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QUANTUM FIELD THEORY OF
NON-EQUILIBRIUM STATES


This text introduces the real-time approach to non-equilibrium statistical mechanics
and the quantum ¬eld theory of non-equilibrium states in general. After a lucid
introduction to quantum ¬eld theory and Green™s functions, Schwinger™s closed
time path technique is developed, followed eventually by the real-time formulation
and its Feynman diagram technique. The formalism is employed to derive quantum
kinetic equations by using the quasi-classical Green™s function technique, and is
applied to study renormalization effects, non-equilibrium superconductivity, and
quantum effects in disordered conductors.
The book offers two ways of learning how to study non-equilibrium states
of many-body systems: the mathematical, canonical way, and an intuitive way
using Feynman diagrams. The latter provides an easy introduction to the powerful
functional methods of ¬eld theory. The usefulness of Feynman diagrams, even in a
classical context, is shown by studies of classical stochastic dynamics such as vor-
tex dynamics in disordered superconductors. The book demonstrates that quantum
¬elds and Feynman diagrams are the universal language for studying ¬‚uctuations,
be they of quantum or thermal origin, or even purely statistical.
Complete with numerous exercises to aid self-study, this textbook is suitable for
graduate students in statistical mechanics, condensed matter physics, and quantum
¬eld theory in general.

J ø r g e n R a m m e r is a professor in the Department of Physics at Ume˚ Univer-
a
sity, Sweden. He has also worked in Denmark, Germany, Norway, Canada and the
USA. His past research interests are partly re¬‚ected in the topics of this book; his
main current interests are in decoherence and charge transport in nanostructures.
QUANTUM FIELD THEORY OF
NON-EQUILIBRIUM STATES

J ˜ RGEN RAMMER
Ume˚ University, Sweden
a
CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo

Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521874991

© J. Rammer 2007


This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2007

eBook (NetLibrary)
ISBN-13 978-0-511-29656-7
ISBN-10 0-511-29656-8 eBook (NetLibrary)

hardback
ISBN-13 978-0-521-87499-1
hardback
ISBN-10 0-521-87499-8




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Contents

Preface xi

1 Quantum ¬elds 1
1.1 Quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 N -particle system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Identical particles . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Kinematics of fermions . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.3 Kinematics of bosons . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.4 Dynamics and probability current and density . . . . . . . . . 13
1.3 Fermi ¬eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 Bose ¬eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4.1 Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4.2 Quantizing a classical ¬eld theory . . . . . . . . . . . . . . . . 26
1.5 Occupation number representation . . . . . . . . . . . . . . . . . . . . 29
1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2 Operators on the multi-particle state space 33
2.1 Physical observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 Probability density and number operators . . . . . . . . . . . . . . . . 37
2.3 Probability current density operator . . . . . . . . . . . . . . . . . . . 40
2.4 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4.1 Two-particle interaction . . . . . . . . . . . . . . . . . . . . . . 42
2.4.2 Fermion“boson interaction . . . . . . . . . . . . . . . . . . . . . 45
2.4.3 Electron“phonon interaction . . . . . . . . . . . . . . . . . . . . 45
2.5 The statistical operator . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3 Quantum dynamics and Green™s functions 53
3.1 Quantum dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.1.1 The Schr¨dinger picture . . . . . . . . . .
o . . . . . . . . . . . . 54
3.1.2 The Heisenberg picture . . . . . . . . . . . . . . . . . . . . . . 56
3.2 Second quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3 Green™s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3.1 Physical properties and Green™s functions . . . . . . . . . . . . 62
3.3.2 Stable of one-particle Green™s functions . . . . . . . . . . . . . 64

