The world according to Bohmian mechanics
The world according to Bohmian mechanics
Is it really worth saving physical reality at the expense of real physics?
In 2002 Chris Fuchs1 wrote that
not a year has gone by in the last 30 when there was not a meeting or conference devoted to some aspect of the quantum foundations. ... Go to any meeting devoted to some aspect of the quantum foundations, and it is like being in a holy city in great tumult. You will find all the religions with all their priests pitted in holy war — the Bohmians, the Consistent Historians, the Transactionalists, the Spontaneous Collapseans, the Einselectionists, the Contextual Objectivists, the outright Everettics, and many more beyond that. They all declare to see the light, the ultimate light. Each tells us that if we will accept their solution as our savior, then we too will see the light.
In this mailing we will look at the Bohmian persuasion. It will be helpful to begin with a discussion of the two-slit experiment with electrons. According to a poll conducted by Physics World in 2002, it ranks as “the most beautiful experiment in physics.” To Richard Feynman it “has in it the heart of quantum mechanics” and is “impossible, absolutely impossible, to explain in any classical way.”
The setup features an electron gun situated at G, a plate with two slits (L and R) equidistant from G, and a screen consisting of an array of detectors, one of which (situated at D) is shown in the following schematic. To prevent the electrons from scattering light or interacting with air molecules, the experiment is carried out in darkness and in a decent vacuum.
We are interested in how the number of electrons detected at the screen varies across the screen. As the number of detected electrons increases, this number (as a function of D) will more and more closely resemble the probability of detection predicted by quantum mechanics (QM). If it is possible to find out, for each electron, through which slit it went, the following probability function is predicted.
The solid hump is more or less what common sense leads us to expect. If it is not possible to find out, for each electron, through which slit it went, the following probability function is predicted.
Let’s assume that
each electron goes through a particular slit (either L or R),
the behavior of the electrons that go through L does not depend on whether R is open or shut, and vice versa.
If the first assumption is true and both slits are open then the distribution of detections across the screen is the sum of two distributions, one due to the electrons that went through L and one due to the electrons that went through R. If the second assumption is true, then we can observe the first distribution by keeping the right slit shut, and we can observe the second distribution by keeping the left slit shut. What we observe when the right slit is shut is the left dotted hump of the first graph, and what we observe when the left slit is shut is the right dotted hump of that graph. If both assumptions are true, we thus expect to observe the sum of these two humps represented by the solid hump. Yet if it is not possible to find out, for each electron, through which slit it went, what is predicted and what we observe is the second graph with its maxima and minima. So at least one of the two assumptions must be false.
According to an interpretive strategy proposed by David Bohm2 (and in fact much earlier by Louis de Broglie), it is the second assumption that is false: the behavior of the electrons that go through L does depend on whether R is open or shut (and vice versa). Like classical particles, the electrons follow precise paths, but unlike classical paths, these wiggle in a peculiar manner and cluster at the screen so as to produce the distribution of detections predicted by QM:
What causes the wiggles? Bohmians explain this by positing the existence of a “pilot wave,” which is derived from the quantum-mechanical wave function, and which affects the motion of the electrons by exerting on them some kind of force. If both slits are open, this wave passes through both slits, and if it is not possible to find out through which slit each electron went, the secondary waves emerging from the slits interfere, with the result that the electrons are steered along wiggly paths.
The reason electrons emerging from the same source or the same slit arrive in different places, is that they start out in slightly different directions and/or with slightly different speeds. If we had precise knowledge of these initial values, we would be in a position to predict each electron’s future motion with classical precision. Since QM implies that such knowledge cannot be had, and since Bohmian mechanics (BM) was designed to put a realistic spin on QM, Bohmians must insist that the trajectories of electrons are hidden; they cannot be observed.3
What jumps out from the guidance equation of BM (in the form introduced by de Broglie, which most Bohmians today prefer) is that the velocity of each particle can depend on the wave function of the entire system of particles — which is to say, the wave function of the universe — evaluated at the simultaneous positions of all other particles. If two particles are correlated — which in technical terms means that their wave function is not the product of two one-particle wave functions, and which is the case whenever the two particles have interacted in the past — then a change in the motion of the first particle drastically alters the motion of the second particle instantly and in a manner that does not diminish with the distance between the two particles.
