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I have always had my difficulties (being bored, impatient) with the theoretical considerations around quantum nonlocality, especially Bell's inequalities. What makes it cumbersome for me is that these analyses try to prove something about the world without exactly knowing what "world" means (i.e. what type of quantum theory to consider in particular).

Since the path integral formalism of quantum field theory is pretty tolerant with respect to the question how the world is working in detail (the kind of tolerance the proofs of Bell's have to work out in the first place), but nonetheless captures everything about being "quantum" in a very vivid way, I have asked myself, whether it is possible to understand quantum nonlocality in the path integral formalism more elegantly.

So given a partition function $$Z = \int e^\frac{i\mathcal{S}[\mathbf{x}]}{\hbar}\, \mathcal{D}\mathbf{x}$$ and a classical action $$\mathcal{S}[\mathbf{x}]=\int_0^T L[\mathbf{x}(t),\dot{\mathbf{x}}(t)]\, dt$$ the partition function represents kind of a "wave function" for all possible paths (especially those that are not classical "action minimizers", e.g. paths with jumps etc.). Classical paths become most probable (a form of Ehrenfest theorem so to say) because their action is minimal, which is why they contribute only slow phases to $Z$, whereas non-classical paths with higher action tend to contribute fast, cancelling phases (I hope I recall this right from my memory).

Then, how could one prove generically (i.e. without reference to the specific Lagrangian $L$), that it is not possible to describe the system classically with "local hidden variables", i.e. prove quantum nonlocality?

I know that for bosons-only theories (Yang-Mills-Theory) there exists a classical statistical interpretation of $Z$ (which is obtained by Wick rotation $it=\tau$), which I think is close to already representing local hidden variables (just as classical statistical mechanics has hidden variables for all the particle positions and velocities we do not know macroscopically). So I would assume, that one has to bring fermions into play (where $Z$ includes Grassmann variables) in order to get true quantum non-locality. Is that picture correct?

Edit: If it serves the purpose, feel free to limit the scope to quantum mechanics instead of QFT. But my feeling is that by excluding special relativity (and hence, inevitably, field theory) the possible relation between nonlocality and proper causality could get lost. Don't know exactly, because, as mentioned, I don't understand Bell's arguments.

oliver
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    Suggestion to the post (v2): Limit the scope to QM rather than QFT. – Qmechanic Feb 07 '21 at 21:01
  • https://en.wikipedia.org/wiki/Von_Neumann_entropy?wprov=sfla1 – user220348 Feb 07 '21 at 21:26
  • https://physics.stackexchange.com/q/327773/ plus hamiltonian flow which describes the evolution of the system similar to how normal incompressible flow functions. – user220348 Feb 07 '21 at 21:29
  • @user220348: interesting references, but to be honest, I don't see the connection to my question. I know how classical statistics is derived from entropy. Do you mean applying entropy somehow to the wick rotated path integral? I am confused. – oliver Feb 07 '21 at 21:33
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    I believe you are dazzled by the glamor of path integrals and attribute features to them they lack. They are an intuitive and elegant method to calculate transition amplitudes (propagators) for states, efficiently folding in all quantum interference effects. Any correlation, collapse, measurement, etc... issues are strictly identical to those of the standard Hilbert space formulation. Do you have something specific or trenchant in mind? – Cosmas Zachos Feb 07 '21 at 21:36
  • Which hidden variable postulate do you wish to test. We can see that the H flow follows a maximum velocity constraint meaning that destruction of our entangled state via entropy must also follow that rule – user220348 Feb 07 '21 at 21:38
  • @CosmasZachos: I know that they are identical to Hilbert space formulation because the are derived from it. But I find path integrals more intuitive. Independent of that, you're probably right in that I believe that there could be more to path integrals than is apparent in Hilbert space. – oliver Feb 07 '21 at 21:40
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    @user220348: I am afraid you're assuming much more knowledge about quantum nonlocality on my side than there actually is... – oliver Feb 07 '21 at 21:42
  • What does non locality mean for you exactly? – user220348 Feb 07 '21 at 21:58
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    Regarding the need "to bring fermions into play": quantum nonlocality is well established for systems with only photons, no need for fermions. – fqq Feb 07 '21 at 22:09
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    It seems to me that, at least intuitively, the path integral formulation implies non-locality to start with, since by construction it amounts to an instantaneous scan of a full domain within some boundary conditions. If that is not a non-local way of doing things, I don't know what is... – Stéphane Rollandin Feb 08 '21 at 00:18
  • @fqq: maybe I am misunderstanding my superficial knowledge of quantum field theory, but can photons interact with each other? AFAIK not, unless there is at least pair creation. But If photon's don't interact, how can a theory of only photons describe the measurement process that is supposed to be the root of quantum nonlocality? – oliver Feb 08 '21 at 15:55
  • @StéphaneRollandin: that may well be, but can you put "instantaneous scan of a full domain within some boundary conditions" into some grossly mathematical form? I am afraid I don't know what you mean by this phrase? – oliver Feb 08 '21 at 15:59
  • @user220348: that is the very question. What should nonlocality mean for me? As stated, I don't know what people mean by that currently. All I know is that the naive nonlocality, where one was allegedly able to communicate faster than light (which Einstein feared), has long been debunked. So what is left of nonlocality then? My impression was that it's the Bell inequalities, so how do these translate into path integral language, so that a quantum DIY'er like me can understand it. – oliver Feb 08 '21 at 16:04
  • @user220348: maybe what you said about the maximum velocity constraint of H flow can answer my question. I don't know how to follow that hint myself, but If that's well established theory maybe you have the reference to a paper that discusses it? – oliver Feb 08 '21 at 16:13
  • No, I can't (what why I said 'intuitively'). What I meant by 'scaning a full domain' (forget about the 'instantaneous', it's only a figure of speech - better say 'at once') is that, just as its name indicates, a path integral is an integral on all admissible (another can of worms) paths. This means the correct result relating to a specific physical problem described by boundary conditions is computed by taking into account all possibilities within those boundaries. This is nonlocal in that it yields non-causal correlations - see my answer here: https://physics.stackexchange.com/a/498060/109928 – Stéphane Rollandin Feb 08 '21 at 19:34
  • @StéphaneRollandin: your referenced answer tells me neither how correlations over space-like distances nor hidden variables can be understood in the path integral method. Another try: why couldn't "all admissible paths" in the path integral method represent actual scattering processes at yet unknown/hidden particles (while only the classical path represents movement strictly according to the known particles)? I cannot understand how such hypotheses can be excluded without even knowing what the nature of those hidden scattering centers are. But that, as I understand it, does Bell claim. – oliver Feb 08 '21 at 22:37

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