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A simple observation that any insertions within the path integral are classical variables (Not operators) and hence, objects inside the path integral "commute" (is symmetric under exchange). Hence, any correlation function computed via the path integral procedure must be time ordered in nature.

My question is two-fold:

  1. Is there any method which allows us to compute OTOC's using the path integral formalism?

  2. Are there any examples in which path integral has been used to study Thermalization or Non-equilibrium Thermodynamics?

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  1. The correspondence between the operator formalism and the path integral formalism is, in general, not fully understood and the issue of ordering prescriptions remains something that one may understand in specific cases but for which no general method is known. The "equivalence" of the path integral and operator formalisms is of a formal nature and has to discard/arbitrarily deal with several terms related to ordering ambiguities along the way, see also Qmechanic's answers here and here for more details and examples.

  2. The path integral in the Keldysh-Schwinger formalism is a standard tool in non-equilibrium QFT, cf. e.g. Berges' "Introduction to non-equilibrium QFT".

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I think this is what you are looking for:Schwinger-Keldysh formalism. Part I: BRST symmetries and superspace

As mentioned the Keldysh formalism is the way to go. In this work they aim to provide a more mathematically elegant formulation of such formalism by means of a set of underlying BRST symmetries. It also has a chapter on computation of Out-of-time-order correlators.

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