1.0 Book
title
NONEQUILIBRIUM QUANTUM FIELD THEORY
2.0 Authors
ESTEBAN CALZETTA; BEI-LOK B. HU
3.0
Publisher
CAMBRIDGE UNIVERSITY PRESS (2007)
4.0
Description
This is the first book that presents a coherent, self-contained and comprehensive description of the concepts, methodology and practice of quantum field theory and nonequilibrium statistical mechanics applied to the analysis of the dynamics of fundamental fields such as for the nonabelian gauge fields and gravitational fields, as well as the collective behavior of second-quantized many-body system, such as Bose- Einstein condensation (BEC). It begins with the foundational aspects of the theory, presents the most important and useful techniques, discusses issues of basic interest (decoherence, entropy generation) and shows how the thermal field, linear response kinetic and hydrodynamical theories emerge. It then illustrates the applications of these concepts and methodology with current research topics such as phase transitions, thermalization in relativistic heavy ion collisions, the dynamics of BEC and the generation of structures from quantum fluctuations in the Early Universe.
The book is divided in five sections. Each
section addresses a particular stage in the conceptual and technical
development of the subject at graded levels of difficulty. A guide to the flow
of topics in the Preface aids the reader to select material for teaching or
self-instruction. We attempt to make each chapter as self-contained as possible,
minimizing the need to go back and forth between chapters. We have also made
efforts to present full derivations of or detailed plausibility arguments for
all the statements we make, with only a handful of exceptions in the whole
book. We provide an extensive
bibliography, which represents the state of the art of the various aspects of
the title thesis as of June 2007. This would be a valuable resource in itself
for researchers entering the field.
The book is designed to lead a student with some background in quantum field theory (for example, at the level of M. Peskin and D. Schroeder, An Introduction to Quantum Field Theory (Addison-Wesley, New York, 1995)) and statistical mechanics (at the level of K. Huang, Statistical Physics (John Wiley & Sons, New York, 1987)) from the fundamentals of the theory to cutting edge applications. This is the only place where a coherent and comprehensive account on nonequilibrium quantum field theory can be found. There are also extensive discussions of alternative approaches and pointers to the literature, so by the end of the book the student will have developed a firm grasp of the concepts and methodology of the theory and the way how they are applied to a range of important topics, as well as gaining a sense of the directions and open problems of current research.
This book will be of interest to scholars and students keen on the foundational aspects of quantum and statistical physics, quantum field theory, the foundations and the philosophy of physics. We have extensive discussions of fundamental issues (see e. g. chapters 1 and 9), and have taken care to show how these issues bear on the other discipline of physics and are of essence to the concrete applications of the theory.
The book is ready for adoption as a regular or special topics course on nonequilibrium quantum field theory suitable for second year postgraduate students or above. We believe it contains enough material for two full terms, but can comfortably be used as a single term course with some selection of topics. We gave in the Preface several possible tracks, according to the main focus of the course and the interests of the lecturer.
5.0 Contents
and flow of topics
The book has
five parts: The first part comprising Chapters 1-3 deals with the basics. After
an introduction chapter to basic notions and issues in Nonequilibrium Statistical
Mechanics, two chapters are devoted to the basic ideas and techniques of
nonequilibrium systems. The second part comprising Chapters 4-6 begins with
Chapter 4 on quantum field processes in dynamical backgrounds. Chapters 5-6
form the backbone of this book, in establishing the real-time quantum field
theory framework based on the so-called closed time path (CTP or
Schwinger-Keldysh) effective action and the influence functional (IF, or
Feynman-Vernon) formalisms. This is followed by three chapters in Part III to
illustrate the use of these formalisms for addressing issues like gauge
invariance, dissipation, entropy, noise and decoherence. From these formalisms
