Part I. Many-Body Systems and Classical Field Theory 1. Classical and Quantum Mechanics of Particle Systems 3 1.1 Introduction. 3 1.2 Classical Mechanics of Mass Points 4 1.3 Quantum Mechanics: The Harmonic Oscillator 6 1.3.1 The Harmonic Oscillator 8 1.4 The Linear Chain (Classical Treatment) 10 1.5 The Linear Chain (Quantum Treatment) 18 2. Classical Field Theory 31 2.1 Introduction 1 31 2.2 The Hamilton Formalism 34 2.3 Functional Derivatives 36 2.4 Conservation Laws in Classical Field Theories..., 39-2.5 The Generators of the Poincar6 Group 49 Part II. Canonical Quantization 3. Nonrelativistic Quantum Field Theory...!.. 57 3.1 Introduction 57 3.2 Quantization Rules for Bose Particles : 58 3.3 Quantization Rules for Fermi Particles 65 4. Spin-0 Fields: The Klein-Gordon Equation 75 4.1 The Neutral Klein-Gordon Field 75 4.2 The Charged Klein-Gordon Field 91 4.3 Symmetry Transformations 95 4.4 The Invariant Commutation Relations 100 4.5 The Scalar Feynman Propagator... 106 4.6 Supplement: The A Functions 109 5. Spin- Fields: The Dirac Equation 117 5.1 Introduction 117 5.2 Canonical Quantization of the Dirac Field 123 http://d-nb.info/971674655
Contents 5.3 Plane-Wave Expansion of the Field Operator 124 5.4 The Feynman Propagator for Dirac Fields 132 6. Spin-1 Fields: The Maxwell and Proca Equations 141 6.1 Introduction 141 6.2 The Maxwell Equations 141 6.2.1 The Lorentz Gauge 144 ' 6.2.2 The Coulomb Gauge 144 6.2.3 Lagrange Density and Conserved Quantities 145 6.2.4 The Angular-Momentum Tensor 148 6.3 The Proca Equation 152 6.4 Plane-Wave Expansion of the Vector Field 154 6.4.1 The Massive Vector Field 154 6.4.2 The Massless Vector Field 156 6.5 Canonical Quantization of the Massive Vector Field 158 7. Quantization of the Photon Field 171 7.1 Introduction 171 7.2 The Electromagnetic Field in Lorentz Gauge 172 7.3 Canonical Quantization in the Lorentz Gauge 176 7.3.1 Fourier Decomposition of the Field Operator 177 7.4 The Gupta-Bleuler Method 180 7.5 The Feynman Propagator for Photons 185 7.6 Supplement: Simple Rule for Deriving Feynman Propagators.. 188 7.7 Canonical Quantization in the Coulomb Gauge 196 7.7.1 The Coulomb Interaction 200 8. Interacting Quantum Fields 211 8.1 Introduction 211 8.2 The Interaction Picture 211 8.3 The Time-Evolution Operator 215 8.4 The Scattering Matrix 219 8.5 Wick's Theorem 225 8.6 The Feynman Rules of Quantum Electrodynamics 233 8.7 Appendix: The Scattering Cross Section 267 9. The Reduction Formalism 269 9.1 Introduction 269 9.2 In and Out Fields 270 9.3 The Lehmann-Kallen Spectral Representation 278 9.4 The LSZ Reduction Formula 282 9.5 Perturbation Theory for the n-point Function 290 10. Discrete Symmetry Transformations 301 10.1 Introduction 301 10.2 Scalar Fields 301 10.2.1 Space Inversion 301 10.2.2 Charge Conjugation 305 10.2.3 Time Reversal 306
