On the single-particle Bohmian account

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    Ken Wharton

    My main concern with Bohmian mechanics (along with the Everettian viewpoint, and any other psi-ontic approach) is that some of its ontological entities live in configuration space. (To me, at least, such spaces seem to be obvious epistemic constructions: Quantum States as Ordinary Information) Still, I have had great respect for the single-particle Bohmian account, where it appears that no such configuration space is needed.

    But is this really true? As I first learned about Bohmian mechanics, focusing on the single-particle story, I had trouble wrapping my head around non-position measurements. (Velocity, energy, angular momentum, etc.) The Bohmian account of these measurements looks solid on paper, but I could never intuit how it translated to my favored single-particle examples.

    Finally, it dawned on me: Such measurements *do not* easily translate to single-particle cases. It is a strict necessity that the wavefunction of the non-position-measurement device lives in configuration space (encompassing both the particles in the device, and the measured particle itself). Maybe this is an obvious fact to everyone in the field, but I have never seen it noted explicitly.

    My conclusion, then, is that the single-particle Bohmian account is not as problem-free as it is often presented. Only for position measurements can such an account avoid the need for configuration space.

    Aurelien Drezet

    Dear Ken Wharton, In my view there is no contradiction between your first impression concerning the single particle theory and the many-particles approach acting in the configuration space. Indeed, actually you are confusing the measurement problem with the ontological problem. The most basic part of Bohmian theory is of course the ontological problem and following this view the single particle indeed follow a simple path in the 3D space. As you mentionned this is the most intuitive part of the theory. However, when you need to consider interactions and measurements in most cases you get something non local due to entanglement. This makes sense in the configuration space since you need then a larger system. It is only with some very simple measurements that your intuition can be recovered and in most cases you have to be prudent, has it was pointed out many times by Bohm, Bell and others.


    Ken Wharton

    Thanks, Aurelien… It sounds like we’re mostly in agreement about how it all fits together; I just wish this point was clearer in presentations of the “intuitive” single-particle examples. One sometimes gets the impression that Bohmian mechanics has a spacetime-based resolution of any single-particle problem.

    But I think there are lots of outstanding single-particle puzzles in QM. For example: why is it only Gaussian distributions that have non-negative Wigner functions? This is a single-particle puzzle, but it doesn’t seem like the Bohmian single-particle account can help address it, because velocity measurements turn out to be so problematic in such a context.

    (Okay, now I’m expecting Travis to point me to where he’s already explained all this in great detail… 🙂


    Hi Ken, I’m not really sure what needs explaining here. I mean, one should really understand (“orthodox”??) Bohmian mechanics as a theory of the entire universe, whose ontology is (a) all of the particle positions and (b) a single universal wave function evolving according to Schroedinger’s equation. And that universal wave function, to be sure, is a function on the very high-dimensional configuration space of the whole universe. So while of course you can apply the theory to sub-systems within the universe, such as individual particles that are suitably decoupled from the rest of the universe, and hence get by with a one-particle wave function (that can be thought of as a field in physical space) in this kind of context, it should be kind of obvious from the outset that you can’t do this in general. There is, according to Bohmian mechanics, such a thing as entanglement, and so in general one is never going to get rid of configuration space wave functions completely. And so to whatever extent one cannot accept configuration space wave functions as part of the ontology of a theory, one just won’t accept Bohmian mechanics, and it doesn’t really matter exactly how many different kinds of situations there are where one can avoid this thing one doesn’t like.

    That said, as you know, I share your sense that there is something a bit indigestible about physically real configuration space wave functions. Here I am unusual among people who like Bohmian mechanics — most others I think tend to view the (universal) wave function as more like a “law” than a “field” and so the idea that it is mathematically a function on configuration space doesn’t bother them much… whereas I, influenced in no small part by the very sorts of one-particle phenomena you raise here for discussion, where there is a really clear and intuitive physical story taking place exclusively in physical space, tend to think of the wave function as more “field”-like, which means I am quite bothered indeed by the idea of a physically real field on (what seems obviously like) an abstract, non-physical, space. So I remain ultimately not-fully-satisfied by the “orthodox” version of Bohmian mechanics I meant to be talking about in the previous paragraph. But where you seem inclined to just dismiss the whole theory on the basis of its having configuration space wave functions, I actually see it as by far the most promising jumping-off point for people (like me/us) who are concerned about configuration space ontology issues and want a theory whose ontology is exclusively in physical space.

