The purpose of this book is to provide an introduction to the applications of quantum field theoretic methods to systems out of equilibrium. The reason for adding a book on the subject of quantum field theory is two-fold: the presentation is, to my knowledge, the first to extensively present and apply to non-equilibrium phenomena the real-time approach originally developed by Schwinger, and subsequently applied by Keldysh and others to derive transport equations. Secondly, the aim is to show the universality of the method by applying it to a broad range of phenomena. The book should thus not just be of interest to condensed matter physicists, but to physicists in general as the method is of general interest with applications ranging the whole scale from high-energy to soft condensed matter physics.

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

11.2.5 Quantum interference in a time-dependent field

 11.3 Phase breaking in weak localization

11.3.1 Election p~onon interaction

11.3.2 Electron-electron interaction

 11.4 Anomalous magneto-resistance

11.4.1 Magneto-resistance in thin films

 11.5 Conlomb interaction in a disordered conductor

 11.6 Mesoscopic fluctnations

 11.7 Summary

12 Classical statistical dynamics

 12.1 Field theory of stochastic dynamics

12.1.1 Langevin dynamics

12.1.2 Fluctuating linear oscillator

12.1.3 Quenched disolder

12.1.4 Dynamical index notation

12.1.5 Quenched disoider and diagrammatics

12.1.6 Over-damped dynamics and the Jacobian

 12.2 Magnetic properties of type-H superconductors

12.2.1 Abrikosov vortex state

12.2.2 Vortex lattice dynamics

 12.3 Field theory of pinning

12.3.1 Effective sction

 12.4 Self-consistent theory of vortex dynamics

12.4.1 Hartree approximation

 12.5 Single vortex

12.5.1 Perturbation theoiy

12.5.2 Self-consistent theory

12.5.3 Simulations

12.5.4 Numerical results

12.5.5 Hail force

 12.6 Vortex lattice

12.6.1 High-velocity limit

12.6.2 Numerical results

12.6.3 Ha]] force

 12.7 Dynamic melting

 12.8 Summary

Appendices

A Path integrals

B Path integrals and symmetries

C Retarded and advanced Green's functions

D Analytic properties of Green's functions