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This is something I've been wondering about. For those who don't know, Bohmian mechanics is an interpretation of quantum mechanics that is in the class of what are known as "hidden variable theories", which are theories that posit the probabilistic and unreal wave functions of quantum mechanics are not the whole story of what is going on but that there is hidden information behind that scenes and not accessible to our measuring devices which provides a deterministic and well-defined physical state to the particle at each point in time, allowing us to speak of events proper, as opposed to just measurements with no idea of what is going on in between or even saying it doesn't exist at all or is meaningless despite our intuition not barfing at it in the way it would when presented with a truly meaningless phrase like "Colorless green ideas sleep furiously". It accomplishes this, at least in the context of simple fixed-particle non-relativistic QM with classical forces ("semiclassical mechanics"), by positing first a universal wavefunction, like in Everett's many worlds theories, and then on top of that a hidden particle position for each particle in its universe that evolves according to a "guiding equation" that results in the particle positions having a Born rule distribution at every point in time, and which effectively adds a "tracer" to the "many worlds" which is taken as describing the actual world we see and why we don't see the others. (Some people wonder about the fate of the other "worlds" in the wave function which then become so-called "empty waves", but I won't get into that here. I'm more interested in the pure structure of the theory itself, for the point of this inquiry.) It provides a rather neat account of the quantum measurement problem that both subsumes decoherence and addresses the uniqueness of outcomes: the measurement causes a local collapse through the same Everett-style decoherence process, and then the particles making up both the measured and measurer are shunted down a single branch of the universal wave function, thus explaining why we never see the rest. It has no invocation of nondeterminstic behavior at some kind of unspecifiable and ill-defined when-time (e.g. "when a measurement" ... but when is that? When in the process does the nondeterministic collapse strike? Or "when in consciousness"? But when then? When in the brain does this happen? Etc.) which can plague some other interpretations. Or in terms I like, you can write a computer program to play a Universe game with the physics engine running Bohmian mechanics with no ambiguity at all as to where to place things like calls to the random number generator to collapse wavefunctions, something made not at all clear in the Copenhagen Non-Interpretation.

The trick with this and the price to all its seeming very tantalizing advantages, however, is, of course, Bell's theorem, which requires that any such theory be explicitly and truly nonlocal - that is, in a relativistic context it demands communication of information - the hidden information, in particular - across a space-like separation, i.e. faster than light, into the "elsewhere" region of the space-time diagram. And indeed it makes not an exception to that, and does indeed involve such a communication of the hidden information. And this poses a serious problem for generalization of it to relativistic contexts like fully Relativistic Quantum Field Theories (RQFTs), including the celebrated jewel Quantum ElectroDynamics (QED), which make up the best theories of nature we have that are good to up to 12-15 decimal places in some cases, truly awesome marvels of scientific understanding and human endeavor indeed.

My question, then, is this. It is often said that when it comes to this kind of thing, you have the following choice:

  • Relativity
  • FTL
  • Causality

and you must "pick two, but not three". Bohmian requires us to pick FTL, so we must then choose one of the other two, and the most common one is to pick causality, and sacrifice relativity, in particular, assert there must be some kind of universal or etheric reference frame, with the caveat this violates the Lorentz transformation symmetry that is considered so fundamental to the Universe.

However, what I'm wondering about is: what if we choose instead to reject causality? What if we bite the bullet and try to build a Lorentz-invariant Bohmian theory where that the hidden information does not have a preferred simultaneity, and can interact with its own past and future in an acausal manner? Now some will say "but then grandfather paradox" - yet that's not specific enough for me. What happens if we try to write down the math for such a thing and solve the equations? What does it predict? As I've heard of the idea of "self-consistency principles" ("Novikov's self consistency principle") and was wondering if that could work here. After all, the information is hidden, and it doesn't let us predict anything other than regular QM would let us have access to as stated. If the four-dimensional evolution equations only permit self-consistent solutions, how does that affect the theory? How exactly does it fail, which might explain why it's not done? Like for example does the restriction of Novikov consistency end up resulting in, say, no consistent solutions existing, or does it cause something else, like a dramatic deviation from the Born rule statistics that would result in the theory's predictions then coming to be at variance with regular QM and with observed reality?

