:8px;padding-right:12px;padding-bottom:8px;padding-left:12px;font-weight:thin;line-height:1.4;text-align:start;border-radius:4px\" type=\"number\" value=\"100000\"\u003e\u003c/input\u003e\u003cp id=\"label-Units\" 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style=\"display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:24px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\"\u003e\u003cp id=\"text\" style=\"font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eState\u003c/p\u003e\u003cstretch id=\"indicator-stretch\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:0.2em;min-height:0.2em;width:0.2em;height:0.2em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/stretch\u003e\u003csvg id=\"indicator\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:18px;min-height:18px;width:18px;height:18px;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M480-344 240-584l43-43 197 197 197-197 43 43-240 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style=\"background:#C5C6CF;display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1px;margin:6px;font-weight:thin;text-align:start\"\u003e\u003c/hr\u003e\u003chr id=\"wavesim-toolbars-go-82\" style=\"background:#C5C6CF;display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1px;margin:6px;font-weight:thin;text-align:start\"\u003e\u003c/hr\u003e\u003cinput id=\"minSwitch\" style=\"display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:4px;padding-right:4px;padding-bottom:4px;padding-left:4px;font-weight:thin;text-align:center;border-radius:8px\" type=\"checkbox\" value=\"true\"\u003e\u003cdiv id=\"space\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:0.1ch;min-height:1em;width:0.1ch;height:1em;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"font-weight:normal;line-height:1.5;text-align:center\"\u003eMin\u003c/p\u003e\u003c/input\u003e\u003cinput id=\"minSpin\" style=\"background:var(--surface-container-color);display:flex;flex-direction:row;justify-content:start;align-items:center;min-width:15ch;min-height:1.1em;max-width:15ch;padding-top:8px;padding-right:12px;padding-bottom:8px;padding-left:12px;font-weight:thin;line-height:1.4;text-align:start;border-radius:4px\" type=\"number\" value=\"-0.8\"\u003e\u003c/input\u003e\u003ccolor-map-button id=\"cmap\" style=\"background:#00FFFF;display:flex;flex-direction:row;flex-grow:1;justify-content:center;align-items:center;min-width:10em;min-height:1.2em;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\"\u003e\u003c/color-map-button\u003e\u003cinput id=\"maxSwitch\" style=\"display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:4px;padding-right:4px;padding-bottom:4px;padding-left:4px;font-weight:thin;text-align:center;border-radius:8px\" type=\"checkbox\" value=\"true\"\u003e\u003cdiv id=\"space\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:0.1ch;min-height:1em;width:0.1ch;height:1em;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"font-weight:normal;line-height:1.5;text-align:center\"\u003eMax\u003c/p\u003e\u003c/input\u003e\u003cinput id=\"maxSpin\" style=\"background:var(--surface-container-color);display:flex;flex-direction:row;justify-content:start;align-items:center;min-width:15ch;min-height:1.1em;max-width:15ch;padding-top:8px;padding-right:12px;padding-bottom:8px;padding-left:12px;font-weight:thin;line-height:1.4;text-align:start;border-radius:4px\" type=\"number\" value=\"0.8\"\u003e\u003c/input\u003e\u003cinput id=\"zeroCtrSwitch\" style=\"display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:4px;padding-right:4px;padding-bottom:4px;padding-left:4px;font-weight:thin;text-align:center;border-radius:8px\" type=\"checkbox\" value=\"true\"\u003e\u003cdiv id=\"space\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:0.1ch;min-height:1em;width:0.1ch;height:1em;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"font-weight:normal;line-height:1.5;text-align:center\"\u003eZeroCtr\u003c/p\u003e\u003c/input\u003e\u003cbutton id=\"overflow-menu\" style=\"display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:16px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\"\u003e\u003csvg id=\"icon\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M479.858-160Q460-160 446-174.142q-14-14.141-14-34Q432-228 446.142-242q14.141-14 34-14Q500-256 514-241.858q14 14.141 14 34Q528-188 513.858-174q-14.141 14-34 14Zm0-272Q460-432 446-446.142q-14-14.141-14-34Q432-500 446.142-514q14.141-14 34-14Q500-528 514-513.858q14 14.141 14 34Q528-460 513.858-446q-14.141 14-34 14Zm0-272Q460-704 446-718.142q-14-14.141-14-34Q432-772 446.142-786q14.141-14 34-14Q500-800 514-785.858q14 14.141 14 34Q528-732 513.858-718q-14.141 14-34 14Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003c/button\u003e\u003c/toolbar\u003e\u003cdiv id=\"midframe\" style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"vars\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:10em;font-weight:thin;text-align:start\"\u003e\u003cinput id=\"curprv\" style=\"display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:4px;padding-right:4px;padding-bottom:4px;padding-left:4px;font-weight:thin;text-align:center;border-radius:8px\" type=\"checkbox\" value=\"true\"\u003e\u003cdiv id=\"space\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:0.1ch;min-height:1em;width:0.1ch;height:1em;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"font-weight:normal;line-height:1.5;text-align:center\"\u003eCurrent\u003c/p\u003e\u003c/input\u003e\u003cbutton id=\"Pos\" style=\"color:#0B1B36;background:#D8E2FF;display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:24px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\"\u003e\u003cp id=\"text\" style=\"color:#0B1B36;font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003ePos\u003c/p\u003e\u003c/button\u003e\u003cbutton id=\"Vel\" style=\"display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:24px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\"\u003e\u003cp id=\"text\" style=\"font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eVel\u003c/p\u003e\u003c/button\u003e\u003cbutton id=\"Force\" 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style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:512px;min-height:384px;font-weight:thin;text-align:start\"\u003e\u003c/plot\u003e\u003c/editor\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"frame-3\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis simulation runs the 1D wave equation starting with a moving wave packet initial state.\u003c/p\u003e\u003c/div\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cstretch id=\"content-buttons-go-167\" style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1ch;min-height:1em;font-weight:thin;text-align:start\"\u003e\u003c/stretch\u003e\u003ca id=\"content-buttons-go-168\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"waves-simulator\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M686-450H160v-60h526L438-758l42-42 320 320-320 320-42-42 248-248Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Waves simulator","URL":"waves-simulator","Title":"Waves simulator","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Simulations"],"Specials":{},"Description":"The \u003cstrong\u003ewaves\u003c/strong\u003e simulator is an open source wave and physics simulation system. See the \u003ca href=\"https://github.com/WaveReality/waves\" target=\"_blank\"\u003egithub repository\u003c/a\u003e for the source code and implementational details. The software is written in \u003ca href=\"https://go.dev/\" target=\"_blank\"\u003eGo\u003c/a\u003e, and runs on \u003ca href=\"https://webgpu.org/\" target=\"_blank\"\u003eWebGPU\u003c/a\u003e both on the web browser and locally using native libraries, so the web equations even for large state spaces can run at high speed.","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eWaves simulator\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-40\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe \u003cstrong\u003ewaves\u003c/strong\u003e simulator is an open source wave and physics simulation system. See the \u003ca href=\"https://github.com/WaveReality/waves\" target=\"_blank\"\u003egithub repository\u003c/a\u003e for the source code and implementational details. The software is written in \u003ca href=\"https://go.dev/\" target=\"_blank\"\u003eGo\u003c/a\u003e, and runs on \u003ca href=\"https://webgpu.org/\" target=\"_blank\"\u003eWebGPU\u003c/a\u003e both on the web browser and locally using native libraries, so the web equations even for large state spaces can run at high speed.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe Simulations page shows all pages that include running versions of this software, which is integrated with the framework that displays this content, all of which operates under the \u003ca href=\"https://cogentcore.org\" target=\"_blank\"\u003eCogent Core\u003c/a\u003e framework.\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"waves-simulation\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"m274-450 248 248-42 42-320-320 320-320 42 42-248 248h526v60H274Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003ePrevious\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Waves","URL":"waves","Title":"Waves","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{"eq":["eq_force","eq_a","eq_","eq_","eq_speed","eq_m","eq_acc","eq_vel","eq_vel","eq_kinetic","eq_potential","eq_total","eq_midpoint","eq_ke","eq_spatial","eq_temporal","eq_x-partial","eq_t-partial","eq_x-2","eq_x-2-2","eq_r-f","eq_acc","eq_acc-2","eq_wave","eq_3D","eq_laplacian","eq_grad","eq_div","eq_wave-2","eq_faces","eq_faces-2","eq_all26","eq_wt","eq_pe"],"figure":["figure_terms","figure_eq","figure_pendulum","figure_superposition","figure_integer","figure_derivative","figure_derivative-1d","figure_laplacian-1d","figure_gradient","figure_cubes","figure_scaling"]},"Description":"Waves are the foundation of the wave electrodynamics (WELD) model, so we start here by exploring the basic properties of the simplest kind of waves, produced by a \u003cstrong\u003esecond-order wave equation\u003c/strong\u003e. This is an idealization of the kinds of waves that are familiar to you in everyday life: water waves, sound waves, guitar strings and drum surfaces vibrating, etc. There are a few key physical properties of these systems that produce wave behavior:","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eWaves\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-41\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"figure_terms\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_terms\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 1:\u003c/b\u003e \u003cp\u003eTerminology for wave properties: wavelength is the distance between repeating elements of the wave, amplitude is the height of the wave, and frequency (not shown) is how many oscillations of the wave take place per unit time.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWaves are the foundation of the wave electrodynamics (WELD) model, so we start here by exploring the basic properties of the simplest kind of waves, produced by a \u003cstrong\u003esecond-order wave equation\u003c/strong\u003e. This is an idealization of the kinds of waves that are familiar to you in everyday life: water waves, sound waves, guitar strings and drum surfaces vibrating, etc. There are a few key physical properties of these systems that produce wave behavior:\u003c/p\u003e\u003cul id=\"frame-3\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• Stuff in one location can move in some way relative to stuff in neighboring locations, creating a \u003cstrong\u003edisturbance\u003c/strong\u003e: water molecules, guitar string, and drum skin can move up and down.