v
vi CONTENTS


3.4 Equilibrium Green™s functions . . . . . . . . . . . . . . . . . . . . . . 70
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4 Non-equilibrium theory 79
4.1 The non-equilibrium problem . . . . . . . . . . . . . . . . . . . . . . . 79
4.2 Ground state formalism . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.3 Closed time path formalism . . . . . . . . . . . . . . . . . . . . . . . . 84
4.3.1 Closed time path Green™s function . . . . . . . . . . . . . . . . 87
4.3.2 Non-equilibrium perturbation theory . . . . . . . . . . . . . . . 90
4.3.3 Wick™s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.4 Non-equilibrium diagrammatics . . . . . . . . . . . . . . . . . . . . . . 103
4.4.1 Particles coupled to a classical ¬eld . . . . . . . . . . . . . . . . 104
4.4.2 Particles coupled to a stochastic ¬eld . . . . . . . . . . . . . . . 106
4.4.3 Interacting fermions and bosons . . . . . . . . . . . . . . . . . 107
4.5 The self-energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.5.1 Non-equilibrium Dyson equations . . . . . . . . . . . . . . . . . 116
4.5.2 Skeleton diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5 Real-time formalism 121
5.1 Real-time matrix representation . . . . . . . ........ . . . . . . . 121
5.2 Real-time diagrammatics . . . . . . . . . . . ........ . . . . . . . 123
5.2.1 Feynman rules for a scalar potential ........ . . . . . . . 123
5.2.2 Feynman rules for interacting bosons and fermions . . . . . . . 125
5.3 Triagonal and symmetric representations . . ........ . . . . . . . 127
5.3.1 Fermion“boson coupling . . . . . . . ........ . . . . . . . 129
5.3.2 Two-particle interaction . . . . . . . ........ . . . . . . . 131
5.4 The real rules: the RAK-rules . . . . . . . . ........ . . . . . . . 133
5.5 Non-equilibrium Dyson equations . . . . . . ........ . . . . . . . 135
5.6 Equilibrium Dyson equation . . . . . . . . . ........ . . . . . . . 138
5.7 Real-time versus imaginary-time formalism ........ . . . . . . . 140
5.7.1 Imaginary-time formalism . . . . . . ........ . . . . . . . 140
5.7.2 Imaginary-time Green™s functions . . ........ . . . . . . . 142
5.7.3 Analytical continuation procedure . ........ . . . . . . . 143
5.7.4 Kadano¬“Baym equations . . . . . . ........ . . . . . . . 148
5.8 Summary . . . . . . . . . . . . . . . . . . . ........ . . . . . . . 149

6 Linear response theory 151
6.1 Linear response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.1.1 Density response . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.1.2 Current response . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.1.3 Conductivity tensor . . . . . . . . . . . . . . . . . . . . . . . . 158
6.1.4 Conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
6.2 Linear response of Green™s functions . . . . . . . . . . . . . . . . . . . 159
6.3 Properties of response functions . . . . . . . . . . . . . . . . . . . . . . 164
6.4 Stability of the thermal equilibrium state . . . . . . . . . . . . . . . . 165
CONTENTS vii


6.5 Fluctuation“dissipation theorem . . . . . . . . . . . . . . . . . . . . . 169
6.6 Time-reversal symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.7 Scattering and correlation functions . . . . . . . . . . . . . . . . . . . 174
6.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

7 Quantum kinetic equations 179
7.1 Left“right subtracted Dyson equation . . . . . . . . . . . . . . . . . . 179
7.2 Wigner or mixed coordinates . . . . . . . . . . . . . . . . . . . . . . . 181
7.3 Gradient approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 184
7.3.1 Spectral weight function . . . . . . . . . . . . . . . . . . . . . . 185
7.3.2 Quasi-particle approximation . . . . . . . . . . . . . . . . . . . 186
7.4 Impurity scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
7.4.1 Boltzmannian motion in a random potential . . . . . . . . . . . 192
7.4.2 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . 193
7.5 Quasi-classical Green™s function technique . . . . . . . . . . . . . . . . 198
7.5.1 Electron“phonon interaction . . . . . . . . . . . . . . . . . . . . 200
7.5.2 Renormalization of the a.c. conductivity . . . . . . . . . . . . . 206
7.5.3 Excitation representation . . . . . . . . . . . . . . . . . . . . . 207
7.5.4 Particle conservation . . . . . . . . . . . . . . . . . . . . . . . . 209
7.5.5 Impurity scattering . . . . . . . . . . . . . . . . . . . . . . . . . 211
7.6 Beyond the quasi-classical approximation . . . . . . . . . . . . . . . . 211
7.6.1 Thermo-electrics and magneto-transport . . . . . . . . . . . . . 215
7.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

