Quantum Field Theory - An Introduction

This textbook is intended to be used in an introductory course in quantum field theory. It assumes the standard undergraduate education of a physics major and it is designed to appeal to a wide array of physics graduate students, from those studying theoretical and experimental high energy physics to those interested in condensed matter, optical, atomic, nuclear and astrophysicists. It includes a thorough development of the field theoretic approach to nonrelativistic many-body physics as a step in developing a broad-based working knowledge of some of the basic aspects of quantum field theory. It presents a logical, step by step systematic development of relativistic field theory and of functional techniques and their applications to perturbation theory with Feynman diagrams, renormalization, and basic computations in quantum electrodynamics. Contents 7 1 Prologue 11 2 Many Particle Physics as a Quantum Field Theory 15 2.1 Introduction 15 2.2 Non-relativistic Particles 15 2.2.1 Identical and Indistinguishable Particles 17 2.2.2 The Example of Weakly Interacting Particles 19 2.2.3 Hamiltonian and Stationary States 21 2.2.4 Particles with Spin 23 2.3 Second Quantization in the Schrödinger Picture 25 2.4 Second Quantization in the Heisenberg Picture 30 3 Degenerate Fermi and Bose Gases 33 3.1 The Limit of Weakly Interacting Particles 33 3.2 Degenerate Fermi Gas 38 3.2.1 The Ground State |mathcalO> 38 3.2.2 Particles and Holes 40 3.2.3 The Grand Canonical Free Energy 43 3.3 Degenerate Bose Gas 45 3.3.1 Landau's Criterion for Superfluidity 48 3.3.2 Vacuum Expectation Value 50 3.4 Spontaneous Symmetry Breaking 56 4 The Action Principle and Noether's Theorem 58 4.1 The Action 60 4.1.1 The Euler–Lagrange Equations 61 4.2 Canonical Momenta, Poisson Brackets and Commutation Relations 64 4.3 Noether's Theorem 66 4.3.1 Conservation Laws and Continuity Equations 66 4.3.2 Definition of Symmetry 67 4.3.3 Examples of Symmetries 68 4.3.4 Proof of Noether's Theorem 70 4.4 Phase Symmetry and the Conservation of Particle Number 72 5 Non-relativistic Space–Time Symmetries 74 5.1 Translation Invariance and the Stress Tensor 74 5.2 Galilean Symmetry 77 5.3 Scale Invariance 82 5.3.1 Improving the Stress Tensor 85 5.3.2 The Consequences of Scale Invariance 86 5.4 Special Schrödinger Symmetry 88 5.5 Summary 89 6 Space–Time Symmetry and Relativistic Field Theory 94 6.1 Quantum Mechanics and Special Relativity 94 6.2 Coordinates 101 6.3 Scalars, Vectors, Tensors 104 6.4 The Metric 105 6.5 Symmetry of Space–Time 107 6.6 The Symmetries of Minkowski Space 108 6.7 Natural Units 111 6.8 Relativistic Fields 112 7 The Real Scalar Quantum Field Theory 114 7.1 Constructing a Relativistic Lagrangian Density 114 7.2 Field Equation and Commutation Relations 116 7.3 Noether's Theorem and Poincare Symmetry 117 7.4 Correlation Functions of the Real Scalar Field 120 7.5 The Free Scalar Field 121 7.6 Consequences of Spacetime Symmetry 126 7.7 Spectral Theorem 129 7.8 Normalization of the Spectral Function 132 7.9 Analyticity 132 7.9.1 The Reeh–Schlieder Theorem 135 7.10 Conformal Symmetry 139 8 Emergent Relativistic Symmetry 146 8.1 Phonons 146 8.2 The Debye Theory of Solids 152 8.3 Relativistic Fermions in Graphene 156 9 The Dirac Field Theory 165 9.1 The Dirac Equation 165 9.2 Solving the Dirac Equation 169 9.3 Lorentz Invariance of the Dirac Equation 173 9.4 Spin of the Dirac Field 175 9.5 Phase Symmetry and the Conservation of Charge 177 9.5.1 Conserved Number Current 178 9.5.2 Relativistic Noether's Theorem for the Dirac Equation 179 9.5.3 Alternative Proof of Noether's Theorem 180 9.6 Spacetime Symmetry 182 9.6.1 Translation Invariance and the Stress Tensor 182 9.6.2 Lorentz Transformations 184 9.6.3 Stress Tensor and Killing Vectors 187 10 Photons 188 10.1 Relativistic Classical Electrodynamics 189 10.2 Quantization 194 10.2.1 Negative Normed States 197 10.2.2 Physical State Condition 197 10.2.3 Null States and the Equivalence Relation 201 10.3 Space–Time Symmetries of the Photon 208 10.4 Massive Photon 210 10.5 Quantum Electrodynamics 214 10.5.1 C, P and T 216 11 Functional Methods 218 11.1 Functional Derivative 219 11.2 Functional Integral 221 11.3 Generating Functional for Free