  We have developed a time propagation scheme for the Kadanoff-Baym equations
for general inhomogeneous systems. These equations describe the time evolution
of the nonequilibrium Green function for interacting many-body systems in the
presence of time-dependent external fields. The external fields are treated
nonperturbatively whereas the many-body interactions are incorporated
perturbatively using Phi-derivable self-energy approximations that guarantee
the satisfaction of the macroscopic conservation laws of the system. These
approximations are discussed in detail for the time-dependent Hartree-Fock, the
second Born and the GW approximation.
