The Feynman path integrals are becoming increasingly important in the applications of quantum mechanics and field theory. The path integral formulation of quantum anomalies, (i.e.: the quantum breaking of certain symmetries), can now cover all the known quantum anomalies in a coherent manner. In this book the authors provide an introduction to the path integral method in quantum field theory and its applications to the analysis of quantum anomalies. No previous knowledge of field theory beyond the advanced undergraduate quantum mechanics is assumed. The book provides the first coherent introductory treatment of the path integral formulation of chiral and Weyl anomalies, with applications to gauge theory in two and four dimensions, conformal field theory and string theory. Explicit and elementary path integral calculations of most of the quantum anomalies covered are given. The conceptual basis of the path integral bosonization in two-dimensional theory, which may have applications to condensed matter theory, for example is clarified. The book also covers the recent interesting developments in the treatment of fermions and chiral anomalies in lattice gauge theory.
Author(s): Kazuo Fujikawa, Hiroshi Suzuki
Year: 2004
Language: English
Pages: 296
Contents......Page 10
1.1 Introduction......Page 14
1.2 Is the photon massless?......Page 15
1.3 The discovery of the quantum anomaly......Page 16
2.1 Quantum theory of a harmonic oscillator......Page 20
2.2 Path integral for the harmonic oscillator......Page 21
2.3 Quantization of a scalar field......Page 24
2.4 Path integral for fermions......Page 28
2.5 Path integral for Dirac particles......Page 33
2.6 Feynman path integral and Schwinger's action principle......Page 36
3.1 Canonical quantization of the electromagnetic field......Page 44
3.2 Path integral quantization of the electromagnetic field......Page 48
3.3 Photon phase operator and the notion of index......Page 53
3.4 Is there a hermitian phase operator?......Page 55
3.5 Index theorem for a harmonic oscillator......Page 57
4.1 Current conservation and Ward Takahashi identities......Page 60
4.2 Self-energy of the photon......Page 62
4.3 Quantum breaking of chiral symmetry......Page 70
4.4 Adler–Bardeen theorem......Page 75
5.1 The chiral Jacobian in quantum electrodynamics......Page 78
5.2 Ward–Takahashi identities in quantum electrodynamics......Page 84
5.3 Chiral anomaly in QCD-type theory......Page 87
5.4 Instantons......Page 92
5.5 Atiyah–Singer index theorem......Page 97
5.6 Nambu–Goldstone theorem......Page 99
6.1 Gauge theory with axial-vector gauge fields......Page 105
6.2 Pauli–Villars regularization......Page 109
6.3 Chiral gauge theory and the quantum anomaly......Page 111
6.4 Covariant anomaly......Page 115
6.5 Anomaly cancellation in Weinberg–Salam theory......Page 122
6.6 The Wess–Zumino integrability condition......Page 125
6.7 Quantum anomalies and anomalous commutators......Page 134
7.1 Scale transformation in field theory......Page 136
7.2 Identities for the Weyl transformation and Weyl anomalies......Page 137
7.3 Identities related to coordinate transformations......Page 140
7.4 Weyl anomalies and β functions in QED and QCD......Page 145
7.5 The Weyl anomaly in curved space-time......Page 153
7.6 The Weyl anomaly in two-dimensional space-time......Page 157
7.7 Other applications of Weyl anomalies......Page 160
8.1 Chiral anomalies in two-dimensional theory......Page 162
8.2 Abelian bosonization of fermions......Page 170
8.3 Non-Abelian bosonization of fermion theory......Page 181
8.4 Kac Moody algebra and Virasoro algebra......Page 187
8.5 Quantum theory of strings and Liouville action......Page 201
8.6 Ghost number anomaly and the Riemann–Roch theorem......Page 207
9.1 Lattice gauge theory......Page 209
9.2 Lattice Dirac fields and species doubling......Page 211
9.3 Representation of the Ginsparg–Wilson algebra......Page 215
9.4 Atiyah–Singer index theorem on the lattice and the chiral anomaly......Page 219
9.5 The operator D satisfying the Ginsparg–Wilson relation......Page 223
9.6 Some characteristic features of lattice chiral theory......Page 231
10.1 Chiral U(1) gravitational anomalies......Page 236
10.2 Evaluation by a quantum mechanical path integral......Page 241
10.3 Chern character and Dirac genus......Page 243
10.4 Anomaly in general coordinate transformations......Page 244
10.5 General properties of gravitational anomalies......Page 247
10.6 Explicit examples of gravitational anomalies......Page 249
11 Concluding remarks......Page 253
A.1 Quantum electrodynamics......Page 259
A.2 Interaction representation and perturbation formulas......Page 262
B.1 Coordinate transformation and energy-momentum tensor......Page 266
B.2 Path integral measure in gravitational theory......Page 271
C.1 Genesis of quantum anomalies......Page 275
C.3 Quantum theory of photons and the phase operator......Page 276
C.4 Regularization of field theory and chiral anomalies......Page 277
C.5 The Jacobian in path integrals and quantum anomalies......Page 278
C.6 Quantum breaking of gauge symmetry......Page 279
C.7 The Weyl anomaly and renormalization group......Page 283
C.8 Two-dimensional field theory and bosonization......Page 285
C.9 Index theorem on the lattice and chiral anomalies......Page 288
C.10 Gravitational anomalies......Page 290
C.11 Concluding remarks......Page 292
F......Page 295
P......Page 296
Z......Page 297
