Table of Contents
1 How to Build an Action Functional 1
1.1 The Action Functional: Elementary Considerations 1
1.2 The Lorentz Group (A Cursory Look) 4
1.3 The Poinearé Group 10
1.4 Behavior of Local Fields under the Poinearé Group 13
1.5 General Properties of the Action 23
1.6 The Action for Scalar Fields 29
1.7 The Action for Spinor Fields 34
1.8 An Action with Scalar and Spinor Fields and Supersymmetry 38
2 The Action Functional in Quantum Mechanics 45
2.1 Canonical Transformations in Classical and Quantum Mechanics 45
2.2 The Feynman Path Integral 51
2.3 The Path Integral and the Forced Harmonic Oscillator 56
3 The Feynman Path Integral in Field Theory 63
3.1 The Generating Functional 63
3.2 The Feynman Propagator 66
3.3 The Effective Action 70
3.4 Saddle Point Evaluation of the Path Integral 74
3.5 First Quantum Effects: ζ-Function Evaluation of Determinants 81
3.6 Scaling of Determinants: Scale Dependent Coupling Constant 85
3.7 Finite Temperature Field Theory 88
4 Perturbative Evaluation of the FPI: φ4 THEORY 101
4.1 Feynman Rules for λφ4 Theory 101
4.2 Divergences of Feynman Diagrams 108
4.3 Dimensional Regularization of Feynman Diagrams 115
4.4 Evaluation of Feynman Integrals 119
4.5 Renormalisation 127
4.6 Renormalisation Prescriptions 136
4.7 Prescription Dependence of Renormalization Group Coefficients 146
4.8 Continuation to Minkowski Space; Analyticity 148
4.9 Cross-Sections and Unitarity 152
5 Path Integral Formulation with Fermions 161
5.1 Integration over Grassmann numbers 161
5.2 Path Integral of Free Fermi Fields 165
5.3 Feynman Rules for Spinor Fields 171
5.4 Evaluation and Scaling of Fermion Determinants 175
6 Gauge Symmetries : Yang-Mills and Gravity 183
6.1 Global and Local Symmetries 183
6.2 Construction of Locally Symmetric Lagrangians 192
6.3 The Pure Yang-Mills Theory 197
6.4 Gravity as a Gauge Theory 206
7 Path Integral Formulation of Gauge Theories 223
7.1 Hamiltonian Formalism of Gauge Theories: Abelian Case 223
7.2 Hamiltonian Formalism of Gauge Theories: Non-Abelian Case 231
7.3 The Faddeev-Popov Procedure 238
8 Perturbative Evaluation of Gauge Theories 241
8.1 Feynman Rules for Gauge Theories 241
8.2 QED: One-Loop Structure 248
8.3 QED: Ward Identities 259
8.4 QED: Applications 264
8.5 Yang-Mills Theory: Preliminaries 269
8.6 Yang-Mills Theory: One-Loop Structure 273
8.7 Yang-Mills Theory: Slavnov-Taylor Identities 284
8.8 Yang-Mills Theory; Asymptotic freedom 290
8.9 Anomalies 294
Appendix A Gaussian Integration 311
Appendix B Integration over Arbitrary Dimensions 315
Appendix C Euclidean Space Feynman Rules 319
Bibliography 321
Index 327
