Table Of Content

In The Quantum Theory of Fields Nobel Laureate Steven Weinberg com bines his exceptional physical insight with his gift for clear exposition to provide a self-contained, comprehensive, and up-to-date introduction to quantum field theory. Volume I introduces the foundations of quantum field theory. The development is fresh and logical throughout, with each step carefully motivated by what has gone before, and emphasizing the reasons why such a theory should describe nature. After a brief historical outline, the book begins anew with the principles about which we are most certain, relativity and quantum mechanics, and the properties of particles that follow from these principles. Quantum field theory then emerges from this as a natural consequence. The classic calculations of quantum electrodynamics are presented in a thoroughly modern way, showing the use ofpath integrals and dimensional regularization. The account of renormalization theory reflects the changes in our view of quantum field theory since the advent of effective field theories. The book's scope extends beyond quantum electrodynamics to ele· mentary particle physics and nuclear physics. It contains much original material, and is peppered with examples and insights drawn from the author's experience as a leader of elementary particle research. Problems are included at the end of each chapter. A second volume will describe the modern applications of quantum field theory in today's standard model of elementary particles, and in some areas of condensed matter physics. This will be an invaluable reference work for all physicists and mathe maticians who use quantum field theory, as well as a textbook appropriate to graduate courses. The Quantum Theory of Fields Volume I Foundations i '-ff --") f"~ y of Fields The Qua - Volume I Foundations Steven Weinberg University of Texas at Austin 1~ ABR. 1996 . CAMBRIDGE UNIVERSITY PRESS • Published by the Press Syndicate of the Univenity of Cambridge The Pitt Building, Trurnpington Street, Cambridge CBl IRP 40 West 20th Street, New York, NY 10011-4211, USA 10 Stamford Road, Oaldeigh, Melbourne 3166, Australia @ Steven Weinberg 1995 First published 1995 Printed in the United States ofAmerica A catalogue record for this book is availableIrom the British Library Librar)' ofCongress cataloguing In publication dara Weinberg, Steven, 1933- I The quantum theory of fields Steven Weinberg. p. em. Includes bibliographical reference and index. Contents: v. L Foundations. ISBN 0-521-55001-7 1. Quantum field theory. L Title. QC174.4SW45 1995 530.1'43-dc20 95-2782 CIP ISBN 0 521 55001 7 TAG l : To Louise g !- r ) ~ i j Contents Sections marked with an asterisk are somewhat out of the book's main line of development and may be omitted in a first reading. PREFACE xx NOTATION xxv 1 HISTORICAL INTRODUCTION 1 1.1 Relativistic Wave Mechanics 3 De Broglie waves D Schrodinger-Klein-Gordon wave equation 0 Fine structure D Spin D Dirac equation D Negative energies D Exclusion principle D Positrons o Dirac equation reconsidered 1.2 The Birth of Quantum Field Theory 15 Born, Heisenberg, Jordan quantized field D Spontaneous emission D Anticom mutators D Heisenberg-Pauli quantum field theory 0 Furry--Oppenheimer quan tization of Dirac field 0 Pauli-Weisskopf quantization of scalar field DEarly calculations in quantum electrodynamics D Neutrons D Mesons 1.3 The Problem of Infinities 31 Infinite electron energy shifts D Vacuum polarization D Scattering oflight by light D Infrared divergences D Search for alternatives D Renormalization D Shelter Island Conference D Lamb shift D Anomalous electron magnetic moment D Schwinger, Tomonaga, Feynman, Dyson formalisms D Why not earlier? Bibliography 39 References 40 2 RELATIVISTIC QUANTUM MECHANICS 49 2.1 Quantum Mechanics 49 Rays D Scalar products D Observables D Probabilities IX x Contents 2.2 Symmetriell 50 Wigner's theorem 0 Antilinear and antiunitary operators 0 ObselVables 0 Group structure 0 Representations up to a phase 0 Superselection rules 0 Lie groups o Structure constants 0 Abelian symmetries 2.3 Quantum Lonmtz Transformations 55 Lorentz transformations 0 Quantum operators 0 Inversions 2.4 The Poincare Algebra 58 JI" and pI' 0 Transformation properties 0 Commutation relations 0 ConselVed and non-conserved generators 0 Finite translations and rotations 0 Inonii Wigner contraction 0 Galilean algebra 25 One-Particle States 62 Transformation rules 0 Boosts 0 Little groups 0 Normalization 0 Massive particles 0 Massless particles 0 Helicity and polarization 2.6 Space Inversion and Time-Reversal 74 Transfonnation of JI" and pI' 0 P is unitary and T is antiunitary 0 Massive particles 0 Massless particles 0 Kramers degeneracy 0 Electric dipole moments 2.7 Projective Representations' 81 Two-cocyles 0 Central charges 0 Simply connected groups 0 No central charges in the Lorentz group 0 Double connectivity of the Lorentz group 0 Covering groups 0 Superselection rules reconsidered Appendix A The Symmetry Representation Theorem 91 Appendix B Group Operators and Homotopy Classes 96 Appendix C Inversions and Degenerate Multiplets 100 Problems 104 References 105 3 SCATTERING THEORY 107 3.1 'In' and 'Out' States 107 Multi-particle states 0 Wave packets 0 Asymptotic conditions at early and late times 0 Lippmann- Schwinger equations 0 Principal value and delta functions 3.2 The S-matrix 113 Definition of the S-matrix. 