ńņš. 1 |

NON-EQUILIBRIUM STATES

This text introduces the real-time approach to non-equilibrium statistical mechanics

and the quantum ļ¬eld theory of non-equilibrium states in general. After a lucid

introduction to quantum ļ¬eld theory and Greenā™s functions, Schwingerā™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 Greenā™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 ļ¬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

ļ¬elds and Feynman diagrams are the universal language for studying ļ¬‚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

ļ¬eld theory in general.

J Ćø r g e n R a m m e r is a professor in the Department of Physics at UmeĖ Univer-

a

sity, Sweden. He has also worked in Denmark, Germany, Norway, Canada and the

USA. His past research interests are partly reļ¬‚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 Ć˜ RGEN RAMMER

UmeĖ University, Sweden

a

CAMBRIDGE UNIVERSITY PRESS

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Ā© J. Rammer 2007

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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

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ISBN-13 978-0-521-87499-1

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Contents

Preface xi

1 Quantum ļ¬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 ļ¬eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4 Bose ļ¬eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.4.1 Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.4.2 Quantizing a classical ļ¬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 Fermionā“boson interaction . . . . . . . . . . . . . . . . . . . . . 45

2.4.3 Electronā“phonon interaction . . . . . . . . . . . . . . . . . . . . 45

2.5 The statistical operator . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3 Quantum dynamics and Greenā™s functions 53

3.1 Quantum dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.1.1 The SchrĀØdinger picture . . . . . . . . . .

o . . . . . . . . . . . . 54

3.1.2 The Heisenberg picture . . . . . . . . . . . . . . . . . . . . . . 56

3.2 Second quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.3 Greenā™s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.3.1 Physical properties and Greenā™s functions . . . . . . . . . . . . 62

3.3.2 Stable of one-particle Greenā™s functions . . . . . . . . . . . . . 64

v

vi CONTENTS

3.4 Equilibrium Greenā™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 Greenā™s function . . . . . . . . . . . . . . . . 87

4.3.2 Non-equilibrium perturbation theory . . . . . . . . . . . . . . . 90

4.3.3 Wickā™s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.4 Non-equilibrium diagrammatics . . . . . . . . . . . . . . . . . . . . . . 103

4.4.1 Particles coupled to a classical ļ¬eld . . . . . . . . . . . . . . . . 104

4.4.2 Particles coupled to a stochastic ļ¬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 Fermionā“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 Greenā™s functions . . ........ . . . . . . . 142

5.7.3 Analytical continuation procedure . ........ . . . . . . . 143

5.7.4 Kadanoļ¬ā“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 Greenā™s functions . . . . . . . . . . . . . . . . . . . 159

6.3 Properties of response functions . . . . . . . . . . . . . . . . . . . . . . 164

6.4 Stability of the thermal equilibrium state . . . . . . . . . . . . . . . . 165

CONTENTS vii

6.5 Fluctuationā“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 Leftā“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 Greenā™s function technique . . . . . . . . . . . . . . . . 198

7.5.1 Electronā“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 particleā“hole space . . . . . . . . . . . . . . . . . . . 225

8.1.2 Equations of motion in Nambuā“Keldysh space . . . . . . . . . 228

8.1.3 Greenā™s functions and gauge transformations . . . . . . . . . . 231

8.2 Quasi-classical Greenā™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 Greenā™s functions . . . . . . . . . . . . . . . . . . . . . . . . 238

8.4 Kinetics in a dirty superconductor . . . . . . . . . . . . . . . . . . . . 242

8.4.1 Kinetic equation . . . . . . . . . . . . . . . . . . . . . . . . . . 244

8.4.2 Ginzburgā“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 diļ¬erentiation . . . . . . . . . . . . . . . . . . . . . 272

9.2.2 From diagrammatics to diļ¬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 Greenā™s functions and one-particle irreducible vertices . . . . . 302

9.7 Diagrammatics and action . . . . . . . . . . . . . . . . . . . . . . . . . 306

9.8 Eļ¬ective action and skeleton diagrams . . . . . . . . . . . . . . . . . . 307

9.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

10 Eļ¬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 Wickā™s theorem and functionals . . . . . . . . . . . . . . . . . . 324

10.3 Generators and 1PI vacuum diagrams . . . . . . . . . . . . . . . . . . 330

10.4 1PI loop expansion of the eļ¬ective action . . . . . . . . . . . . . . . . 333

10.5 Two-particle irreducible eļ¬ective action . . . . . . . . . . . . . . . . . 339

10.5.1 The 2PI loop expansion of the eļ¬ective action . . . . . . . . . . 346

10.6 Eļ¬ective action approach to Bose gases . . . . . . . . . . . . . . . . . 351

10.6.1 Dilute Bose gases . . . . . . . . . . . . . . . . . . . . . . . . . . 351

10.6.2 Eļ¬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 ļ¬eld . . . . . . . . . . . . 402

CONTENTS ix

11.2.5 Quantum interference in a time-dependent ļ¬eld . . . . . . . . . 404

11.3 Phase breaking in weak localization . . . . . . . . . . . . . . . . . . . . 408

11.3.1 Electronā“phonon interaction . . . . . . . . . . . . . . . . . . . . 410

11.3.2 Electronā“electron interaction . . . . . . . . . . . . . . . . . . . 416

11.4 Anomalous magneto-resistance . . . . . . . . . . . . . . . . . . . . . . 423

11.4.1 Magneto-resistance in thin ļ¬lms . . . . . . . . . . . . . . . . . 424

11.5 Coulomb interaction in a disordered conductor . . . . . . . . . . . . . 428

11.6 Mesoscopic ļ¬‚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 Eļ¬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 Greenā™s functions 513

D Analytic properties of Greenā™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

ļ¬eld theoretic methods to systems out of equilibrium. The reason for adding a

book on the subject of quantum ļ¬eld theory is two-fold: the presentation is, to my

knowledge, the ļ¬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 testiļ¬ed by the range of topics covered, reveals that the language of quantum

ļ¬elds is the universal description of ļ¬‚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 ļ¬elds of physics, providing a universal tool for describing

non-equilibrium states.

Chapter 1 introduces the basic notions of quantum ļ¬eld theory, the bose and

fermi quantum ļ¬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 ļ¬elds and their correla-

tion functions, the Greenā™s functions. In Chapter 4, the key formal tool to describe

non-equilibrium states is introduced: Schwingerā™s closed time path formulation of

non-equilibrium quantum ļ¬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; ļ¬rst in terms of matrices

and eventually in terms of two diļ¬erent types of Greenā™s functions. The diagram

representation of non-equilibrium perturbation theory is constructed in a way that

the diļ¬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

coeļ¬cients due to interactions. In Chapter 8 we consider non-equilibrium supercon-

ductivity. In particular we introduce the quasi-classical Greenā™s function technique

so eļ¬cient for the description of superļ¬‚uids. We derive the quantum kinetic equation

describing elastic and inelastic scattering in superconductors. The time-dependent

Ginzburgā“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 ļ¬gment of the imagination, its usefulness for developing phys-

ical intuition has amply proved its value, as witnessed ļ¬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

