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.

  • 1
    $\begingroup$ Suggestion to the post (v2): Limit the scope to QM rather than QFT. $\endgroup$ – Qmechanic Feb 7 at 21:01
  • 1
    $\begingroup$ 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? $\endgroup$ – Cosmas Zachos Feb 7 at 21:36
  • 1
    $\begingroup$ @user220348: I am afraid you're assuming much more knowledge about quantum nonlocality on my side than there actually is... $\endgroup$ – oliver Feb 7 at 21:42
  • 2
    $\begingroup$ Regarding the need "to bring fermions into play": quantum nonlocality is well established for systems with only photons, no need for fermions. $\endgroup$ – fqq Feb 7 at 22:09
  • 1
    $\begingroup$ 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... $\endgroup$ – Stéphane Rollandin Feb 8 at 0:18

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.