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• When stuff in one location does move relative to its neighbors, it experiences a \u003cstrong\u003erestoring force\u003c/strong\u003e pulling it back in line with the neighbors: the guitar string or drum skin stretches from the disturbance, and pulls back against this stretch.\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• The stuff also has \u003cstrong\u003einertia\u003c/strong\u003e that keeps it moving \u003cem\u003epast\u003c/em\u003e the point of equality with the neighbors, creating a new disturbance in the opposite direction, which is then subject to a new restoring force, causing the process to repeat again.\u003c/p\u003e\u003c/ul\u003e\u003cp id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWe can capture these properties in a simple linear equation, which constitutes the standard second-order wave equation. It is “second order” in order to capture the inertia property.\u003c/p\u003e\u003cp id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAside from looking completely mesmerizing, these simple waves can exhibit some important properties. At the most basic level, we can measure things like \u003cstrong\u003efrequency\u003c/strong\u003e, \u003cstrong\u003ewavelength\u003c/strong\u003e, and \u003cstrong\u003ephase\u003c/strong\u003e of the wave vibrations (\u003ca href=\"waves#figure_terms\" target=\"_blank\"\u003eFigure 1\u003c/a\u003e). And critically for understanding quantum mechanics, waves exhibit the amazing property of \u003cstrong\u003esuperposition\u003c/strong\u003e (Figure 2) — two different waves can pass right through each other and come out the other side unscathed, due to the linearity of the wave equation. This ability to encode many different things all added up together into one complex wave disturbance is leveraged in QM to capture a combination of different uncertain possibilities all wrapped up in one inscrutable package.\u003c/p\u003e\u003ch2 id=\"the-wave-equation-newtonian-version\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eThe Wave Equation: Newtonian Version\u003c/h2\u003e\u003cdiv id=\"figure_eq\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_eq\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 2:\u003c/b\u003e \u003cp\u003eKey elements of the basic wave equation, in a discretized space and time system, where each discrete location (across the horizontal axis, indexed by coordinate “x”) has a discrete state value “y” at a given time step “t”, indicated as \u003cspan class=\"math inline\"\u003e\\(y^t_x\\)\u003c/span\u003e. The restoring force \u003cem\u003ef\u003c/em\u003e pulling back on a given point is the total disturbance of that point relative to its two neighbors on either side, which is the sum of the differences between the state value at that point compared to the each of the two neighbors (\u003cspan class=\"math inline\"\u003e\\(f = f_l + f_r\\)\u003c/span\u003e). This force \u003cem\u003ef\u003c/em\u003e creates an acceleration \u003cspan class=\"math inline\"\u003e\\(a=f/m\\)\u003c/span\u003e, which in turn updates the velocity, which in turn drives a change in the state value for the next time step. That’s all there is to it. This system has a built-in inertia (due to the basic Newtonian physics of force, acceleration, and velocity), so it will end up overshooting the average of its neighbors (who meanwhile are on the move themselves). This all creates the fascinating wave dynamics.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cp id=\"text-9\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWe start by deriving the second order wave equation from basic Newtonian physical principles, in a simple one-dimensional case with discrete uniform cells each having a “state value” that represents the “stuff” that moves around in the wave (e.g., the height of water) (\u003ca href=\"waves#figure_eq\" target=\"_blank\"\u003eFigure 2\u003c/a\u003e). Time is also discretized, with everything computed in discrete time steps, consistent with the \u003ca href=\"cellular-automaton\" target=\"_blank\"\u003ecellular automaton\u003c/a\u003e (CA) framework. In the notation of the figure, we label the location of each cell using the index \u003cem\u003ex\u003c/em\u003e, and each time step with the index \u003cem\u003et\u003c/em\u003e, and the current state value of a given cell as \u003cspan class=\"math inline\"\u003e\\(y^t_x\\)\u003c/span\u003e. The restoring force is proportional to the difference in state values between a given point and its two neighbors on either side:\u003c/p\u003e\u003cdiv id=\"inline-container-11\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_force\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_force\"\u003e\u003cb\u003eEq 1:\u003c/b\u003e restoring force\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-12\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nf = f_l + f_r = \\left( y^t_{x-1} - y^t_x \\right) + \\left( y^t_{x+1} - y^t_x \\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-13\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nf = \\left( y^t_{x-1} + y^t_{x+1} \\right) - 2 y^t_x\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-14\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eUsing the standard equations of Newtonian physics, e.g., \u003cspan class=\"math inline\"\u003e\\(f=ma\\)\u003c/span\u003e, this force then drives an acceleration \u003cem\u003ea\u003c/em\u003e:\u003c/p\u003e\u003cdiv id=\"inline-container-16\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_a\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_a\"\u003e\u003cb\u003eEq 2:\u003c/b\u003e acceleration\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-17\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\na = \\frac{f}{m}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-18\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn the discrete time framework, we can simply increment a new \u003cem\u003evelocity\u003c/em\u003e term \u003cem\u003ev\u003c/em\u003e by this acceleration:\u003c/p\u003e\u003cdiv id=\"inline-container-20\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_\"\u003e\u003cb\u003eEq 3:\u003c/b\u003e new velocity\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-21\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nv^{t+1}_x = v^t_x + a^t_x\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-22\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhere the time index for this new velocity is given by \u003cem\u003et+1\u003c/em\u003e, which is the next step in time, to distinguish it from the current velocity value, designated at time \u003cem\u003et\u003c/em\u003e. One could just as well refer to \u003cem\u003et\u003c/em\u003e as the new value and \u003cem\u003et-1\u003c/em\u003e as the previous value — it is just a matter of taste or convention as to what is used. The notation used here emphasizes the process of computing a new value by putting it into the future relative to the present values that drive the calculation of this new value. We also provide the full time and space indices of the acceleration just for clarity.\u003c/p\u003e\u003cp id=\"text-23\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eNext, this new velocity is used to update the state value, completing a given time step of updating for a given location:\u003c/p\u003e\u003cdiv id=\"inline-container-25\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_\"\u003e\u003cb\u003eEq 3:\u003c/b\u003e new state\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-26\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\ny^{t+1}_x = y^t_x + v^{t+1}_x\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-27\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn the \u003cstrong\u003esynchronous update\u003c/strong\u003e mode used in the CA framework, each state location is updated at the same time (synchronously), and the new states then all become the current states for the next iteration, and so on. In a computer program, one typically has to actually iterate sequentially through all the states one at a time, and the “current” and “next” states must both be maintained for each location to achieve the proper synchronous computation without order of update effects.\u003c/p\u003e\u003cp id=\"text-28\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn simplified computer programming notation, using semantically-labeled temporary variables, the wave equation is:\u003c/p\u003e\u003cdiv id=\"frame-29\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ceditor id=\"editor-0\" style=\"background:var(--surface-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:10em;padding-top:0.5em;padding-right:0.5em;padding-bottom:0.5em;padding-left:0.5em;font-weight:thin;line-height:1.3;text-align:start;border-radius:16px\"\u003e\u003c/editor\u003e\u003c/div\u003e\u003cp id=\"text-30\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIt is important to be clear about the so-called \u003cstrong\u003eunique degrees of freedom\u003c/strong\u003e of this system at each point in space: what is the truly minimal number of variables one would need to maintain at each point in space? It is \u003cem\u003etwo\u003c/em\u003e for this system: either the current state and velocity, or the current and prior state values (from which velocity can be computed as in the above example) must be stored. A given computer implementation may store more than this minimal number to manage the synchronous updating, and for display and analysis purposes, etc. Because there are two unique degrees of freedom per point, a full specification of the initial state of the system requires specifying these two values for each state.\u003c/p\u003e\u003ch2 id=\"mass-and-speed-of-light\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eMass and Speed of Light\u003c/h2\u003e\u003cp id=\"text-32\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe mass term showing up in the basic wave equation arises because we need to translate the restoring force into an acceleration, and the basic laws of physics dictate that mass is what mediates this translation: informally, it is how much resistance or inertia the system has in the face of forces acting upon it. Intuitively, it makes sense that this mass would determine how fast waves will propagate in the system: with a higher mass, waves will move more slowly because they will resist the neighborhood forces more strongly. Indeed, it turns out that the speed of wave propagation is inversely proportional to this mass, but with a squared term:\u003c/p\u003e\u003cdiv id=\"inline-container-34\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_speed\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_speed\"\u003e\u003cb\u003eEq 5:\u003c/b\u003e speed of wave propagation, squared\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-35\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nc^2 = \\frac{1}{m}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cdiv id=\"inline-container-37\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_m\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_m\"\u003e\u003cb\u003eEq 6:\u003c/b\u003e mass in terms of c squared\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-38\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nm = \\frac{1}{c^2}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-39\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhere we are using \u003cem\u003ec\u003c/em\u003e to represent the \u003cstrong\u003espeed of light\u003c/strong\u003e in the system. Waves using this simple second-order wave equation always travel at the same speed, and this speed is effectively the speed of light if these waves are electromagnetic waves (which we’ll see in the next chapter can indeed be computed using this simple wave equation).