ļ¬eld theory in terms of the powerful functional methods; ļ¬rst in terms of the gen-

erating functional and functional diļ¬erentiation technique. In Chapter 10 we then

introduce the ļ¬nal tool in the functional arsenal: functional integration, and arrive at

the eļ¬ective action description of general non-equilibrium states. As an application

of the eļ¬ective action approach we consider the dilute Bose gas, and the case of a

trapped Boseā“Einstein condensate. In Chapter 11 we consider quantum transport

properties of disordered conductors, weak localization and interaction eļ¬ects. In par-

ticular we show how the quasi-classical Greenā™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 eļ¬ects

and interactions. Finally, in Chapter 12, we consider the classical limit of the devel-

oped general non-equilibrium quantum ļ¬eld theory. We consider classical stochastic

dynamics and show that ļ¬eld theoretic methods and diagrammatics are useful tools

even in the classical context. As an example we consider the ļ¬‚ux ļ¬‚ow properties of

the Abrikosov lattice in a type-II superconductor. We thus demonstrate the fact that

quantum ļ¬eld theory, through its diagrammatics and functional formulations, is the

universal language for describing ļ¬‚uctuations whatever their nature.

Readersā™ 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 ļ¬eld theories, the book oļ¬ers the possibility to jump directly to Chapter 9

where a quantum ļ¬eld theory is deļ¬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

general quantum ļ¬eld theory applicable to non-equilibrium states.

The scope of the book is not so much to dwell on a detailed application of the non-

equilibrium theory to a single topic, but rather to show the versatility and universality

of the method by applying it to a broad range of core topics of physics. One purpose

of the book is to demonstrate the utility of Feynman diagrams in non-equilibrium

quantum statistical mechanics using an approach appealing to physical intuition. The

real-time description of non-equilibrium quantum statistical mechanics is therefore

adopted, and the diagrammatic technique for systems out of equilibrium is developed

systematically, and a representation most appealing to physical intuition applied.

Though most examples are taken from condensed matter physics, the book is intended

to contribute to the cross-fertilization between all the ļ¬elds of physics studying the

inļ¬‚uence of ļ¬‚uctuations, be they quantum or thermal or purely statistical, and to

establish that the convenient technique to use is in fact that of non-equilibrium

quantum ļ¬eld theory. The book should therefore be of interest to a wide audience of

physicists; in particular the book is intended to be self-contained so that students of

physics and physicists in general can beneļ¬t from its detailed expositions. It is even

contended that the method is of importance for other ļ¬elds such as chemistry, and

of course useful for electrical engineers.

A complete allocation of the credit for the progress in developing and applying

the real-time description of non-equilibrium states has not been attempted. However,

the references, in particular the cited review articles, should make it possible for the

interested reader to trace this information.

The book is intended to be suļ¬ciently broad to serve as a text for a one- or

two-semester graduate course on non-equilibrium statistical mechanics or condensed

matter theory. It is also hoped that the book can serve as a useful reference book

for courses on quantum ļ¬eld theory, physics of disordered systems, and quantum

transport in general. It is hoped that this attempt to make the exposition as lucid as

possible will be successful to the point that the book can be read by students with

only elementary knowledge of quantum and statistical mechanics, and read with

beneļ¬t on its own. Exercises have been provided in order to aid self-instruction.

I am grateful to Dr. Joachim Wabnig for providing ļ¬gures.

JĆørgen Rammer

1

Quantum ļ¬elds

Quantum ļ¬eld theory is a necessary tool for the quantum mechanical description

of processes that allow for transitions between states which diļ¬er in their particle

content. Quantum ļ¬eld theory is thus quantum mechanics of an arbitrary number of

particles. It is therefore mandatory for relativistic quantum theory since relativistic

kinematics allows for creation and annihilation of particles in accordance with the

formula for equivalence of energy and mass. Relativistic quantum theory is thus in-

herently dealing with many-body systems. One may, however, wonder why quantum

ļ¬eld theoretic methods are so prevalent in condensed matter theory, which consid-

ers non-relativistic many-body systems. The reason is that, though not mandatory,

it provides an eļ¬cient way of respecting the quantum statistics of the particles,

i.e. the states of identical fermions or bosons must be antisymmetric and symmet-

ric, respectively, under the interchange of pairs of identical particles. Furthermore,

the treatment of spontaneously symmetry broken states, such as superļ¬‚uids, is fa-

cilitated; not to mention critical phenomena in connection with phase transitions.

Furthermore, the powerful functional methods of ļ¬eld theory, and methods such as

the renormalization group, can by use of the non-equilibrium ļ¬eld theory technique

be extended to treat non-equilibrium states and thereby transport phenomena.

It is useful to delve once into the underlying mathematical structure of quantum

ļ¬eld theory, but the upshot of this chapter will be very simple: just as in quan-

tum mechanics, where the transition operators, |Ļ Ļ|, contain the whole content of

quantum kinematics, and the bra and ket annihilate and create states in accordance

with

(|Ļ Ļ|) |Ļ = Ļ|Ļ |Ļ (1.1)

we shall ļ¬nd that in quantum ļ¬eld theory two types of operators do the same job.

One of these operators, the creation operator, aā , is similar in nature to the ket in

the transition operator, and the other, the annihilation operator, a, is similar to

the action of the bra in Eq. (1.1), annihilating the state it operates on. Then the

otherwise messy obedience of the quantum statistics of particles becomes a trivial

matter expressed through the anti-commutation or commutation relations of the

creation and annihilation operators.

1

2 1. Quantum ļ¬elds

1.1 Quantum mechanics

A short discussion of quantum mechanics is ļ¬rst given, setting the scene for the

notation. In quantum mechanics, the state of a physical system is described by a

vector, |Ļ , providing a complete description of the system. The description is unique

modulo a phase factor, i.e. the state of a physical system is properly represented by

a ray, the equivalence class of vectors eiĻ• |Ļ , diļ¬ering only by an overall phase factor

of modulo one.

We consider ļ¬rst a single particle. Of particular intuitive importance are the

states where the particle is deļ¬nitely at a given spatial position, say x, the corre-

sponding state vector being denoted by |x . The projection of an arbitrary state onto

such a position state, the scalar product between the states,

Ļ(x) = x|Ļ , (1.2)

speciļ¬es the probability amplitude, the so-called wave function, whose absolute square

is the probability for the event that the particle is located at the position in question.1

The states of deļ¬nite spatial positions are delta normalized

= Ī“(x ā’ x ) . (1.3)

x|x

Of equal importance is the complementary representation in terms of the states

of deļ¬nite momentum, the corresponding state vectors denoted by |p . Analogous to

the position states they form a complete set or, equivalently, they provide a resolution

Ė

of the identity operator, I, in terms of the momentum state projection operators

Ė

dp |p p| = I . (1.4)

The appearance of an integral in Eq. (1.4) assumes space to be inļ¬nite, and the

(conditional) probability amplitude for the event of the particle to be at position x

given it has momentum p is speciļ¬ed by the plane wave function

1

e pĀ·x ,

i

= (1.5)

x|p 3/2

(2Ļ )

the transformation between the complementary representations being Fourier trans-

formation. The states of deļ¬nite momentum are therefore also delta normalized2

= Ī“(p ā’ p ) . (1.6)

p|p

The possible physical momentum values are represented as eigenvalues, p|p =

Ė

p|p , of the operator

dp p |p p|

Ė

p= (1.7)

1 Treating space as a continuum, the relevant quantity is of course the probability for the particle

being in a small volume around the position in question, P (x)Ī”x = |Ļ(x)|2 Ī”x, the absolute square

of the wave function denoting a probability density.

