Part I. Many-Body Systems and Classical Field Theory

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1 Part I. Many-Body Systems and Classical Field Theory 1. Classical and Quantum Mechanics of Particle Systems Introduction Classical Mechanics of Mass Points Quantum Mechanics: The Harmonic Oscillator The Harmonic Oscillator The Linear Chain (Classical Treatment) The Linear Chain (Quantum Treatment) Classical Field Theory Introduction The Hamilton Formalism Functional Derivatives Conservation Laws in Classical Field Theories..., The Generators of the Poincar6 Group 49 Part II. Canonical Quantization 3. Nonrelativistic Quantum Field Theory...! Introduction Quantization Rules for Bose Particles : Quantization Rules for Fermi Particles Spin-0 Fields: The Klein-Gordon Equation The Neutral Klein-Gordon Field The Charged Klein-Gordon Field Symmetry Transformations The Invariant Commutation Relations The Scalar Feynman Propagator Supplement: The A Functions Spin- Fields: The Dirac Equation Introduction Canonical Quantization of the Dirac Field 123

2 Contents 5.3 Plane-Wave Expansion of the Field Operator The Feynman Propagator for Dirac Fields Spin-1 Fields: The Maxwell and Proca Equations Introduction The Maxwell Equations The Lorentz Gauge 144 ' The Coulomb Gauge Lagrange Density and Conserved Quantities The Angular-Momentum Tensor The Proca Equation Plane-Wave Expansion of the Vector Field The Massive Vector Field The Massless Vector Field Canonical Quantization of the Massive Vector Field Quantization of the Photon Field Introduction The Electromagnetic Field in Lorentz Gauge Canonical Quantization in the Lorentz Gauge Fourier Decomposition of the Field Operator The Gupta-Bleuler Method The Feynman Propagator for Photons Supplement: Simple Rule for Deriving Feynman Propagators Canonical Quantization in the Coulomb Gauge The Coulomb Interaction Interacting Quantum Fields Introduction The Interaction Picture The Time-Evolution Operator The Scattering Matrix Wick's Theorem The Feynman Rules of Quantum Electrodynamics Appendix: The Scattering Cross Section The Reduction Formalism Introduction In and Out Fields The Lehmann-Kallen Spectral Representation The LSZ Reduction Formula Perturbation Theory for the n-point Function Discrete Symmetry Transformations Introduction Scalar Fields Space Inversion Charge Conjugation Time Reversal 306

3 Contents 10.3 Dirac Fields Space Inversion Charge Conjugation Time Reversal The Electromagnetic Field Invariance of the S Matrix The CPT Theorem 326 Part III. Quantization with Path Integrals 11. The Path-Integral Method Introduction Path Integrals in Nonrelativistic Quantum Mechanics Feynman's Path Integral The Multi-Dimensional Path Integral The Time-Ordered Product and n-point Functions The Vacuum Persistence Amplitude W[J\ Path Integrals in Field Theory The Path Integral for Scalar Quantum Fields Euclidian Field Theory The Feynman Propagator Generating Functional and Green's Function Generating Functional for Interacting Fields Green's Functions in Momentum Space One-Particle Irreducible Graphs and the Effective Action Path Integrals for Fermion Fields Generating Functional and Green's Function for Fermion Fields Generating Functional and Feynman Propagator for the Free Dirac Field 421 Index.. ; 433

4 Contents of Examples and Exercises 1.1 Normal Coordinates The Linear Chain Subject to External Forces.... ^ The Baker-Campbell-Hausdorff Relation The Symmetrized Energy-Momentum Tensor The Poincare Algebra for Classical Fields The Normalization of Fock States...' Interacting Particle Systems: The Hartree-Fock Approximation Commutation Relations for Creation and Annihilation Operators Commutation Relations of the Angular-Momentum Operator The Field Operator in the Spherical Representation The Charge of a State Commutation Relations Between Field Operators and Generators The Function z^i(x - y) for Equal Time Arguments The Symmetrized Dirac Lagrange Density The Symmetrized Current Operator The Momentum Operator Helicity States General Commutation Relations and Microcausality The Lagrangian of the Maxwell Field Coupled Maxwell and Dirac Fields Fourier Decomposition of the Proca Field Operator Invariant Commutation Relations and the Feynman Propagator of the Proca Field The Energy Density of the Photon Field in the Lorentz Gauge Gauge Transformations and Pseudo-photon States The Feynman Propagator for Arbitrary Values of the Gauge Parameter The Transverse Delta Function General Commutation Rules for the Electromagnetic Field The Gell-Mann-Low Theorem Proof of Wick's Theorem Disconnected Vacuum Graphs Moller Scattering and Compton Scattering The Feynman Graphs of Photon-Photon Scattering Scalar Electrodynamics Theory Derivation of the Yang-Feldman Equation The Reduction Formula for Spin- Particles 288

5 Contents of Examples and Exercises 9.3 The Equation of Motion for the Operator U(t) Green's Functions and the S Matrix of <f> A Theory The Operators V and C for Scalar Fields The Classification of Positronium States Transformation Rules for the Bilinear Covariants The Relation Between Particles and Antiparticles The Path Integral for the Propagation of a Free Particle Weyl Ordering of Operators Gaussian Integrals in D Dimensions Construction of the Field-Theoretical Path Integral Series Expansion of the Generating Functional A Differential Equation for W[J] The Perturbation Series for the ip 4 Theory Connected Green's Functions Grassmann Integration Yukawa Coupling 424

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