\u003c/p\u003e\u003cp id=\"text-40\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThus, we can re-write our basic equations using this speed-of-light factor, and while we’re at it, we’ll also introduce the “dot” and “double-dot” notation for the velocity and acceleration terms, reflecting the fact that the velocity is the first temporal derivative (one dot), and acceleration is the second temporal derivative (two dots). As we discuss in greater detail in the next section, the velocity is the slope or rate of change in the state value, while acceleration is the slope or rate of change of the velocity — a double-slope or double-rate-of-change of the state value.\u003c/p\u003e\u003cdiv id=\"inline-container-42\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_acc\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_acc\"\u003e\u003cb\u003eEq 7:\u003c/b\u003e acceleration\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-43\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\na^t_x = \\ddot y^t_x = c^2 \\left( y^t_{x-1} + y^t_{x+1} - 2 y^t_x \\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cdiv id=\"inline-container-45\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_vel\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_vel\"\u003e\u003cb\u003eEq 8:\u003c/b\u003e velocity\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-46\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nv^{t+1}_x = \\dot y^{t+1}_x = \\dot y^t_x + \\ddot y^t_x\n\\]\u003c/span\u003e\u003c/p\u003e\u003cdiv id=\"inline-container-48\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_vel\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_vel\"\u003e\u003cb\u003eEq 8:\u003c/b\u003e state\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-49\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\ny^{t+1}_x = y^t_x + \\dot y^{t+1}_x\n\\]\u003c/span\u003e\u003c/p\u003e\u003ch2 id=\"energy\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eEnergy\u003c/h2\u003e\u003cp id=\"text-51\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAs we progress to more complex wave equations, the concept of the energy associated with the wave will become critical. Energy must be strictly conserved over time, or otherwise the universe quickly gets out of balance — exploding with too much energy or fading away into nothingness. Much of physics involves accounting for where all the energy is and how it gets transformed into different forms over time.\u003c/p\u003e\u003cp id=\"text-52\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eEinstein’s famous equation \u003cspan class=\"math inline\"\u003e\\(E=mc^2\\)\u003c/span\u003e, which we saw some hint of above, is so important because it shows that there is energy trapped inside of matter, which can be liberated in various ways (e.g., nuclear bombs, or solar fusion reactions), which release great quantities of energy. This very fact of the ability to convert energy into different forms like this is suggestive of some kind of underlying common currency where matter and energy are fundamentally the same thing — this is what the wave framework provides. Indeed, we will derive our matter wave equation by using Einstein’s equation, showing exactly how matter and other forms of energy are all bound up together in single wave equation, whose undulations over time result in the constant conversion of energy into different forms.\u003c/p\u003e\u003cdiv id=\"figure_pendulum\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_pendulum\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 3:\u003c/b\u003e \u003cp\u003eTwo forms of energy in a simple harmonic oscillator such as a pendulum: kinetic energy from the speed (velocity) of motion, and potential energy from the displacement relative to the resting position (in the pendulum case, this is gravitational energy that is then transformed into kinetic energy as the ball drops). Each cell in our discrete wave system is a simple harmonic oscillator with both of these forms of energy — potential energy is a function of the difference between the neighbor state values — the same thing that drives the restoring force.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cp id=\"text-55\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eTo ground our understanding of energy right at the start, we derive the energy equation for the simplest form of wave equation, given above. To figure out the correct equation, we start with an analogy to the \u003cstrong\u003esimple harmonic oscillator\u003c/strong\u003e exemplified by a swinging \u003cstrong\u003ependulum\u003c/strong\u003e (\u003ca href=\"waves#figure_pendulum\" target=\"_blank\"\u003eFigure 3\u003c/a\u003e). As a pendulum swings, there is a continuous transformation between \u003cstrong\u003ekinetic energy\u003c/strong\u003e, which is a function of the speed of the swinging bob, and \u003cstrong\u003epotential energy\u003c/strong\u003e, which is a function of the height of the bob above its minimal height, which has potential energy because gravitational forces will push it down to its minimal height, producing kinetic energy in the process.\u003c/p\u003e\u003cp id=\"text-56\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eEach cell in our discrete wave equation acts just like this simple harmonic oscillator. For example as a sine wave passes through a given point, the state at that point experiences a trajectory that is identical to the simple harmonic oscillator: it moves at different velocities going from 0 at the extreme top or bottom of the wave, to the fastest right in the middle of the wave disturbance. Thus, we can compute the kinetic and potential energy for a given point along the wave as simply the sum of these two energy components. Instead of the gravitational force, the potential energy arises from the restoring force, which depends on the local curvature of the wave.\u003c/p\u003e\u003cp id=\"text-57\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWe can start with the kinetic energy, as it is the most straightforward, having a standard equation in Newtonian physics:\u003c/p\u003e\u003cdiv id=\"inline-container-59\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_kinetic\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_kinetic\"\u003e\u003cb\u003eEq 10:\u003c/b\u003e kinetic energy\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-60\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nE_k = \\frac{1}{2} m v^2 = \\frac{1}{2 c^2} v^2\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-61\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ein the second version we replace the mass with the speed of light squared per the above conversion equation.\u003c/p\u003e\u003cp id=\"text-62\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe potential energy turns out to be a function of the squared differences between neighboring state values:\u003c/p\u003e\u003cdiv id=\"inline-container-64\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_potential\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_potential\"\u003e\u003cb\u003eEq 11:\u003c/b\u003e potential energy\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-65\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nE_p = \\frac{1}{2} \\left( \\left( y^t_{x-1} - y^t_x \\right)^2 + \\left( y^t_{x+1} - y^t_x \\right)^2 \\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-66\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn general, energy values are squared, because they tend to be positive numbers, and they almost always have this \u003csup\u003e1\u003c/sup\u003e⁄\u003csub\u003e2\u003c/sub\u003e multiplier. It is interesting that each neighborhood difference (to the left and right) is squared separately and then added together, instead of squaring the sum of these differences — this makes sense because the energy is really in each of these differences, and even if they happened to cancel out across the left and right sides (e.g., the left neighbor was as low as the right neighbor is high relative to the cell’s state value), there is a lot of energy latent in those two differences, compared to a case where left and right were actually each the same value as the cell’s current state.\u003c/p\u003e\u003cp id=\"text-67\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe total energy, which should be a constant over time, is then just the sum of these two forms of energy:\u003c/p\u003e\u003cdiv id=\"inline-container-69\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_total\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_total\"\u003e\u003cb\u003eEq 12:\u003c/b\u003e total energy\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-70\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nE_t = E_k + E_p\n\\]\u003c/span\u003e\u003c/p\u003e\u003ch2 id=\"discrete-time-adjustment\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eDiscrete Time Adjustment\u003c/h2\u003e\u003cp id=\"text-72\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThere is a further wrinkle to the energy calculation in the discrete space and time framework, having to do with the exact point at which a given velocity and state values are both valid for capturing the total energy of the system. It turns out that you have to interpolate the velocity value half-way between its current and new values to get a more accurate energy measure:\u003c/p\u003e\u003cdiv id=\"inline-container-74\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_midpoint\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_midpoint\"\u003e\u003cb\u003eEq 13:\u003c/b\u003e midpoint velocity\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-75\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nv_{mid} = \\frac{1}{2} \\left( v^t + v^{t+1} \\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cdiv id=\"inline-container-77\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_ke\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_ke\"\u003e\u003cb\u003eEq 14:\u003c/b\u003e kinetic energy\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-78\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nE_k = \\frac{1}{2 c^2} v_{mid}^2\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-79\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIntuitively, the current velocity by itself doesn’t coincide with the current state values, because it was really driven by the prior state values, so it needs to be updated with some influence of the velocity that was driven by the current state values (i.e., the new velocity) — but this new velocity by itself goes too far, and thus the half-way point works out best.