Introduction to Relativistic Transport Theory

Bibliography 43 3 Contents 4 Preface The present manuscript are lecture notes on an introduction to relativistic transport theory, held at the SERC School on Modern Theories of Nuclear Reactions in Roorkee, India. The aim of these lectures is to provide a self-contained presentation of the general foundations of rela-tivistic transport theory with the aim to provide the theoretical tools for the application of such methods to relativistic heavy-ion collisions. Here, the main challenge lies in the fact that one has to describe strongly coupled systems of hadrons that mostly have a considerable finite width already in the vacuum. In addition they are strongly interacting and thus their properties change drastically in the medium. This leads to the necessity to develop equations for the propagation of particles off their mass shell, which is still a challenging problem although a lot of progress in this respect has been made in the recent years. These notes are structured as follows. In Chapter 1 we develop the basic ideas of kinetic theory for classical relativistic particles. After a short reminder on special relativity we consider the Boltzmann equation for a many-body system with one species of indistinguishable particles with a collision term considering elastic two-body collisions. Then we shall discuss the most important general applications of this equation, as the derivation of the Boltzmann-H theorem, the equilibrium distributions, the validity of conservation laws, and hy-drodynamical equations. We conclude this chapter with a brief discussion of the extension to take into account quantum statistics (i.e., the case of degenerate gases) in the Boltzmann-Uehling-Uhlenbeck approach. In Chapter 2 we turn our attention to relativistic quantum-field theory, making the connection with the previous Lectures of the School on fundamental interactions, but extending the quantum-field theoretical techniques to the case of many-body theory. In these notes I concentrate on the Schwinger-Keldysh real-time formalism which is suitable for both equilibrium and non-equilibrium situations. We shall introduce the path-integral formulation for the most simple relativistic quantum field theory , the φ 4 model and develop the important technique of generating functionals for the derivation of self-consistent equations for the one-particle Green's functions, the Kadanoff-Baym equations, which provide the starting point for the derivation of transport equations in Chapter 3, where we discuss various levels of approximations for the off-shell kinetics of broad resonances and their properties regarding fundamental issues as the validity of the conservation laws and the H-theorem. …