This theory is therefore nonlocal in a way that unadulterated QM cannot aspire to be. And that means that there can’t possibly be any such thing as a relativistic version of BM. Bohm’s theory will invariably require a preferred inertial reference frame. It will invariably require an absolute standard of simultaneity, the abolition of which was the major achievement of Einstein’s theory of relativity. On the other hand, if all we know of a system of particles is at most its wave function, which must be the case since BM is designed to reproduce the statistical predictions of QM, it will be impossible to find out which inertial reference frame is the preferred one. A relativistic version of BM will therefore be relativistic for all practical purposes, even though in reality it won’t be.
But in order to make the statistical predictions of BM come out right, God (in the beginning) must distribute all existing particles across the universe in a typical way. Because this typical distribution is roughly analogous to the distribution of particles in classical statistical mechanics, which leads to thermodynamic equilibrium, this assumption is known as the “quantum equilibrium hypothesis.”
Bohmian mechanics, moreover, even in its effectively relativistic version, isn’t a replacement for quantum theory in general but only for those quantum theories which happen to be theories of persistent particles. The trouble is that in reality — at any rate, in the relativistic regime — there are no persistent particles. As was mentioned in this mailing, what is conserved is not things but only such quantities as electric charge, baryon number, and lepton number. Relativistic quantum theory, which takes this into account, is technically a theory of fields, and it is far from clear that a Bohm-type replacement for quantum field theory can be cooked up.4 In his entry on “Philosophy of Quantum Mechanics” in The Ashgate Companion to Contemporary Philosophy of Physics,5 David Wallace characterizes BM as an interesting research program rather than an actual theory:
If the Everett interpretation is incoherent for one reason or another (as is probably the mainstream view among philosophers of physics, if not among physicists) then currently we have no realist solutions to the measurement problem. There are interesting research programs, which (disregarding their potential conceptual problems) have successfully reproduced the predictions of non-relativistic physics, but a research program is not a theory.
(I intend to look at the Everett interpretation in a future mailing.) Moving on, the contextuality of standard QM, as we have seen, implies that measurements do not simply reveal the pre-existent properties of quantum systems or the pre-existent values of quantum observables but actually create the properties or values they indicate. Bohmians, intend on preserving a semblance of common-sense realism (“classicality”), assert on the contrary that positions enjoy a measurement-independent existence. Even if this is true as far as the outcome-indicating positions of apparatus pointers are concerned, it is highly questionable when it comes to the position of the electron in the ground state of the hydrogen atom. For there, according to BM, the electron has a precise position, it has a precise velocity, and the latter is precisely zero. And the electron doesn’t fall into the nucleus either. Common-sense realism indeed!
All measurable quantities other than position are as contextual in BM as they are in QM. In fact, they are more contextual, for not only are they defined by the experimental context in which they are measured but they also depend on which other quantities are measured at the same time. (This kind of contextuality, an illustration of which can be found in this mailing, is also known as “co-measurement contextuality.”) Regarding all measurable properties other than position, Detlef Dürr (a staunch advocate of BM) writes, “[t]he best way to understand the status of these observables — and to better appreciate the minimality of Bohmian mechanics6 — is Bohr’s way: What are called quantum observables obtain meaning only through their association with specific experiments.” (It is quite amazing how often Bohr is invoked to help his detractors out of some impasse.)
Bohmians pride themselves on dispensing with the observer. BM has even been characterized by its advocates as “quantum mechanics without observers”: measurements produce definite outcomes (as long as outcomes are indicated by the positions of pointers); there is no need for observers to take cognizance of them. Why not? Because the apparatus pointer is really somewhere, whether or not it is observed. How so? Because the electrons and nuclei the pointer is composed of are really somewhere, assembled in a pointerlike shape. But if, as Bohmians will concede, the electron’s position in the ground state of atomic hydrogen cannot be observed, how is it that the positions of the electrons and nucleons the pointer is composed of can be observed? And if these cannot be observed, how is it that the position of the pointer can be observed? Bohmians leave it to standard QM to work this out to the extent that standard QM is capable of working this out, which is for all practical purposes only.
Because the expression “for all practical purposes” occurs rather frequently in discussion of the quantum measurement problem, John Bell has invented for it the acronym FAPP.7 Most attempts to demonstrate the emergence of a reality that includes measurement outcomes begin by quantifying the difference between an incoherent mixture and a coherent superposition. Thereupon they proffer some argument designed to demonstrate that whenever an apparatus is involved, that difference will be so small that it can be ignored FAPP — in blatant disregard of the fact that conceptually the difference is considerable and independent of its “size.”
