The differences between the standard Copenhagen interpretation vs. the pilot-wave model 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, the Coulomb gauge for Maxwell’s equations of electromagnetic (EM) waves provides a convenient way to calculate the electrostatic force between two charged particles in terms of their respective charges (\(q_1\), \(q_2\)) and distance r between the two particles:
\[ F = k_e \frac{q_1 q_2}{r^2} \]
But this is not a plausible physical model because the math requires the use of the distances between relevant particles, and their respective aggregate charges. 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 ends up being exponentially explosive.
By contrast, a plausible physical model should compute EM forces directly from the collective effects of each individual charged particle everywhere, and propagate these forces via local mechanisms at the speed of light. Indeed, precisely such a model is available, in the form of the Lorenz gauge version of Maxwell’s equations, which propagates EM forces in an entirely local manner via standard wave equations.
The seemingly obvious conclusion is:
If you’re trying to understand how Nature actually operates (i.e., in terms of a physical model), you definitely want to use the Lorenz gauge version, and not the Coulomb one.
But critically:
Both gauges describe exactly the same physics, so you can’t run an experiment to tell you which gauge Nature is actually using.
Some people will conclude from this last point that it therefore is entirely irrelevant which gauge Nature is using, because all you care about is having any complete description of the physical phenomenon, and everything else is just “aesthetics”. Furthermore, you should definitely have as many tools in your toolkit to conveniently solve different types of problems with a minimum of effort. Indeed, according to this criterion, the Coulomb gauge is typically much more convenient that the Lorenz gauge (although you can get into trouble by using this gauge incorrectly; Brill & Goodman, 1967; Jackson, 2002; Onoochin, 2001).
However, it is a fact that, historically, many people were disturbed by the Newtonian gravitational version of the Eq 1 inverse-square law, because of its implied action-at-a-distance, prior to Einstein’s development of general relativity. Likewise, a considerable amount of ink has been spilled wrestling with the seemingly inexplicable nature of the measurement process in the Copenhagen interpretation of quantum physics.
Furthermore, the popularity of the otherwise fantastical many worlds interpretation (MWI, Everett, 1957) is largely due to the fact that it avoids the measurement problem entirely. But at what cost? By forking off an entire new Universe at each possible point of measurement!? This is supposed to be a more plausible physical model?
Therefore, to reiterate:
People should not waste their time worrying about the physical plausibility of something that is obviously a calculational tool.
Specifically, with the availability of the Bohmian version of pilot-wave model, which entirely eliminates this issue of the inexplicable nature of the measurement process (Norsen et al., 2015), it does not make sense to continue to try to make sense of the Copenhagen framework as a physical model. For example, the pilot-wave framework makes intuitive sense of the otherwise-mysterious double-slit experiment results: the particle goes through one slit, while the wave goes through both and gives rise to the interference effects. Done. See Bush et al. (2024) for discussion of the physical, macroscopic realization of these and many other quantum phenomena, using vibrating oil droplets.
The Bohmian pilot-wave model is directly analogous to the Lorenz gauge framework for EM, while the Copenhagen interpretation is likewise analogous to the Coulomb gauge. Both produce entirely the same mathematical results (so you cannot run an experiment to determine which one Nature is using), but the pilot-wave model provides a more plausible physical model, while the Copenhagen framework is typically more convenient for calculating results.
It is also important to emphasize that, unlike the Lorenz gauge of EM (or general relativity), the Bohmian pilot-wave model has a number of significant limitations as a plausible physical model. However, all of the issues with the pilot-wave model are actually present in the standard Hilbert space formulation as well: it is just that the pilot-wave model makes these issues more obvious, by making stronger ontological commitments. For example, the incompatibility of the pilot-wave model with special relativity is also entirely present in the configuration space framework used in the standard Hilbert space formulation (Bell, 1975; Aharonov & Albert, 1981; Peres, 2000; Peres & Terno, 2004; Norsen, 2011).
Do we really want to reward the foundational evasiveness of the Copenhagen or MWI frameworks, instead of recognizing and confronting the fundamental issues in the standard framework head on? According to surveys, most working physicists endorse either the Copenhagen or MWI frameworks, while none endorsed the pilot-wave framework (Tegmark, 2010). Thus, it does appear that evasiveness is winning at this point.
See universal basis for a way of formalizing what makes a physical model more intuitively appealing as such, without appealing to macroscopic physical intuition or other such vague constraints. That page also has further discussion about the limitations of the standard frameworks for providing such a universal basis representation, and highlights the ways in which the existing Standard Model is incomplete, despite its ability to generate spectacularly precise predictions of various experimental results.
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. See conservation for more discussion.
Another potential example is the idea of using only interacting wave equations and somehow avoiding the need for any hard little particles entirely (Hobson, 2013; Sebens, 2022). Wave equations are unambiguously at the foundation of the Standard Model, and even such an exquisitely simple equation like Klein-Gordon provides such incredible explanatory scope of physics phenomena. But this simplicity diminishes once you need to account for interactions with the EM force field, leading to the much more “complex” Dirac equations, via gauge theory. And then the electroweak introduces even more mind-boggling complexity!
Evidently, Nature is just as complex as it needs to be, and putting a strong prior on mathematical simplicity and elegance ends up running the risk of searching under the proverbial street light. Indeed, the strong appeal of the Bohmian formulation of the pilot-wave theory is that it is mathematically extremely simple, and thus can be easily proven to be identical to the standard model. But we may need to venture out into rougher terrain to fix all of the remaining issues.