Scalar Fields 225 11.3.1 Wick's Theorem for Scalar Fields 229 11.3.2 Generating Functional as a Functional Integral 229 11.4 The Interacting Real Scalar Field 232 12 More Functional Integrals 236 12.1 Functional Integrals for the Photon Field 237 12.2 Functional Methods for Fermions 243 12.3 Generating Functionals for Non-relativistic Fermions 249 12.3.1 Interacting Non-relativistic Fermions 253 12.4 The Dirac Field 254 12.4.1 2 Point Function for the Dirac Field 255 12.4.2 Generating Functional for the Dirac Field 257 12.4.3 Functional Integral for the Dirac Field 259 12.5 Functional Quantum Electrodynamics 260 13 The Weakly Coupled Real Scalar Field 264 13.1 Counterterms 269 13.2 Computation of the 2 Point Function 274 13.3 Feynman Diagrams 277 13.4 Simplifications of Feynman Diagrams 280 13.5 Computation of a One-Loop Feynman Integral 287 13.5.1 Dimensional Regularization 288 13.5.2 Wick Rotation 288 13.5.3 Feynman Parameters 289 13.5.4 Integration in 2ω-Dimensions 290 13.5.5 Asymptotic Expansion at 2ωsim4 291 13.5.6 Inverse Wick Rotation 291 13.5.7 The Mass Tadpole 292 13.5.8 Euclidean Quantum Field Theory 293 13.5.9 The 2 Point and 4 Point Functions 294 13.6 Subtraction Schemes 295 13.7 Renormalization Group 299 13.8 Appendix: Integration Formulae 308 13.8.1 Euler's Gamma Function 308 13.8.2 Feynman Parameter Formula 310 13.8.3 Dimensional Regularization Integral 312 14 More Theory of the Real Scalar Field 314 14.1 The S Matrix 314 14.1.1 The T Matrix 317 14.2 The LSZ Formula 318 14.3 Elastic Two-Particle Scattering 322 14.4 Connected and Irreducible Generating Functionals 326 14.4.1 Connected Correlation Functions and the Linked Cluster Theorem 327 14.4.2 Connected Correlation Functions 328 14.4.3 Cancelation of Vacuum Diagrams 332 14.4.4 Irreducible Correlation Function 332 14.5 Derivation of the LSZ Formula 342 15 Perturbative Quantum Electrodynamics 346 15.1 Counterterms 347 15.2 The Generating Functional in Perturbation Theory 350 15.2.1 Wick's Theorem for Photons and Electrons 351 15.3 Feynman Diagrams 353 15.4 Feynman Rules 357 15.5 The Electron 2 Point Function 358 15.6 Feynman Rules in Momentum Space 360 15.7 The Photon 2 Point Function 364 15.8 Quantum Corrections of the Coulomb Potential 372 15.9 The Electron 2 Point Function 377 15.10 Radiative Correction of the Vertex 380 15.10.1 Electromagnetic form Factors 384 15.10.2 Anomalous Magnetic Moment 391 15.11 Photon Production, the Soft Photon Theorem 395 15.12 Furry's Theorem 400 15.13 The Ward–Takahashi Identities 402 16 Epilogue 406 Index 407 This book is a pedagogical introduction to quantum field theory, suitable for a students'first exposure to the subject. It assumes a minimal amount of technical background and it is intended to be accessible to a wide audience including students of theoretical and experimental high energy physics, condensed matter, optical, atomic, nuclear and gravitational physics and astrophysics. It includes a thorough development of second quantization and the field theoretic approach to nonrelativistic many-body physics as a step in developing a broad-based working knowledge of the basic aspects of quantum field theory. It presents a logical and systematic first principles development of relativistic field theory and of functional techniques and perturbation theory with Feynman diagrams, renormalization, and basic computations in quantum electrodynamics. This book is a pedagogical introduction to quantum field theory, suitable for a students' first exposure to the subject. It assumes a minimal amount of technical background and it is intended to be accessible to a wide audience including students of theoretical and experimental high energy physics, condensed matter, optical, atomic, nuclear and gravitational physics and astrophysics. It includes a thorough development of second quantization and the field theoretic approach to nonrelativistic many-body physics as a step in developing a broad-based working knowledge of the basic aspects of quantum field theory. It presents a logical and systematic first principles development of relativistic field theory and of functional techniques and perturbation theory with Feynman diagrams, renormalization, and basic computations in quantum electrodynamics