QUANTUM FIELD THEORY. A Modern Introduction MICHIO KAKU. Department of Physics City College of the City University of New York

Transcription

1 QUANTUM FIELD THEORY A Modern Introduction MICHIO KAKU Department of Physics City College of the City University of New York New York Oxford OXFORD UNIVERSITY PRESS 1993

2 Contents Quantum Fields and Renormalization 1. Why Quantum Field Theory? Historical Perspective Strong Interactions Weak Interactions Gravitational Interaction Gauge Revolution Unification Action Principle From First to Second Quantization Noether's Theorem Exercises Symmetries and Group Theory Elements of Group Theory SO(2) Representations of SO(2) and U(1) Representations of SO(3) and SU (2) Representations of SO (N) Spinors Lorentz Group Representations of the Poincar6 Group Master Groups and Supersymmetry Exercises Spin-O and Z Fields Quantization Schemes Klein Gordon Scalar Field Charged Scalar Field Propagator Theory Dirac Spinor Field 77 3,6 Quantizing the Spinor Field 86

3 xiv Contents 3.7 Weyl Neutrinos Exercises Quantum Electrodynamics Maxwell's Equations Relativistic Quantum Mechanics Quantizing the Maxwell Field Gupta-Bleuler Quantization C, P, and T Invariance Parity Charge Conjugation Time Reversal CPT Theorem Exercises Feynman Rules and LSZ Reduction Cross Sections Propagator Theory and Rutherford Scattering LSZ Reduction Formulas Reduction of Dirac Spinors Time Evolution Operator Wick's Theorem Feynman's Rules Exercises Scattering Processes and the S Matrix Compton Effect Pair Annihilation M011er Scattering Bhabha Scattering Bremsstrahlung Radiative Corrections Anomalous Magnetic Moment Infrared Divergence Lamb Shift Dispersion Relations Exercises Renormalization of QED The Renormalization Program Renormalization Types Nonrenormalizable Theories Renormalizable Theories 215

4 Contents xv Super-renormalizable Theories Finite Theories Overview of Renormalization in 0 4 Theory Overview of Renormalization in QED Types of Regularization Ward Takahashi Identities Overlapping Divergences Renormalization of QED Step One Step Two Step Three Step Four Exercises 256 II Gauge Theory and the Standard Model 8. Path Integrals Postulates of Quantum Mechanics Postulate I Postulate II Derivation of the Schrödinger Equation From First to Second Quantization Generator of Connected Graphs Loop Expansion Integration over Grassmann Variables Schwinger Dyson Equations Exercises Gauge Theory Local Symmetry Faddeev Popov Gauge Fixing Feynman Rules for Gauge Theory Coulomb Gauge The Gribov Ambiguity Equivalence of the Coulomb and Landau Gauge Exercises The Weinberg Salam Model Broken Symmetry in Nature The Higgs Mechanism Weak Interactions Weinberg Salam Model Lepton Decay 338

5 xvi Contents 10.6 Re Gauge 't Hooft Gauge Coleman Weinberg Mechanism Exercises The Standard Model The Quark Model QCD Spin-Statistics Problem Pair Annihilation Jets Absence of Exotics Pion Decay Asymptotic Freedom Confinement Chiral Symmetry No Anomalies Jets Current Algebra PCAC and the Adler Weisberger Relation CVC PCAC Adler Weisberger Relation Mixing Angle and Decay Processes Purely Leptonic Decays Semileptonic Decays Nonleptonic Decays GIM Mechanism and Kobayashi Maskawa Matrix Exercises Ward Identities, BRST, and Anomalies Ward Takahashi Identity Slavnov Taylor Identities BRST Quantization Anomalies Non-Abelian Anomalies QCD and Pion Decay into Gamma Rays Fujikawa's Method Exercises BPHZ Renormalization of Gauge Theories Counterterms in Gauge Theory Dimensional Regularization of Gauge Theory 436

6 Contents xvii 13.3 BPHZ Renormalization Forests and Skeletons Does Quantum Field Theory Really Exist? Exercises QCD and the Renormalization Group Deep Inelastic Scattering Parton Model Neutrino Sum Rules Product Expansion at the Light-Cone Renormalization Group Asymptotic Freedom Callan-Symanzik Relation Minimal Subtraction Scale Violations Renormalization Group Proof Step One Step Two Step Three Exercises 499 III Nonperturbative Methods and Unification 15. Lattice Gange Theory The Wilson Lattice Scalars and Fermions an the Lattice Confinement Strong Coupling Approximation Monte Carlo Simulations Hamiltonian Formulation Renormalization Group Exercises Solitons, Monopoles, and Instantons Solitons Example: Example: Sine-Gordon Equation Example: Nonlinear 0(3) Model Monopole Solutions 't Hooft-Polyakov Monopole WKB, Tunneling, and Instantons Yang-Mills Instantons Vacua and the Strong CP Problem Exercises 566

7 xviii Contents 17. Phase Transitions and Critical Phenomena Critical Exponents The Ising Model X Y Z Heisenberg Model IRF and Vertex Models Yang Baxter Relation Mean-Field Approximation Scaling and the Renormalization Group Step One Step Two Step Three Step Four e Expansion Exercises Grand Unified Theories Unification and Running Coupling Constants S U (5) Anomaly Cancellation Fermion Representation Spontaneous Breaking of S U (5) Hierarchy Problem S 0 (10) Beyond GUT Technicolor Preons or Subquarks Supersymtnetry and Superstrings Exercises Quantum Gravity Equivalence Principle Generally Covariant Action Vierbeins and Spinors in General Relativity GUTs and Cosmology Inflation Cosmological Constant Problem Kaluza Klein Theory Generalization to Yang Mills Theory Quantizing Gravity Counterterms in Quantum Gravity Exercises 660

8 Contents xix 20. Supersymmetry and Supergravity Supersymmetry Supersymmetric Actions Superspace Supersymmetric Feynman Rules Nonrenormalization Theorems Finite Field Theories Super Groups Supergravity Exercises Superstrings Why Strings? Points versus Strings Quantizing the String Gupta Bleuler Quantization Light-Cone Gauge BRST Quantization Scattering Amplitudes ; Superstrings Types of Strings Type I Type IIA Type IIB Heterotic String Higher Loops Phenomenology Light-Cone String Field Theory BRST Action Exercises 736 Appendix 741 A.1 SU(N) 741 A.2 Tensor Products 743 A.3 SU(3) 747 A.4 Lorentz Group 749 A.5 Dirac Matrices 751 A.6 Infrared Divergences to All Orders 755 A.7 Dimensional Regularization 760 Notes 763 References 775 Index 779