The differences between the standard Copenhagen interpretation vs. the pilot-wave model reviewed in history nicely exemplify the broader distinctions between calculational tools vs. physical models. 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.
For 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 (\(m_1\), \(m_2\)) and distance r between the centers of mass of two bodies:
\[ F = G \frac{m_1 m_2}{r^2} \]
But 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.
By 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.
Indeed, Einstein’s general relativity 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”.
In 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.
However, 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 many worlds interpretation (MWI, Everett, 1957) attempts to avoid the measurement problem entirely by forking off an entire new Universe at each possible point of measurement!
According to surveys, a relatively large number of working physicists endorse this perspective (Tegmark, 2010)! 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 configuration space 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.
Instead 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).
Thus, 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.
In this context, a somewhat smaller subset of physicists have continued to pursue the de Broglie-Bohm pilot-wave 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 possible 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) (Norsen et al., 2015).
This 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.
The 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 (Norsen et al., 2015; Norsen, 2022).
Other examples of calculational tools
Before 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.
For 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.
Coulomb’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 Brill & Goodman, 1967; Jackson, 2002; Onoochin, 2001).
In 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.
As explored in Hilbert space, 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.
In short, here’s a useful mantra: Don’t confuse the math for the physics! 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.
The seduction of elegant models
An important corollary of the “don’t confuse the math for the physics” mantra is: don’t be seduced by elegant mathematical models. For example, there is an overarching framework based on symmetry groups 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 generative principle that allows one to get deeper insight than the hard-won, empirically-based standard model.
An 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 Klein-Gordon equations, which are then inherited by the Dirac 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.