The following manuscript aims at an introduction to modern methods in relativistic quantum many-
body theory. In this manuscript we introduce many-body techniques, using relativistic quantum field theory, em-
phasizing the socalled
real-time formulation
, which allows the investigation of
dynamical properties
of quantum systems. We start with a thorough discussion of
thermal equilibrium
, where the
closed
real time-path formalism of Schwinger and Keldysh
is extended by an
imaginary-time path
, lead-
ing to a unification of the real-time methods with the older imaginary-time (Matsubara) formalism via
analytic continuition. Later, we shall consider more generall
off-equilibrium situations
, concentrat-
ing on the derivation of semiclassical
transport equations
from the underlying microscopic quantum-
field theoretical models. Despite its great practical applicability to the analysis of many-body systems
off thermal equilibrium, it also helps to answer the fundamental question, how the classical descrip-
tion of matter and fields (mostly the electromagnetic field) emerges from the underlying fundamental
quantum theory. |