2 If the particle is conļ¬ned in space, say conļ¬ned in a box as often assumed, the momentum states

are Kronecker normalized, p|p = Ī“p,p .

1.1. Quantum mechanics 3

representing the physical quantity momentum. Similarly for the position of a particle.

The average value of a physical quantity is thus speciļ¬ed by the matrix element of

its corresponding operator, say the average position in state |Ļ is given by the three

real numbers composing the vector Ļ|Ė |Ļ . In physics it is customary to interpret a

x

scalar product as the value of the bra, a linear functional on the state vector space,

on the vector, ket, in question.3

The complementarity of the position and momentum descriptions is also expressed

by the commutator, [Ė , p] ā” x p ā’ p x, of the operators representing the two physical

xĖ ĖĖ ĖĖ

quantities, being the c-number speciļ¬ed by the quantum of action

xĖ

[Ė , p] = i . (1.8)

The fundamental position and momentum representations refer only to the kine-

matical structure of quantum mechanics. The dynamics of a system is determined

Ė pĖ

by the Hamiltonian H = H(Ė , x), the operator speciļ¬ed according to the correspon-

pĖ

dence principle by Hamiltonā™s function H(Ė , x), i.e. for a non-relativistic particle of

mass m in a potential V (x) the Hamiltonian, the energy operator, is

Ė2

Ė = p + V (Ė ) .

H (1.9)

x

2m

It can often be convenient to employ the eigenstates of the Hamiltonian

Ė

H| |

= . (1.10)

Ī» Ī» Ī»

The completeness of the states of deļ¬nite energy, | , is speciļ¬ed by their resolution

Ī»

of the identity

Ė

| Ī» Ī»| = I (1.11)

Ī»

here using a notation corresponding to the case of a discrete spectrum.

At each instant of time a complete description is provided by a state vector, |Ļ(t) ,

thereby deļ¬ning an operator, the time-evolution operator connecting state vectors at

diļ¬erent times

Ė

|Ļ(t) = U (t, t ) |Ļ(t ) . (1.12)

Conservation of probability, conservation of the length of a state vector, or its nor-

malized scalar product Ļ(t)|Ļ(t) = 1, under time evolution, determines the evo-

lution operator to be unitary, U ā’1 (t, t ) = U ā (t, t ). The dynamics is given by the

SchrĀØdinger equation

o

d|Ļ(t) Ė

= H |Ļ(t)

i (1.13)

dt

and for an isolated system the evolution operator is thus the unitary operator

U (t, t ) = eā’ H(tā’t ) .

iĖ

Ė (1.14)

Here we have presented the operator calculus approach to quantum dynamics, the

equivalent path integral approach is presented in Appendix A.

3 For a detailed introduction to quantum mechanics we direct the reader to chapter 1 in reference

[1].

4 1. Quantum ļ¬elds

In order to describe a physical problem we need to specify particulars, typically in

the form of an initial condition. Such general initial condition problems can be solved

through the introduction of the Greenā™s function. The Greenā™s function G(x, t; x , t )

represents the solution to the SchrĀØdinger equation for the particular initial condition

o

where the particle is deļ¬nitely at position x at time t

lim Ļ(x, t) = Ī“(x ā’ x ) = x, t |x , t . (1.15)

t t

The solution of the SchrĀØdinger equation corresponding to this initial condition there-

o

fore depends parametrically on x (and t ), and is by deļ¬nition the conditional prob-

ability density amplitude for the dynamics in question4

Ė ā” G(x, t; x , t ) .

Ļx ,t (x, t) = x, t|x , t = x|U (t, t )|x (1.16)

The Greenā™s function, deļ¬ned to be a solution of the SchrĀØdinger equation, satis-

o

ļ¬es

ā‚

ā’ H(ā’i āx , x) G(x, t; x , t ) = 0

i (1.17)

ā‚t

where, according to Eq. (1.3), the Hamiltonian in the position representation, H, is

speciļ¬ed by the position matrix elements of the Hamiltonian

Ė = H(ā’i āx , x) Ī“(x ā’ x ) .

x|H|x (1.18)

The Greenā™s function, G, is the kernel of the SchrĀØdinger equation on integral

o

form (being a ļ¬rst order diļ¬erential equation in time)

Ļ(x, t) = dx G(x, t; x , t ) Ļ(x , t ) (1.19)

as identiļ¬ed in terms of the matrix elements of the evolution operator by using the

resolution of the identity in terms of the position basis states

Ė x |Ļ(t ) .

x|Ļ(t) = dx x|U (t, t )|x (1.20)

The Greenā™s function propagates the wave function, and we shall therefore also refer

to the Greenā™s function as the propagator. It completely speciļ¬es the quantum

dynamics of the particle.

We note that the partition function of thermodynamics and the trace of the

evolution operator are related by analytical continuation:

= Tr eā’H/kT = dx x|eā’H/kT |x

Ė Ė Ė

Z = Tr U(ā’i /kT, 0)

dx G(x, ā’i /kT ; x, 0)

= (1.21)

4 In

the continuum limit the Greenā™s function is not a normalizable solution of the SchrĀØdinger

o

equation, as is clear from Eq. (1.15).

1.2. N-particle system 5

showing that the partition function is obtained from the propagator at the imaginary

time Ļ„ = ā’i /kT . The formalisms of thermodynamics, i.e. equilibrium statistical

mechanics, and quantum mechanics are thus equivalent, a fact we shall take advan-

tage of throughout. The physical signiļ¬cance is the formal equivalence of quantum

and thermal ļ¬‚uctuations.

Quantum mechanics can be formulated without the use of operators, viz. using

Feynmanā™s path integral formulation. In Appendix A, the path integral expressions

for the propagator and partition function for a single particle are obtained. Various

types of Greenā™s functions and their properties for the case of a single particle are

discussed in Appendix C, and their analytical properties are considered in Appendix

D.

N -particle system

1.2

Next we consider a physical system consisting of N particles. If the particles in an

assembly are distinguishable, i.e. diļ¬erent species of particles, an orthonormal basis

in the N -particle state space H (N ) = H1 ā— H2 ā— Ā· Ā· Ā· ā— HN is the (tensor) product

states, for example speciļ¬ed in terms of the momentum quantum numbers of the

particles

|p1 , p2 , . . . , pN ā” |p1 ā— |p2 ā— Ā· Ā· Ā· ā— |pN ā” |p1 |p2 Ā· Ā· Ā· |pN . (1.22)

We follow the custom of suppressing the tensorial notation.

Formally everything in the following, where an N -particle system is considered,

is equivalent no matter which complete set of single-particle states are used. In prac-

tice the choice follows from the context, and to be speciļ¬c we shall mainly explicitly

employ the momentum states, the choice convenient in practice for a spatially trans-

lational invariant system.5 These states are eigenstates of the momentum operators

pi |p1 , p2 , . . . , pN = pi |p1 , p2 , . . . , pN ,

Ė (1.23)

where tensorial notation for operators are suppressed, i.e.

