Path Integral Quantization and Stochastic Quantization

by Masujima, Michio

• ISBN13: 9783540878506
• ISBN10: 3540878505
• Edition: 2nd
• Format: Paperback
• Copyright: 2008-12-01
• Publisher: Springer Verlag

Summary

The book gives an overview of path integral quantization and stochastic quantization of classical mechanics and field theory. The non-Abelian gauge field, the gravitational field, and the path integral representation of quantum statistical mechanics and stochastic quantization are described.

Table of Contents

Path Integral Representation of Quantum Mechanics	p. 1
Quantum Mechanics in the Lagrangian Formalism	p. 3
Contact Transformation in Analytical Mechanics and Quantum Mechanics	p. 3
The Lagrangian and the Action Principle	p. 8
The Feynman Path Integral Formula	p. 13
The Time-Dependent Schrodinger Equation	p. 16
The Principle of Superposition and the Composition Law	p. 20
Path Integral Representation of Quantum Mechanics in the Hamiltonian Formalism	p. 22
Review of Quantum Mechanics in the Hamiltonian Formalism	p. 23
Phase Space Path Integral Representation of the Transformation Function	p. 25
Matrix Element of a Time-Ordered Product	p. 27
Wave Function of the Vacuum	p. 31
Generating Functional of the Green's Function	p. 34
Configuration Space Path Integral Representation	p. 35
Weyl Correspondence	p. 40
Weyl Correspondence	p. 41
Path Integral Formula in a Cartesian Coordinate System	p. 43
Path Integral Formula in a Curvilinear Coordinate System	p. 45
Bibliography	p. 51
Path Integral Representation of Quantum Field Theory	p. 55
Path Integral Quantization of Field Theory	p. 56
Review of Classical Field Theory	p. 57
Phase Space Path Integral Quantization of Field Theory	p. 59
Configuration Space Path Integral Quantization of Field Theory	p. 62
Covariant Perturbation Theory	p. 67
Generating Functional of Green's Functions of the Free Field	p. 68
Generating Functional of Full Green's Functions of an Interacting System	p. 70
Feynman-Dyson Expansion and Wick's Theorem	p. 71
Symanzik Construction	p. 72
Equation of Motion of the Generating Functional	p. 73
Method of the Functional Fourier Transform	p. 78
External Field Problem	p. 80
Schwinger Theory of the Green's Function	p. 83
Definition of the Green's Function and the Equation of Motion	p. 83
Proper Self-Energy Parts and the Vertex Operator	p. 87
Equivalence of Path Integral Quantization and Canonical Quantization	p. 91
Feynman's Action Principle	p. 91
The Operator, Equation of Motion and Time-Ordered Product	p. 93
Canonically Conjugate Momentum and Equal-Time Canonical (Anti-) Commutators	p. 96
Bibliography	p. 100
Path Integral Quantization of Gauge Field	p. 103
Review of Lie Groups	p. 107
Group Theory	p. 107
Lie Groups	p. 108
Non-Abelian Gauge Field Theory	p. 113
Gauge Principle: "U(1) [implies] SU(2) Isospin"	p. 113
Non-Abelian Gauge Field Theory	p. 116
Abelian Gauge Fields vs. Non-Abelian Gauge Fields	p. 121
Examples	p. 123
Path Integral Quantization of Gauge Fields	p. 127
The First Faddeev-Popov Formula	p. 128
The Second Faddeev-Popov Formula	p. 134
Choice of Gauge-Fixing Condition	p. 136
The Ward-Takahashi-Slavnov-Taylor Identity and Gauge Independence of the Physical S-Matrix	p. 146
Spontaneous Symmetry Breaking and the Gauge Field	p. 150
Goldstone's Theorem	p. 151
Higgs-Kibble Mechanism	p. 154
Path Integral Quantization of the Gauge Field in the R[subscript xi]-Gauge	p. 158
Ward-Takahashi-Slavnov-Taylor Identity and the [xi]-Independence of the Physical S-Matrix	p. 167
Bibliography	p. 168
Path Integral Representation of Quantum Statistical Mechanics	p. 175
Partition Function of the Canonical Ensemble and the Grand Canonical Ensemble	p. 176
The Canonical Ensemble and the Bloch Equation	p. 177
Extension to the Grand Canonical Ensemble	p. 179
Fradkin Construction	p. 184
Density Matrix of Relativistic Quantum Field Theory at Finite Temperature	p. 185
Functional Differential Equation of the Partition Function of the Grand Canonical Ensemble	p. 188
Schwinger-Dyson Equation	p. 194
The Schwinger-Dyson Equation	p. 195
Nonrelativistic Limit	p. 197
Methods of the Auxiliary Field	p. 198
Method of the Auxiliary Field in the Lagrangian Formalism	p. 199
Stratonovich-Hubbard Transformation: Gaussian Method	p. 201
Bibliography	p. 205
Stochastic Quantization	p. 209
Review of the Theory of Probability and Stochastic Processes	p. 210
Random Variables and the Notion of Convergence	p. 211
Stochastic Processes	p. 216
Evolution Equation of Quantum Mechanics and Quantum Field Theory	p. 219
Stochastic Quantization of Non-Abelian Gauge Field	p. 223
Parisi-Wu Equation and Fokker-Planck Equation	p. 223
Stochastic Quantization of Abelian and Non-Abelian Gauge Fields	p. 230
Covariant Nonholonomic Gauge-Fixing Condition and Stochastic Quantization of the Non-Abelian Gauge Field	p. 235
Bibliography	p. 240
Appendices
Gaussian Integration	p. 245
Fermion Number Integration	p. 251
Functional Integration	p. 255
Gauge Invariance of D[A subscript alpha mu Delta subscript F A subscript alpha mu]	p. 263
Minkowskian and Euclidean Spinors	p. 267
Multivariate Normal Analysis	p. 269
Bibliography	p. 273
Index	p. 275
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