we proceed to Part IV, including chapters 10-
Below is a
quick guide to the use of this book for readers of different backgrounds. Readers
with
some good understanding of NEqSM may go directly to Part
II while readers familiar with the CTP-IF formalism may start with Part I and
go to Part III. Readers more interested in the structure
of the kinetic and hydrodynamic theories can delve into Part IV after Part
II, while readers more
interested in statistical mechanical issues manifested in
quantum field theory may want to focus
on Chapters 1, 8, 9. Recognizing that the readers may come from different
disciplines with solid
knowledge of their own field who want to learn nonequilibrium
quantum field theory for applications
to their own problems, we suggest the following streams:
1)
Atomic-Optical and condensed matter physics : Chapters
2, 3, 5, 6, 8, 10, 11, 13
2)
Nuclear-particle physics : Chapters 2, 3, 4, 5, 6, 7,
8, 9, 10, 11, 12, 13, 14
4) Cosmology: Chapters 1, 2, 3, 4, 5, 6, 8, 9, 10, 11,
12, 15
6.0 Table of
contents
Part 1: Fundamentals of
nonequilibrium statistical mechanics 9
1 Basic issues in nonequilibrium statistical mechanics 11
1.1 Macroscopic description of
physical processes 12
1.2 Microscopic
characterization from dynamical systems behavior 16
1.2.1 Ergodicity describes a
system in equilibrium 16
1.2.2 Mixing system is time-reversible,
weak sense of approach to equilibrium 17
1.2.3 Dissipative system:
coarse-grained mixing permits approach to equilibrium 18
1.2.4 Nonequilibrium
thermodynamics and chaotic dynamics 19
1.3 Physical conditions 20
1.3.1 Large systems: fluctuations,
Poincare recurrence and thermodynamic limit 20
1.3.2 Initial conditions:
specific, randomized, dynamical correlations 21
1.3.3 Time scales and
interaction 22
1.3.4 Coarse-graining 23
1.4 Coarse-graining and
persistent structure in the physical world 24
1.5 Physical systems: closed,
open, effectively closed and effectively open 27
1.5.1 Quantum open systems:
coarse-graining and backreaction 27
1.5.2 From closed to
effectively open systems 28
1.5.3 Two major paradigms of nonequilibrium
statistical mechanics 29
1.6 Appendix A: Stochastic
processes and equations in a (tiny) nutshell 30
1.6.1 Probability, random
variables and stochastic processes 30
1.6.2 Markov processes 33
1.6.3 Kramers-Moyal and
Fokker-Planck equations 34
2 Relaxation,
dissipation, noise and fluctuations 36
2.1.1 The linear oscillator
model 37
2.1.2 Fluctuation-dissipation
theorem 39
2.2 The Fokker-Planck and
Kramers-Moyal equations 41
2.3 The Boltzmann equation 44
2.3.1 Slaving of higher
correlations in the Boltzmann equation 46
2.3.2 Corrections from quantum
statistics 48
2.3.3 Relativistic kinetic
theory 48
2.3.4 Fluctuations in the
Boltzmann equation 50
3 Quantum open systems
53
3.1.1 Wigner functions 55
3.1.2 Closed time path (CTP)
integrals 58
3.2 Influence functional 60
3.2.1 Some properties of the
influence action 61
3.2.2 The linear bath model 62
3.3 The master equation 62
3.3.1 The linear system model
63
3.4 The Langevin equation 64
3.4.1 The linear bath model 67
3.5 The Kramers-Moyal equation 68
3.5.1 The linear system model
70
3.6 Detailed derivation of the
propagator and the master equation 71
3.6.1 Evolution of the reduced
density matrix 72
3.6.2 Master equation 73
3.7 Consistent histories and
decoherence functional 74
Part 2: Basics of
nonequilibrium quantum field theory 79
4 Quantum fields on
time-dependent backgrounds: particle creation 81
4.1 Basic field theory 82
4.1.1 Classical fields 82
4.1.2 Quantum fields 83
4.1.3 Free fields 84
4.1.4 Particle creation 85
4.1.5 Adiabatic vacua 86
4.1.6 Hamiltonian mean field dynamics and general Gaussian ansatz 88
4.2 Particle production in an
external field 91
4.2.1 Particle creation in a
constant electric field 93
4.3 Spontaneous and stimulated
production 95
4.3.1 Spontaneous production
96
4.3.2 Stimulated production 97
4.4 Quantum Vlasov equation 97
4.4.1 Adiabatic number state
98
4.4.2 Number and correlation
98
4.4.3 Current and energy
density 99
4.4.4 Quantum Vlasov equation
100
4.5 Periodically driven fields
100
4.5.1 Broad resonance 101
4.5.2 Narrow resonance 102
4.6 Particle creation in a
dynamical spacetime 103
4.6.1 Wave equations in curved
spacetimes 103
4.6.2 Conformal vacuum in
conformally-static spacetimes 105
4.6.3 Thermal radiance 106
4.6.4 Conformal stress-energy
tensor 106
4.6.5 Adiabatic regularization
107
4.6.6 A simple model of a
cosmological phase transition 108
4.7 Particle creation as
squeezing 111