Contents 10.3 Dirac Fields 311 10.3.1 Space Inversion 312 10.3.2 Charge Conjugation 313 10.3.3 Time Reversal 315 10.4 The Electromagnetic Field 318 10.5 Invariance of the S Matrix 324 10.6 The CPT Theorem 326 Part III. Quantization with Path Integrals 11. The Path-Integral Method 337 11.1 Introduction 337 11.2 Path Integrals in Nonrelativistic Quantum Mechanics 337 11.3 Feynman's Path Integral 343 11.4 The Multi-Dimensional Path Integral 350 11.5 The Time-Ordered Product and n-point Functions 356 11.6 The Vacuum Persistence Amplitude W[J\ 360 12. Path Integrals in Field Theory 365 12.1 The Path Integral for Scalar Quantum Fields 365 12.2 Euclidian Field Theory 371 12.3 The Feynman Propagator 375 12.4 Generating Functional and Green's Function 380 12.5 Generating Functional for Interacting Fields 384 12.6 Green's Functions in Momentum Space 391 12.7 One-Particle Irreducible Graphs and the Effective Action... 400 12.8 Path Integrals for Fermion Fields 408 12.9 Generating Functional and Green's Function for Fermion Fields 419 12.10 Generating Functional and Feynman Propagator for the Free Dirac Field 421 Index.. ; 433
Contents of Examples and Exercises 1.1 Normal Coordinates. 15 1.2 The Linear Chain Subject to External Forces.... ^ 20 1.3 The Baker-Campbell-Hausdorff Relation 27 2.1 The Symmetrized Energy-Momentum Tensor.. 47 2.2 The Poincare Algebra for Classical Fields 51 3.1 The Normalization of Fock States...' 68 3.2 Interacting Particle Systems: The Hartree-Fock Approximation... 69 4.1 Commutation Relations for Creation and Annihilation Operators.. 83 4.2 Commutation Relations of the Angular-Momentum Operator 85 4.3 The Field Operator in the Spherical Representation 86 4.4 The Charge of a State 94 4.5 Commutation Relations Between Field Operators and Generators.. 99 4.6 The Function z^i(x - y) for Equal Time Arguments 105 5.1 The Symmetrized Dirac Lagrange Density 121 5.2 The Symmetrized Current Operator 134 5.3 The Momentum Operator 135 5.4 Helicity States 136 5.5 General Commutation Relations and Microcausality 138 6.1 The Lagrangian of the Maxwell Field 149 6.2 Coupled Maxwell and Dirac Fields 150 6.3 Fourier Decomposition of the Proca Field Operator 160 6.4 Invariant Commutation Relations and the Feynman Propagator of the Proca Field 167 7.1 The Energy Density of the Photon Field in the Lorentz Gauge... 175 7.2 Gauge Transformations and Pseudo-photon States 183 7.3 The Feynman Propagator for Arbitrary Values of the Gauge Parameter 189 7.4 The Transverse Delta Function 202 7.5 General Commutation Rules for the Electromagnetic Field 205 8.1 The Gell-Mann-Low Theorem 220 8.2 Proof of Wick's Theorem 231 8.3 Disconnected Vacuum Graphs 243 8.4 Moller Scattering and Compton Scattering 245 8.5 The Feynman Graphs of Photon-Photon Scattering 249 8.6 Scalar Electrodynamics 251 8.7 0 4 Theory 261 9.1 Derivation of the Yang-Feldman Equation 276 9.2 The Reduction Formula for Spin- Particles 288
Contents of Examples and Exercises 9.3 The Equation of Motion for the Operator U(t) 294 9.4 Green's Functions and the S Matrix of <f> A Theory 295 10.1 The Operators V and C for Scalar Fields 309 10.2 The Classification of Positronium States 320 10.3 Transformation Rules for the Bilinear Covariants 330 10.4 The Relation Between Particles and Antiparticles 331 11.1 The Path Integral for the Propagation of a Free Particle 345 11.2 Weyl Ordering of Operators 347 11.3 Gaussian Integrals in D Dimensions 353 12.1 Construction of the Field-Theoretical Path Integral 368 12.2 Series Expansion of the Generating Functional 386 12.3 A Differential Equation for W[J] 387 12.4 The Perturbation Series for the ip 4 Theory 392 12.5 Connected Green's Functions 398 12.6 Grassmann Integration 415 12.7 Yukawa Coupling 424