8 Non-equilibrium superconductivity 217
8.1 BCS-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
8.1.1 Nambu or particle“hole space . . . . . . . . . . . . . . . . . . . 225
8.1.2 Equations of motion in Nambu“Keldysh space . . . . . . . . . 228
8.1.3 Green™s functions and gauge transformations . . . . . . . . . . 231
8.2 Quasi-classical Green™s function theory . . . . . . . . . . . . . . . . . . 232
8.2.1 Normalization condition . . . . . . . . . . . . . . . . . . . . . . 235
8.2.2 Kinetic equation . . . . . . . . . . . . . . . . . . . . . . . . . . 236
8.2.3 Spectral densities . . . . . . . . . . . . . . . . . . . . . . . . . . 236
8.3 Trajectory Green™s functions . . . . . . . . . . . . . . . . . . . . . . . . 238
8.4 Kinetics in a dirty superconductor . . . . . . . . . . . . . . . . . . . . 242
8.4.1 Kinetic equation . . . . . . . . . . . . . . . . . . . . . . . . . . 244
8.4.2 Ginzburg“Landau regime . . . . . . . . . . . . . . . . . . . . . 246
8.5 Charge imbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

9 Diagrammatics and generating functionals 253
9.1 Diagrammatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
9.1.1 Propagators and vertices . . . . . . . . . . . . . . . . . . . . . . 255
9.1.2 Amplitudes and superposition . . . . . . . . . . . . . . . . . . . 258
9.1.3 Fundamental dynamic relation . . . . . . . . . . . . . . . . . . 261
9.1.4 Low order diagrams . . . . . . . . . . . . . . . . . . . . . . . . 265
viii CONTENTS


9.2 Generating functional . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
9.2.1 Functional di¬erentiation . . . . . . . . . . . . . . . . . . . . . 272
9.2.2 From diagrammatics to di¬erential equations . . . . . . . . . . 274
9.3 Connection to operator formalism . . . . . . . . . . . . . . . . . . . . . 281
9.4 Fermions and Grassmann variables . . . . . . . . . . . . . . . . . . . . 282
9.5 Generator of connected amplitudes . . . . . . . . . . . . . . . . . . . . 284
9.5.1 Source derivative proof . . . . . . . . . . . . . . . . . . . . . . . 284
9.5.2 Combinatorial proof . . . . . . . . . . . . . . . . . . . . . . . . 290
9.5.3 Functional equation for the generator . . . . . . . . . . . . . . 294
9.6 One-particle irreducible vertices . . . . . . . . . . . . . . . . . . . . . . 296
9.6.1 Symmetry broken states . . . . . . . . . . . . . . . . . . . . . . 301
9.6.2 Green™s functions and one-particle irreducible vertices . . . . . 302
9.7 Diagrammatics and action . . . . . . . . . . . . . . . . . . . . . . . . . 306
9.8 E¬ective action and skeleton diagrams . . . . . . . . . . . . . . . . . . 307
9.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

10 E¬ective action 313
10.1 Functional integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
10.1.1 Functional Fourier transformation . . . . . . . . . . . . . . . . 314
10.1.2 Gaussian integrals . . . . . . . . . . . . . . . . . . . . . . . . . 315
10.1.3 Fermionic path integrals . . . . . . . . . . . . . . . . . . . . . . 319
10.2 Generators as functional integrals . . . . . . . . . . . . . . . . . . . . 320
10.2.1 Euclid versus Minkowski . . . . . . . . . . . . . . . . . . . . . . 323
10.2.2 Wick™s theorem and functionals . . . . . . . . . . . . . . . . . . 324
10.3 Generators and 1PI vacuum diagrams . . . . . . . . . . . . . . . . . . 330
10.4 1PI loop expansion of the e¬ective action . . . . . . . . . . . . . . . . 333
10.5 Two-particle irreducible e¬ective action . . . . . . . . . . . . . . . . . 339
10.5.1 The 2PI loop expansion of the e¬ective action . . . . . . . . . . 346
10.6 E¬ective action approach to Bose gases . . . . . . . . . . . . . . . . . 351
10.6.1 Dilute Bose gases . . . . . . . . . . . . . . . . . . . . . . . . . . 351
10.6.2 E¬ective action formalism for bosons . . . . . . . . . . . . . . . 352
10.6.3 Homogeneous Bose gas . . . . . . . . . . . . . . . . . . . . . . . 356
10.6.4 Renormalization of the interaction . . . . . . . . . . . . . . . . 359
10.6.5 Inhomogeneous Bose gas . . . . . . . . . . . . . . . . . . . . . . 363
10.6.6 Loop expansion for a trapped Bose gas . . . . . . . . . . . . . . 365
10.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372

