Figure 1:
Illustration 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.
Figure 2:
Neighborhood interactions in regular cubic tiling of space in three-dimensions — these interactions are used to compute the wave equation locally.
As noted earlier, the computational modeling approach here is essentially a complex version of a cellular automaton (CA), 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 cells, each of which has one or more state values, and each cell interacts only with its nearest neighbors (i.e., locally) to update its state value over time (Figure 1; Figure 2).
Such 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 Gardner, 1970). 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 > 3 or < 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.
The CA framework provides the simplest kinds of answers to fundamental questions about space, time, and the basic nature of physical laws (Zuse, 1970; Fredkin & Toffoli, 1982; Fredkin, 1990). Space is real 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.
One 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 within 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.
Time 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 speed of light 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.
Furthermore, 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 Maxwell’s equations for the electromagnetic field and Dirac’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.
Note 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.
In 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 work 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.
Another 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 Pauli exclusion principle is strongly suggestive of a discrete CA-like state. This principle posits that only one fermion (electron, quark, etc, with a quantum spin of 1⁄2) can occupy the same quantum state, including position, at a time.
Thus, 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 boson 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, mentioned earlier).
The notion of autonomy 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.
When you look at the examples of plausible physical models discussed earlier, 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.
In 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.
However, two important objections are typically raised about such a framework: isn’t it just like the aether 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 non-locality of QM?