Functional Methods in Quantum Field Theory and Statistical Physics

327 Pages·2019·107.185 MB·English

by  Vasiliev, A.N

Functional Methods in Quantum Field
Theory and Statistical Physics

CRC Press
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CONTENTS
Foreword xi
Preface xiii
Chapter 1 THE BASIC FORMALISM OF FIELD THEORY
1. 1 Fields and Products
1.1.1 Canonical quantization 1
1.1.2 The classical free theory 3
1.1.3 Anticommuting fields 4
1.1.4 The normal-ordered product of free-field operators 7
1.2 Functional Formulations of Wick's Theorem 9
1.2.1 Wick's theorem for a simple product 9
1.2.2 The Sym-product and the T-product 12
1.2.3 Wick's theorem for symmetric products 15
1.2.4 Reduction formulas for operator functionals 18
1.2.5 The Wick and Dyson T-products 21
1.3 The S·Matrix and Green Functions 23
1.3.1 Definitions 23
1.3.2 Transformation to the interaction picture
in the evolution operator 25
1.3.3 Transformation to the interaction picture for
the Green functions 27
1.3.4 Interactions containing time derivatives of the field 31
1.3.5 Generating functionals for the S-matrix and Green
functions 34
1.4 Graphs 37
1.4.1 Pertuibation theory 37
1.4.2 Some concepts from graph theory 39
1.4.3 Symmetry coefficients 39
1.4.4 Recursion relation for the symmetry coefficients 41
1.4.5 Transformation to Mayer graphs for the exponential
interaction 42
1.4.6 Graphs for the Yukawa-type interaction 43
1.4.7 Graphs for a pair interaction 48
1.4.8 Connectedness of the logarithm of R(<p) 49
1.4.9 Graphs for the Green functions 50
1.5 Unitarity of the S Matrix 53
1.5.1 The conjugation operation 53
1.5.2 Formal unitarity of the off-shell S matrix 55
v
vi CONTENTS
1.6 Functional Integrals 58
1.6.1 Gaussian integrals 58
1.6.2 Integrals on a Grassmann algebra 61
~.6.3 Gaussian integrals on a Grassmann algebra 63
1.6.4 Gaussian integrals in field theory 65
1.6.5 Representations of generating functionals
for the S-matrix and Green functions by
functional integrals . 68
1.6.6 The stationary-phase method 70
1.6.7 The Dominicis-Englert theorem 73
1.7 Equations in Variational Derivatives 75
1.7.1 The Schwinger equations 75
1.7 .2 Linear equations for connected Green functions 77
1.7.3 General method of deriving the equations 78
1. 7.4 Iteration solution of the equations 82
1.8 One-Irreducible Green Functions 84
1.8.1 Definitions 84
1.8.2 The equations of motion fQr r 87
1.8.3 Iteration solution of the equations and proof
of !-irreducibility 88
1.9 Renormalization Transformations 91
1.10 Anomalous Green Functions and Spontaneous
Symmetry Breaking 94
Chapter 2 SPECIFIC SYSTEMS 97
2.1 Quantum Mechanics 97
2.1.1 The oscillator 97
2.1.2 The free particle 99
2.2 Nonrelativistic Field Theory 104
2.2.1 The quantum Bose and Fermi gases 108
2.2.2 The atom 109
2.2.3 Electrons in solids and phonons 110
2.3 Relativistic Field Theory 111
2.4 Integral Representations of the Transition Amplitude 114
2.5 The Space E(A) for Various Systems 120
2.6 Functional Integrals Over Phase Space ·122
Chapter 3 THE MASSLESS YANG-MILLS FIELD 127
3.1 Quantization of the Yang-Mills Field 127
3.1.1 The classical theory 127
CONTENTS vii
3.1.2 A general recipe for quantization 128
=
3.1.3 Perturbation theory for gauges nB + c 0 131
3.1.4 The generalized Feynman gauge 134
3 .1.5 The S-matrix generating functional 134
3.2 Gauge Invariance 136
