Preface
1 Quantum fields
1.1 Quantum mechanics
1.2 N-particle system
1.2.1 Identical particles
1.2.2 Kinematics of fermions
1.2.3 Kinematics of bosons
1.2.4 Dynamics and probability current and density
1.3 Fermi field
1.4 Bose field
1.4.1 Phonons
1.4.2 Quantizing a classical field theory
1.5 Occupation number representation
1.6 Summary
2 Operators on the multi-particle state space
2.1 Physical observables
2.2 Probability density and number operators
2.3 Probability current density operator
2.4 Interactions
2.4.1 Two-particle interaction
2.4.2 Fermio boson interaction
2.4.3 Electron-phonon interaction
2.5 The statistical operator
2.6 Summary
3 Quantum dynamics and Green's functions
3.1 Quantum dynamics
3.1.1 The SchrSdinger picture
3.1.2 The Heisenberg picture
3.2 Second quantization
3.3 Green's functions
3.3.1 Physical properties and Green's functions
3.3.2 Stable of one-particle Green's functions
3.4 Equilibrium Green's functions
3.5 Summary
4 Non-equilibrium theory
4.1 The non-equilibrium problem
4.2 Ground state formalism
4.3 Closed time path formalism
4.3.1 Closed time path Green's function
4.3.2 Non-equilibrium perturbation theory
4.3.3 Wick's theorem
4.4 Non-equilibrium diagrammatics
4.4.1 Particles coupled to a classical field
4.4.2 Particles coupled to a stochastic field
4.4.3 Interacting fermions and bosons
4.5 The self-energy
4.5.1 Non-equilibrium Dyson equations
4.5.2 Skeleton diagrams
4.6 Summary
5 Real-time formalism
5.1 Real-time matrix representation
5.2 Real-time diagrammatics
5.2.1 Feynman rules for a scalar potential
5.2.2 Feynman rules for interacting bosons and fermions
5.3 Triagonal and symmetric representations
5.3.1 Fermion-boson coupling
5.3.2 Two-particle interaction
5.4 The real rules: the RAK-rules
5.5 Non-equilibrium Dyscn equations
5.6 Equilibrium Dyscn equation
5.7 Real-time versus imaginary-time formalism
5.7.1 Imaginary-time formalism
5.7.2 Imaginary-time Green's functions
5.7.3 Analytical continuation procedure
5.7.4 Kadanoff-Baym equations
5.8 Summary
6 Linear response theory
6.1 Linear response
6.1.1 Density re~,ponse
6.1.2 Current response
6.1.3 Ccnductivity tensor
6.1.4 Ccnductance
6.2 Linear response cf Green's functions
6.3 Properties cf respone hmctions
6.4 Stability cf the thermal equilibrium ,tate
6.5 Fluctuation-dissipation theorem
6.6 Time-reversal symmetry
6.7 Scattering and correlation functions
6.8 Summary
7 Quantum kinetic equations
7.1 Left-right subtracted Dyson equation
7.2 Wigner or mixed coordinates
7.3 Gradient approximation
7.3.1 Spectral weight function
7.3.2 Quasi-particle approximation
7.4 Impurity scattering
7.4.1 Boltzmannian motion in a random potential
7.4.2 Brownian motion
7.5 Quasi-classical Green's function technique
7.5.1 Electron-phonon interaction
7.5.2 Renormalization of the a.c. conductivity
7.5.3 Excitation representation
7.5.4 Particle conservation
7.5.5 Impurity scattering
7.6 Beyond the quasi-classical approximation
7.6.1 Thermo-electrics and magneto-transport
7.7 Summary
8 Non-equilibrium superconductivity
8.1 BCS-theory
8.1.1 Nambu or particle-hole space
8.1.2 Equations of motion in Nambu Keldysh space
8.1.3 Green's functions and gauge transformations
8.2 Quasi-classical Green's function theory
8.2.1 Normalization condition
8.2.2 Kinetic equation
8.2.3 Spectral densities
8.3 Trajectory Green's functions
8.4 Kinetics in a dirty superconductor
8.4.1 Kinetic equation
8.4.2 Ginzburg-Landau regime
8.5 Charge imbalance
8.6 Summary
9 Diagrammatics and generating functionals
9.1 Diagrammatics
9.1.1 Propagators and vertices
9.1.2 Amplitudes and superposition
9.1.3 Fundamental dynamic relation
9.1.4 Low order diagrams
9.2 Generating functional
9.2.1 Fhnctional differentiation
9.2.2 From diagrammatics to differential equations
9.3 Connection to operator formalism
9.4 Fermions and Grassmann variables
9.5 Generator of connected amplitudes
9.5.1 Source derivative proof
9.5.2 Combinatorial proof
9.5.3 Functional equation for the generator
9.6 One-particle irreducible vertices
9.6.1 Symmetry broken states
9.6.2 Green's functions and one-particle irreducible vertices
9.7 Diagrammatics and action
9.8 Effective action and skeleton diagrams
9.9 Summary
10 Effective action
10.1 Functional integration
10.1.1 Functional Fourier transformation
10.1.2 Gaussian integrals
10.1.3 Fermionic path integrals
10.2 Generators as functional integrals
10.2.1 Euclid versus Minkowski
10.2.2 Wick's theorem and functionals
10.3 Generators and 1PI vacuum diagrams
10.4 1PI loop expansion of the effective action
10.5 Two-particle irreducible effective action
10.5.1 The 2PI loop expansion of the effective action
10.6 Effective action approach to Bose gases
10.6.1 Dilute Bose gases
10.6.2 Effective action formalism for bosons
10.6.3 Homogeneous Bose gas
10.6.4 Renormalization of the interaction
10.6.5 Inhomogeneous Bose gas
10.6.6 Loop expansion for a trapped Bose gas
10.7 Summary
11 Disordered conductors
11.1 Localization
11.1.1 Scaling theory of localization
11.1.2 Coherent backscattering
11.2 Weak localization
11.2.1 Quantum correction to conductivity
11.2.2 Cooperon equation
11.2.3 Quantum interference and the Cooperon
11.2.4 Quantum interference in a magnetic field
……
12 Classical Statistical Dynamics
Appendices
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