11 Disordered conductors 373
11.1 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
11.1.1 Scaling theory of localization . . . . . . . . . . . . . . . . . . . 374
11.1.2 Coherent backscattering . . . . . . . . . . . . . . . . . . . . . . 377
11.2 Weak localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
11.2.1 Quantum correction to conductivity . . . . . . . . . . . . . . . 388
11.2.2 Cooperon equation . . . . . . . . . . . . . . . . . . . . . . . . . 392
11.2.3 Quantum interference and the Cooperon . . . . . . . . . . . . . 398
11.2.4 Quantum interference in a magnetic ¬eld . . . . . . . . . . . . 402
CONTENTS ix


11.2.5 Quantum interference in a time-dependent ¬eld . . . . . . . . . 404
11.3 Phase breaking in weak localization . . . . . . . . . . . . . . . . . . . . 408
11.3.1 Electron“phonon interaction . . . . . . . . . . . . . . . . . . . . 410
11.3.2 Electron“electron interaction . . . . . . . . . . . . . . . . . . . 416
11.4 Anomalous magneto-resistance . . . . . . . . . . . . . . . . . . . . . . 423
11.4.1 Magneto-resistance in thin ¬lms . . . . . . . . . . . . . . . . . 424
11.5 Coulomb interaction in a disordered conductor . . . . . . . . . . . . . 428
11.6 Mesoscopic ¬‚uctuations . . . . . . . . . . . . . . . . . . . . . . . . . . 437
11.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448

12 Classical statistical dynamics 449
12.1 Field theory of stochastic dynamics . . . . . . . . . . . . . . . . . . . . 450
12.1.1 Langevin dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 450
12.1.2 Fluctuating linear oscillator . . . . . . . . . . . . . . . . . . . . 451
12.1.3 Quenched disorder . . . . . . . . . . . . . . . . . . . . . . . . . 454
12.1.4 Dynamical index notation . . . . . . . . . . . . . . . . . . . . . 455
12.1.5 Quenched disorder and diagrammatics . . . . . . . . . . . . . . 457
12.1.6 Over-damped dynamics and the Jacobian . . . . . . . . . . . . 459
12.2Magnetic properties of type-II superconductors . . . . . . . . . . . . . . 460
12.2.1 Abrikosov vortex state . . . . . . . . . . . . . . . . . . . . . . . 460
12.2.2 Vortex lattice dynamics . . . . . . . . . . . . . . . . . . . . . . 462
12.3 Field theory of pinning . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
12.3.1 E¬ective action . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
12.4 Self-consistent theory of vortex dynamics . . . . . . . . . . . . . . . . 469
12.4.1 Hartree approximation . . . . . . . . . . . . . . . . . . . . . . . 470
12.5 Single vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
12.5.1 Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . 473
12.5.2 Self-consistent theory . . . . . . . . . . . . . . . . . . . . . . . 474
12.5.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476
12.5.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . 476
12.5.5 Hall force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482
12.6 Vortex lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
12.6.1 High-velocity limit . . . . . . . . . . . . . . . . . . . . . . . . . 488
12.6.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . 489
12.6.3 Hall force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492
12.7 Dynamic melting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493
12.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500

Appendices 501

A Path integrals 503

B Path integrals and symmetries 511

C Retarded and advanced Green™s functions 513

D Analytic properties of Green™s functions 517
x CONTENTS


Bibliography 523

Index 531
Preface

The purpose of this book is to provide an introduction to the applications of quantum
¬eld theoretic methods to systems out of equilibrium. The reason for adding a
book on the subject of quantum ¬eld theory is two-fold: the presentation is, to my
knowledge, the ¬rst 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. The universality of the method,
as testi¬ed by the range of topics covered, reveals that the language of quantum
¬elds is the universal description of ¬‚uctuations, be they of quantum nature, thermal
or classical stochastic. The book is thus intended as a contribution to unifying the
languages used in separate ¬elds of physics, providing a universal tool for describing
non-equilibrium states.
Chapter 1 introduces the basic notions of quantum ¬eld theory, the bose and
fermi quantum ¬elds operating on the multi-particle state spaces. In Chapter 2, op-
erators on the multi-particle space representing physical quantities of a many-body
system are constructed. The detailed exposition in these two chapters is intended
to ensure the book is self-contained. In Chapter 3, the quantum dynamics of a
many-body system is described in terms of its quantum ¬elds and their correla-
tion functions, the Green™s functions. In Chapter 4, the key formal tool to describe
non-equilibrium states is introduced: Schwinger™s closed time path formulation of
non-equilibrium quantum ¬eld theory, quantum statistical mechanics. Perturbation
theory for non-equilibrium states is constructed starting from the canonical operator
formalism presented in the previous chapters. In Chapter 5 we develop the real-time
formalism necessary to deal with non-equilibrium states; ¬rst in terms of matrices
and eventually in terms of two di¬erent types of Green™s functions. The diagram
representation of non-equilibrium perturbation theory is constructed in a way that
the di¬erent aspects of spectral and quantum kinetic properties appear in a physi-
cally transparent and important fashion for non-equilibrium states. The equivalence
of the real-time and imaginary-time formalisms are discussed in detail. In Chap-
ter 6 we consider the coexistence regime between equilibrium and non-equilibrium
states, the linear response regime. In Chapter 7 we develop and apply the quantum
kinetic equation approach to the normal state, and in particular consider electrons

