**
In this talk I discuss the idea of nonequilibrium Green's functions
(NEGF) and recent theoretical and computational results. I will start by
introducing the real-time (Keldysh) Green's functions for a many-body
system of bosons or fermions and their equations of motion - the
Keldysh-Kadanoff-Baym equations (KBE). Their main advantage is that they
guarantee the relevant conservation laws, are applicable to fields of
arbitrary intensity as well as to arbitrarily short pulses. There are
two main lines of research:
**

**1. Use the KBE to derive general quantum kinetic equations for the
Wigner function or density matrix. These equations contain collisions
including nonlinear field effects, finite collision durations etc. I
will illustrate this by examples from dense laser plasmas.
**

**2. Direct solution of the KBE for the two-time NEGF. Here numerical
results were obtained for the correlated electron gas and for
electron-hole plasmas in semiconductors in excited by optical pulses.
These results can directly be extended to small atoms interacting with
short x-ray pulses. Here first results have been obtained by our group,
in particular by my affiliate Karsten Balzer. I will give a brief
overview and discuss the capabilities of this method.
**

The presentation contains a summary of the talk and a few additional applications with references which were skipped due to time limitations, in particular:

- transparancies 21-24 an example how to derive quantum kinetic equations from the two-time NEGF. The example shows a gauge-invariant derivation of the collision integral of an electron-ion plasma in the presence of a monochromotic field (optical to x-ray). The result includes multiphoton absorption and inverse bremsstrahlung heating of electrons.
- 25-27: numerical solution of the derived kinetic equation for an electron-ion plasma in a strong field, including harmonics generation.
- 28 demonstration of total energy conservation - observed by full solution of the two-time Keldysh-Kadanoff-Baym equations (KBE) for the NEGF. This is in contrast to Boltzmann type (Markovian) kinetic equations which conserve only kinetic energy, transparancy 29.
- 30-31 Solution of the KBE for a two band semiconductor under optical excitation.
- 32-34 Solution of the KBE for an inhomogeneous electron gas which yields the dynamic structure factor including correlations and vertex corrections in a sum rule preserving fashion.

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**

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