- Aim: give a consistent description of electron relaxation and energy renormalization
- In contrast to free electrons which are characterized by an undamped electron DeBroglie wave,
correlations lead to damping
- To see the damped spectral functions,
Kadanoff-Baym equations for an electron gas
- Equations of motion for the two-time correlation functions.
- The g's have to obey the adjoint equation (second equation) also.
- The I's are the scattering (collision) integrals containing the influence of all other particles.
Relation to the center of mass and difference time, t and tau
Relation of g< to the electron (Wigner) distribution function
- On the time-diagonal, g is essentially the Wigner function.
- Consider the difference of the two KB equations above
- The equal time limit (second equation below) is just the kinetic equation for the
Wigner distribution (third equation below) - you only need to identify f and I.
Additional information in the Green's functions
- Due to the two-time structure, the g's contain essentially more information than the
single-time Wigner distributions.
- Besides this statistical information, they contain the full dynamic properties -
the energy spectrum - which is reflected by the behavior away from the time-diagonal
- Consider the sum of the two KB equations above:
- Solution for free particles (I=0):
- For interacting particles expect the generalization:
- which corresponds to an equation with an "effective" (complex) energy -
which is a simple special case of the above equation.
Thus, the Kadanoff-Baym equations yield the renormalized single-particle spectrum,
fully selfconsistently. To see the corresponding spectral functions,
Idea of nonequilibrium Greens functions
See time-dependent solutions |
Results for optically excited electrons in semiconductors
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