\u003c/p\u003e\u003ch2 id=\"lack-of-strict-conservation\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eLack of Strict Conservation\u003c/h2\u003e\u003cp id=\"text-81\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWhen total energy is computed in the above way across all cells, it often does \u003cem\u003enot\u003c/em\u003e remain strictly constant over time, and instead varies in ways that reflect the wave dynamics operating within the system. Over time, the \u003cem\u003eaverage\u003c/em\u003e value converges on a constant value, and it is clear that there is no risk of the system exploding or dying out, but from one time step to the next, there can actually be a fairly substantial variability in the total energy computed. You will observe this first-hand in the following exploration.\u003c/p\u003e\u003cp id=\"text-82\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003etodo: potential implications for gravitation, etc?\u003c/p\u003e\u003ch2 id=\"exploration-of-1d-waves\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eExploration of 1D Waves\u003c/h2\u003e\u003cdiv id=\"figure_superposition\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_superposition\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 4:\u003c/b\u003e \u003cp\u003eSuperposition of two wave packets, shown initially separate moving toward each other in the first (left) panel, then in a state of superposition where it is hard to imagine that two complete and separate wave packets could be latent within such a wave pattern (middle panel), and finally the two waves re-emerge fully intact after passing right through each other. This mind-boggling property of linear wave equations plays a crucial role in standard quantum mechanics (QM), and also represents a major limitation of QM, in that it eliminates any way for such wave packets to interact with each other.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"figure_integer\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_integer\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 5:\u003c/b\u003e \u003cp\u003eA wave within fixed bounds, having an integer number of wavelengths within the fixed width. This “bound state” wave provides a basic model of the quantized atomic system — the quantization comes entirely from the fact that the only stable wave configuration within such boundaries must have an integral number of wavelengths. This is the quantum in quantum mechanics.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cp id=\"text-88\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAt this point, it is useful to use the \u003ca href=\"waves-simulator\" target=\"_blank\"\u003ewaves simulator\u003c/a\u003e to see how the above equations give rise to wave behavior, so you have a solid understanding of that, along with some basic wave behavior, before continuing. Follow the directions given in the \u003ca href=\"waves-simulation\" target=\"_blank\"\u003ewaves simulation\u003c/a\u003e, which gets you familiar with the basic use of the simulator and observing the resulting wave behavior.\u003c/p\u003e\u003cp id=\"text-89\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOne of the most fascinating and amazing properties that you’ll explore in this simulation is \u003cstrong\u003esuperposition\u003c/strong\u003e, as shown in \u003ca href=\"waves#figure_superposition\" target=\"_blank\"\u003eFigure 4\u003c/a\u003e — it is even more compelling seeing it happen in full motion in the simulation.\u003c/p\u003e\u003cp id=\"text-90\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe absolutely critical point about superposition is this: \u003cstrong\u003elinear superposition means that the wave packets have no possible way of interacting with each other — they just pass right through each other like ghosts\u003c/strong\u003e — the entire edifice of standard quantum mechanics, being based on the linear Schrödinger wave equation, is thus fundamentally incapable of capturing any form of possible interaction among wave packets — all interaction must be mediated either through an external potential, or more typically, by putting the waves into high-dimensional spaces where the interaction becomes an unfolding dynamic of wave propagation across these different dimensions.\u003c/p\u003e\u003cp id=\"text-91\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eAnother critical property of waves is that there must be an integer (discrete, quantized) number of wavelengths within a fixed-width boundary for the system to be stable (Figure 5). You’ll see this in the exploration by comparing the behavior of the system when this condition is true and when it is not true. This constraint is the \u003cem\u003eonly\u003c/em\u003e source of quantization in quantum mechanics, according to the WELD framework. Everything is actually fully continuous waves, but the atomic nucleus creates a potential well that traps electron waves, and these trapped electron waves thus must have only integer (quantized) wavelengths. This quantization of wavelength creates a corresponding quantization of the energy of the electron, giving rise to the discrete spectrum that Planck initially conceived of in his 1901 paper that started quantum mechanics. Critically, this discretization of wavelength makes the atomic system stable: it will radiate away energy until the electron wavelength achieves a stable quantized fit within the atomic system, and at this point, it will go no further because any perturbation away from this perfect quantized wavelength will be unstable, and push it back toward the stable quantized state.\u003c/p\u003e\u003ch2 id=\"second-order-derivative-formulation-and-the-laplacian\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eSecond-order Derivative Formulation and the Laplacian\u003c/h2\u003e\u003cdiv id=\"figure_derivative\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_derivative\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 6:\u003c/b\u003e \u003cp\u003eA derivative is just the slope of a function at a given point. This figure shows the slopes at three different points on the function. It is computed in discrete space and time in terms of the change in y divided by the change in x.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"figure_derivative-1d\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_derivative-1d\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 7:\u003c/b\u003e \u003cp\u003eDiscrete space derivatives for our wave equation, computed as difference in y values divided by space between “cells”.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"figure_laplacian-1d\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_laplacian-1d\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 8:\u003c/b\u003e \u003cp\u003eThe second-order spatial derivative, which is the difference of the first-order derivatives around the central point, divided as we must do in a derivative by the distance between points. This is the source of the restoring force in the wave equation, which should be familiar from the Newtonian version of the wave equation as shown in Figure 2. When this equation is properly generalized to multiple spatial dimensions, it is called the \u003cem\u003eLaplacian\u003c/em\u003e.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cp id=\"text-99\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eNow that you understand how waves emerge from the simple wave equation, we can work in a bit more mathematical notation based on \u003cstrong\u003ederivatives\u003c/strong\u003e, moving beyond the dot and double-dot introduced above (\u003ca href=\"waves#figure_derivative\" target=\"_blank\"\u003eFigure 6\u003c/a\u003e). This will help in obtaining a more complete understanding of the wave equation, and provides a simpler, more compact notation that will be used in developing more complex wave equations later.\u003c/p\u003e\u003cp id=\"text-100\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eA derivative is just the slope of the function at a given point. In our discrete space and time CA framework, this is easily computed as the difference in state values \u003cem\u003ey\u003c/em\u003e divided by the spacing between cells, which we label by convention with the greek character “epsilon” \u003cspan class=\"math inline\"\u003e\\(\\epsilon\\)\u003c/span\u003e (\u003ca href=\"waves#figure_derivative-1d\" target=\"_blank\"\u003eFigure 7\u003c/a\u003e):\u003c/p\u003e\u003cdiv id=\"inline-container-102\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_spatial\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_spatial\"\u003e\u003cb\u003eEq 15:\u003c/b\u003e spatial derivative (slope)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-103\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{dy}{dx} = \\frac{\\rm{change} \\rm{in} y \\rm{values}}{\\rm{change} \\rm{in} x \\rm{values}}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-104\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eTechnically speaking, the true derivative involves shrinking the epsilon infinitely small, and thus achieving a truly continuous derivative value. This is the mind boggling aspect of calculus, which gives people so much trouble, and we can avoid all that by sticking with discrete space and time, in which case our simple \u003cstrong\u003edifference equations\u003c/strong\u003e can be used as discrete approximations to the true derivatives. Where necessary, we can leverage the analytical power of the true continuum, but any actual computational solution to differential equations always uses a discretization: the continuum is an idealization that cannot be realized in practice.\u003c/p\u003e\u003cp id=\"text-105\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn addition to the spatial derivative (slope) shown in , there is also a \u003cstrong\u003etemporal derivative\u003c/strong\u003e that involves computing the change in the state values over a discrete increment in \u003cem\u003etime\u003c/em\u003e, instead of over a discrete distance in \u003cem\u003espace\u003c/em\u003e. These temporal derivatives have the special dot notation that we introduced earlier:\u003c/p\u003e\u003cdiv id=\"inline-container-107\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_temporal\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_temporal\"\u003e\u003cb\u003eEq 16:\u003c/b\u003e temporal derivative (rate of change)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-108\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\dot y = \\frac{dy}{dt} = \\frac{\\rm{change} \\rm{in} y \\rm{values}}{\\rm{change} \\rm{in} \\rm{time}}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-109\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eTo be fully mathematically correct, whenever you have multiple different variables at work (such as space and time), you should use a \u003cstrong\u003epartial derivative\u003c/strong\u003e notation \u003cspan class=\"math inline\"\u003e\\(\\partial\\)\u003c/span\u003e instead of just the regular letter “d”:\u003c/p\u003e\u003cdiv id=\"inline-container-111\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_x-partial\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_x-partial\"\u003e\u003cb\u003eEq 17:\u003c/b\u003e spatial partial derivative\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-112\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial y}{\\partial x}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-113\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e(change in y over change in x).