| 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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