This talk is based on the recent paper nucl-th/9909080. It is motivated by the finding that e.g. in more energetic heavy ion collisions the (decay and/or scattering) width of the involved states might get comparable to or larger than the typical temperature. Traditionally for the description of such systems at lower energies the quasi-particle approximation is used to reduce the field theoretical equations to a tractable transport equation of Vlasov-Botzmann type. For the case described above where the states become large width, however, the quasi-particle picture is no longer reasonable. Therefore a scheme has to be developed to deduce and solve a transport equation beyond the quasi-particle regime.

Basically the talk is devided into two parts. In the first part I describe how to derive in first order gradient approximation from the underlying quantum field theoretical Kadanoff-Baym equations an off-shell transport equation. No further approximations have to be made and especially the quasi-particle approximation is avoided. Nonetheless the presented transport equation is still tractable since an extended test particle description can be used to solve the off-shell transport equation.

In the second part I describe some conceptually important details of this test particle description. The corner stone is the identification of an exactly conserved quantity which does not coincide with the particle number of the full quantum field theory. This conserved quantity is interpreted as a pseudo-particle number obtained by coarse graining. Using a test particle ansatz for this conserved quantity allows to rewrite the transport equation into equations of motion for test particles. Finally the two-body collision terms are formulated in terms of the test particles which gain non-trivial renormalization factors due to the coarse graining process.