39. Schwinger-Keldysh: Quasiclassical Keldysh action (non-equilibrium diagrammatic methods 1)
Save This Post as PDFThe is a continuation of the series of posts devoted to the discussion of non-equilibrium diagrammatic methods. Last time we discussed what information is carried by different Keldysh Green’s functions. Today I want to finally start alking about dynamics and will determine saddle points of the Schwinger-Keldysh action.
The trivial saddle point of the generating functional
![Z[J_{\chi}^{c}]=\int{\cal D}\chi^{a}\exp(-i(S[\chi^{a}]+\int d^{4}x\sqrt{-g(x)}J_{\chi}^{a}(x)\chi^{a}(x))](http://www.nonequilibrium.net/latexrender/pictures/39e643f12d43a9c5d4b2459b1566ea0d.gif "Z[J_{\chi}^{c}]=\int{\cal D}\chi^{a}\exp(-i(S[\chi^{a}]+\int d^{4}x\sqrt{-g(x)}J_{\chi}^{a}(x)\chi^{a}(x))")
rewritten in terms of quantum and classical Keldysh fields
is determined by the equations
and
![\frac{\delta S}{\delta\bar{\chi}_{q}}=2{\cal O}_{R}[\chi_{cl}]\chi_{cl}=0](http://www.nonequilibrium.net/latexrender/pictures/8dd14a2d7e18cbc316f8c25a5a0d3a8a.gif "\frac{\delta S}{\delta\bar{\chi}_{q}}=2{\cal O}_{R}[\chi_{cl}]\chi_{cl}=0")
,
where 
is retarded operator desribing the dynamics of the classical Keldysh field. One can immediately see that 
on the trivial saddle point, and 
. In fact, the latter remains true even if one considers fluctuations near this saddle point.
To consider the quasiclassical limit, fluctuations of 
should be allowed near the classical trajectory. Let us keep only terms up to the second order in in the Keldysh action. The semiclassical action will have the following form
![S_{scl}=2\int\int dtdt'\left[\bar{\chi}_{q}[{\cal G}^{-1}]^{K}\chi_{q}+(\bar{\chi}_{q}{\cal O}^{R}[\chi_{cl}]\chi_{cl}+c.c.)\right],](http://www.nonequilibrium.net/latexrender/pictures/ea0ed4e01dd152ae218de778f95473c5.gif "S_{scl}=2\int\int dtdt'\left[\bar{\chi}_{q}[{\cal G}^{-1}]^{K}\chi_{q}+(\bar{\chi}_{q}{\cal O}^{R}[\chi_{cl}]\chi_{cl}+c.c.)\right],")
where 
denotes the complex conjugation.
One simple way to treat this semiclassical theory is to use the Hubbard–Stratonovich transformation introducing the auxiliary stochastic field 
and decoupling the quadratic term in the quasiclasical equation. One will find that the resulting action is linear with respect to , i.e., the integration over leads to the functional 
-function and to the stochastic Langevin equation
![{\cal O}^{R}[\phi_{cl}]\phi_{cl}(t)=\xi(t),](http://www.nonequilibrium.net/latexrender/pictures/0daedc51acd52621ba25f5e529c4b1cd.gif "{\cal O}^{R}[\phi_{cl}]\phi_{cl}(t)=\xi(t),")
(1)
where
![\langle\xi(t)\bar{\xi}(t')\rangle=\frac{i}{2}[{\cal G}^{-1}]^{K}(t,t').](http://www.nonequilibrium.net/latexrender/pictures/21a602902364cd42432760142d4e772d.gif "\langle\xi(t)\bar{\xi}(t')\rangle=\frac{i}{2}[{\cal G}^{-1}]^{K}(t,t').")
Another way to deal with the semiclassical theory is to integrate the field out completely since its contribution into the action is quadratic. The result is the theory of the classical Keldysh field
![S[\chi_{cl}]=2\int\int_{-\infty}^{\infty} dtdt'\bar{\chi}_{cl}({\cal O}^{A}[\bar{\chi}_{cl}]{\cal G}^{A})[{\cal G}^{K}]^{-1}({\cal G}^{R}{\cal O}^{R}[\chi_{cl}])\chi_{cl}.](http://www.nonequilibrium.net/latexrender/pictures/ad6ee2e401e9bc43d647440b374e0d57.gif "S[\chi_{cl}]=2\int\int_{-\infty}^{\infty} dtdt'\bar{\chi}_{cl}({\cal O}^{A}[\bar{\chi}_{cl}]{\cal G}^{A})[{\cal G}^{K}]^{-1}({\cal G}^{R}{\cal O}^{R}[\chi_{cl}])\chi_{cl}.")
If non-linearities with respect to 
are neglected, the theory is nothing else but the free field theory with a very complicated propagator. Using first quantized version of this theory, one can show that dynamics of the probability 
to find a particle excitation in the point 
at time 
is governed by the Fokker-Planck equation. Of course, this is not suprprising since, as we have shown, variation of the classical limit of the Schwinger-Keldysh equation gives the the Langevin equation (1).
In fact, the opposite procedure is 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. This procedure was first introduced by Martin, Siggia and Rose and we will discuss it the next time.
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