Ė Ė Ė Ė

pi = I1 ā— Ā· Ā· Ā· Iiā’1 ā— pi ā— Ii+1 ā— Ā· Ā· Ā· IN ,

Ė Ė (1.24)

each operating in the one-particle subspace dictated by its index. In particular the

N -particle momentum states are eigenstates of the total momentum operator

N

Ė Ė

PN = (1.25)

pi

i=1

5 In the next sections we shall mainly use the momentum basis, and refer in the following to

the quantum numbers labeling the one-particle states as momentum, although any complete set

of quantum numbers could equally well be used. The N -tuple (p1 , p2 , . . . , pN ) is a complete

description of the N -particle system if the particles do not posses internal degrees of freedom. In

the following, where we for example have electrons in mind, we suppress for simplicity of notation

the spin labeling and simply assume it is absorbed in the momentum labeling. If the particles

have additional internal degrees of freedom, such as color and ļ¬‚avor, these are included in a similar

fashion. If more than one type of species is to be considered simultaneously the species type, say

quark and gluon, must also be indicated.

6 1. Quantum ļ¬elds

corresponding to the total momentum eigenvalue

N

P= pi . (1.26)

i=1

The position representation of the momentum states is speciļ¬ed by the plane wave

functions, Eq. (1.5), the scalar product of the momentum states and the analogous

N -particle states of deļ¬nite positions being

N

x1 , x2 , . . . , xN |p1 , p2 , . . . , pN xi |pi

Ļp1 ,...,pN (x1 , . . . , xN ) = =

i=1

N

1 p1 Ā·x1 p2 Ā·x2 pN Ā·xN

i i i

Ā·Ā·Ā· e

= e e . (1.27)

(2Ļ )3/2

1.2.1 Identical particles

For an assembly of identical particles a profound change in the above description

is needed. In quantum mechanics true identity between objects are realized, viz.

elementary particle species, say electrons, are profoundly identical, i.e. there exists

nothing in Nature which can distinguish any two electrons. Identical particles are in-

distinguishable. States which diļ¬er only by two identical particles being interchanged

are thus described by the same ray.6 As a consequence of their indistinguishability,

assemblies of identical particles are described by states which with respect to inter-

change of pairs of identical particles are either antisymmetric or symmetric

|p1 , p2 , . . . , pN = Ā± |p2 , p1 , . . . , pN , (1.28)

this leaving the probability for a set of momenta of the particles, P (p1 , p2 , . . . , pN ), a

function symmetric with respect to interchange of any pair of the identical particles.

A word on notation: the particle we call the ļ¬rst particle is in the momentum

state speciļ¬ed by the ļ¬rst argument, and the particle we call the N th particle is

in the momentum state speciļ¬ed by the N th argument. Particles whose states are

symmetric with respect to interchange are called bosons , and for the antisymmetric

case called fermions.7

6 The quantum state with all of the electrons in the Universe interchanged will thus be the same

as the present one. A radical invariance property of systems of identical particles!

7 Quantum statistics and the spin degree of freedom of a particle are intimately connected as

relativistic quantum ļ¬eld theory demands that bosons have integer spin, whereas particles with

half-integer spin are fermions. This so-called spin-statistics connection seems in the present non-

relativistic quantum theory quite mysterious, i.e. unintelligible. It only gets its explanation in

the relativistic quantum theory, which we usually connect with high energy phenomena, where for

any particle relativity, through Lorentz invariance, requires the existence of an anti-particle of the

same mass and opposite charge (some neutral particles, such as the photon, are their own anti-

particle). Then, in fermion anti-fermion pair production the particles must be antisymmetric with

already existing particles as unitarity, i.e. conservation of probability, requires such a minus sign

[2]. Historically, the exclusion principle, which is a direct consequence of Fermi statistics, was

discovered by Pauli before the advent of relativistic quantum theory as a vehicle to explain the

periodic properties of the elements. Pauli was also the ļ¬rst to show that the spin-statistics relation

is a consequence of Lorentz invariance, causality and energy and norm positivity.

1.2. N-particle system 7

Any N -particle state |p1 , p2 , . . . , pN can be mapped into a state which is either

symmetric or antisymmetric with respect to interchange of any two particles. To

Ė

obtain the symmetric state we simply apply the symmetrization operator S which

symmetrizes an N -particle state according to

1

Ė

S |p1 , p2 , . . . , pN |pP 1 |pP 2 Ā· Ā· Ā· |pP N

= (1.29)

N!

P

Ė

and the antisymmetrization operator A antisymmetrizes according to

1

Ė

A |p1 , p2 , . . . , pN (ā’1)Ī¶P |pP 1 |pP 2 Ā· Ā· Ā· |pP N .

= (1.30)

N!

P

The summations are over all permutations P of the particles. Permutations form

a group, and any permutation can be build by successive transpositions which only

permute a pair. In the case of antisymmetrization, each term appears with the sign

of the permutation in question

jā’i

sign(P ) = . (1.31)

Pj ā’ Pi

1ā¤i<jā¤N

We have written this in terms of the number Ī¶P which counts the number of trans-

positions needed to build the permutation P , since sign(P ) = (ā’1)Ī¶P .

If the single-particle state labels in the N -particle state to be symmetrized on the

left in Eq. (1.29) are permuted, the same symmetrized state results, since if P can

be any of the N ! permutations, then P P for ļ¬xed permutation P will run through

Ė Ė

them all, S |pP 1 , pP 2 , . . . pP N = S |p1 , p2 , . . . , pN .

We note that the sign of a product of permutations, Q = P P , equals the product

of the signs of the two permutations, sign(Q) = sign(P ) Ā· sign(P ), and a permutation

and its inverse have the same sign (owing to their equal number of transpositions),

Ī¶P ā’1 = Ī¶P . Antisymmetrization of a permuted state gives the same antisymmetric

state multiplied by the sign of the permutation permuting the original N -particle

state since

1

Ė

A |pP 1 , pP 2 , . . . pP N (ā’1)Ī¶P |pQ 1 |pQ 2 Ā· Ā· Ā· |pQ N

= (1.32)

N!

P

and as P runs through all the permutations so does Q = P P , and therefore

1

Ė

A |pP 1 , pP 2 , . . . pP N (ā’1)Ī¶Q |pQ 1 |pQ 2 Ā· Ā· Ā· |pQ N

= (ā’1)Ī¶P

N!

Q

Ė

= (ā’1)Ī¶P A |p1 , p2 , . . . , pN . (1.33)

Therefore, if any two single-particle states are identical, the antisymmetrized state

vector equals the zero vector, since the two states obtained by permuting the two

identical labels are identical and yet upon antisymmatrization they diļ¬er by a minus

sign, i.e. Pauliā™s exclusion principle for fermions: no two fermions can occupy the

same state.

8 1. Quantum ļ¬elds

Further, according to Eq. (1.33), applying the antisymmetrization operator twice

Ė1

Ė

A2 |p1 , p2 , . . . , pN =A (ā’1)Ī¶P |pP 1 |pP 2 Ā· Ā· Ā· |pP N

N!

