Contents
I Equilibrium Theory
Introduction to quantum-field theory: Free fields
PoincarΓ© invariance
Free scalar bosons
Free Dirac fermions
The classical Dirac field
Quantization of the free Dirac field
PoincarΓ© symmetry of the quantized Dirac theory
The discrete symmetry transformations P, C and T
Sesquilinear forms of Dirac-field operators
The real-time formulation of equilibrium quantum-field theory
The general Schwinger-Keldysh contour
States and observables in the Heisenberg picture
The interaction picture
The entropy principle
Thermal equilibrium and thermodynamic potentials
Perturbation theory in thermal equilibrium
The canonical statistical operator
Thermal perturbation theory
The generating functional for Green's functions
The free contour propagator
Feynman rules
Renormalization
Path-integral formulation
Definition of the path integral
Two-point functions along the real-time contour
The two-point Green's function in equilibrium
The free equilibrium propagator
Thermodynamics of ideal Bose gases
Path-integral evaluation of the partition sum
The partition sum as functional determinant and the heat-kernel method
Functional treatment of Bose-Einstein condensation
Interacting field theory
Generating functionals
Loop expansion and effective action
Perturbative evaluation of the effective potential and renormalization
Fermions
Path integrals for Dirac fermions
Partion sum for non-interacting Dirac fermions
The free propagator
Gauge models
The electromagnetic field
Self-consistent Phi-derivable approximations
Necessity of resummations
Phi-derivable approximations
Renormalization of Phi-derivable approximations
Order lambda, m2>0
Order lambda, m2>0
Symmetry analysis of the 1PI and 2PI action functionals
Linear O(N)-sigma model
II Nonequilibrium Theory
Classical transport theory
Hamiltonian dynamics and Liouville's theorem
The BBGKY Hierarchy
The Boltzmann equation for a dilute gas
The entropy and the H theorem
Local equilibrium
Noether's Theorem
Symmetries of the action
Noether currents and conserved quantities
Imaginary-time formalism
Bosons
Bosonic Matsubara sums
Imaginary time Green's function
Mills representation in imaginary time
Finite chemical potential
Imaginary-time Feynman rules
Fermions
Fermionic Matsubara sums
Wigner representation of two-point functions
Definition and analytical properties
Convolution theorem for Wigner transforms
Bibliography