xi
xii Preface


in metals and semiconductors. As applications we consider the Boltzmann limit, and
then phenomena beyond the Boltzmann theory, such as renormalization of transport
coe¬cients due to interactions. In Chapter 8 we consider non-equilibrium supercon-
ductivity. In particular we introduce the quasi-classical Green™s function technique
so e¬cient for the description of super¬‚uids. We derive the quantum kinetic equation
describing elastic and inelastic scattering in superconductors. The time-dependent
Ginzburg“Landau equation is obtained for a dirty superconductor. As an applica-
tion of the quasi-classical theory, we consider the phenomena of conversion of normal
currents to supercurrents and the corresponding charge imbalance.
Unlike Schwinger, not stooping to the paganism of using diagrams, we shall, like
the boys in the basement, take heavy advantage of using Feynman diagrams. By
introducing Feynman diagrams, the most developed of our senses can become func-
tional in the pursuit of understanding quantum dynamics, an addition that shall
make its pursuit easier also for non-equilibrium situations. Though the picture of
reality that the representation of perturbation theory in terms of Feynman diagrams
inspires might be a ¬gment of the imagination, its usefulness for developing phys-
ical intuition has amply proved its value, as witnessed ¬rst in elementary particle
physics. We develop the diagrammatics for non-equilibrium states, and show that
the additional rules for the universal vertex display the two important features of
quantum statistics and spectral properties of the interacting particles in an explicit
fashion. In Chapter 9 we shall take the stand of formulating the laws of physics in
terms of propagators and vertices and their Feynman diagrams representing prob-
ability amplitudes as dictated by the superposition principle. In fact, we take the
Shakespearian approach and construct quantum dynamics in terms of Feynman di-
agrams by invoking the only two options for a particle: to act or not to interact.
From this diagrammatic starting point, and employing the intuitive appeal of dia-
grammatic arguments, we then construct the formalism of non-equilibrium quantum
¬eld theory in terms of the powerful functional methods; ¬rst in terms of the gen-
erating functional and functional di¬erentiation technique. In Chapter 10 we then
introduce the ¬nal tool in the functional arsenal: functional integration, and arrive at
the e¬ective action description of general non-equilibrium states. As an application
of the e¬ective action approach we consider the dilute Bose gas, and the case of a
trapped Bose“Einstein condensate. In Chapter 11 we consider quantum transport
properties of disordered conductors, weak localization and interaction e¬ects. In par-
ticular we show how the quasi-classical Green™s function technique used in describing
non-equilibrium properties of a dirty superconductor can be utilized to describe the
destruction of phase coherence in the normal state due to non-equilibrium e¬ects
and interactions. Finally, in Chapter 12, we consider the classical limit of the devel-
oped general non-equilibrium quantum ¬eld theory. We consider classical stochastic
dynamics and show that ¬eld theoretic methods and diagrammatics are useful tools
even in the classical context. As an example we consider the ¬‚ux ¬‚ow properties of
the Abrikosov lattice in a type-II superconductor. We thus demonstrate the fact that
quantum ¬eld theory, through its diagrammatics and functional formulations, is the
universal language for describing ¬‚uctuations whatever their nature.
Readers™ guide. Firstly, readers bothered by the old-fashioned habit of footnotes
can simply skip them; they are either quick reminders or serve the purpose of pro-
Preface xiii


viding a general perspective. The book can be read chronologically but, like any fox
hole, it has two entrances. For the reader whose interest is the general structure of
quantum ¬eld theories, the book o¬ers the possibility to jump directly to Chapter 9
where a quantum ¬eld theory is de¬ned in terms of its propagators and vertices and
their resulting Feynman diagrams as dictated by the superposition principle. The
powerful methods of generating functionals are then constructed from the diagram-
matics. However, the reader acquainted with Chapter 4 will then have at hand the

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