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:
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.