    Here briefly is how/why I see it as a promising jumping-off point. First, in Bohmian mechanics, what we think of in everyday experience as the physical world (the tables and chairs and cats and trees and planets we see around us) are made of *particles*, not wave function. And of course the particles in Bohmian mechanics live in ordinary 3D physical space. So basically (and unlike a lot of other extant candidate quantum theories) Bohmian mechanics already gives a really clear and coherent (and empirically adequate) account of the comings and goings of material objects in 3D space. The worrisome thing — the config space wave function — is a kind of secondary, behind-the-scenes player (unlike in, e.g., Everett’s theory, in which whatever worries one has about config space ontology are completely front and center). And then the second point: Bohmian mechanics, uniquely and under-appreciatedly, makes it possible to define single-particle wave functions (and I mean in general — not just for unentangled particles) and it is then possible to understand the Bohmian guidance formula as defining each particle’s motion in terms of its associated single-particle-wave function. These single-particle wave functions (technically called “conditional wave functions” in the literature) can be understood as something like fields in physical space, and so it is possible — in a sense — to recover the intuitively sensible stories that you and I both like so much in certain simple one-particle situations, in complete generality. That is, one can tell the story of the universe (according to Bohmian mechanics) by describing the motion of each individual particle, and regarding each particle’s motion as determined/guided/piloted by its associated “conditional wave function” (thought of as a field in 3D physical space). The problem is, the set of N conditional wave functions doesn’t contain all the information that’s in the universal wave function. (Obviously!) So the particles and the set of conditional wave functions can’t really be regarded as a complete ontology — this would not constitute a closed dynamical system. But to me this is not so much cause for despair, as a promising research program: find some other stuff (that can also be understood as living in 3D physical space) that contains the “missing entanglement information” (i.e., the residue of the universal wave function that fails to be captured by the set of N conditional wave functions) to supplement the ontology with, to produce a closed dynamical system that reproduces the particle trajectories of “orthodox” Bohmian mechanics, but without the universal wave function (on configuration space) being, as such at least, part of the ontology. As you know, I wrote a paper a few years ago showing one (ugly/implausible) way this can be done, as a kind of “proof of principle”, and have been doing some work with Xavier Oriols and other people trying to push the idea forward. It remains very much a work in progress, but, again, I think there is room for optimism here to whatever extent one’s goal is to get rid of config space ontology. Rejecting Bohm’s theory out of hand (just on the grounds that it has, in general, a config space wave function in it), and starting over from scratch in some totally different way, seems unwise if one just wants a “theory of exclusively local beables”, since Bohmian mechanics is, in some sense, really close to what one wants already.

    (But of course you probably wouldn’t agree with that last sentence since Bohmian mechanics has what you would regard as an additional problem, namely the *dynamical* nonlocality. My project of searching for a re-formulation which gets rid of the nonlocal *ontology*, in favor of exclusively local beables, will certainly not get rid of the dynamical nonlocality. So if for you that’s a deal-breaker, then none of this will be convincing. But for me, it’s again unwise to regard dynamical nonlocality as a deal-breaker since, I think, we know, with certainty, from Bell’s theorem and the associated experiments that we need dynamical nonlocality no matter what. Of course, you dispute that because Bell and I don’t allow retrocausality — or more precisely, classify retrocausality as just an example of dynamical nonlocality — whereas you think that one can save locality by embracing retrocausality…)

    And then finally, going back to your original post and following up a comment I made to you last week, I think the class of “experiments/phenomena that can be understood in Bohmian terms without any config space wave function” is a bit broader than you acknowledge in your post. Basically anything which ends with a position measurement of the particle in question, can be understood in the way you have in mind. This of course involves things like the 2-slit experiment where, literally, one just measures the position of the particle when it hits the screen. But one can also, for example, understand spin measurements in this way: you shoot a particle through a Stern-Gerlach device and assign a value to its spin based on … where it hits a screen behind the magnets. (So “measuring spin” is, or at least can be, nothing but “measuring position” after some suitable external fields are applied.) And velocity/momentum can also be measured in this same way using the so-called “time of flight” technique: if the particle whose velocity you want to measure is initially localized (by some potential well, say), just turn off the potential, let the particle evolve freely for a long time, and then measure its position; the distance it went (from where the well was, to where it was later detected) divided by the time you let it evolve for, is its velocity, and it can be shown that (in the large-time limit) this constitutes a perfectly valid way of “measuring the velocity”. Same for energy, etc. As I said at the beginning, I’m not sure this really matters, since you really can’t avoid entanglement forever (unless you get on board with the project I outlined in the previous couple of paragraphs), but maybe it’s helpful to realize that a perhaps-surprising number of different types of measurements really do (or can) just come down to a position measurement of the particle in question, and can hence be understood in the perfectly intuitive config-space-free way that we both like.

