(Note: parts of this post appear in the comment section of the article being discussed. In that article the author uses the term Bohmian mechanics or just Bohm to refer to Pilot Wave Theory which is a more illustrative name for what is also sometimes called de Broglie-Bohm Pilot Wave Theory. I opt for Pilot Wave Theory (PWT) throughout this post. No matter the name it is the same theory that is being discussed.)

This recent article (Bohmian mechanics has a big problem) published by Tim Andersen, Ph.D. at The Infinite Universe presents an excellent illustration of a simple fact:

Math + Philosophy ≠ Physics

What is on display in Andersen’s article is a painfully contorted effort to dismiss Pilot Wave Theory on mathematical and philosophical grounds. But PWT is at root a phenomenological assertion about the physical nature of quantum systems. It is a physical description with ontologically relevant math. That math may have some shortcomings, though they are hardly as damning as the author would have you believe but the physical description it provides stands head and shoulders above the metaphysical nonsense generated by the standard Wavefunction-Only interpretation.

The author also displays considerable confusion about the nature of PWT, saying correctly at one point with regard to the status of a particle in the theory, “The particle itself is what we measure, and it is always there, whether we measure it or not.” A few paragraphs later, in the context of discussing locality he describes the status of the particle quite differently saying, “This is the price you pay for giving up locality. Because particles are not localized, they cannot be treated as free agents.” But then again we are told, “Bohm’s mechanics accepted nonlocality as the price to have definite particles (realism).”

The only thing that can be pointed out here is that in PWT particles always have a definite “local” position while in the standard WO interpretation particles are not localized, but smeared out. Andersen, for some reason, appears to be confused about this distinction which suggests his critique here is not based on a clear understanding of the Pilot Wave Theory.

The author also asserts that PWT “gives up” locality. It does not. What PWT does is add a non-local component explicitly to the model with it’s invocation of a pilot wave. It is the pilot (or guiding) wave that is, like all waves, a non-local phenomenon. The pilot wave is also distinct from the Schrodinger wavefunction which is another point that Andersen seems confused about:

“Bohm’s theory has a downside in that the wavefunction can guide as many particles as you like, and it does so instantaneously, allowing one particle to influence another also instantaneously.”

On the subject of non-locality, the author admits to being philosophically indisposed to the idea. This despite the clear evidence, that non-local effects are part of the phenomenological world. The problem seems to be an inclination to see it as matter of having to choose between mutually exclusive possibilities — either the world is local or it is non-local. That is, however, a false dichotomy.

Physical reality has both local (particle) and non-local (electromagnetic wave) components. Another way to say this is, particles are 3-dimensional (local) and electromagnetic waves are 4-dimensional (non-local) phenomena. That is typical of the complementary dualities often found in nature; it is a fundamental aspect of physical reality that the common reductionist viewpoint tends to obscure in its search for the One. It’s not a case of either/or; it’s both. Complementary dualities are the engines of physical reality.

In PWT the statistical outcomes of quantum experiments are attributed to the interaction of particles with waves. This is consistent with the rest of physical reality where waves and particles can be observed as distinct, interacting phenomena.

In contrast the, standard Wavefunction-Only interpretation of QM attributes the outcome of quantum experiments to a superposition of states, wherein a particle is said to be “smeared out” with no definite location until it is observed, at which point it is always found at a specific, though only statistically predicted, position. This miraculous turn of events, from smeared out to definite location is attributed to a “collapse of the wavefunction”. That miraculous and inexplicable and unobservable collapse is then said to present a “measurement problem”. It could also be thought of as a credibility problem.

So this strained, empirically baseless account of quantum behavior requires a non-local event, the “wavefunction collapse” which happens instantly at the moment the particle is detected. The author criticizes PWT for its explicit non-locality while failing to note that non-locality is also prominent in the standard WO interpretation. The reason non-locality is unavoidable is that non-local phenomena (waves) are as much a part of physical reality as local phenomena (particles). The standard interpretation does not acknowledge that duality and consequently it has a problem when non-locality makes its unwelcome appearance in a model that only recognizes locality.

PWT is consistent with the observed wave and particle nature of physical reality on all other scales, whereas the WO model produces a borderline irrational account of particles being spread out like waves only to become particles when you look at them.

Interestingly there are tabletop experiments conducted at MIT that produce directly observable quantum-like effects that are visually quite striking. The images produced in these clever experiments can be helpful to visualize and understand the mechanics of PWT: https://thales.mit.edu/bush/index.php/4801-2/

Throughout his diatribe Andersen teases a “Fatal Flaw” that will be revealed eventually, as if it were some thrilling conclusion to a schlock horror movie. When the momentous revelation arrives it doesn’t fail to disappoint. The Fatal Flaw requires taking this extravagant mathematicist-fantasyland conceit into consideration:

“This is relativity, but not in ordinary 4D spacetime like Einstein’s relativity. It is relativity in an infinite-dimensional space called a Hilbert space. Hilbert space is where the wavefunction actually lives and moves.”

Apparently the Fatal Flaw is that the PWT math is not mathematically equivalent to Heisenberg’s math which is equivalent to Schrodinger’s math somewhere over there in Hilbert space (in all its infinite glory). Unfortunately, for this FF theory, PWT has an additional guiding equation which describes a particle’s interaction with a pilot wave. 

So the criticism that PWT isn’t equivalent mathematically to the Heisenberg formalism that is equivalent to Schrodinger’s math is a fatuous mathematicist argument that has nothing to do with physics. Not being equivalent to S&H is a feature not a bug.

PWT does what the Schrodinger’s and Heisenberg’s math doesn’t do, describe a physical system in physical and mathematical terms that are consistent with empirical reality. Implying as Andersen does that a plausible, coherent, physical and mathematical model like PWT should be set aside because it doesn’t fit well with some arbitrary mathematicist conventions does not constitute a reasonable scientific argument.

PWT is not equivalent to those models – because it is more mathematically complete than either of them. What makes it more complete is the fact that it actually describes the physical interaction that produces the observed results — WO does not do that.

PWT provides a plausible physical account of a quantum physical process, with mathematics, that produces the observed outcomes of this type of quantum experiment while neither Heisenberg nor Schrodinger describe a process that is coherent or rational in scientific terms. All you get is statistics and absurd metaphysical handwaving. It is the Wavefunction-Only version of Quantum Mechanics that has a big problem: the model doesn’t make physical sense — it’s just some math. It gets the right answers on the test but it doesn’t know why the answers are right.