33. Schwinger-Keldysh: information carried by Keldysh Green’s functions (nonequilibrium diagrammatic methods 1)

Posted by Dmitry
May 1, 2008

Last time we introduced Keldysh Green’s functions and generating functional for non-equilibrium field theory of a single scalar field. Let us now discuss the information carried by them (and hopefully it will become clear why we need more than one Green’s function in the situation out of equilibrium).

For a free massive scalar field \chi(x) one has

G^{++}(k)=(k^{2}-m^{2}+i\epsilon)^{-1}-2\pi in(k)\delta(k^{2}-m^{2}), (1)

G^{–}(k)=-(k^{2}-m^{2}+i\epsilon)^{-1}-2\pi in(k)\delta(k^{2}-m^{2}), (2)

G^{+-}(k)=-2\pi i(\theta(k^{0})+n(k))\delta(k^{2}-m^{2}). (3)

Probably, the simplest way to see this is to use the WKB representation. For example, the limit x\to x' of the G^{++} Green’s function (neglecting the vacuum contribution) is given by

\langle\chi^{+}(x)\chi^{+}(x)\rangle=\int\frac{d^{3}k}{(2\pi)^{3}2\omega_{k}}n_{k}=
=2\pi\int\frac{d^{4}k}{(2\pi)^{4}}n_{k}\delta(\omega_{k}^{2}-k^{2}-m^{2}).

Recalling that \langle\chi^{+}(x)\chi^{+}(x)\rangle=iG^{++}(x,x) we come to the expression (1).

After the Keldysh rotation one has

G^{K}(k)=-2\pi i(1+2n_{k})\delta(k^{2}-m^{2}),
G^{R}(k)=(k^{2}-m^{2}+i0)^{-1},
G^{A}(k)=(k^{2}-m^{2}-i0)^{-1}.

We conclude that the Keldysh Green function G^{K} carries the information about distribution function n_{k}, while the retarded and advanced Green functions define spectrum of particles (and are independent of the distribution function). This separation is only true for systems sufficiently close to the thermal equilibrium.

The reason why we need more than a single Green’s function is now clear. Away for equilibrium vacuum state gets excited, and the number of excitations in a given mode k depends on time (how in particular? we will discuss it later). Advanced and retarded Green’s functions carry information only about the spectrum, so we need something else to describe dynamics of the system completely. This “something else” is the Keldysh Green’s function G^K.

Let us discuss a bit the issue of the spectrum. In principle, a physical system is completely described by its Hamiltonian (and initial density matrix). The latter, generally speaking, contains information bout all the interactions between the relevant degrees of freedom in the system, and these interactions may be strong. How to understand which degrees of freedom are relevant for the dynamics? One needs to diagonalize the Hamiltonian, and this can be done only approximately in the general case. (Approximate) eigenstates of the Hamiltonian will correspond to relevant degrees of freedom - and that is what G^A and G^R - they show the basis of eigenstates of the diagonalized Hamiltonian (in the case of free field theory, as above, these states are Fourier harmonics).

Generally speaking, since diagonalization is not exact, the states may have some width (related to the strength of interactions). Only if their width is much smaller than the difference between the energies of nearest eigenstates, description in terms of separate degrees of freedom is possible.

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Condensed Matter, Fellow Craft, Non-equilibrium diagrammatic methods, Quantum field theory

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43. Schwinger-Keldysh: Martin-Siggia-Rose diagrammatics (non-equilibrium diagrammatic methods 2)
31. Schwinger-Keldysh: brief review (Nonequilibrium diagrammatic methods 1)
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