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In non-equilibrium statistical mechanics, there is this formidable formalism, called the Schwinger-Keldysh formalism. I have read about it, and I understand what it is.

However, what I what to know is if there are any concrete applications of this formalism that give answers that cannot be obtained by other techniques? For example, in calculating transport coefficients, are there problems that can be solved by Schwinger-Keldysh that cannot be solved by using the Boltzmann equation or the Kubo formula?

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In my opinion, the reason to be of the Keldysh formalism is that it is the way to write a path integral for non-equilibrium quantum systems. It provides an action which can be used to sample paths for out-of-equilibrium systems. This is great because it opens the door to the huge toolbox of equilibrium quantum field theory to non-equilibrium problems. I can give four examples that I know well. I am sure that there are many more.

  • The 2PI formalism was used in this, this and more recently this work. Non-equilibrium steady states are studied. The 2PI formalism makes it possible to re-sum an infinite series of diagrams and extract some kind of non-perturbative quantum Boltzmann equation. The collision integrals are then analysed and non-trivial scaling relations are extracted for the steady-state.

  • A very interesting parallel is made in between the Keldysn and Martin-Siggia-Rose / Janssen-DeDominicis (MSR) formalisms here. MSR tell us how to write a path integral for classical stochastic systems of differential equations. In this work the term that makes the Keldysh path integral different from the action obtained within MSR is identified. Then one can tell if a system is quantum or classical by finding out if this terms plays an important role or not.

  • In this paper, the property of being at thermal equilibrium is identified as a symmetry of the Keldysh action. The Keldysh formalism provides an action that is very general and can be used in any non-equilibrium situation. If the system reaches a steady state, this one may or may not follow a Gibbs distribution. It may or may not obey a fluctuation-dissipation theorem. These properties are hard to test for since they involve all the correlation functions of the system. Instead this work tells us that if the Keldysh action has the appropriate symmetry, then it is in thermal equilibrium. The fluctuation-dissipation relations even come out as Ward identities.

  • Renormalisation Group methods become available. In particular, the Functional Renormalisation Group is used here, here and more recently here. This makes it possible to study quantitatively far-from-equilibrium criticality. In particular, non-equilibrium critical exponents are computed.

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