For those who continue to be unsatisfied with Quantum Mechanics (QM), Bohmian Mechanics (BM) is an alternative worth considering. It is sometimes claimed that BM is equivalent to QM, but Lubos Motl recently argued on his blog that this is true only for a limited class of quantum phenomena, namely those that do not belong to Quantum Field Theory (QFT) including loop corrections. I'm curious if anybody can provide any insight into how Bohmian Mechanics could incorporate what we know from QFT.
For the specific case of a fixed number of interacting spinless point particles, there is a Bohmian recipe that works fine: you start with solutions to the Schrodinger equation, construct trajectories from the gradient of the probability current, and assign a probability measure to those trajectories according to the Born rule. That gives you a "classical" theory equivalent to the quantum theory.
For the specific case of a UV-complete, nonrelativistic quantum field theory of interacting spinless fields, I think exactly the same recipe should work, but so far as I know, no-one has actually done the work to demonstrate this. It should be demonstrated, because fields have an infinite number of degrees of freedom and that might cause technical problems not present in the finite-dimensional case (e.g. it might be necessary to work on a compact manifold); and since a handful of interacting QFTs in lower dimensions have been exactly solved, the raw material should be there for someone to prove, for at least one QFT, a genuine equivalence between the Bohmian recipe, and the usual perturbative approach (which is the context in which "loop corrections" appear).
But it's the scarcity of exactly solved QFTs which is the immediate obstacle to the development of Bohmian field theory, even for the case of nonrelativistic spinless fields, at anything more than a formal level. Most practical applications of QFT are motivated as approximations to idealized exact QFTs which mathematically are not completely specified. Physics has a philosophy, effective field theory, which explains why this is OK, and it also has the concept of a "UV-complete field theory", for which a mathematically rigorous definition should exist. But that's an area of mathematical research; one of the million-dollar Millennium Prizes is in this area - that should tell you how much work remains to be done!
And yet I find it hard to see how Bohmian field theory can deliver substantial independent results, except for field theories that have been defined with this rare and difficult degree of exactitude. To me, that seems to be required by the nature of the Bohmian recipe. Without it, one seems to be reduced to purely formal manipulations, and qualitative reasoning of the form, "if the field theory exists mathematically, then the Bohmian recipe should be able to reproduce perturbation theory".
So the remainder of what I have to say falls into that category of qualitative reasoning. The more realistic field theories that we would wish to discuss, simply have not been defined mathematically in a way which would allow concrete Bohmian calculations to be exhibited. Practical QFT can ignore that constraint because of the EFT philosophy, but Bohmian field theory cannot.
Nonetheless, one can try to reason about whether a Bohmian reconstruction of various phenomena of practical QFT is possible, even in principle; this is what Lubos does in his article. Here's my take. Compared to the starting point (nonrelativistic spinless fields) where I think the Bohmian recipe should work, the problems I see for extending Bohmian field theory to, say, the standard model, are special relativity, gauge symmetry, and nonzero spin.
Special relativity is a problem because the Bohmian recipe employs a preferred time-slicing of space-time. I know of no way around this except to accept the necessity of it. So you would end up with a theory like that before Einstein, where there is an ontologically preferred frame of reference, an objective universal time, but there's no way to experimentally identify which reference frame that is. Obviously this is contrary to the spirit of relativity, even if it gives the same predictions.
Gauge symmetry might be a more serious problem. Bohmian mechanics can deal with the problem of special relativity by just picking a reference frame and saying that's the real one. That's a kind of gauge-fixing and it's going to work. I have no similar confidence that gauge-fixing would work for "Bohmian gauge field theory". Maybe it would; I just lack the insight to say one way or another.
Nonzero spins... This is an area where I know that some work has been done - I'm thinking of Peter Holland in his book "The quantum theory of motion", where he proposes to define a spin 1/2 field as a Bohmian field whose local degrees of freedom are the same as a type of rotor - but I can't vouch for its correctness, and I think it must exist only on that formal plane where you can write formulas and do algebra but not to the point of calculating anything, because we don't know how to solve the resulting equations.
I know that Lubos blogged on another occasion that Bohmian mechanics absolutely couldn't deal with fermion fields, because they are based on Grassmann variables and you can't have "Grassmann beables". I don't know if that argument is valid; if it is, maybe you could still get by just with beables for the bosonic fields; but for the sake of completeness, I mention this further claim of his.
For some counterarguments against Lubos Motl's argumentation against de Broglie-Bohm theory see http://ilja-schmelzer.de/forum/forumdisplay.php?fid=6 and http://ilja-schmelzer.de/realism/Motl.php
The first proposal for a Bohmian variant of a relativistic quantum field theory has been made in Bohm's original 1952 paper, for the EM field. For a possibility to handle fermion fields, see http://ilja-schmelzer.de/forum/showthread.php?tid=36
deBroglie developed double solution theory, which is the most relevant description of photon and the Bohm / orthodox quantum mechanics are just high/low energy limits of that model. It should be pointed out that de Broglie disagreed with Bohmian mechanics.
protected by Qmechanic♦ May 15 '16 at 4:09
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