P

1 Ė

(ā’1)Ī¶P (ā’1)Ī¶P A |p1 , p2 , . . . , pN

=

N!

P

Ė

= A |p1 , p2 , . . . , pN (1.34)

gives the same state as applying it only once, i.e. the symmetrization operators are

Ė ĖĖ Ė

projectors, A2 = A, S 2 = S. The presence of the factor 1/N ! in the deļ¬nitions,

Eq. (1.29) and Eq. (1.30), is thus there to ensure the operators are normalized pro-

jectors. Representing mutually exclusive symmetry properties, they are orthogonal

projectors, their product is the operator that maps any vector onto the zero vector

ĖĖ 0 ĖĖ

AS =Ė=SA (1.35)

since symmetrizing an antisymmetric state, or vice versa, gives the zero vector.

The symmetrization operators are hermitian, Aā = A, S ā = S, as veriļ¬ed for

Ė ĖĖ Ė

Ė

example for A by ļ¬rst noting that according to the deļ¬nition of the adjoint operator

p1 , . . . , pN |Aā |p1 , p2 , . . . , pN ā—

Ė Ė

p1 , . . . , pN |A|p1 , p2 , . . . , pN

=

1 ā— ā—

(ā’1)Ī¶P p1 |pP 1 Ā· Ā· Ā· pN |pP N

=

N!

P

(ā’1)Ī¶S

pS 1 |p1 Ā· Ā· Ā· pS N |pN

= (1.36)

N!

the matrix element being nonzero only if the set {pi }i=1,...,N is a permutation of the

set {pi }i=1,...,N , S being the permutation that brings the set {pi }i=1,...,N into the set

{pi }i=1,...,N , pS i = pi . Permuting both sets of indices by the inverse permutation

S ā’1 of S, and using that Ī¶S ā’1 = Ī¶S , we get

1

p1 , . . . , pN |Aā |p1 , p2 , . . . , pN

Ė (ā’1)Ī¶S ā’1 p1 |pS ā’1 Ā· Ā· Ā· pN |pS ā’1

=

N! 1 N

1

(ā’1)Ī¶P p1 , . . . , pN |pP1 , . . . , pPN

=

N!

P

Ė

p1 , . . . , pN |A|p1 , . . . , pN .

= (1.37)

Exercise 1.1. Show that the adjoint of a product of linear operators A and B is the

product of their adjoint operators in opposite sequence

(A B)ā = B ā Aā (1.38)

and generalize to the case of an arbitrary number of operators.

1.2. N-particle system 9

Exercise 1.2. The vector space of state vectors, the kets, and the dual space of

linear functionals on the state space, the bras, are isomorphic vector spaces, which

we express by the adjoint operation, |Ļ ā = Ļ| and Ļ|ā = |Ļ . This mapping

is anti-linear and isomorphic, and we use the same symbol as for the adjoint of an

operator.

Show that for arbitrary state vectors and operators on the state space the rela-

tionship (X|Ļ )ā = Ļ|X ā . An operator being its own adjoint, X ā = X, is said to

Ė Ė Ė Ė

be a hermitian operator and its eigenvalues are real, such operators being of primary

importance in quantum mechanics.

Ė

Exercise 1.3. Show that the symmetrization operator, S, is hermitian.

Ė Ė

The linear operators S and A project any state onto either of the two orthogonal

subspaces of symmetric or antisymmetric states.8 The state space for a physical

system consisting of N identical particles is thus not H (N ) , the N -fold product of

the one-particle state space, but either the symmetric subspace, B (N ) , for bosons,

or antisymmetric subspace, F (N ) , for fermions, obtained by projecting the states of

H (N ) by either type of symmetrization operator depending on the statistics of the

particles in question.

1.2.2 Kinematics of fermions

Let us introduce the orthogonal, normalized up to a phase factor, antisymmetric

basis states in the antisymmetric N -particle state space F (N )

ā

Ė

|p1 ā§ p2 ā§ Ā· Ā· Ā· ā§ pN ā” N ! A |p1 , p2 , . . . , pN

1

ā (ā’1)Ī¶P |pP 1 ā— |pP 2 ā— Ā· Ā· Ā· ā— |pP N

=

N! P

1

ā (ā’1)Ī¶P |pP 1 |pP 2 Ā· Ā· Ā· |pP N

=

N! P

1

ā (ā’1)Ī¶P |pP 1 , pP 2 , . . . , pP N .

= (1.39)

N! P

We demonstrate that they are orthogonal by using the properties of the antisym-

metrization operator (we ļ¬rst for simplicity of the Kronecker function assume box

normalization, i.e, the momentum values are discrete)

N ! p1 , . . . , pN |Aā A|p1 , . . . , pN

ĖĖ

p1 ā§ Ā· Ā· Ā· ā§ pN |p1 ā§ Ā· Ā· Ā· ā§ pN =

Ė

N ! p1 , . . . , pN |A|p1 , . . . , pN

=

8 Only for the case of two particles do the two subspaces of symmetric and antisymmetric states

span the original state space, H (2) = H ā— H. In general, the other subspaces for the case of more

than two particles do not seem to be state spaces for systems of identical particles.

10 1. Quantum ļ¬elds

p1 , . . . , pN | (ā’1)Ī¶P |pP 1 , . . . , pP

=

N

P

ā§

{p }i ā” {p}i

āØ (ā’1)Ī¶S

= (1.40)

ā©

0 otherwise

where {pi }i=1,...,N ā” {pi }i=1,...,N is short for the labels {pi }i=1,...,N being a permuta-

tion of the labels {pi }i=1,...,N , and S the permutation that takes the set {pi }i=1,...,N

into {pi }i=1,...,N , pS i = pi . Or simply in words, only if the primed set of momenta

is a permutation of the unprimed set is the scalar product of the states nonzero (we

have of course assumed that all momenta are diļ¬erent since otherwise for fermions

the vector is the zero-vector).

Incidentally we have

ā§1

āØ āN ! (ā’1) S {p }i ā” {p}i

Ī¶

p1 ā§ p2 , ā§ Ā· Ā· Ā· ā§ pN | p1 , p2 , .., pN = (1.41)

ā©

0 otherwise

expressing that additional permutations are redundant, for example an additional an-

tisymmetrization is redundant as expressed by the second equality sign in Eq. (1.40),

or equivalently that the symmetrization operators are hermitian projectors.

The scalar product of antisymmetric states is the determinant of the N Ć— N

matrix with entries pi |pj

p1 ā§ Ā· Ā· Ā· ā§ pN |p1 ā§ Ā· Ā· Ā· ā§ pN det( pi |pj )

=

(ā’1)Ī¶P p1 |pP 1 Ā· Ā· Ā· pN |pP

= , (1.42)

N

P

the Slater determinant.