\u003c/p\u003e\u003cdiv id=\"inline-container-115\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_t-partial\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_t-partial\"\u003e\u003cb\u003eEq 18:\u003c/b\u003e temporal partial derivative\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-116\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial y}{\\partial t}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-117\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e(change in y over change in t).\u003c/p\u003e\u003cp id=\"text-118\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis partial letter is more difficult to draw in the figures, so we often just use \u003cspan class=\"math inline\"\u003e\\(d\\)\u003c/span\u003e in the figures, but it means essentially the same thing, and the difference is really just for sticklers.\u003c/p\u003e\u003cp id=\"text-119\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eBecause the wave equation deals with \u003cem\u003eacceleration\u003c/em\u003e, it is \u003cem\u003esecond order\u003c/em\u003e and this means that we need to deal with \u003cstrong\u003esecond order derivatives\u003c/strong\u003e, which just mean doing the derivative of the derivative — just do it twice! This means computing the slope of the slope, or the rate of change of the rate of change:\u003c/p\u003e\u003cdiv id=\"inline-container-121\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_x-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_x-2\"\u003e\u003cb\u003eEq 19:\u003c/b\u003e second order spatial derivative\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-122\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2y}{\\partial x^2} = \\frac{\\partial}{\\partial x} \\frac{\\partial y}{\\partial x}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-123\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e(change in change in y over change in \u003cspan class=\"math inline\"\u003e\\(x^2\\)\u003c/span\u003e).\u003c/p\u003e\u003cp id=\"text-124\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis shows how this second order spatial derivative is computed in the discrete space and time CA framework — you literally just take the difference between the two derivatives on either side of the central point. You have to keep dividing by the distance between cells \u003cspan class=\"math inline\"\u003e\\(\\epsilon\\)\u003c/span\u003e every time you do a derivative, so that ends up being squared in the denominator:\u003c/p\u003e\u003cdiv id=\"inline-container-126\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_x-2-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_x-2-2\"\u003e\u003cb\u003eEq 20:\u003c/b\u003e second order spatial derivative\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-127\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2y}{\\partial x^2} = \\frac{\\left( \\frac{y^t_{x+1} - y^t_x}{\\epsilon} \\right) - \\left( \\frac{y^t_x - y^t_{x-1}}{\\epsilon} \\right)}{\\epsilon} = \\frac{1}{\\epsilon^2} (y^t_{x+1} + y^t_{x-1}) - 2 y^t_x\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-128\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eInterestingly, this second order spatial derivative is effectively the same as the restoring force in the wave equation — there is just a factor of the epsilon squared difference between them:\u003c/p\u003e\u003cdiv id=\"inline-container-130\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_r-f\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_r-f\"\u003e\u003cb\u003eEq 21:\u003c/b\u003e restoring force\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-131\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nf = \\left( \\frac{y^t_{x-1} + y^t_{x+1}}{2} \\right) - 2 y^t_x = \\epsilon^2 \\frac{\\partial^2y}{\\partial x^2} = \\left( y^t_{x+1} + y^t_{x-1} \\right) - 2 y^t_x\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-132\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWe normally set this epsilon constant to be 1 in the native units of the simulation, so that it effectively disappears from the computation. Nevertheless, understanding these constants is important when trying to get all the units right, but they don’t affect the core conceptual basis of what is going on, which is that the restoring force is driven by the curvature of the curvature (slope of the slope) of the wave medium.\u003c/p\u003e\u003cp id=\"text-133\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eTo complete our new second-order derivative based wave equation, we can write the acceleration as a second-order temporal derivative:\u003c/p\u003e\u003cdiv id=\"inline-container-135\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_acc\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_acc\"\u003e\u003cb\u003eEq 7:\u003c/b\u003e acceleration\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-136\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\na = \\ddot y^t_x = \\frac{\\partial^2 y}{\\partial t^2}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-137\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e(rate of change of rate of change).\u003c/p\u003e\u003cp id=\"text-138\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ePutting this all together, we can now transform our previous equation for the wave acceleration:\u003c/p\u003e\u003cdiv id=\"inline-container-140\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_acc-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_acc-2\"\u003e\u003cb\u003eEq 23:\u003c/b\u003e acceleration\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-141\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\ddot y^t_x = c^2 \\left( y^t_{x-1} + y^t_{x+1} - 2 y^t_x \\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-142\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003einto the following elegant expression based on second-order derivatives:\u003c/p\u003e\u003cdiv id=\"inline-container-144\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_wave\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_wave\"\u003e\u003cb\u003eEq 24:\u003c/b\u003e the wave equation\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-145\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2 y}{\\partial t^2} = c^2 \\frac{\\partial^2 y}{\\partial x^2}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-146\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eNote that this equation directly implies the velocity and state update equations given above, which are really definitional in terms of what a velocity is: a velocity is updated by an acceleration, and it updates the state value. Thus, this one equation captures everything needed to produce wave dynamics.\u003c/p\u003e\u003cp id=\"text-147\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe wave equation has a very nice symmetry, where the second-order temporal derivative is equal to the second-order spatial derivative. In other words, the rate of change of the rate of change over time is driven by the rate of change of the rate of change over space.\u003c/p\u003e\u003cp id=\"text-148\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eComplementing this abstract mathematical elegance is the extreme simplicity of the discrete version of the equation, where each point in space is just trying to keep up with its neighbors. From the CA perspective, we want an update rule that involves only the immediate neighbors, and produces interesting overall behavior. Just about the simplest thing you can compute on the neighbors is the average of their state values, and the wave equation is pretty much the simplest way of using this neighbor average that produces interesting behavior — lots of other possible permutations end up just causing the system to settle very quickly into a uniform overall state.\u003c/p\u003e\u003cp id=\"text-149\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFor example, if you just directly update the new state value of a given cell by how different the average of its neighbors is from your current state value, then nothing interesting happens: all disturbances are very quickly eliminated, and the system assumes a completely uniform state across space and time. This is known as the \u003cem\u003ediffusion equation.\u003c/em\u003e\u003c/p\u003e\u003cp id=\"text-150\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThus, this wave equation seems like a highly auspicious starting point, if we want to come up with the simplest possible physical mechanisms for fundamental physics. It is nothing short of mind blowing that this very simplest of wave equations can drive the updating of the electromagnetic field: a few other equations are also required to compute the effects of the EM field on electric charges, but the EM field by itself can be fully realized with only this simple wave equation.\u003c/p\u003e\u003cp id=\"text-151\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eNext, we extend this wave equation to the full three-dimensional case, which just requires a little bit more mathematical notation, and some interesting ways of integrating across the 26 neighbors in 3D space.\u003c/p\u003e\u003ch2 id=\"waves-in-three-dimensions\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eWaves in Three Dimensions\u003c/h2\u003e\u003cp id=\"text-153\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe second-order wave equation in three-dimensional space is not too different at an abstract mathematical level from the one dimension case — you basically just have to add extra terms for each of the additional dimensions. One minor complication is that we conventionally use \u003cem\u003ex,y,z\u003c/em\u003e for the spatial dimensions, and we’ve been previously using \u003cem\u003ey\u003c/em\u003e to represent the state value, so now we’ll switch over to the notation that is typically used in quantum physics, based on the Greek symbols “psi” \u003cspan class=\"math inline\"\u003e\\(\\psi\\)\u003c/span\u003e, “phi” \u003cspan class=\"math inline\"\u003e\\(\\phi\\)\u003c/span\u003e, and a variant of phi: \u003cspan class=\"math inline\"\u003e\\(\\varphi\\)\u003c/span\u003e.\u003c/p\u003e\u003cp id=\"text-154\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eTo keep things consistent and clear, we establish the following convention for these state variables:\u003c/p\u003e\u003cul id=\"frame-155\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003cspan class=\"math inline\"\u003e\\(\\varphi\\)\u003c/span\u003e = varphi = simple scalar state value — one single real-valued number, like we’ve been considering already.\u003c/li\u003e\u003cli id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003cspan class=\"math inline\"\u003e\\(\\phi\\)\u003c/span\u003e = phi = complex-number state value — two independent real-valued numbers per state.\u003c/li\u003e\u003cli id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003cspan class=\"math inline\"\u003e\\(\\psi\\)\u003c/span\u003e = psi = full Dirac state value, which has two separate complex numbers, and captures the property of \u003cem\u003espin\u003c/em\u003e.