4.7.1 Evolutionary operator,
squeezing and rotation 111
4.7.2 Dynamics of the
squeezing parameters 114
4.7.3 Number, coherence and
initial states 116
4.7.4 Fluctuations in number
118
4.8 Squeezed quantum open
systems 120
4.8.1 Dissipation and noise
kernels 120
4.8.2 u1 →
v2 functions 121
4.8.3 a11 →
b4 functions 122
5 Open systems of
interacting quantum fields 124
5.1 Influence functional: two
interacting quantum fields 125
5.1.1 Perturbation theory 126
5.1.2 Noise and fluctuations
130
5.1.3 Langevin equation and
fluctuation-dissipation relation 131
5.2 Quantum functional master
equation 132
5.3 The closed time path coarse grained effective action 135
5.3.1 Stochastic equations 139
6 Functional methods in
nonequilibrium QFT 142
6.1 Propagators 143
6.1.1 Interacting fields 145
6.2 Functional methods 146
6.2.1 The generating
functional and the effective action. 147
6.2.2 Not quite beyond
equilibrium 148
6.2.3 Trouble in the gφ3 theory 148
6.3 The closed time path
effective action 151
6.3.1 An example 152
6.3.2 The structure of the
closed time path effective action 154
6.4 Computing the closed time
path effective action 156
6.4.1 The background field
method 156
6.4.2 The loop expansion 158
6.4.3 The one loop closed time
path effective action for the g_3 theory 158
6.4.4 The large N expansion
161
6.5 The two particle
irreducible (2PI) effective action 162
6.5.1 The 2PI effective action
in the g_3 theory
165
6.5.2 Large N expansion (suite)
167
6.6 Handling divergences 169
6.6.1 Ultraviolet divergences 169
6.6.2 Initial time
singularities 171
6.6.3 Other divergences 172
Part 3: Gauge
invariance, dissipation, entropy, noise and decoherence 173
7 Closed time path
effective action for gauge theories 175
7.1 Path integral quantization
of gauge theories - an overview 177
7.1.1 Gauge theories 177
7.1.2 Gauge symmetries and
constraints 178
7.1.3 The measure of
integration 179
7.1.4 BRST invariance 180
7.1.5 Physical states 182
7.1.6 Initial conditions for
non-vacuum states 183
7.2 The 2PI formalism applied
to gauge theories 184
7.2.1 The 2PI effective action
184
7.2.2 The 2PI Schwinger-Dyson
equations 185
7.2.3 The reduced 2PI
effective action 186
7.3 Gauge dependence and
propagator structure 187
7.3.1 The Zinn-Justin equation
187
7.3.2 Gauge dependence of the propagators
188
7.3.3 Transverse and
longitudinal gluon propagators. 190
8 Dissipation and noise
in mean field dynamics 191
8.1 Preliminaries 193
8.2 Dissipation in the mean
field dynamics. 193
8.3 Dissipation and particle
creation 194
8.4 Particle creation and
noise 196
8.5 Full quantum correlations
from the Langevin approach 198
8.6 The
fluctuation-dissipation theorem 199
8.7 Particle creation and
decoherence 200
8.8 The nonlinear regime 201
8.9 Final remarks 205
9 Entropy generation
and decoherence of quantum fields 207
9.1 Entropy generation from
particle creation 207
9.1.1 Choice of
representations and initial conditions 207
9.1.2 Coarse-graining the
environment in an open system 208
9.1.3 Differences in various
definitions of entropy 209
9.2 Entropy of quantum fields
210
9.2.1 Entropy special to
choice of representation and initial conditions 210
9.2.2 Entropy from projecting
out irrelevant variables 211
9.2.3 Entropy from slaving of
higher correlations 211
9.3 Entropy from the
(apparent) damping of the mean field 212
9.3.1 Time scales 212
9.3.2 Density matrix 213
9.3.3 Entropy generation 213
9.3.4 Decoherence functional
214
9.4 Entropy of squeezed
quantum open systems 215
9.4.1 Entropy from the
evolutionary operator for reduced density matrix 215
9.4.2 Measures of fluctuations
and coherence 216
9.4.3 Entropy and uncertainty
functions of an inverted oscillator 217
9.4.4 Entropy from graviton
production in de Sitter spacetime 219
9.4.5 Discussion 221
9.5 Decoherence in a quantum
phase transition 221
9.6 Spinodal decomposition of
an interacting quantum field 224
9.6.1 The quench transition
time 225
9.6.2 Decoherence time 226
9.7 Decoherence of the
inflaton field 229
9.7.1 Noise from interacting
quantum fields 230
9.7.2 Decoherence in two
interacting fields model 231
9.7.3 Partitioning One interacting field: noise from high frequency modes 232
Part 4: Thermal,
kinetic and hydrodynamic regimes 235
10 Thermal field and
linear response theory 237
10.1 The thermal generating
functional 237
10.2 Linear response theory
238
10.3 The Kubo-Martin-Schwinger
theorem 239
10.4 Thermal self-energy:
screening. 241
10.5 Landau damping 243
10.5.1 Landau damping in a relativistic collisionless plasma 243
