Basic idea of nonequilibrium Greens functions

(Kadanoff-Baym equations)

Aim: give a consistent description of electron relaxation and energy renormalization
Expectation: due to correlations and scattering, the electron spectrum is modified, "renormalized", giving rise to a finite life time of the electron states
In contrast: free electrons are characterized by an undamped electron DeBroglie wave (see schematic figure below)
To see the Kadanoff-Baym equations, click here

Shown is the real part of A(t,t') for a constant momentum value
This is the result of full solution of the Kadanoff-Baym equations
One clearly sees the decay away from the diagonal, which is the result of e-e-scattering (and other scattering mechanisms)

Shown is the real part of the spectral function together with its absolute value (dots)
Left column: Spectral function versus difference time tau=t-t'
Right column: Spectral function versus frequency (energy), i.e. after Fourier transform with respect to tau
First line: Spectral function of free (quasi-) particles - it is sharply localized in energy, i.e. it has zero width, corresponding to infinite lifetime
Second line: interacting particles - heuristic account of correlations and finite life time effects by introduction of an exponential damping factor gamma. In frequency space (right fig.), this gives rise to a Lorentzian function. (These factors can be derived from the electron selfenergy: gamma arises from its imaginary part and Delta is related to the real part)
The Lorentzian has fundamental problems at large frequencies: it decays too slowly. This leads to violation of energy conservation. This can be "cured" by correcting the behavior on the time-diagonal.
Third line: Spectral function from a full Kadanoff-Baym calculation. Notice the zero slope of A at the time diagonal. This translates into a fast decay at large frequency.

Picture from M. Bonitz, Quantum Kinetic Theory