\u003c/li\u003e\u003c/ul\u003e\u003cp id=\"text-156\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThus, for now, we’ll be using the “varphi” value to represent our wave states. The wave equation in abstract differential form is:\u003c/p\u003e\u003cdiv id=\"inline-container-158\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_3D\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_3D\"\u003e\u003cb\u003eEq 25:\u003c/b\u003e wave equation in 3D\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-159\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2 \\varphi}{\\partial t^2} = c^2 \\left( \\frac{\\partial^2 \\varphi}{\\partial x^2} + \\frac{\\partial^2 \\varphi}{\\partial y^2} + \\frac{\\partial^2 \\varphi}{\\partial z^2} \\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-160\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe second-order spatial derivatives on the right-hand side of this equation might look like a straightforward 3D generalization of the second-order spatial derivative from our 1D equation, but actually a true second-order spatial derivative in 3D has a bunch of cross-terms involving the slope in one dimension relative to the slope in another dimension — it is a much more complicated beast. The proper name for the right-hand side of the 3D wave equation is the \u003cstrong\u003eLaplacian\u003c/strong\u003e, which can be written in various ways:\u003c/p\u003e\u003cdiv id=\"inline-container-162\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_laplacian\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_laplacian\"\u003e\u003cb\u003eEq 26:\u003c/b\u003e Laplacian\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-163\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\Delta \\varphi = \\nabla^2 \\varphi = \\nabla \\cdot \\nabla \\varphi = \\frac{\\partial^2 \\varphi}{\\partial x^2} + \\frac{\\partial^2 \\varphi}{\\partial y^2} + \\frac{\\partial^2 \\varphi}{\\partial z^2}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cdiv id=\"figure_gradient\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_gradient\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 9:\u003c/b\u003e \u003cp\u003eThe gradient, which is a vector consisting of the local slope along each of the different dimensions (two-dimensional case shown here).\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cp id=\"text-166\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe symbol \u003cspan class=\"math inline\"\u003e\\(\\nabla\\)\u003c/span\u003e (called a “nabla”; \u003ca href=\"http://http://en.wikipedia.org/wiki/Nabla_symbol\" target=\"_blank\"\u003ewikipedia link\u003c/a\u003e) indicates the \u003cstrong\u003egradient\u003c/strong\u003e, which just takes the first-order spatial derivatives along each dimension (\u003ca href=\"waves#figure_gradient\" target=\"_blank\"\u003eFigure 9\u003c/a\u003e):\u003c/p\u003e\u003cdiv id=\"inline-container-168\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_grad\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_grad\"\u003e\u003cb\u003eEq 27:\u003c/b\u003e gradient\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-169\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\nabla \\varphi = \\left( \\frac{\\partial \\varphi}{\\partial x}, \\frac{\\partial \\varphi}{\\partial y}, \\frac{\\partial \\varphi}{\\partial z} \\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-170\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003enote that the result of the gradient is a \u003cem\u003evector\u003c/em\u003e with 3 values in it (a \u003cem\u003ethree vector\u003c/em\u003e), one for each dimension, with each value giving the local slope along each of the dimensions.\u003c/p\u003e\u003cp id=\"text-171\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe idea that the Laplacian is the gradient squared (\u003cspan class=\"math inline\"\u003e\\(\\nabla^2\\)\u003c/span\u003e) must be taken as a short-hand for the third expression of the four shown for the Laplacian above, where \u003cspan class=\"math inline\"\u003e\\(\\nabla \\cdot\\)\u003c/span\u003e is another thing entirely, called the \u003cstrong\u003edivergence\u003c/strong\u003e, which operates on fields of vectors and turns them back into a single scalar value:\u003c/p\u003e\u003cdiv id=\"inline-container-173\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_div\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_div\"\u003e\u003cb\u003eEq 28:\u003c/b\u003e divergence\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-174\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\nabla \\cdot F = \\frac{\\partial F}{\\partial x} + \\frac{\\partial F}{\\partial y} + \\frac{\\partial F}{\\partial z}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-175\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhere \u003cstrong\u003eF\u003c/strong\u003e is a vector field, e.g., of the sort that would be generated by taking the gradient of a 3D scalar field. It is hopefully at least somewhat clear how taking the divergence of the gradient of our scalar field results in the expression for the Laplacian: each of the 3 separate derivatives in the gradient gets the second-order treatment by virtue of the derivatives in the divergence, and the result gets added up into a single overall number as shown in the divergence equation.\u003c/p\u003e\u003cp id=\"text-176\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe conceptual bottom line for the Laplacian is the same as before: it measures the overall curvature of the curvature (slope of the slope) of the local neighborhood around the central point, and this is the total restoring force that drives acceleration. The Laplacian symbol just allows us to write the overall equation in an even simpler form:\u003c/p\u003e\u003cdiv id=\"inline-container-178\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_wave-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_wave-2\"\u003e\u003cb\u003eEq 29:\u003c/b\u003e Laplacian wave equation\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-179\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\frac{\\partial^2 \\varphi}{\\partial t^2} = c^2 \\nabla^2 \\varphi\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-180\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eWe prefer the \u003cspan class=\"math inline\"\u003e\\(\\nabla^2\\)\u003c/span\u003e version of the Laplacian because it conveys its essential second-order nature.\u003c/p\u003e\u003cp id=\"text-181\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eEmbedded in all this math is an absolutely critical point that emerges only in this 3D version of the wave equation, compared to the 1D version: \u003cstrong\u003ethe Laplacian adds up all the curvature around it into one single number, causing an inevitable blending of signals coming from different directions.\u003c/strong\u003e\u003c/p\u003e\u003cp id=\"text-182\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe net result of this is that \u003cem\u003ea wave packet that is initially localized will inevitably end up spreading out over space,\u003c/em\u003e due to this mixing of curvature across different directions. This spreading of the wave packet represents a critical problem for the pure wave model, and we can see that it enters very directly and inexorably into even the most basic wave equation. All the subsequent wave equations we will develop share this core Laplacian spreading behavior, and thus inherit this important problem.\u003c/p\u003e\u003ch2 id=\"the-discrete-3d-laplacian\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eThe Discrete 3D Laplacian\u003c/h2\u003e\u003cdiv id=\"figure_cubes\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_cubes\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 10:\u003c/b\u003e \u003cp\u003eThe 26 neighbors of a given cell in 3D space, each of which is weighted differently according to its distance — faces are 1 unit away, edges are \u003cspan class=\"math inline\"\u003e\\(\\sqrt{2}\\)\u003c/span\u003e away, and corners are \u003cspan class=\"math inline\"\u003e\\(\\sqrt{3}\\)\u003c/span\u003e away.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cp id=\"text-186\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eNow we consider how to compute the 3D Laplacian in the discrete space and time CA framework. To start, we only consider the 6 faces adjacent to the central point (\u003ca href=\"waves#figure_cubes\" target=\"_blank\"\u003eFigure 10\u003c/a\u003e), which are relatively easy because each pair of opposing faces can be treated just like a separate one-dimensional second-order derivative like we computed before, so we just have three times the number of terms as before:\u003c/p\u003e\u003cdiv id=\"inline-container-188\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_faces\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_faces\"\u003e\u003cb\u003eEq 30:\u003c/b\u003e discrete 3D Laplacian, faces only\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-189\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\nabla^2 \\varphi = \\frac{1}{\\epsilon^2} \\left( \\varphi_{(1,0,0)} + \\varphi_{(-1,0,0)} + \\varphi_{(0,1,0)} + \\varphi_{(0,-1,0)} + \\varphi_{(0,0,1)} + \\varphi_{(0,0,-1)} - 6 \\varphi_{(0,0,0)} \\right)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-190\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhere the subscript indicates the relative offset along the \u003cem\u003e(x,y,z)\u003c/em\u003e dimensions from the central point, which is then at \u003cem\u003e(0,0,0)\u003c/em\u003e. We can simplify this expression by just computing a sum of pairwise differences for each face element:\u003c/p\u003e\u003cdiv id=\"inline-container-192\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_faces-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_faces-2\"\u003e\u003cb\u003eEq 31:\u003c/b\u003e discrete 3D Laplacian, faces only\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-193\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\nabla^2 \\varphi = \\frac{1}{\\epsilon^2} \\sum_{j \\in N_{faces}} (\\varphi_j - \\varphi_0)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-194\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003ewhere \u003cem\u003ej\u003c/em\u003e is just an index into the set of 6 different faces and \u003cspan class=\"math inline\"\u003e\\(\\varphi_0\\)\u003c/span\u003e is the central point.\u003c/p\u003e\u003cp id=\"text-195\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe problem with only using the 6 face neighbors is that it misses all the curvature present in the other neighboring points (edges and corners; \u003ca href=\"waves#figure_cubes\" target=\"_blank\"\u003eFigure 10\u003c/a\u003e) and as a result, the wave propagation is very \u003cstrong\u003eanisotropic\u003c/strong\u003e — it is not the same in every possible direction. Waves flowing along one of the 3 primary dimensions work great, wave disturbances in other directions propagate very differently. If this was how nature worked, then we would easily be able to tell that the rules of physics are different in different directions, which is definitely not the case.