- #1

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- Summary:
- The thermal interpretation of QM contains both local and nonlocal beables. Taking the local ones for granted, I argue that nonlocal beables are either inconsistent with the local ones or consistent but superfluous.

I have already discussed the ontology problem of thermal interpretation (TI) of quantum mechanics (QM) several times in the main thread on TI. The following is supposed to be the final refined version of my argument, so I don't want it to be lost among other posts in the main TI thread. Therefore I open a new thread about it.

Suppose that Alice measures the field ##\phi(x)## at the point ##x_A## and Bob measures the field at the point ##x_B##. The values of the field obtained by Alice and Bob are denoted by ##\varphi_A## and ##\varphi_B##, respectively. After that Alice and Bob meet and tell their results of measurements to each other. Then they compute the product of those results, the numerical value of which is

$$\varphi_{AB}=\varphi_A\varphi_B$$

But according to TI we have

$$\varphi_A=\langle\phi(x_A)\rangle, \;\;\; \varphi_B=\langle\phi(x_B)\rangle$$

$$\varphi_{AB}=\langle\phi(x_A)\phi(x_B)\rangle$$

which implies

$$\langle\phi(x_A)\rangle\langle\phi(x_B)\rangle = \langle\phi(x_A)\phi(x_B)\rangle \;\;\;\;\; (1)$$

On the other hand, in QM (or QFT, to be more precise) in general we have

$$\langle\phi(x_A)\rangle\langle\phi(x_B)\rangle \neq \langle\phi(x_A)\phi(x_B)\rangle$$

so it seems that QM is in contradiction with TI. In other words, the nonlocal TI beable ##\langle\phi(x_A)\phi(x_B)\rangle## seems inconsistent with the local TI beables ##\langle\phi(x_A)\rangle## and ##\langle\phi(x_B)\rangle##.

A possible way out of this conundrum is to take into account quantum contextuality. Perhaps the equality (1) is not always satisfied, but only at the time of measurement. And perhaps at that time the state ##\rho## is something close to ##|\psi\rangle\langle\psi|## where ##|\psi\rangle## is an eigenstate of both ##\phi(x_A)## and ##\phi(x_B)##. If so, then (1) would be at least approximately valid, which might resolve the problem in the FAPP sense.

But there are problems. First, in my opinion, it is not clear how can ##\rho## in TI evolve in such a way to provide the FAPP equality (1). I have discussed this in a somewhat different context in https://www.physicsforums.com/threads/proof-that-the-thermal-interpretation-of-qm-is-wrong.970038/ . Second, even if I ignore this problem, or assume that it is somehow solved in a way I don't understand, there still remains another problem. If FAPP equalities of the form (1) are valid contextually, i.e. whenever the corresponding experiments are performed, then, in the FAPP sense, all measurements can be reduced to measurements of local beables such as ##\langle\phi(x_A)\rangle## and ##\langle\phi(x_B)\rangle##. The nonlocal beables such as ##\langle\phi(x_A)\phi(x_B)\rangle## become superfluous.

Finally a comment on nonlocal action at a distance. If only local beables exist, then the Bell theorem implies that there must be a nonlocal action at a distance between them. If, on the other hand, a measurement of correlation is really a measurement of a nonlocal beable such as ##\langle\phi(x_A)\phi(x_B)\rangle##, one might think that it can avoid a nonlocal action at a distance. But it is true only if the problem of evolution of ##\rho## discussed in the paragraph above is solved without introducing a nonlocal mechanism such as an objective collapse. But, as I argued in the link above, I think the problem of evolution of ##\rho## cannot be solved in that way.

Suppose that Alice measures the field ##\phi(x)## at the point ##x_A## and Bob measures the field at the point ##x_B##. The values of the field obtained by Alice and Bob are denoted by ##\varphi_A## and ##\varphi_B##, respectively. After that Alice and Bob meet and tell their results of measurements to each other. Then they compute the product of those results, the numerical value of which is

$$\varphi_{AB}=\varphi_A\varphi_B$$

But according to TI we have

$$\varphi_A=\langle\phi(x_A)\rangle, \;\;\; \varphi_B=\langle\phi(x_B)\rangle$$

$$\varphi_{AB}=\langle\phi(x_A)\phi(x_B)\rangle$$

which implies

$$\langle\phi(x_A)\rangle\langle\phi(x_B)\rangle = \langle\phi(x_A)\phi(x_B)\rangle \;\;\;\;\; (1)$$

On the other hand, in QM (or QFT, to be more precise) in general we have

$$\langle\phi(x_A)\rangle\langle\phi(x_B)\rangle \neq \langle\phi(x_A)\phi(x_B)\rangle$$

so it seems that QM is in contradiction with TI. In other words, the nonlocal TI beable ##\langle\phi(x_A)\phi(x_B)\rangle## seems inconsistent with the local TI beables ##\langle\phi(x_A)\rangle## and ##\langle\phi(x_B)\rangle##.

A possible way out of this conundrum is to take into account quantum contextuality. Perhaps the equality (1) is not always satisfied, but only at the time of measurement. And perhaps at that time the state ##\rho## is something close to ##|\psi\rangle\langle\psi|## where ##|\psi\rangle## is an eigenstate of both ##\phi(x_A)## and ##\phi(x_B)##. If so, then (1) would be at least approximately valid, which might resolve the problem in the FAPP sense.

But there are problems. First, in my opinion, it is not clear how can ##\rho## in TI evolve in such a way to provide the FAPP equality (1). I have discussed this in a somewhat different context in https://www.physicsforums.com/threads/proof-that-the-thermal-interpretation-of-qm-is-wrong.970038/ . Second, even if I ignore this problem, or assume that it is somehow solved in a way I don't understand, there still remains another problem. If FAPP equalities of the form (1) are valid contextually, i.e. whenever the corresponding experiments are performed, then, in the FAPP sense, all measurements can be reduced to measurements of local beables such as ##\langle\phi(x_A)\rangle## and ##\langle\phi(x_B)\rangle##. The nonlocal beables such as ##\langle\phi(x_A)\phi(x_B)\rangle## become superfluous.

Finally a comment on nonlocal action at a distance. If only local beables exist, then the Bell theorem implies that there must be a nonlocal action at a distance between them. If, on the other hand, a measurement of correlation is really a measurement of a nonlocal beable such as ##\langle\phi(x_A)\phi(x_B)\rangle##, one might think that it can avoid a nonlocal action at a distance. But it is true only if the problem of evolution of ##\rho## discussed in the paragraph above is solved without introducing a nonlocal mechanism such as an objective collapse. But, as I argued in the link above, I think the problem of evolution of ##\rho## cannot be solved in that way.