Table of Contents XV Part III.Relativistic Fields 12.Quantization of Relativistic Fields Coupled Oscillators,the L …249 12. n,Lattice Vibrations....249 12.1.1 Linear Chain of Coupled Oscillators................249 12.1.2 Continuum Limit,Vibrating String.................255 12.1.3 Generalization to Three Dimensions, Relationship to the Klein-Gordon Field.............258 12.2 Classical Field Theory 261 12.2.1 Lagrangian and Euler-Lagrange Equations of Motion.261 12.3 Canonical Quantization 12.4S nd Cone Noether's Tho ·.266 .266 12.4.1 The Energy-Momentum Tensor,Continuity Equations, and Conservation Laws .....266 12.4.2 Derivation from Noether's Theorem of the Conservation Laws for Four-Momentum. Angular Momentum,and Charge...................268 Problems,.... .275 13.Free Fields. 13.1 The Real Klein-Gordon Field... .277 13.1.1 The Lagrangian Density,Commutation Relations. and the hamiltonian 27 1312Pr 9只1 13.2 The Co 85 13.3 ntiz of tho -Gordon Field e Dirac Field 2 281 13.3.2 Conserved Quantities............................ 2ǒ 13.3.3 Quantization.................................... 290 13.3.4 Charge.... 293 13.3.5 The Infinite-Volume Limit. 205 13.4 The Spin Statistics Theorem 908 13.4.1Pro orem 296 r Propertie of Antic utators and Propagators of the Dirac Field Problems… 14.Quantization of the Radiation Field 307 41 a sical Electrodynamics 307 14.1.1 Maxwell Equations .............................. 307 14.1.2 Gauge Transformations ......................... 309 14.2 The Coulomb Gauge....... 3309 14.3 The Lagrangian Density for the Electromagnetic Field......311 14.4 The Free Electromagnatic Field and its Quantization .312
Table of Contents XV Part III. Relativistic Fields 12. Quantization of Relativistic Fields ........................ 249 12.1 Coupled Oscillators, the Linear Chain, Lattice Vibrations. . . . 249 12.1.1 Linear Chain of Coupled Oscillators . . . . . . . . . . . . . . . . 249 12.1.2 Continuum Limit, Vibrating String . . . . . . . . . . . . . . . . . 255 12.1.3 Generalization to Three Dimensions, Relationship to the Klein–Gordon Field . . . . . . . . . . . . . 258 12.2 Classical Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 12.2.1 Lagrangian and Euler–Lagrange Equations of Motion . 261 12.3 Canonical Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 12.4 Symmetries and Conservation Laws, Noether’s Theorem . . . . . 266 12.4.1 The Energy–Momentum Tensor, Continuity Equations, and Conservation Laws. . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 12.4.2 Derivation from Noether’s Theorem of the Conservation Laws for Four-Momentum, Angular Momentum, and Charge . . . . . . . . . . . . . . . . . . . 268 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 13. Free Fields ............................................... 277 13.1 The Real Klein–Gordon Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 13.1.1 The Lagrangian Density, Commutation Relations, and the Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 13.1.2 Propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 13.2 The Complex Klein–Gordon Field . . . . . . . . . . . . . . . . . . . . . . . . 285 13.3 Quantization of the Dirac Field . . . . . . . . . . . . . . . . . . . . . . . . . . 287 13.3.1 Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 13.3.2 Conserved Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 13.3.3 Quantization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 13.3.4 Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 ∗13.3.5 The Infinite-Volume Limit . . . . . . . . . . . . . . . . . . . . . . . . . 295 13.4 The Spin Statistics Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 13.4.1 Propagators and the Spin Statistics Theorem . . . . . . . . 296 13.4.2 Further Properties of Anticommutators and Propagators of the Dirac Field . . . . . . . . . . . . . . . . . 301 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 14. Quantization of the Radiation Field ...................... 307 14.1 Classical Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 14.1.1 Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 14.1.2 Gauge Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 14.2 The Coulomb Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 14.3 The Lagrangian Density for the Electromagnetic Field . . . . . . 311 14.4 The Free Electromagnatic Field and its Quantization . . . . . . . 312