\u003c/p\u003e\u003cp id=\"text-196\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe anisotropy problem can be fixed by including all 26 neighbors, in a relatively simple generalization of the last sum-based expression:\u003c/p\u003e\u003cdiv id=\"inline-container-198\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_all26\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_all26\"\u003e\u003cb\u003eEq 32:\u003c/b\u003e discrete 3D Laplacian, all 26 neighbors\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-199\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\n\\nabla^2 \\varphi = \\frac{3}{13 \\epsilon^2} \\sum_{j \\in N_{26}} k_j (\\varphi_j - \\varphi_0)\n\\]\u003c/span\u003e\u003c/p\u003e\u003cp id=\"text-200\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe key to making this work is to have different weighting factors \u003cspan class=\"math inline\"\u003e\\(k_j\\)\u003c/span\u003e for the different neighbors, depending on their Euclidian distance \u003cem\u003ed\u003c/em\u003e from the central point:\u003c/p\u003e\u003cdiv id=\"inline-container-202\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_wt\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_wt\"\u003e\u003cb\u003eEq 33:\u003c/b\u003e neighbor weight\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-203\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nk_j = \\frac{1}{d^2}\n\\]\u003c/span\u003e\u003c/p\u003e\u003cul id=\"frame-204\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003cstrong\u003efaces:\u003c/strong\u003e \u003cspan class=\"math inline\"\u003e\\(k_j = 1 \\)\u003c/span\u003e\u003c/li\u003e\u003cli id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003cstrong\u003eedges:\u003c/strong\u003e \u003cspan class=\"math inline\"\u003e\\(k_j = \\frac{1}{2} \\)\u003c/span\u003e\u003c/li\u003e\u003cli id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003cstrong\u003ecorners:\u003c/strong\u003e \u003cspan class=\"math inline\"\u003e\\(k_j = \\frac{1}{3} \\)\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cp id=\"text-205\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe full mathematical justification for this equation, and the demonstration of its isotropic behavior relative to the standard 6-face version, is given in the following unpublished manuscript:\u003c/p\u003e\u003cul id=\"frame-206\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• O’Reilly, R. C. \u0026 Beck, J. M. (2006/unpublished). A Family of Large-Stencil Discrete Laplacian Approximations in Three Dimensions.\u003c/li\u003e\u003c/ul\u003e\u003cp id=\"text-207\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThis is the equation we use for all 3D simulations. As we emphasized for the 1D equation before, this 3D equation also provides an appealingly simple mechanism for nature to compute: it is just an average over all the neighbors of a given point, weighted by the relative distance. It is difficult to imagine a simpler kind of neighborhood interaction.\u003c/p\u003e\u003ch2 id=\"wave-energy-in-3d\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eWave Energy in 3D\u003c/h2\u003e\u003cp id=\"text-209\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe equation for the wave energy for the 3D version just requires an update to the potential energy component — the kinetic energy component is identical, as it only involves the temporal derivative which remains the same regardless of the dimensionality. The one-dimensional equation generalizes in a straightforward manner to 3D, where we just add up the squared neighborhood differences, using the same weighting factors as in computing the Laplacian:\u003c/p\u003e\u003cdiv id=\"inline-container-211\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cspan id=\"eq_pe\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan id=\"eq_pe\"\u003e\u003cb\u003eEq 34:\u003c/b\u003e potential energy in 3D, all 26 neighbors\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cp id=\"text-212\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cspan class=\"math display\"\u003e\\[\nE_p = \\frac{3}{13 \\epsilon^2} \\sum_{j \\in N_{26}} k_j (\\varphi_j - \\varphi_0)^2\n\\]\u003c/span\u003e\u003c/p\u003e\u003ch2 id=\"dealing-with-edges\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eDealing with Edges\u003c/h2\u003e\u003cp id=\"text-214\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOne of the trickiest problems in making numerical simulations of wave phenomena is how to manage behavior at the edge of your simulation. This is perhaps one indication that nature has no edges — great way to avoid this problem. However, we have no such luxury with finite computational resources, and have to wrestle with this problem. There are basically three solutions:\u003c/p\u003e\u003cul id=\"frame-215\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003cstrong\u003eFixed edges:\u003c/strong\u003e the edge values are permanently fixed to specific values — this causes waves to bounce off of them and reflect back into the middle. It is not a good solution for any kind of force field (e.g., EM waves) because these require energy to dissipate over space.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003cstrong\u003eDissipative edges:\u003c/strong\u003e the edges essentially absorb the wave energy, producing the effect of the waves just continuing to propagate outward, without reflecting back etc. This is the best solution for EM waves. It is achieved by just getting rid of the acceleration term in the wave equation, so that the force directly drives the current velocity, without incrementing a prior velocity — this is also equivalent to the diffusion equation. It is also known as a Sommerfield boundary condition.\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• \u003cstrong\u003eWrap-around edges:\u003c/strong\u003e you just connect the two ends of your space back upon each other, creating a weird doughnut-like topology — this is good for exploring the motion of free wave packets, working somewhat like a treadmill where the wave keeps looping around through the same simulation space, without ever apparently hitting any edges. It is particularly useful for examining the spreading of wave packets over time. Computationally it can be a bit more challenging to implement the wrap-around, but that is just a coding problem.\u003c/p\u003e\u003c/ul\u003e\u003cp id=\"text-216\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThere is a further question as to what overall shape the simulated universe is — the two basic options are a cube and a sphere — we typically use spheres for things like atomic systems, which are naturally spherical, so that preserves the radial symmetry of the system, whereas a cube is much better for the wrap-around case (doing wrap-around in a sphere is more difficult and not supported in \u003ca href=\"EmeWave\" target=\"_blank\" title=\"wikilink\"\u003eEmeWave\u003c/a\u003e).\u003c/p\u003e\u003ch2 id=\"exploration-of-3d-waves\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eExploration of 3D Waves\u003c/h2\u003e\u003cdiv id=\"figure_scaling\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cimg id=\"figure_scaling\"\u003e\u003c/img\u003e\u003cp id=\"text-1\" style=\"max-width:8in;font-weight:normal;line-height:1.5;text-align:start\"\u003e\u003cb\u003eFigure 11:\u003c/b\u003e \u003cp\u003eScaling behavior of the discrete Laplacian wave equation — the bottom row of images are from a simulation that is 4 times larger than the one on the top row, and the snapshots are spaced 4 times further apart — it is difficult to tell the difference visually, although there is some roughness on the high-curvature edges of the lower resolution simulation compared to the milky smoothness of the higher-resolution one, the overall wave propagation is identical. This also shows the inevitable spread of the wave packet, which quickly becomes distributed widely over space. Due to the use of wrap-around edges, the wave ends up superposing upon itself many times over, producing what looks like interference patterns.\u003c/p\u003e\n \u003cbr\u003e\u003cbr\u003e \u003c/p\u003e\u003c/div\u003e\u003cp id=\"text-220\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eNow we return to the \u003ca href=\"EmeWave\" target=\"_blank\" title=\"wikilink\"\u003eEmeWave\u003c/a\u003e simulator to see how the 3D waves behave, including the spreading behavior resulting from the Laplacian as described above. Follow the directions given in the \u003ca href=\"waves_sim.md\" target=\"_blank\"\u003eWaves\u003c/a\u003e exploration, picking up where you left off, at the 3D Waves section. Running these simulations yourself is very strongly recommended — you really get to see the dynamics unfolding over time, and gain considerable insight into the overall behavior of the waves. You can also see how incredibly smooth and symmetric (isomorphic in all directions) the 3D Laplacian approximation is — there is no hint of any preferred directions or other kinds of numerical artifacts.\u003c/p\u003e\u003cp id=\"text-221\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOne question you might have is how the discrete laplacian scales, and whether what we observe in a small-scale 3D simulation that fits on our laptop will generalize to a higher-resolution case, ultimately to whatever resolution that nature actually operates at (which could be extremely high resolution indeed, as we discuss later). shows that the scaling behavior is extremely good, in the sense that two simulations differing in resolution by a factor of 4 (along each dimension — the larger one requires 64 times more total cells) behave essentially identically. Thus, we can generally be reasonably confident that our results will generalize at least qualitatively to higher resolution systems.\u003c/p\u003e\u003cp id=\"text-222\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOn a technical note about actually running these larger simulations: you need to use a computing cluster where different chunks of the model are distributed across different processors — \u003ca href=\"EmeWave\" target=\"_blank\" title=\"wikilink\"\u003eEmeWave\u003c/a\u003e supports this through the standard MPI protocol. To run a simulation of size 512 on each dimension requires a total of 30Gb of main state memory — we break this up into 8 chunks of just under 4Gb each, which then run much faster when distributed across 8 nodes, using 6 processors per node to divide the load using threads.\u003c/p\u003e\u003cp id=\"text-223\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eHere are some movies of the larger simulations in action:\u003c/p\u003e\u003cul id=\"frame-224\" style=\"display:flex;flex-direction:column;justify-content:start;align-items:start;gap:0.5em;padding-left:4ch;font-weight:thin;text-align:start\"\u003e\u003cli id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• The smaller model from : \u003ca href=\"Media:wavebg_u128_wv16_movie.mp4\" target=\"_blank\" title=\"wikilink\"\u003eMovie: Matrix=128, initial wavelength=16\u003c/a\u003e\u003c/li\u003e\u003cli id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003e• This same sized initial wave, in a matrix of size 512 \u003ca href=\"Media:wavebg_u512_wv16_movie.mp4\" target=\"_blank\" title=\"wikilink\"\u003eMovie: Matrix=512, initial wavelength=16\u003c/a\u003e — this should help you see better how the wave front becomes curved and stretched out, and it is more obvious how the wrap-around edges enable a limited sized simulation to capture a much larger effective size, due to the linear superposition property.