Quantum Field Theory. Kerson Huang. Second, Revised, and Enlarged Edition WILEY- VCH. From Operators to Path Integrals

Transcription

1 Kerson Huang Quantum Field Theory From Operators to Path Integrals Second, Revised, and Enlarged Edition WILEY- VCH WILEY-VCH Verlag GmbH & Co. KGaA

2 I vh Contents Preface XIII 1 Introducing Quantum Fields / 1.1 The Classical String Renormalization The Quantum String Second Quantization Creation and Annihilation Operators Bose and Fermi Statistics 11 2 Scalar Fields Klein Gordon Equation Real Scalar Field Energy and Momentum Particle Spectrum Continuum Normalization Complex Scalar Field Charge and Antiparticle Microcausality The Feynman Propagator The Wave Functional Functional Operations Vacuum Wave Functional The 0 4 Theory 34 3 Relativistic Fields Lorentz Transformations Minimal Representation: SL(2C) The Poincare Group Scalar, Vector, and Spinor Fields Relativistic Quantum Fields One-Particle States 48 4 Canonical Formalism Principle of Stationary Action 55 Quantum Field Theoryfrom Operators to Path Integrals. K. Huang Copyright 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN:

3 VIII 1 Contents 4.2 Noether's Theorem Transla tional Invariance Lorentz Invariance Symmetrized Energy-Momentum Tensor Gauge Invariance 65 5 Electromagnetic Field Maxwell's Equations Covariance of the Classical Theory Canonical Formalism Quantization in Coulomb Gauge Spin Angular Momentum Intrinsic Parity Transverse Propagator Vacuum Fluctuations The Casimir Effect The Gauge Principle 88 6 Dirac Equation Dirac Algebra Wave Functions and Current Density Plane Waves Lorentz Transformations Interpretation of Dirac Matrices External Electromagnetic Field Non-Relativistic Limit Thomas Precession Hole Theory Charge Conjugation Massless Particles The Dirac Field Quantization of the Dirac Field Feynman Propagator Normal Ordering Electromagnetic Interactions Isospin Parity Charge Conjugation Time Reversal Dynamics of Interacting Fields Time Evolution Interaction Picture Adiabatic Switching Correlation Functions in the Interaction Picture S-Matrix and Scattering 147

4 Contents IIX 8.6 Scattering Cross Section Potential Scattering Adiabatic Theorem Feynman Graphs Perturbation Theory Time-Ordered and Normal Products Wick's Theorem Feynman Rules for Scalar Theory Types of Feynman Graphs Vacuum Graph Self-Energy Graph Connected Graph Wick Rotation Regularization Schemes Linked-Cluster Theorem Vacuum Graphs Vacuum Correlation Functions Feynman Rules Reduction Formula The Generating Functional Connected Correlation Functions Lehmann Representation Dyson-Schwinger Equations Bound States Bethe-Salpeter Equation Quantum Electrodynamics Interaction Hamiltonian Photon Propagator Feynman Graphs Feynman Rules Properties of Feynman Graphs Processes in Quantum Electrodynamics Compton Scattering Electromagnetic Form Factors Anomalous Magnetic Moment Charge Distribution Perturbative Renormalization Primitive Divergences in QED Electron Self-Energy Vacuum Polarization Running Coupling Constant Full Vertex 245

5 X I Contents 13.6 Ward Identity Renormalization to Second Order Renormalization to All Orders Callan-Symanzik Equation Triviality Path Integrals Path integrals in Quantum Mechanics Time-ordering Imaginary Time Path Integrals in Quantum Field Theory Euclidean Space-time Vacuum Amplitudes Statistical Mechanics Gaussian Integrals Perturbation Theory The Loop Expansion Boson and Fermion Loops Grassmann Variables Broken Symmetry Why Broken Symmetry Ferromagnetism Spin Waves Breaking Gauge Invariance Superfluidity Ginsburg-Landau Theory Effective Action Effective Potential Renormalization The Cutoff as Scale Parameter RG Trajectories Fixed points Momentum Space RG Real Space RG Renormalization of Correlation Functions Relevant and Irrelevant Parameters The Free Field IR Fixed Point and Phase Transition Crossover Perturbative Renormalization Revisited Critical Evaluation Why Correct Theories Are Beautiful The Gaussian Fixed Point Stability of the Free Field 339

6 Contents IXI 17.2 General Scalar Field Feynman Graphs Wegner-Houghton Equation Renormalized Couplings The RG Matrix Non-Triviality and Asymptotic Freedom The Case d = In two Dimensions Absence of Long-Range Order Topological Order XY Model Kosterlitz-Thouless Transition Vortex Model D Superfluidity RG Trajectories Universal Jump of Superfluid Density Topological Excitations Topological Soliton Instanton and Tunneling Depinning of Charge Density Waves Nonlinear Sigma Model The Skyrmion The Hopf Invariant Fractional Spin Monopoles, Vortices, and Anomalies 393 Appendix A Background Material 399 A.1 Notation 399 A.2 Classical Mechanics 400 A.3 Quantum Mechanics 401 Appendix B Superfluidity 405 B.1 Linear Response 405 B.2 Normal Fluid and Superfluid 406 B.3 Superfluid Density 408 Appendix C Polchinski's Renormalization Equation 411 C.1 Renormalization Scheme 411 C.2 The Equation 412 C.3 Asymptotically Free Scalar Field 414 Index 417