10.5.2 A nonequilibrium
problem with fermions: the case of QED 244
10.5.3 KMS and thermal Fermi
propagators 246
10.5.4 Induced charge density
from a finite temperature Dirac quantum field 247
10.6 Hard thermal loops 248
10.6.1 The model 248
10.6.2 Hard thermal loops 249
10.6.3 Hard thermal loops from
the 2PI CTP effective action 250
10.6.4 The Vlasov equation for
hard modes 251
10.6.5 Ultrasoft modes and
Boltzmann equation 252
10.6.6 Langevin dynamics of
ultrasoft modes 253
10.6.7 A note on the
literature 255
11
Quantum kinetic field theory 256
11.1 The Kadanoff-Baym
equations 256
11.1.1 The model 256
11.1.2 Density of states and
distribution function 258
11.1.3 The dissipation and
noise kernels 259
11.1.4 The retarded and
advanced propagators 260
11.1.5 The off-shell kinetic
equation 261
11.1.6Weakly coupled theories
and the Boltzmann equation 263
11.1.7 The Vlasov equation 265
11.1.8 Time reversal
invariance 266
11.1.9 The limits of the
kinetic approach 267
11.2 Quantum kinetic field
theory on nontrivial backgrounds 268
11.2.1 The scalar Wigner
function in scalar quantum electrodynamics (SQED) 268
11.2.2 Scalar Wigner functions
on non-Abelian backgrounds 271
11.2.3 Quantum kinetic theory
in curved spacetimes 274
11.2.4 A note on the
literature 279
12 Hydrodynamics and
thermalization 280
12.1 Classical relativistic
hydrodynamics 280
12.1.1 A primer on
thermodynamics 280
12.1.2 Covariant hydrostatics
281
12.1.3 Ideal and real fluids
283
12.1.4 Stability and the
Landau-Lifshitz theory 285
12.2 Quantum fields in the
hydrodynamic limit 286
12.2.1 Quantum hydrodynamic
models 286
12.2.2 Thermal equilibrium
states 289
12.2.3 Local equilibria 290
12.3 Transport functions in
the hydrodynamic limit 292
12.3.1 The collision term 292
12.3.2 The linearized
transport equation 293
12.3.3 The temperature shift
and the bulk stress 295
12.3.4 Shear stress and bulk
viscosity 296
12.3.5 Transport functions for
non-Abelian plasmas 296
12.4 Transport functions from
linear response theory 297
12.4.1 The spin diffusion
coefficient 298
12.4.2 The bulk and shear
viscosity coefficients 300
12.5 Thermalization 303
12.5.1 A toy model of
thermalization 304
12.5.2 Thermalization of
isolated fields 306
12.5.3 The stages of
thermalization 308
12.5.4 Coda 313
Part 5: Applications to
selected current research 315
13 Nonequilibrium
Bose-Einstein condensates 317
13.1 The closed time path
integral approach to BECs 318
13.1.1 The coherent state
representation 319
13.1.2 The closed time path
boundary conditions 321
13.2 The symmetry breaking
approach to BECs 321
13.2.1 A relationship between ηA and _AB 324
13.2.2 Gaplessness and phase
invariance 325
13.2.3 Conserving and Phi−
derivable theories
326
13.2.4 The full 2PI effective
action as a Phi − derivable
approach 328
13.2.5 Varieties of theories
from truncations of the 2PI effective action 329
13.2.6 Higher gapless
approximations 332
13.2.7 Damping 335
13.2.8 The stochastic
Gross-Pitaevskii equation 337
13.2.9 The hydrodynamic and
quantum kinetic approach to BECs 338
13.3 The particle number
conserving formalism 341
13.3.1 Problems with the
symmetry breaking approach 341
13.3.2 The one-body density
matrix and long range coherence 342
13.3.3 The particle number
conserving approach 343
13.3.4 Particle number
conserving functional approach 346
14 Nonequilibrium
issues in RHICs and DCCs 348
14.1 Relativistic heavy ion
collisions (RHICs) 348
14.1.1 In
the beginning 348
14.1.2 The Bjorken scenario
349
14.1.3 Break-up 350
14.1.4Measuring the fireball
352
14.1.5 Insights from
nonequilibrium quantum field theory 354
14.2 Disoriented chiral
condensates (DCCs) 355
14.2.1 Self consistent mean
fields in the large N approximation 357
14.2.2 The quantum pion field
359
14.2.3 Adiabatic modes and
renormalization 360
15 Nonequilibrium
quantum processes in the early universe 362
15.1 Quantum fluctuations and
noise in inflationary cosmology 362
15.1.1 Inflationary cosmology
362
15.1.2 Noise in stochastic
inflation 366
15.2 Structure formation:
effect of colored noise 369
15.2.1 Colored noise from
smooth window functions 371
15.2.2 Curvature perturbations
and blue tilt 374
15.2.3 Structures from
coarse-graining an interacting field 375
15.2.4 Structures from
interaction with other fields, 376
15.2.5 Primordial spectrum
from nonequilibrium renormalization group 379
15.3 Reheating in the
inflationary universe 382
15.3.1 Case study I:
backreaction of Fermi fields during preheating 383
15.3.2 Case study II:
primordial magnetic field generation 388
References 394
Index 436