Reinhard, Could you elaborate on your statement that (even) position is not “directly observable” in BM? I don’t think you’re right, but then I’m not entirely sure what you mean. I would say that position is directly observable in BM in the sense that it is possible, according to the theory, to find out the actual position of some particle at some time. I’m guessing you wouldn’t dispute that (but I’m really not sure, because your ideas/criticisms tend to be obscured by a frothy layer of polemical rhetoric), but would consider it insufficient. That is, I think you assume that genuinely “directly observing” particle positions somehow has to mean monitoring them over time, with perfect accuracy, but without changing the trajectory from what it would have been in the absence of such monitoring. I would of course agree that if *that’s* what you mean by “directly observing” a Bohmian particle position, then you are right, you cannot do that. But I wouldn’t consider that an appropriate definition/standard for “genuinely observing particle positions”.
(And then similarly, vis a vis the “surreal trajectories” business, I think the situation there is just that, some experimental setup which you might have naively expected to constitute a “genuine observation of the particle position”, according to BM, actually turns out, according to BM, *not* to be that. I again wouldn’t consider that as refuting the claim that it is possible to genuinely observe particle positions in BM: “anything that seems to me, naively, without really considering in detail what the theory says, like it should constitute a valid position measuring device, must actually be a valid position measuring device” is not the appropriate standard. What’s relevant is just that it is in fact possible, according to the theory, to observe particle positions — and you cannot ignore what the theory itself says about exactly under what conditions, and with what accuracy, and for which kinds of setups, etc., this is possible.)
But all of this seems like the kind of stuff that you’ll just pounce on and denounce as worthless loose talk. So maybe it would be helpful to instead try to argue in the context of some simple but concrete example. Let’s take the “Einstein’s Boxes” setup, where there’s some particle that can either be on the left (psi_L) or on the right (psi_R), or perhaps a superposition of the two. Now imagine a position measuring apparatus with a pointer (initially in state phi_0) that is intended to swing to the left/right (phi_L/phi_R) to indicate the presence of the particle on the left/right. Let’s assume that the detector is ideal/perfect in the sense that, if the particle is prepared in the initial state psi_L, then the Schroedinger evolution of the combined particle/pointer system goes like this:
psi_L(x) phi_0(y) –> psi_L(x) phi_L(y)
and then similarly for the other case where the outcome should be certain:
psi_R(x) phi_0(y) –> psi_R(x) phi_R(y).
Now of course according to BM there is, in addition to the wave function, the actual particle positions X and Y. If we assume that the initial configuration X(0), Y(0) is random and |Psi|^2 distributed (where Psi = psi phi is the “Universal” wave function), then it is a theorem that this remains true over time. OK, so in the case that the particle is prepared in state psi_L(x), the final position of the pointer Y(t_f) ends up in the support of phi_L(y) with certainty. That is, the pointer definitely ends up veering left if the particle definitely started out on the left. And similarly for the other possibility.
Now what about the case where the initial quantum state has the particle in a superposition of psi_L and psi_R? Well in this case, the actual position X(0) of the particle will be either in the support of psi_L or in the support of psi_R. (I assume that these supports don’t overlap.) Then the Schroedinger evolution of the universal wf is as follows:
[ psi_L(x) + psi_R(x) ] phi_0(y) –> psi_L(x) phi_L(y) + psi_R(x) phi_R(y)
which can be understood as two disjoint lumps in configuration space. The actual particle/pointer positions at the end, X(t_f) and Y(t_f), will end up in the support of one lump or the other, just depending on whether X(0) was initially in the support of psi_L or psi_R. So the pointer indeed registers the actual pre-measurement (“Bohmian”) position of the particle, and with perfect faithfulness/fidelity.
That’s the kind of thing I have in mind when I say that “the position of a particle can be genuinely measured in BM”. I would be very interested to understand better exactly which parts of this you find wrong and/or so fraught with dubious handwaving as to be worthless. Is it for example that the measuring device is treated so schematically (as basically just a single particle, the pointer, with all the real details of the physical structure and operation of the device mocked up with some special interaction Hamiltonian between the “particle” and “pointer” which, to annoy you maximally, I didn’t even bother to write down)? Or is it the assumption that the two wave packets under consideration don’t overlap at all? So that, in a more realistic treatment in which there is some small overlap, it could occasionally happen that the pointer moves Right when the particle actually starts out on the Left, and vice versa, so the measurement is less than 100% faithful? Or is the problem that you think there’s some infinite regress, since the pointer position will be just as observable/unobservable as the original particle, so that this kind of schematic analysis just moves the problem back one level without really solving it? Or what? I’d really like to understand better exactly what you see as problematic.
And then let me also clarify (again in the context of this simple example) what I meant to be saying about ordinary/operationalist/orthodox QM, when I accused it of just making up ad hoc rules on the fly. I meant, specifically, the need to apply “measurement postulates” (such as the collapse rule). That is, instead of treating the particle+pointer in a fully quantum way, which would obviously produce the entangled post-measurement state I wrote above, in which there is no particular fact about which direction the pointer is pointing (which I would say contradicts, or at least appears to contradict, what we see with our eyes in actual labs in this kind of situation), we treat the particle-pointer interaction as a “measurement” in which the normal dynamical rules (namely Schroedinger’s equation) momentarily fail to apply. And so we just say — in flat contradiction to what the unitary Schroedinger dynamics would apply — that the pointer, being classical rather than quantum, just magically ends up pointing Left or Right, with 50/50 probability, as a result of its interaction with the particle, which interaction also results in the wave function of the particle collapsing to either psi_L (if the particle magically points Left) or psi_R (if instead the particle magically points Right). Now maybe you will disavow this kind of “orthodox” treatment as not capturing what you, Reinhard, think is going on. If so, I’d love to understand better what you think is going on. But what I just described really is the orthodoxy — it’s what all the textbooks say, for example. And I don’t think anybody could dispute the claim that, compared to this orthodox treatment, the Bohmian analysis really adds something. It explains how the pointer ends up pointing in a definite direction, which (under the admittedly idealized setup assumed) correlates perfectly with the pre-measurement position of the particle, purely as a result of the two basic dynamical postulates, without any need for additional ad hoc “measurement postulates”. So let me re-assert my point from before this way: anybody who accepts this orthodox account, but who also criticizes BM for the allegedly hand-wavy and un-rigorous and unconvincing and ad hoc character of its analysis of this kind of measurement, is a hypocrite. Hopefully you can clarify exactly why this does not apply to you. =)
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