Reply To: Why Bohmian theory?

Robert Griffiths

A good understanding of quantum theory using a precise formulation free of conceptual problems, such as measurements and observables, no superluminal influences, no tension with relativity, a decent ontology, a Schrodinger cat taught not to bite, etc., should make it possible to provide a detailed critique of Bohmian mechanics: what it gets right, where it goes wrong, and the like. I haven’t written up such a detailed critique, but I have some ideas on how it might go, and would be interested to hear the reaction of others who have been engaged in this conversation.

Bohmian mechanics (BM) as I understand it makes important use of two concepts found in SQM = standard (i.e., textbook) quantum mechanics: the unitarily developing wavefunction, in principle of the entire universe; I call it the ‘uniwave’, and the probability current in position space, thought of as traced out in a fairly literal sense by a single particle or a collection of particles. Both the uniwave and the probability current have their uses, and I employ both when I teach a course (though I don’t discuss the entire universe).

A concept in SQM which is not used in BM is the notion that you can employ different representations for the quantum state: the position representation, the momentum representation, the energy eigenstate representation, etc. Different representations are useful for discussing different things, but distinct representations are typically incompatible and cannot be combined, e.g., position and momentum, apart from approximations using coarse grainings (beware of Heisenberg uncertainty!). As an example, harmonic oscillator energy eigenstates are useful when counting photons, but coherent states have the advantage that you can “see” the oscillator oscillating.

In BM the liberty of choosing different representations is no longer available: you must use the position representation. One advantage is that Hilbert space projectors for position (think of them as indicating the particle is definitely in some small region of space) commute with one another, as in classical mechanics, and so one avoids quantum mysteries associated with the noncommutation of projectors that correspond to incompatible quantum properties.

The trouble comes when one wants to discuss properties other than positions, e.g., how can we measure momentum? or energy? Here the discussions seem (to me) a bit odd, and one is likely to be told that all measurements are, ultimately, position measurements. Maybe so, but there are cases in which experimentalists claim that their pointer positions enable them to measure other things; are they mistaken? And might it not be the case that too much emphasis on position is behind the difficulty in constructing a clean relativistic version, since Lorentz transformations tend to mix up position and momentum?

But even if we stick to positions, there is still a serious problem arising from the fact that unitary time development typically transforms the position representation into some other representation, so what were position projectors at an earlier time are mapped into something else. As long as one simply uses the uniwave to calculate the probability that a particle at an earlier time in some region R will later be in some region S, the SQM result (using the Born rule) and BM agree. On the other hand, BM is in trouble when it comes to making sense of position at three (or more) successive times, as pointed out many years ago by critics (see [1] for references) who called the Bohmian trajectories “surrealistic”. In reply, defenders of BM pointed out, with some justification, that SQM is also unable to tell you where the particle was, and therefore there is no reason to doubt the Bohmian claim that a particle can trigger a detector without ever going near it. My response in [1] noted that SQM can be made more precise by a consistent treatment of stochastic time development, and when this is done detectors designed by competent experimentalists do what they are designed to do: they are triggered when particles pass through them, and not otherwise. (I might add that in more recent work [2] I noted a situation in which BM does better than either spontaneous localization (GRW) or many worlds in addressing what I call the second measurement problem, but in general BM seems unreliable.)

It seems that [1] has been ignored by the Bohmian community, apart from a preprint [3] that appeared many years ago. I was awaiting the published version before writing a response, but subsequent correspondence with Basil Hiley indicated he was no longer interested. Would someone else (Shelly? Travis? Aurelien?) like to take up the cudgels?

Bob Griffiths

[1] R. B. Griffiths, “Bohmian mechanics and consistent histories”, Phys. Lett. A 261 (1999) 227. arXiv:quant-ph/9902059

[2] R. B. Griffiths, “Consistent Quantum Measurements”, arXiv:1501.04813. Accepted in Stud. Hist. Phil. Mod. Phys.

[3] B. J. Hiley, O. J. E. Maroney, “Consistent histories and the Bohm approach”, arXiv:quant-ph/0009056.

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