In the operator calculus perturbation theory for a single particle, the resolution

of the identity plays a crucial eļ¬cient role. For an assembly of identical particles

this role will be taken over by the commutation rules for the quantum ļ¬elds we shall

shortly introduce. The resolutions of the identity on the symmetrized subspaces

reļ¬‚ect the redundancy of antisymmetrized or symmetrized states. Though not of

much practical use, we include them for completeness (the resolution of the identity

makes a short appearance in Section 3.1.1). The resolution of the identity on the

antisymmetric state space can be written in terms of the N -state identity operator

since the identity operator commutes with any operator

Aā

ĖĖ Ė ĖĖ Ė Ė Ė Ė

A I (N ) A = A I1 ā— I2 ā— I3 ā— Ā· Ā· Ā· ā— IN

1=

|p1 p1 | ā— |p2 p2 | ā— Ā· Ā· Ā· ā— |pN pN | Aā

Ė Ė

A

=

p1 ,...,pN

|p1 , p2 , . . . , pN p1 , p2 , . . . , pN | Aā

Ė Ė

A

=

p1 ,...,pN

1.2. N-particle system 11

1

|p1 ā§ p2 ā§ Ā· Ā· Ā· ā§ pN p1 ā§ p2 ā§ Ā· Ā· Ā· ā§ pN |

=

N! p

1 ,...,pN

|p1 ā§ p2 ā§ Ā· Ā· Ā· ā§ pN p1 ā§ p2 ā§ Ā· Ā· Ā· ā§ pN | .

= (1.43)

|p1 |<|p2 |< Ā·Ā·Ā· <|pN |

In obtaining the last equality we have used the fact that if the momenta of the

particles are interchanged in the N -particle state to be antisymmetrized, the same

antisymmetric state vector is obtained modulo a phase factor Ā±1, for example

Ė Ė

A |p1 , p2 , . . . , pN = ā’ A |p2 , p1 , . . . , pN . (1.44)

In the sum in the second last expression in Eq. (1.43), there are thus N ! identical

terms.

The symmetrization phase factor in Eq. (1.40) can always be chosen to equal 1 by

considering proper orderings in the deļ¬nition of the basis states, thereby removing

the redundancy in the general deļ¬nition, Eq. (1.39), of the basis states. For example,

if we choose to use only basis vectors where the momenta appear ordered according

to the ordering |p1 | < |p2 | < Ā· Ā· Ā· < |pN |, this restriction on deļ¬ning the set of

basis states |p1 ā§ p2 ā§ Ā· Ā· Ā· ā§ pN results in them forming an orthonormal basis in the

antisymmetric state space FN , as also expressed by the last equality in Eq. (1.43).

1.2.3 Kinematics of bosons

We now turn to a discussion of the state space relevant for N identical bosons. In

the symmetric state space, B (N ) , we introduce the symmetric orthogonal basis states

ā

Ė

|p1 āØ p2 āØ Ā· Ā· Ā· āØ pN ā” N ! S |p1 , p2 , . . . , pN

1

ā |pP 1 ā— |pP 2 ā— Ā· Ā· Ā· ā— |pP N

=

N! P

1

ā |pP 1 |pP 2 Ā· Ā· Ā· |pP N

=

N! P

1

ā |pP 1 , pP 2 , . . . , pP N .

= (1.45)

N! P

All derivations of formulas to be obtained for symmetric basis states runs equivalent

to those for antisymmetric basis states. For example,

p1 āØ Ā· Ā· Ā· āØ pN |p1 āØ Ā· Ā· Ā· āØ pN p1 |pP 1 Ā· Ā· Ā· pN |pP

=

N

P

per( pi |pj ) ,

= (1.46)

where the last equality deļ¬nes the permanent of the N Ć— N matrix which has the

entries pi |pj .

12 1. Quantum ļ¬elds

In fact, the bose and fermi cases, i.e. the symmetric and antisymmetric basis

states, can be treated simultaneously if we introduce the factor ( )Ī¶P inside the

summation sign

1

ā

|p1 3 Ā· Ā· Ā· 3pN ā” ( )Ī¶P |pP 1 , pP 2 , . . . , pP N (1.47)

N! P

since then the fermi case corresponds to = ā’1 and the bose case to = +1, and 3

stands for āØ or ā§ for the bose and fermi cases, respectively.

The states introduced in Eq. (1.45) provide a resolution of the identity in the

symmetric state space, B (N ) , speciļ¬ed by

Sā

ĖĖ Ė ĖĖ Ė Ė Ė Ė

S I (N ) S = S I1 ā— I2 ā— I3 ā— Ā· Ā· Ā· ā— IN

1=

|p1 p1 | ā— |p2 p2 | ā— Ā· Ā· Ā· ā— |pN pN | S ā

Ė Ė

S

=

p1 ,...,pN

|p1 , p2 , . . . , pN p1 , p2 , . . . , pN | S ā

Ė Ė

S

=

p1 ,...,pN

1

|p1 āØ p2 āØ Ā· Ā· Ā· āØ pN p1 āØ p2 āØ Ā· Ā· Ā· āØ pN | .

= (1.48)

N! p

1 ,Ā·Ā·Ā·,pN

The symmetric states introduced in Eq. (1.45) are not normalized in general, since

for bosons the momenta need not diļ¬er. Of course, if the momentum values are all

diļ¬erent, the state |p1 āØ p2 āØ Ā· Ā· Ā· āØ pN is a sum of N ! normalized N -particle states

which are all orthogonal to each other, and the state is therefore normalized in view

of the overall prefactor. However, if say n1 of the momentum values equals p1 and all

the rest are diļ¬erent, the state |p1 āØ p2 āØ Ā· Ā· Ā· āØ pN will be a sum of N !/n1 ! N -particle

states each orthogonal to each other but now appearing with the prefactor n1 ! ,

since permutations among the identical labels produce the same N -particle state. In

general, if ni is the number of times pi occurs among the vectors p1 , p2 , . . . , pN , nj

being equal to 0 if the momentum value pj does not appear, then the set of ordered

vectors, choosing for example the ordering according to |p1 | ā¤ |p2 | ā¤ Ā· Ā· Ā· ā¤ |pN |,

1 N!

ā Ė

|p1 āØ p2 āØ Ā· Ā· Ā· āØ pN S |p1 , p2 , . . . , pN (1.49)

=

n1 !n2 ! Ā· Ā· Ā· nN !

n1 !n2 ! Ā· Ā· Ā· nN !

constitute an orthonormal basis for the symmetric state space.

Equivalently we can state for the scalar product in Eq. (1.46)

ā§

āØ n1 ! n2 ! Ā· Ā· Ā· {p }i ā” {p}i

p1 āØ Ā· Ā· Ā· āØ pN |p1 āØ Ā· Ā· Ā· āØ pN = (1.50)

ā©

0 otherwise.

The resolution of the identity in the symmetric N -particle state space can there-

fore also be expressed in terms of orthonormal states according to

1

Ė(N ) |p1 āØ p2 āØ Ā· Ā· Ā· āØ pN p1 āØ p2 āØ Ā· Ā· Ā· āØ pN |. (1.51)

IS =

n1 !n2 !n3 ! Ā· Ā· Ā·

|p1 |ā¤|p2 |ā¤Ā·Ā·Ā·ā¤|pN |

1.2. N-particle system 13

1.2.4 Dynamics and probability current and density

The quantum dynamics of an N -particle system of identical particles is given by the

SchrĀØdinger equation

o

ā‚Ļ(x1 , x2 , . . . ; t)

i = H Ļ(x1 , x2 , . . . ; t) , (1.52)

ā‚t

where H is the Hamiltonian in the position representation for the N -particle sys-

tem. For example, for the case of N non-relativistic electrons interacting through

instantaneous two-particle interaction the Hamiltonian is

N 2

1 ā‚ 1

V (xi ā’ xj ) .