\u003c/li\u003e\u003c/ul\u003e\u003ch2 id=\"initial-conditions\" style=\"max-width:8in;margin:0.25em;font-size:28px;font-weight:normal;line-height:1.2857143;text-align:start\"\u003eInitial Conditions\u003c/h2\u003e\u003cp id=\"text-226\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eOne quick technical note regarding how the \u003cstrong\u003einitial conditions\u003c/strong\u003e of our simulations are constructed. To construct a moving wave, you just set the current state to be the desired wave shape (e.g., a wave packet), and set the prior state to be that same shape offset by the distance that the wave should travel in one time step (i.e., c). Then, you initialize the velocity to be the difference between these states (current - prior) — this ensures that the velocity is exactly as needed to keep the same wave shape moving along. It turns out that this velocity is \u003cem\u003enot\u003c/em\u003e mathematically identical to a wave packet when the states are wave packets — it is very close to being so, but not exactly.\u003c/p\u003e\u003cp id=\"text-227\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eNow that you have a solid understanding of basic wave behavior in their full three-dimensional glory, we are ready to explore a wide range of electromagnetic phenomena in the next chapter.\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" 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style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Weak","URL":"weak","Title":"Weak","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{},"Description":"The weak force seems to be intimately tied up with the internal dynamics of \u003ca href=\"stochastic-particles\" target=\"_blank\"\u003estochastic particles\u003c/a\u003e, in relationship with the \u003ca href=\"spin\" target=\"_blank\"\u003espin\u003c/a\u003e property, for example in the case of the neutrino, which only interacts via the weak force, and can be thought of as a “pure spin” particle.","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eWeak\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-42\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe weak force seems to be intimately tied up with the internal dynamics of \u003ca href=\"stochastic-particles\" target=\"_blank\"\u003estochastic particles\u003c/a\u003e, in relationship with the \u003ca href=\"spin\" target=\"_blank\"\u003espin\u003c/a\u003e property, for example in the case of the neutrino, which only interacts via the weak force, and can be thought of as a “pure spin” particle.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIt is also intimately connected with the \u003cem\u003eflavors\u003c/em\u003e or \u003cem\u003egenerations\u003c/em\u003e of particles, e.g., muon and tau leptons in addition to the \u003ca href=\"electron\" target=\"_blank\"\u003eelectron\u003c/a\u003e, and is what allows the muon or tau to decay into an electron.\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIt has a very short interaction range, on the scale of a proton charge radius (\u003cspan class=\"math inline\"\u003e\\(10^-15\\)\u003c/span\u003em), as predicted by the massive nature of the force carriers, the \u003cstrong\u003eW\u003c/strong\u003e and \u003cstrong\u003eZ\u003c/strong\u003e bosons. Given that these are massive particles, the same arguments about photons just being synonymous with the \u003ca href=\"maxwell\" target=\"_blank\"\u003eMaxwell\u003c/a\u003e wave functions do not apply.\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"waves\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"m274-450 248 248-42 42-320-320 320-320 42 42-248 248h526v60H274Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003ePrevious\u003c/p\u003e\u003c/a\u003e\u003cstretch id=\"content-buttons-go-167\" style=\"display:flex;flex-direction:row;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-width:1ch;min-height:1em;font-weight:thin;text-align:start\"\u003e\u003c/stretch\u003e\u003ca id=\"content-buttons-go-168\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"zero-point\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"M686-450H160v-60h526L438-758l42-42 320 320-320 320-42-42 248-248Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003eNext\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"},{"Name":"Zero point","URL":"zero-point","Title":"Zero point","Date":"0001-01-01T00:00:00Z","Version":"","Authors":"","Affiliations":"","Abstract":"","Reference":"","Draft":false,"NoURLinPDF":false,"Categories":["Other"],"Specials":{},"Description":"In the \u003ca href=\"qed\" target=\"_blank\"\u003eQED\u003c/a\u003e framework, the vacuum is not actually treated as empty space, but is rather the province of the \u003cstrong\u003ezero point field (ZPF)\u003c/strong\u003e, which has a non-zero level of energy. This can be derived from the uncertainty principle: if a system had zero energy sitting in a confined space (i.e., the bottom of an EM potential well), it would have a definite momentum and position at the same time, which is forbidden by the \u003ca href=\"uncertainty-principle\" target=\"_blank\"\u003euncertainty principle\u003c/a\u003e.","HTML":"\u003ccontent id=\"content-0\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;min-height:10em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-155\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003c/div\u003e\u003cdiv id=\"content-content-go-158\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cdiv id=\"content-content-go-210\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;padding-right:0.5em;padding-left:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"content-content-go-232\" style=\"font-size:36px;font-weight:normal;line-height:1.2222222;text-align:start\"\u003eZero point\u003c/p\u003e\u003cdiv id=\"content-content-go-291\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cmain id=\"frame-43\" style=\"display:flex;flex-direction:column;flex-grow:1;justify-content:start;align-items:start;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003cp id=\"text-0\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn the \u003ca href=\"qed\" target=\"_blank\"\u003eQED\u003c/a\u003e framework, the vacuum is not actually treated as empty space, but is rather the province of the \u003cstrong\u003ezero point field (ZPF)\u003c/strong\u003e, which has a non-zero level of energy. This can be derived from the uncertainty principle: if a system had zero energy sitting in a confined space (i.e., the bottom of an EM potential well), it would have a definite momentum and position at the same time, which is forbidden by the \u003ca href=\"uncertainty-principle\" target=\"_blank\"\u003euncertainty principle\u003c/a\u003e.\u003c/p\u003e\u003cp id=\"text-1\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eEmpirical evidence for this ZPF comes potentially from the Casimir effect, which is a tiny force measured between two parallel neutral metal plates brought very close together — the region between these plates should exclude longer wavelengths of the ZPF, and thus have lower energy than the outside region, producing a net force. However, it is also possible that this force reflects a radiation reaction effect, as it can be derived from QED on that basis alone (\u003ca href=\"ref://Jaffe05\" target=\"_blank\"\u003eJaffe, 2005\u003c/a\u003e).\u003c/p\u003e\u003cp id=\"text-2\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe stochastic electrodynamics (SED) and stochastic optics models (\u003ca href=\"ref://MarshallSantos88\" target=\"_blank\"\u003eMarshall \u0026 Santos, 1988\u003c/a\u003e; \u003ca href=\"ref://MarshallSantos97\" target=\"_blank\"\u003eMarshall \u0026 Santos, 1997\u003c/a\u003e; deLaPenaCetto96) incorporate the ZPF as actual random oscillations in the classical EM field (described by Maxwell’s equations), and show how such a field could produce various phenomena such as photon antibunching statistics, which have been taken as one of the last elements of definitive support of the quantum photon model over the \u003ca href=\"semiclassical\" target=\"_blank\"\u003esemiclassical\u003c/a\u003e approach (a classical EM field interacting with a quantized atomic system).\u003c/p\u003e\u003cp id=\"text-3\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eA major problem associated with all of these ZPF models is that the amount of energy in the ZPF would be astronomically huge. Also, it would seem to predict a higher level of spurious photon detection events than is actually observed, although there may be a reasonable solution to this latter problem (\u003ca href=\"ref://MarshallSantos97\" target=\"_blank\"\u003eMarshall \u0026 Santos, 1997\u003c/a\u003e).\u003c/p\u003e\u003cp id=\"text-4\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eThe semiclassical theorist Jaynes suggested that instead of imagining that this ZPF fills all of space, it may just reflect noise emitted by atomic and molecular systems, which will be most intense in the immediate vicinity of these sources, and fall off dramatically outside of them. This could potentially eliminate the problem of the huge energy level, as it would just be a small additional contribution to the observed mass values of atomic systems.\u003c/p\u003e\u003cp id=\"text-5\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eIn any case, we will be on the lookout for these issues in developing the WELD models. The interaction between a discrete electron point particle and the Maxwell EM field will undoubtedly produce a lot of “ripples” of EM signals as it moves about, hopefully consistent with the spectrum of blackbody thermal radiation. However, there may be additional sources of noise, and additional elements of stochasticity that may need to be added.\u003c/p\u003e\u003cp id=\"text-6\" style=\"max-width:8in;margin:0.25em;font-weight:normal;line-height:1.5;text-align:start\"\u003eFor example, there is a long history of work on the connection between stochastic (brownian) motion and QM wave equations, which was developed by \u003ca href=\"ref://Nelson66\" target=\"_blank\"\u003eNelson (1966)\u003c/a\u003e building on original ideas from Feynmann (see \u003ca href=\"ref://Sciarretta18\" target=\"_blank\"\u003eSciarretta, 2018\u003c/a\u003e for a historical overview and recent developments). It may be that we need to make particle movement stochastic, to avoid strong aliasing effects of the cubic lattice.\u003c/p\u003e\u003c/main\u003e\u003c/div\u003e\u003cdiv id=\"content-buttons-go-152\" style=\"display:flex;flex-direction:row;justify-content:start;align-items:center;gap:0.5em;font-weight:thin;text-align:start\"\u003e\u003ca id=\"content-buttons-go-159\" style=\"color:var(--secondary-on-container-color);background:var(--secondary-container-color);display:flex;flex-direction:row;justify-content:center;align-items:center;padding-top:10px;padding-right:24px;padding-bottom:10px;padding-left:16px;font-size:14px;font-weight:thin;text-align:center;border-radius:1e+09px\" href=\"weak\"\u003e\u003csvg id=\"icon\" style=\"color:var(--secondary-on-container-color);stroke:var(--secondary-on-container-color);fill:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;gap:0.5em;min-width:1.2857143em;min-height:1.2857143em;width:1.2857143em;height:1.2857143em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 -960 960 960\"\u003e\u003cpath d=\"m274-450 248 248-42 42-320-320 320-320 42 42-248 248h526v60H274Z\"/\u003e\u003c/svg\u003e\u003c/svg\u003e\u003cdiv id=\"space\" style=\"color:var(--secondary-on-container-color);display:flex;flex-direction:row;justify-content:start;align-items:start;min-width:1ch;min-height:1em;width:1ch;height:1em;font-size:14px;font-weight:thin;text-align:center\"\u003e\u003c/div\u003e\u003cp id=\"text\" style=\"color:var(--secondary-on-container-color);font-size:14px;font-weight:500;line-height:1.4285715;text-align:center\"\u003ePrevious\u003c/p\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/content\u003e"}]