Or if you have some philosophical quibble about causality, then forget about even the scientific applicability and think of it this way: just as a pure mathematical toy, what exactly happens when you try and construct a causality-violating relativistic Bohmian mechanics? I would try it myself but wouldn't be sure of what a simple space-time version of the guiding equation would be, and I think you'd also need to upgrade the Schrodinger equation to some relativistic form (e.g. Dirac or Klein-Gordon equation) for the trial run.

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    $\begingroup$ Your question is an interesting one. However, because the people who are equipped to answer it already know what Bohmian mechanics is, you might consider replacing the Great Wall of Text which comprises your first paragraph (or two) with a link to the Wikipedia article or a similar reference to improve readability. $\endgroup$
    – J. Murray
    Jun 8, 2018 at 6:30
  • $\begingroup$ I don't know if that's exactly what you ask for, but maybe you are interested in the paper arxiv.org/abs/quant-ph/0105040. $\endgroup$
    – Luke
    Jun 8, 2018 at 11:37

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Clearly no one can say that "Bohmian Mechanics is incompatible with relativity". They can say that "no known reconciliation is known to exist" but even that is questionable. Sheldon Goldstein has been working on ways to make Bohmian mechanics relativistic for awhile with varying success https://arxiv.org/abs/1307.1714

As Luke said in the comment this https://arxiv.org/abs/quant-ph/0105040 is another paper where Goldstein tries to make a backwards causal theory.

There is no widely accepted (by physics community as a whole) reconciliation as of yet between Bohmian mechanics and relativity, but you're not alone in thinking, as Goldstein does, that the answer could lie somewhere with backwards causation. This could also fix up the creation and annihilation problems for Bohmian mechanics since the electron positron creation/annihilation could be viewed as simply a particle that swapped temporal directions as opposed to "disappearing" or "appearing" out of nowhere.

So there is potentially plenty of meat on this bone and hardly anyone working on it. You should try to flesh it out yourself if you are equipped with the knowledge. Might be a real revolutionary possibility here that is being left unattended.

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David's answer referred to Goldstein's work, but you might also be interested in the work of Hrvoje Nikolić, who has also worked on creating a relativistic pilot wave theory (abandoning conventional causality). In particular, this paper of his gives a good summary of his model, including the lack of conventional causality: https://arxiv.org/abs/1002.3226

Short answer: his model obeys the Novikov self-consistency principle, so causality violations do not lead to contradictions.

Edit: Also see this paper for more technical description: https://arxiv.org/abs/quant-ph/0406173

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Yes, for a very basic reason: Relativity has $E = mc^2$ - which sets the stage for particle creation and annihilation - processes where the particle number changes. The whole point of Bohmian mechanics is to give particles deterministic trajectories, under the influence of a guiding field that produces the quantum weirdness. But creation and annihilation totally break that narrative - particularly particle decay. Even if you somehow manage to include particle number changing processes into some extended version of Bohm, you still have the issue of what determines where and when a particle decays or, more generally, where/when particle number changes. The time and point of such an event is random. You're not going to recover a deterministic version of that with any kind of guiding field. The guiding field only molds trajectories for particles, it doesn't issue them birth and death certificates.

Now, you could try to evade the issue by just quietly giving up on trying to make the time and place of number-changing events deterministic, and just concede the issue. But, then, you're giving up on determinism, which undercuts the only thing Bohmian mechanics was meant give you.

So, the next time you encounter Bohm enthusiasts, ask them what part of Bohmian mechanics determines when and where a Geiger counter clicks ... or ... what determines when/where an isolated neutron dies and spits out a proton, electron and anti-neutrino (via the W intermediary)?

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I preface this with the fact that I am no means an expert (I am barely an amateur, as it is). So I hope what I write here isn't completely off base or useless.

But I did come across this ArXiv paper (from the Google query "bohmiam mechanics quantum field theory"): https://arxiv.org/abs/quant-ph/0303156 "Bohmian Mechanics and Quantum Field Theory" (2003-4) by Detlef Duerr, Sheldon Goldstein, Roderich Tumulka, Nino Zanghi. Here's the opening paragraphs (emphasis in bold mine):

Despite the uncertainty principle, the predictions of nonrelativistic quantum mechanics permit particles to have precise positions at all times. The simplest theory demonstrating that this is so is Bohmian mechanics [3, 5, 11]; in this theory the position of a particle cannot be known to macroscopic observers more accurately than the $|\psi|^2$ distribution would allow. A frequent complaint about Bohmian mechanics is that, in the words of Steven Weinberg [14], “it does not seem possible to extend Bohm’s version of quantum mechanics to theories in which particles can be created and destroyed, which includes all known relativistic quantum theories.”