When dealing with an incoherent statistical mixture of possibilities, it is consistent to assume that one of the possibilities corresponds to what is actually the case. (For example, it is consistent to assume that Schrödinger’s cat is either dead or alive.) When dealing with a coherent superposition, on the other hand, it is inconsistent to pretend that one of the possibilities has been actualized. (Nor is it that the cat is both dead and alive, or neither. It’s just not consistent to assume that either is the case, unless or until a measurement indicates which is the case.) But when the difference is small enough, then (so the aforementioned type of argument goes) it is FAPP admissible to replace a coherent superposition by an incoherent mixture. Et voilà: “classicality” has emerged; there is an outcome (albeit only FAPP).
Both QM and BM end up with pointer wave functions that are FAPP statistically distributed over the different outcome-indicating directions in which the pointer may point. Yet even if it is FAPP consistent to assume that there is an outcome, QM requires an observer to determine which of the possible outcomes is the actual one. In BM, the branch of the pointer wave function that corresponds to the actual outcome is assumed to contain the particles the pointer is composed of, while the other branches are “unoccupied.” Therefore no observer is needed. All the same, just like the emergence of a definite outcome in QM, this is true only FAPP. The small but irreducibly non-zero difference between “true full stop” and “true FAPP” depends on the overlap of the different branches of the pointer wave function, and this never vanishes completely.
In the philosophy of science, “realism” has two distinct meanings. It denotes a basic demand of science, which is to check all theoretical claims against what can actually be observed or tested. It also stands for the idea that reality can be carved up into individual systems, and that each individual system has an in principle complete description in terms of its properties (“classicality”). Because QM brings these two into conflict, those insisting on the second kind of realism, like the Bohmian school, thereby lose sight of the first: Bohmian trajectories and the particles following them have no connection to empirical fact. To Bohmians, they are the “reality givers.” To everyone else they are just a piece of fantasy.
I conclude this rant with an excerpt from a rant by Reinhard Werner (returning, as he writes, “from a week full of sect-like behaviour, which I guess was the punishment I deserve for being open enough to go there in the first place”):
On one level the workshop [“Quantum Mechanics without observers III,” Bielefeld, April 2013] felt to me like a fundamentalist congregation. To someone not sharing the belief in that particular brand of realism it was especially striking how this belief was enforced time and again by the usual forms of discourse in a one-faith community. Several speakers enjoyed drawing laughter by exposing supposedly absurd quotes from famous physicists, and the hagiography of John Bell kept an amazingly large number of speakers busy. Of course, this saint was claimed whole for the Bohmian camp, and any subtlety left in his writings thoroughly flattened. We heard a sermon on the question whether upon splitting a box with a quantum particle the particle is truly in one or the other box, and a chairman who actually asked the experimentalist speakers to declare their faith in this matter.
C.A. Fuchs, Quantum mechanics as quantum information (and only a little more), in A. Khrennikov (Ed.), Quantum Theory: Reconsideration of Foundations, pp. 463‒543 (Växjö University Press, 2002).
D. Bohm, A suggested interpretation of the quantum theory in terms of “hidden” variables, Physical Review 85, 166‒193 (1952).
Inspired by a 2007 paper by H.M. Wiseman [Grounding Bohmian mechanics in weak values and bayesianism, New Journal of Physics 9, 165], some advocates of Bohm’s theory have touted weak measurements as a means of actually observing these trajectories. It was recently shown by J. Fankhauser and P.M. Dürr [How (not) to understand weak measurements of velocities, Studies in History and Philosophy of Science 85, 16‒29 (2021)] that such claims are mistaken: weak measurements constitute no new arguments, let alone empirical evidence, in favor of Bohmian particle trajectories.
According to Matthew Leifer (Co-Director of the Institute for Quantum Studies at Chapman University), “it is fair to say that, even amongst Bohmians, the usual position based model is regarded as more of a toy-theory that has to be replaced by something more sophisticated to account for quantum field theory.”
D. Rickles (Ed.), The Ashgate Companion to Contemporary Philosophy of Physics, pp. 16‒98 (Ashgate, 2008).
If “minimality” stands for the smallness of the number N of pre-existent properties postulated, and if minimality is a good thing, as Dürr appears to imply, then Bohr’s understanding of QM actually fares better than Bohm’s: N=0 is smaller than N=1.
J.S. Bell, Against “measurement,” In A.I. Miller (Ed.), 62 Years of Uncertainty, pp. 17‒31 (Plenum, 1990).