H= + (1.53)

2m i ā‚xi 2

i=1 i=j

In non-relativistic quantum mechanics the even or odd character of a wave function

is preserved in time as any Hamiltonian for identical particles is symmetric in the

degrees of freedom, here in the momenta and positions, but as well as other degrees

of freedom in general (this is the meaning of identity of particles, no interaction can

distinguish them). So if even- or oddness of a wave function is the state of aļ¬airs at

one moment in time it will stay this way for all times.9

All physical properties are expressible in terms of the wave function, for example

the average density of the particles, or rather the probability for the event that one

of the particles is at position x, is

N N

1

dxj Ī“(xi ā’ x) |Ļ(x1 , x2 , . . . ; t)|2 = dxi |Ļ(x, x2 , . . . ; t)|2

n(x, t) =

N

i=1 j=1 i=1

(1.54)

where the last equality follows from the symmetry of the wave function for identical

particles.

Taking the time derivative of the probability density, and using the SchrĀØdinger

o

equation, gives the continuity equation

ā‚ n(x, t)

+ āx Ā· j(x, t) = 0 , (1.55)

ā‚t

where the probability current density is10

N N

e 1

dxj Ī“(xi ā’ x) āxi ā’ āxi Ļ(x1 , . . . ; t) Ļ ā— (x1 , ..; t)

j(x, t) =

2mi N

i=1 j=1

xi =xi

e

dxi (āx ā’ āx ) Ļ(x, x2 , . . . ; t) Ļ ā— (x , x2 , . . . ; t)

= , (1.56)

2mi

i=1

x =x

9 For the cases of more than two identical particles there are other time invariant subspaces than

the symmetric and antisymmetric ones. They do not seem to be of physical relevance.

10 In the presence of a vector potential the formula must be amended with the diamagnetic term,

see Exercise 1.4 and Section 2.3.

14 1. Quantum ļ¬elds

again the last equality follows from the symmetry of the wave function for identical

particles. The SchrĀØdinger equation guarantees conservation of probability, i.e. the

o

continuity equation, Eq. (1.55), as a consequence of the Hamiltonian being hermitian.

Exercise 1.4. The Hamiltonian for a charged spinless particle coupled to a vector

potential, A, is11

2

Ė= 1 ā‚

ā’ eA(x, t)

H . (1.57)

2m i ā‚x

Show that the probability current density for the particle in state Ļ is

1

āx ā’ 2eA(x, t) Ļ(x, t) Ļ ā— (x , t)

āx ā’

j(x, t) = . (1.58)

2m i i

x =x

Rarely can the dynamics of an N -particle system of identical particles be solved

exactly. When it comes to performing actual approximate calculations, the quantum

statistics of the particles will even in the non-relativistic quantum theory of an in-

teracting N -particle system cause havoc, and a more ļ¬‚exible vehicle for respecting

the quantum statistics of identical particles is convenient. We now turn to introduce

these, the quantum ļ¬elds. In relativistic quantum theory and conveniently for many-

body systems, the quantum ļ¬elds instead of the wave function become the carriers

of the dynamics, as we will discuss in Chapter 3.

1.3 Fermi ļ¬eld

We introduce the fermion creation operator, aā , corresponding to momentum value

p

p, as the linear mapping of F into F

(N ) (N +1)

deļ¬ned for an arbitrary (not necessarily

12

ordered) basis vector by

aā |p1 ā§ p2 ā§ Ā· Ā· Ā· ā§ pN ā” |p ā§ p1 ā§ p2 ā§ Ā· Ā· Ā· ā§ pN , (1.59)

p

i.e. it maps an antisymmetrized N -particle state into the antisymmetrized (N + 1)-

particle state where an additional fermion has momentum p. The choice of placing

p at the front is, of course, arbitrary. The other popular choice is to place it at the

end. This reļ¬‚ects that a creation operator, like a state vector, is deļ¬ned only modulo

a phase factor.

If in the N -fermion state the momentum state p is already occupied, i.e. exactly

one of the pi s equals p, then owing to the antisymmetric nature of the state

aā |p1 ā§ p2 ā§ Ā· Ā· Ā· ā§ pN = 0N +1 , (1.60)

p

11 The form of the Hamiltonian follows from gauge invariance; i.e. the gauge transformation of

the electromagnetic ļ¬eld, A(x, t) ā’ A(x, t) + āĪ(x, t), Ļ(x, t) ā’ Ļ(x, t) ā’ Ī(x, t), and the trans-

Ė™

ie

formation of the wave function Ļ(x, t) ā’ Ļ(x, t) e Ī(x,t) , leaves all physical quantities invariant.

The gauge invariance of quantum mechanics is a consequence of the wave function obtained by the

above phase transformation equally well represents the probability distribution of the particle.

12 As emphasized, the label on the creation operator could refer to any state; usually though, it

refers to a complete set of single-particle states.

1.3. Fermi ļ¬eld 15

the zero vector of state space H (N +1) . This is the expedience with which the fermion

creation operators respect Pauliā™s exclusion principle.

We introduce the sum of state spaces F (N ) and F (N +1) . For example, F (1) + F (2)

consists of states spanned by 2-tuple states, (|p , |p1 ā§ p2 ), and is equipped with the

scalar product, which is the sum of the scalar products in the subspaces, i.e. for the

above vector and (c1 |p , c2 |p1 ā§ p2 ) the scalar product is

( p|, p1 ā§ p2 |)(c1 |p , c2 |p1 ā§ p2 ) = c1 p|p + c2 p1 ā§ p2 |p1 ā§ p2 .

(1.61)

In order for an operator to represent an observable physical quantity it must

map a state space onto itself. In order to facilitate this experience for the fermion

creation operators,13 the multi-particle space or Fock space F (named after the Soviet

physicist Vladimir Fock), is introduced as the sum of the state spaces14

ā

F= F (N ) , (1.62)

N =0

where by deļ¬nition F (0) is the set of complex numbers.

The inclusion of F (0) is demanded in relativistic quantum theory since relativistic

kinematics predicts the creation and annihilation of particles. The zero vector in Fock

space can not represent the state of absence of any particle.15 Particle species not

present in the initial and ļ¬nal states must, before and after a reaction, be in a state

in their respective Fock spaces so that their scalar products equal one, thereby not

inļ¬‚uencing the probabilities for the various possible reactions. Since none of these

particle states is initially and ļ¬nally occupied, though they may appear virtually

in intermediate states to facilitate the reaction, and since the zero vector does not

respect the above property, the state where particles of a given species are absent,

the vacuum state for these particles is represented by (choosing the simplest phase

choice)

|0 ā” (1, 01 , 02 , . . .) . (1.63)

Even for a non-relativistic system, the vacuum state is a convenient vehicle for gen-

erating all states of the multi-particle space as we will see shortly.