0 The T·matrix 0 Born approximation 0 Unitarity of the S-matrix 3.3 Symmetries of the S-Matrix 116 Lorentz invariance 0 Sufficient conditions 0 Internal symmetries 0 Electric charge, strangeness, isospin, SU(3) 0 Parity conselVation 0 Intrinsic parities 0 Contents Xl Pion parity 0 Parity non-conservation 0 Time-reversal invariance 0 Watson's theorem 0 PT non-conservation 0 C, CP, CPT 0 Neutral K -mesons 0 CP non conservation 3.4 Rates and Cross-Sections 134 Rates in a box 0 Decay rates 0 Cross-sections 0 Lorentz invariance 0 Phase space 0 Dalitz plots 3.5 Perturbation Theory 141 Old-fashioned perturbation theory 0 Time-dependent perturbation theory 0 Time-ordered products 0 The Dyson series 0 Lorentz-invariant theories 0 Dis torted wave Born approximation 3.6 Implications of Unitarily 147 Optical theorem 0 Diffraction peaks 0 CPT relations 0 Particle and antiparticle decay rates 0 Kinetic theory 0 Boltzmann H-theorem 3.7 Partial-Wave Expansions· 151 Discrete basis 0 Expansion in spherical harmonics 0 Total elastic and inelastic cross-sections 0 Phase shifts 0 Threshold behavior: exothermic. endothermic, and elastic reactions 0 Scattering length 0 High-energy elastic and inelastic scattering 3.8 Resonances' 159 Reasons for resonances: weak coupling, barriers, complexity 0 Energy dependence 0 Unitarity 0 Breit-Wigner formula 0 Unresolved resonances 0 Phase shifts at resonance 0 Ramsauer-Townsend effect Problems 165 References 166 4 THE CLUSTER DECOMPOSITION PRINCIPLE 169 4.1 Bosons and Fermions 170 Permutation phases 0 Bose and Fermi statistics 0 Normalization for identical particles 4.2 Creation and Annihilation Operators 173 Creation operators 0 Calculating the adjoint 0 Derivation of commutation! anticommutation relations 0 Representation of general operators 0 Free-particle Hamiltonian 0 Lorentz transformation of creation and annihilation operators 0 C, P, T properties of creation and annihilation operators 4.3 Cluster Decomposilion and Connected Amplitudes 177 Decorrelation of distant experiments 0 Connected amplitudes 0 Counting delta functions Contents XII 4.4 Structure of the Interaction 182 Condition for cluster decomposition 0 Graphical analysis 0 Two-body scattering implies three-body scattering Problems 189 References [89 5 QUANTUM FIELDS AND ANTIPARTICLES 191 5.1 Free Fields 191 Creation and annihilation fields 0 Lorentz transformation of the coefficient func tions 0 Construction of the coefficient functions 0 Implementing cluster decom position 0 Lorentz invariance requires causality 0 Causality requires antiparticles o Field equations 0 Normal ordering 5.2 Causal Scalar Fields 201 Creation and annihilation fields 0 Satisfying causality 0 Scalar fields describe bosuns 0 Antiparticles 0 P, C, T transformations 0 11:0 5.3 Causal Vector Fields 207 Creation and annihilation fields 0 Spin zero or spin one 0 Vector fields describe bosons 0 Polarization vectors 0 Satisfying causality 0 Antiparticles 0 Mass zero limit 0 P, C, T transformations 5.4 The Dirac Formalism 213 Clifford representations ofthe Poincare algebra 0 Transformation of Dirac matri ces 0 Dimensionality ofDirac matrices 0 Explicit matrices 0 ('s 0 Pseudounitarity o Complex conjugate and transpose 5.5 Causal Dirac Fields 219 Creation and annihilation fields 0 Dirac spinors 0 Satisfying causality 0 Dirac fields describe fermions 0 Antiparticles 0 Space inversion 0 Intrinsic parity of particle-antiparticle pairs 0 Charge-conjugation 0 Intrinsic C-phase of particle- antiparticle pairs 0 Majorana fermions 0 Time-reversal 0 Bilinear covariants 0 Beta decay interactions 5.6 General Irreducible Representations of the Homogeneous Lorentz Group' 229 Isomorphism with SU(2) 0 SU(2) 0 (A,B) representation of familiar fields 0 Rarita-Schwinger field 0 Space inversion 5.7 General Causal Fields' 233 Constructing the coefficient functions 0 Scalar Hamiltonian densities 0 Satisfying causality 0 Antiparticles 0 General spin-statistics connection 0 Equivalence of different field types 0 Space inversion 0 Intrinsic parity of general particle antiparticle pairs 0 Charge-conjugation 0 Intrinsic C-phase of antiparticles 0