43. Schwinger-Keldysh: Martin-Siggia-Rose diagrammatics (non-equilibrium diagrammatic methods 2)

This post is again from the series devoted to the discussion of non-equilibrium diagrammatic methods. Last time we have found trivial saddle points of the Schwinger-Keldysh action and discovered that quasi-classical dynamics of a quantum system is actually governed by the Langevin-type (and therefore Fokker-Planck) equations as it should be. I also briefly mentioned in the end that the opposite procedure is in fact possible: one can start from a Langevin/Fokker-Planck equation (or, generally speaking, with any equation describing diffusive dynamics) and construct a classical limit of the correponding Schwinger-Keldysh diagrammatic technique (note that derivation of Langevin, Fokker-Planck and kinetic equations typically includes coarse-graining, so it is impossible to recover the full Schwinger-Keldysh action just, say, knowing Fokker-Planck; it means that different quantum theories can give similar classical dynamics).

The procedure was first introduced by Martin, Siggia and Rose (and independently by DeDominics).
Martin-Siggia-Rose diagrammatic methods are used a lot in applications to the theory of turbulence and various problems involving stochasticity (so, yes, in market studies, too )

Starting with the Fokker-Planck equation

,

one writes down the corresponding “generating functional” as a functional -function

which, after introducing the auxiliary field , acquires the form

.(1)

There exists a remarkable similarity between this generating functional and the Schwinger-Keldysh generating functional describing the quasiclassical approximation of the quantum non-equilibrium dynamics – for example, in the Martin-Siggia-Rose diagrammatic technique the field plays the same role as the quatum field plays in the Schwinger-Keldysh technique.

Of course, as it is, the diagrammatic expansion of the generating functional (1) is not of much use, since we are not interested in correlation functions of or – is itself the probability measure for a random walk/diffusion process governed by the corresponding Fokker-Planck equation.

(There are exceptions for this rule though – for example, in eternal inflation such diagrammatic technique is useful as it is – we will discuss it later).

Observables in the problem under consideration (such as the mean square displacement ) are related to the physics of this random walk. However, the correlation functions of can be determined from the correlation functions in the momentum representation generated by (1) by differentiating over momenta. For example, one has

where is the Fourier component of the Green function . This is due to the correspondence between the Langevin equation describing the dynamics of the observable and the Fokker-Planck equation describing the dynamics of probability distribution for the diffusion process.