Stanis aw Mrówczynski
So tan Institute for Nuclear Studies,
ul. Hoza 69, PL - 00-681 Warsaw, Poland
and Institute of Physics, Pedagogical University,
ul. Konopnickiej 15, PL - 25-406 Kielce, Poland
20-th November 1999
Transport theory is a very convenient tool to study many-body nonequilibrium systems, nonrelativistic and relativistic as well. The kinetic equations which play a central role in the transport approach can be usually derived by means of simple heuristic arguments similar to those which were used by Ludwig Boltzmann over hundred years ago when he introduced his famous equation. However, such arguments are insufficient when one studies a system of very complicated dynamics such as the quark-gluon plasma governed by QCD. Then, one has to refer to a formal scheme which allows one to derive the transport equation directly from the underlying quantum field theory. The formal scheme is also useful to understand the limits of the kinetic approach. The derivation then shows the assumptions and approximations which lead to the transport theory, and hence the domain of its applicability can be established.
Schwinger-Keldysh formulation of the quantum field theory provides a natural basis for the transport equation derivation. Kadanoff and Baym developed the technique for nonrelativistic quantum systems and we, among others, have generalized it to study relativistic quantum fields. At first we considered the self-interacting scalar fields  and then the so-called Walecka model where one deals with the system of spinor fields interacting with the vector and scalar ones . The third paper of the series  is devoted to the fields which are massless as bare ones. It appears that this case demands a specific treatment where one starts with the effective lagrangian of massive quasiparticles. At present we work with the system of Higgs field which, depending on the conditions, exists in the symmetric or asymmetric phase.
The technical aspects of the transport equation derivation vary noticeably with the dynamical model under consideration but the general scheme is always the same. The main steps of the derivation are following. One defines the contour Green function with the time arguments on the contour in a complex time plane. This function is a key element of the Schwinger-Keldysh approach. One writes down the exact equations of motion of the contour Green function which are of the Dyson-Schwinger form. Assuming a macroscopic quasi-homogeneity of the system, we perform the gradient expansion and the Wigner transformation. Then, the pair of Dyson-Schwinger equations is converted into the transport and mass-shell equations both satisfied by the Wigner function. The latter equation provides, in particular, the analog of the gap equation which allows one to determine the dynamically generated effective mass of quasiparticles. One performs the perturbative analysis of the self-energies, which enter the equations, and show how the Vlasov and collisional terms emerge. Then, the distribution function of the finite width quasiparticles is defined. The function is of standard probabilistic interpretatioan. Finally, we find the transport equations satisfied by the distribution functions. There are the mean-field and collision terms. In the case of spinor or vector fields the transport equations include the dynamics of spin degrees of freedom.