3.2.1 The Ward-Slavnov identities 136
3.2.2 Transversality and gauge invariance of
the S-matrix in electrodynamics 137
3.2.3 Transversality and gauge invariance of the
on-shell S-matrix for the Yang-Mills field 139
Chapter 4 EUCLIDEAN FIELD THEORY 141
4.1 The Euclidean Rotation 141
4.1.1 Definitions 141
4.1.2 The formal Euclidean rotation of the action
functional 142
4.1.3 Euclidean rotation of the Green functions 143
4.1.4 Properties of the field (j) and the action Se ({j)) 145
4.1.5 Rotation of the Lorentz group into 0 146
4
4.1.6 Examples 148
4.2 Functional Integral Representations 150
4.3 Convexity Properties 154
4.3.1 Quasiprobabilistic theories 154
4.3.2 Convexity and spectral representations 156
Chapter 5 STATISTICAL PHYSICS 159
5.1 The Quantum Statistics of Field Systems 159
5.1.1 Definitions 159
5.1.2 The free theory 161
5.1.3 The average of an operator in normal-ordered form 162
5 .1.4 Diagrammatic representation of the partition function
and the Green functions 165
5.1.5 Periodic extensions of the Green functions 166
5.1.6 Representations by functional integrals 169
5.1.7 The zero-temperature limit 170
5.1.8 The Feynman-Kac formula 171
5.1.9 Convexity properties 172
5.1. 10 Convexity of the logarithm of the partition function 174
5. 1.11 Representation of the partition function of the
free theory by a functional integral 176
viii CONTENTS
5.2 Lattice Spin Systems 180
5.2.1 The Ising model 180
5.2.2 The Heisenberg quantum ferromagnet 182
5.3 The Nonideal Classical Gas 185
5.3.1 A gas with two-body forces 185
5.3.2 A gas with many-body forces 188
Chapter 6 VARIATIONAL METHODS AND FUNCTIONAL
LEGENDRE TRANSFORMS 191
6.1 Phase Transitions 191
6.1.1 Introduction 191
6.1.2 Transformation to the variational problem
in thermodynamics 192
6.1.3 The infinite-volume limit 197
6.1.4 Singular and critical points 199
6.1.5 Description of pha&e transitions 201
6.1.6 Critical and Goldstone fluctuations 204
6.2 Legendre Transformations of the Generating
Functional of Connected Green Functions 207
6.2.1 Functional formulations of the variational
principle 207
6.2.2 The equations of motion in connected
variables 213
6.2.3 The equations of motion in !-irreducible
variables 220
6.2.4 Linear equations and their general solutions 224
6.2.5 Iteration solution of the equations 229
6.2.6. The second Legendre transform 232
6.2.7 The self-consistent field approximation 235
6.2.8 The third Legendre tran~form 238
6.2.9 The fourth transform 243
6.2.10 Stationarity equations, renormalization, and
parquet graphs 249
6.2.11 Symmetry properties of the complete Legendre
transform and the "spontaneous interaction" 252
6.2.12 The ground-state energy 256
6.2.13 Stability and convexity properties of
functional Legendre transforms 258
6.3 Legendre Transforms of the Logarithm of
the S-Matrix Generating Functional 261
CONTENTS ix
6.3.1 Definitions and general properti~s 261
6.3.2 The classical nonideal gas and the virial expansion 266
6.3.3 The Ising model 269
6.3.4 Analysis of nonstar graphs for the Ising model 272
6.3.5 The second Legendre transform for the classical gas 278
6.3.6 The stationarity equations and the self-consistent
field approximation 284
Appendix 1 NONSTAT IONARY PERTURBATION
THEORY FOR A DISCRETE LEVEL 289
Appendix 2 GRAPHS AND SYMMETRY COEFFICIENTS 301
REFERENCES 305
SUBJECT INDEX 309