To remove the grounds of the concern that such an extension may be impossible, we show how, with (more or less) any regularized quantum field theory (QFT), one can associate a particle theory—describing moving particles—that is empirically equivalent to that QFT. In particular, there is a particle theory that recovers all predictions of regularized QED [15].

However, we will not attempt to achieve full Lorentz invariance; that would lead to quite a different set of questions, orthogonal to those with which we shall be concerned here. But we note that though the theories we present here require a preferred reference frame, there can be no experiment that would allow an observer to determine which frame is the preferred one, provided the corresponding QFTs are such that their empirical predictions are Lorentz invariant.

The final lines of the quote are most relevant here; although there seems to be non-Lorentz invariant things "lurking behind the curtains", though in a way that "on stage" everything still looks Lorentz invariant.

This paper (which I see has already been linked in a previous answer) https://arxiv.org/abs/1307.1714 "Can Bohmian mechanics be made relativistic?" (2013) discusses this notion of "on stage Lorentz invariance" in more detail:

Many proponents of Bohmian ideas have thus become resigned to the notion that relativistic Bohmian theories will be relativistic only at the relatively superficial level of empirical predictions: the theories will make relativistically good predictions (including, for example, the correct kind of prediction for the Michelson-Morley experiment) but will involve something like a hidden, empirically undetectable, notion of absolute simultaneity. Such theories, it is usually conceded, are relativistic only in the sense that the Lorentz-Fitzgerald ether theory (considered here as an interpretation of classical electrodynamics) is relativistic – namely, they are not relativistic, not in a serious or fundamental sense.

The aim of the present paper is to question this perspective by suggesting a rather general strategy for making Bohmian theories compatible with fundamental relativity. In Bohmian theories the dynamics of the particles (or fields) is defined in terms of structures extracted from the wave function. The strategy proposed here involves extracting from the wave function also a foliation of space-time into space-like hypersurfaces, which foliation is used to define a Bohmian dynamics in a manner similar to the way equal- time hyperplanes are used to define the usual Bohmian dynamics. We show how this extraction can itself be Lorentz invariant in an appropriate sense, and argue that virtually any relativistic quantum theory, Bohmian or otherwise, will thus already contain a special space-time foliation, buried in the structure of the wave function. This makes it difficult to imagine how one could question the “seriously relativistic” character of the Bohmian theories to be described, without simultaneously denying that any theory in which something like a universal wave function plays a fundamental role can be a candidate for serious compatibility with relativity.

...

Everything in the type of theory proposed here – the dynamical law for $\Psi$, the rule for defining $\mathscr F = \mathscr F(\Psi)$, and the guidance equation(s) for the local beables – is fully Lorentz covariant and thus seemingly entirely compatible with the space-time structure being exclusively Minkowskian. Is such a theory then fundamentally – and/or seriously – relativistic?

This is not an easy question to answer, because it is not at all clear what, exactly, fundamental/serious compatibility with relativity does, or should, require. Lorentz invariance is clearly one necessary requirement. But it is easy to imagine that someone might dismiss – as clearly incompatible with relativity – the type of theory proposed here, simply on the grounds that it involves a dynamically privileged foliation. Frankly, it would be hard to disagree with this sentiment. But on the other hand, one of the important implications of our proposal is that foliations can be extracted from $\Psi$ and will therefore in a sense be present in any kind of quantum theory, Bohmian or otherwise. So if the mere presence of a foliation renders a theory un-relativistic, it seems hopeless that a viable relativistic theory could ever be constructed.

...

Thus if Bohmian mechanics indeed cannot be made relativistic, it seems likely that quantum mechanics can’t either.

So the authors seem to argue for a Shakespearean "If this be error and upon me proved, I never writ, nor no man ever loved." sort of situation.

Still, from the quote above about matching empirical predictions of in particular "the celebrated jewel" QED (as so named in the OP's question), it seems that Bohmian mechanics type theories can still hold their weight.

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