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I have recently read a couple of papers on lattice QCD and found that there is a well-established connection between Boltzmann distribution and the path integral in QFT (disclaimer: I am not a huge QFT expert).

Indeed, in thermodynamics the probability to find a system (in thermal equilibrium with a heat bath at temperature $T$) in a state $\mathbf{x}$ is: $$ p(\mathbf{x}) \propto e^{-\frac{E(\mathbf{x})}{k_{\rm B}T}}. $$ up to a normalisation constant, where $E(\mathbf{x})$ is the energy of the state $\mathbf{x}$.

At the same time, in Euclidean quantum field theory, the probability to observe a field configuration $\phi$ is: $$ p(\phi) \propto e^{-\frac{S(\phi)}{\hbar}}, $$ where $S(\phi)$ is the Euclidean action corresponding to the field configuration $\phi$.

This undoubtedly looks like a cool correspondence, and I have learned since that it is actually being used in some Monte Carlo simulations.

However, if I am not mistaken, the Boltzmann distribution is not fundamental per se — just like the Second Law, it emerges for a macroscopic system out of interaction between its microscopic constituents under mild assumptions. In this case, temperature just gives us some reference energy value after the system equilibrated, and for different systems it is different.

Now, my question is: can the path integral be emergent too? Can it be that the expression for $p(\phi)$ is not fundamental, but in fact can be derived? (just in case: I am not talking about the textbook derivation of path integral from the Schrödinger equation).

I know everything begins to look very vague now, but it doesn't seem completely impossible to me: if interaction of microscopic subsystems gives us the Boltzmann distribution, then, I guess, for the path integral it can be some sort of interaction between elementary units of space-time hosting the field? Then, I can hypothesise the following:

  1. If I presume that the size of elementary space-time unit is roughly Planck length ($10^{-35}$ m) and time ($10^{-44}$ s), then it is no surprise we see no deviations from the path integral predictions — we just have so many of these units crammed into our region of space-time, that we always see their emergent behaviour.
  2. I can speculate that if we get to probe those lengths and times, the path integral formulation might just break apart.
  3. I can also speculate that the Planck constant is also not that fundamental, but, just like the temperature, is an emergent quantity characterising our region of space-time after its equilibration (even though it is mysterious to me what is the nature this equilibration process).

Finally, let me say that I am neither a QFT expert, nor I dare to claim that anything of the above makes actual sense. I just got a buzzing question in my head, and I would simply appreciate if somebody told me that all this is well-known and studied and provided a ton of references (and/or gentle introductions).

Qmechanic
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Aleksei Malyshev
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    Does this answer your question? What is the physical interpretation of the Wick rotation? – By Symmetry Aug 26 '23 at 14:33
  • If you're not doing Wick rotations, there are two things wrong with your second equation. First of all, it should be exp(iS), not exp(S). Second, it's not a probability, it's an amplitude. To go from amplitudes to probability, you need to sum/integrate all amplitudes for all intermediate field configurations consistent with your measured boundaries, and then take the magnitude square of the result. – Ken Wharton Aug 26 '23 at 14:41
  • You might also be interested in reading about hidden variable theories where the probabilities in quantum mechanics are explained by the existence of degrees of freedom we aren't able to directly observe. In particular, look at what Bell's inequalities imply about the most intuitive version of this idea, "local hidden variables." – Andrew Aug 26 '23 at 14:43
  • Hey @BySymmetry, no, not quite: the Wick rotation deals with the conventional definition of the path integral. My question is more like what if this definition is not a fundamental law, but only an emergent one (which wouldn't always work). But thank you! – Aleksei Malyshev Aug 26 '23 at 14:45
  • Hey @KenWharton, this question appeared in my head after reading this paper and so I followed them in defining the probability of observing the field configuration; I agree that for an amplitude we would need exp(iS). – Aleksei Malyshev Aug 26 '23 at 14:53
  • I have voted to keep your question open, but you should realise that the question you are asking is actually indeed just about the Wick rotation. Whether or not this is an emergent property is something that would have to be explored in a post-Standard-Model hypothesis, and we cannot know until a definitive answer appears in the future. All we can say right now is that the Standard Model does not take it as an emergent property, and the experimental evidence would also be difficult to fit if it were. – naturallyInconsistent Sep 03 '23 at 04:42

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