  The conventional transport of particles in the on-shell quasiparticle limit
is extended to particles of finite life time by means of a spectral function
A(X,\vec{P},M^2) for a particle moving in an area of complex self-energy
\Sigma^{ret}_X = Re \Sigma^{ret}_X -i \Gamma_X/2. Starting from the
Kadanoff-Baym equations we derive in first order gradient expansion equations
of motion for testparticles with respect to their time evolution in \vec{X},
\vec{P} and M^2. The off-shell propagation is demonstrated for a couple of
model cases that simulate hadron-nucleus collisions. In case of nucleus-nucleus
collisions the imaginary part of the hadron self-energy \Gamma_X is determined
by the local space-time dependent collision rate dynamically. A first
application is presented for A + A reactions up to 95 A MeV, where the effects
from the off-shell propagation of nucleons are discussed with respect to high
energy proton spectra, high energy photon production as well as kaon yields in
comparison to the available data from GANIL.