The set of basis states of the Fock space consists of the vacuum state and all

the basis vectors of each N -particle subspace. In the Fock space, states of the type

(0, |p1 , 02 , |p1 ā§ p2 ā§ p3 , . . . , |p1 ā§ p2 , ā§ Ā· Ā· Ā· ā§ pN , . . .) are thus encountered, su-

perposition of states with diļ¬erent number of particles. In accordance with the

deļ¬nition of the scalar product of states in the multi-particle space, it can only

13 Whether a fermi ļ¬eld is an observable, i.e. a measurable quantity, is doubtful. For example, it

does not have a classical limit as states can at most be singly occupied. A bose ļ¬eld (introduced

in Section 1.4) on the other hand is an observable, since any number of bosons can occupy a single

state and the average value of a bose ļ¬eld can thus be nonzero, an example being the classical state

of light, the coherent state, created by a laser.

14 In mathematical terms, the state space is a Hilbert space, and the Fock space is a Hilbert sum

of Hilbert spaces, and itself a Hilbert space.

15 The zero vector in the Fock space is of course (0, 0 , 0 , . . .) ā” 0, for which the obvious short

12

notation has been introduced.

16 1. Quantum ļ¬elds

be nonzero if the states have components with the same number of particles. An

N -particle basis state in the multi-particle space is usually shortened according to

(0, 01 , 02 , . . . , |p1 ā§ p2 ā§ Ā· Ā· Ā· ā§ pN , 0N +1 , 0N +2 , . . .) ā’ |p1 ā§ p2 ā§ Ā· Ā· Ā· ā§ pN .

For the vacuum state, the creation operator operates also in accordance with its

general prescription of adding a particle

aā |0 = |0, |p , 02 , 03 , . . . . (1.64)

p

The state vector (using the abbreviated notation introduced above)

aā 1 Ā· Ā· Ā· aā N |0 = |p1 ā§ p2 ā§ Ā· Ā· Ā· ā§ pN (1.65)

p p

is an antisymmetric N -particle basis state in the multi-particle state space, provided

that all the momenta are diļ¬erent of course, otherwise it is the zero-vector.16 The

bracket notation appears a little clumsy in this context, and the notation

= a ā 1 Ā· Ā· Ā· a ā N Ī¦0

Ī¦p1 ,...,pN (1.66)

p p

is often used, where Ī¦0 denotes the vacuum. For a state which is a superposition

of states with diļ¬erent number of particles we can also express it in terms of the

vacuum state, for example (0, |p1 , |p1 ā§ p2 , 03 , . . . = (aā 1 + aā aā )|0 .

p pp 1 2

By construction, any two fermion creation operators, aā and aā , anti-commute,

p p

i.e.

{aā , aā } ā” aā aā + aā aā = 0 , (1.67)

pp pp pp

meaning that operating with the anti-commutator {aā , aā } on any vector in Fock

pp

space produces the zero vector in Fock space, 0 ā” (0, 01 , 02 , . . .), just as multi-

plying any vector in Fock space by the number 0 does. This follows immediately

from the fact that operating with the anti-commutator on any basis vector, say

(0, 01 , 02 , . . . , |p1 ā§ p2 ā§ Ā· Ā· Ā· ā§ pN , 0N +1 , 0N +2 , . . .), gives the sum of two vectors

which diļ¬er only by a minus sign (or if the momentum labels in the anti-commutator

are equal, the sum of two zero vectors). For the case p = p, the anti-commutation

relation Eq. (1.67) becomes aā aā = ā’ aā aā and therefore by itself aā aā = 0. This is

pp pp pp

Pauliā™s exclusion principle expressed in terms of the creation operator: two fermions

can not be accommodated in the same state.

We then introduce the fermion annihilation operator, ap , as the adjoint of the

fermion creation operator aā . Since the creation operator maps an N -particle state

p

into an (N + 1)-particle state, the annihilation operator, being the adjoint, will map

an N -particle state into an (N ā’ 1)-particle state. To understand its properties we

can restrict attention to the basis vectors of the subspaces F (N ) and F (N ā’1) of the

Fock space, and we have

ā—

p1 ā§ Ā· Ā· Ā· ā§ pN |aā |p2 ā§ Ā· Ā· Ā· ā§ pN

p2 ā§ Ā· Ā· Ā· ā§ pN |ap |p1 ā§ Ā· Ā· Ā· ā§ pN = p

p1 ā§ Ā· Ā· Ā· ā§ pN |p ā§ p2 ā§ Ā· Ā· Ā· ā§ pN ā’1

=

det( pi |pj ) ,

= (1.68)

16 With the chosen ordering convention of the previous section it is the ground state for N non-

interacting fermions.

1.3. Fermi ļ¬eld 17

where, in the last equality, we have introduced the notation p1 = p, and used

Eq. (1.42). Expanding the determinant in terms of its ļ¬rst column we get

N

ā—

p2 ā§ Ā· Ā· Ā· ā§ pN |ap |p1 ā§ Ā· Ā· Ā· ā§ pN (ā’1)nā’1 pn |p det( pi |pj (n)

= )

n=1

N

(ā’1)nā’1 pn |p det( pi |pj (n)

= ),

n=1

(1.69)

where the sub-determinant, det( pi |pj (n) ), is the determinant of the matrix Eq. (1.68),

with row n and the ļ¬rst column removed. Using p|p ā— = p |p we get

N

p2 ā§ Ā· Ā· Ā· ā§ pN |ap |p1 ā§ Ā· Ā· Ā· ā§ pN (ā’1)nā’1 p|pn det( pj |pi (n)

= )

n=1

(1.70)

and using Eq. (1.42) for the (N ā’1)-particle case, the right-hand side can be rewritten

as

N

(ā’1)nā’1 p|pn p2 ā§ Ā· Ā· Ā· ā§ pN |p1 ā§ Ā· Ā· Ā· ( no pn ) .. ā§ pN (1.71)

n=1

and we have

N

ap |p1 ā§ Ā· Ā· Ā· ā§ pN (ā’1)nā’1 p|pn |p1 ā§ Ā· Ā· Ā· ( no pn ) Ā· Ā· Ā· ā§ pN .

=

n=1

(1.72)

Thus operating with the fermion annihilation operator labeled by p on an N -particle

basis state produces the zero vector unless exactly one of the momentum values equals

p, and in that case it equals the (N ā’ 1)-particle state where none of the fermions

occupies the originally occupied momentum state p. The annihilation operator ap

thus annihilates the particle in state p. In the simplest of situations we have

ap |p |0 .

= (1.73)

Annihilating the single-particle state turns it into the vacuum state.

In particular it follows from Eq. (1.68) that operating with any fermion annihila-

tion operator on the vacuum state produces the zero vector

ap |0 = 0. (1.74)

18 1. Quantum ļ¬elds

According to Eq. (1.67), the fermion annihilation operators anti-commute

{ap , ap } = 0 . (1.75)

For the case p = p, the anti-commutation relation Eq. (1.75) has the consequence

ap ap = 0, expressing the exclusion principle: no two identical fermions can occupy

the same momentum state.

Next we inquire into the relations obtained by subsequent operations with fermion

creation and annihilation operators, and calculate, according to Eq. (1.72),

ńņš. 1 |