    Ken Wharton

    Hi Travis,

    Thanks for the thoughtful response! You’re right that not much “needs explaining” concerning my original concern, and your last bit is a good point (where you point out that later position measurements can be used to measure other non-position quantities at earlier times). It somewhat bothers me that (say) the energy measured in such a manner isn’t *really* the energy of the particle, etc. But since it matches what a past-energy-measurement would have made, I guess you make a fair point that the single-particle account is a bit more general than I implied.

    Concerning your preferred ‘jumping off point’, I absolutely agree that the approach you outline here is worth pursuing, but of course there are other jumping off points as well, also, based in 3D space (4D spacetime). Namely, classical fields.

    One of the things that most bothers me about particle-based approaches in general, and dBB in particular, is that one ends up with quite strange stories for straightforward cases that map to classical wave interference. I would imagine there are even cases of classical E+M fields, of just the right intensity, that work perfectly fine even in the few-photon limit (say, perfectly tuned interferometers) but which would would require “dynamical nonlocality” in a dBB framework (assuming a model of particle-like photons, which I think you once told me about…). Of course, my field models look quite strange when applied to cases that clearly look like classical particles! But I at least wanted to mention that there’s more than one reasonable jumping-off-point to get back to spacetime-local beables.

    I think you have my reaction just about right, concerning my dislike of “dynamical nonlocality”, and I don’t think that this is in any way equivalent to retrocausality (at least not the sort of models that interest me). You might take a look at the new afterword of my “The Universe is Not a Computer” essay, where I compare/contrast to Rob Spekkens concern about swapping the nonlocality between the dynamics and the kinematics. Here I tried to make it clear how Rob’s concern doesn’t really apply to all-at-once Lagrangian style accounts, hopefully setting retrocausal stories apart from those with true dynamical nonlocality (fundamental action-at-a-distance).


    Just briefly, re: the point that an “energy measurement” (of the sort discussed above) doesn’t reveal the true pre-measurement energy of the particle… First, I don’t think it’s even clear what the true pre-measurement energy of the particle would be, according to Bohm’s theory. There is, in some sense, no such dynamically meaningful property according to the theory. (Of course, one could just make something up — as people indeed do — and call that “the energy”… but it wouldn’t, for example, be a conserved quantity… so there’s really no particular reason to define it in any particular way, and hence no particular reason to define it, or talk about it, at all.)

    And then the more interesting second point about this: it shouldn’t be surprising that not every measurement can just reveal some pre-existing value of some associated property. We know, from the various no-go theorems (Kochen-Specker, Bell considered in a certain way, etc.) that at least some properties in theories like this will have to be “contextual”. Now often, in discussions of such no-go theorems, the idea of properties being “contextual” is regarded as some very strange kind of thing that would require a lot of obviously-implausible ad hoc put-in-by-hand fix-ups. But one of the really wonderful things about Bohm’s theory is that it shows how exactly the required sort of “contextuality” can come out, trivially, from a kind of brutally obvious dynamics, without anything even remotely resembling “implausible ad hoc put-in-by-hand fix-ups”. The particle is just guided by the wave function in the standard way, and this turns out to imply that, for example, if you “measure the energy” using some “time of flight” type procedure, you’ll get exactly the QM-predicted outcome statistics, and the outcome will be determined (once the completely particle state *and the details of the procedure by which the “measurement” occurs* are specified), even though the “measurement” isn’t really a measurement at all in the sense of revealing some pre-existing value of some dynamically-meaningful quantity.

    My paper “The pilot-wave perspective on spin” discusses this point in some depth in the case of spin measurements, where the “contextuality” manifests itself in the fact that two distinct setups, which would both be regarded as valid ways of “measuring the z-component of the particle’s spin”, can give *different* outcomes even for the exact same particle-state input:


    And then, yeah, I completely agree with you that there are lots of different potentially promising jumping-off points for our shared goal. Perhaps more on that later… =)

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