XVI Table of Contents 14.5 Calculation of the Photon Propagator.....................316 Problems..................................................320 15.Interacting Fields,Quantum Electrodynamics ............321 15.1 Lagrangians,Interacting Fields..... 321 15.1.1 Nonlinear Lagrangians 321 15.1.2 Fermions in an External Field. 322 15.1.3 Interaction of Electrons with the Radiation Field: 322 15.2 The Int Quantu on Rer Pertu bation 292 01 The Int eracti 15.2.2 Perturbation Theory.......... 32 15.3 The S Matrix. 328 15.3.1 General Formulation............................. 328 15.3.2 Simple Transitions. 332 *15.4 Wick's Theorem 335 15.5 Simple Scattering Processes,Feynman Diagrams 220 15.5.1 The First-Order Term .220 341 15.5.3 Second-Order 346 15.5.4 Feynman Rules of Quantum Electrodynamics........356 15.6 Radiative Corrections... 。。。。。。。。。。。。。。。。。。。。。350 15.6.1 The Self-Energy of the Electron....................359 15.6.2 Self-Energy of the Photon,Vacuum Polarization.... 365 15.6.3 Vertex Corrections .... 366 15.6.4 The Ward Identity and Charge renormalization 36R 15.6.5 Anomalous Magnetic Moment of the Electron....... 271 Problem 373 Bibliography for Part III. 375 Appendix. 377 Alternative Derivation of the Dirac Equation. .377 B Dirac Matrice 370 ard Represe ntation 379 Chiral Representation 37 B.3 Majorana Representations ....................... Projection Operators for the Spin.................... Definition.. 380 0.2 Rest frame. 380 03 General Significance of the Projection Operator P(n).381 D The Path-Integral Representation of Quantum Mechanics. .385 of the Electromagnetic Field, -Bleuler Method 3R7 E.1 Quantization and the Feynman Propagator.. 8>
XVI Table of Contents 14.5 Calculation of the Photon Propagator . . . . . . . . . . . . . . . . . . . . . 316 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 15. Interacting Fields, Quantum Electrodynamics ............ 321 15.1 Lagrangians, Interacting Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 15.1.1 Nonlinear Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 15.1.2 Fermions in an External Field . . . . . . . . . . . . . . . . . . . . . . 322 15.1.3 Interaction of Electrons with the Radiation Field: Quantum Electrodynamics (QED) . . . . . . . . . . . . . . . . . . 322 15.2 The Interaction Representation, Perturbation Theory . . . . . . . 323 15.2.1 The Interaction Representation (Dirac Representation) 324 15.2.2 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 15.3 The S Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 15.3.1 General Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 15.3.2 Simple Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 ∗15.4 Wick’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 15.5 Simple Scattering Processes, Feynman Diagrams . . . . . . . . . . . 339 15.5.1 The First-Order Term. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 15.5.2 Mott Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 15.5.3 Second-Order Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 15.5.4 Feynman Rules of Quantum Electrodynamics . . . . . . . . 356 ∗15.6 Radiative Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 15.6.1 The Self-Energy of the Electron . . . . . . . . . . . . . . . . . . . . 359 15.6.2 Self-Energy of the Photon, Vacuum Polarization . . . . . . 365 15.6.3 Vertex Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 15.6.4 The Ward Identity and Charge Renormalization . . . . . . 368 15.6.5 Anomalous Magnetic Moment of the Electron . . . . . . . . 371 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 Bibliography for Part III ..................................... 375 Appendix ..................................................... 377 A Alternative Derivation of the Dirac Equation . . . . . . . . . . . . . . . 377 B Dirac Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 B.1 Standard Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 379 B.2 Chiral Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 B.3 Majorana Representations . . . . . . . . . . . . . . . . . . . . . . . . . 380 C Projection Operators for the Spin . . . . . . . . . . . . . . . . . . . . . . . . 380 C.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 C.2 Rest Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 C.3 General Significance of the Projection Operator P(n) . 381 D The Path-Integral Representation of Quantum Mechanics . . . . 385 E Covariant Quantization of the Electromagnetic Field, the Gupta–Bleuler Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 E.1 Quantization and the Feynman Propagator . . . . . . . . . . 387
Table of Contents XVII E.2 The Physical Significance of Longitudinal and Scalar Photons E …389 F Coupling of Charged Scalar Mesons to the Electromagnetic Field.............................394 Index...397
Table of Contents XVII E.2 The Physical Significance of Longitudinal and Scalar Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 E.3 The Feynman Photon Propagator . . . . . . . . . . . . . . . . . . 392 E.4 Conserved Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 F Coupling of Charged Scalar Mesons to the Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 Index ......................................................... 397
Part I Nonrelativistic Many-Particle Systems
Part I Nonrelativistic Many-Particle Systems
1.Second Quantization In this first part.we shall consider nonrelativistic systems consisting of a large numbe e,we will introduc a particularl efficient formalism,namely,th ethod of cond quantiza tion.Nature has given us two types of particle,bosons and fermions.These have states that are,respectively,completely symmetric and completely an- tisymmetric.Fermions possess half-integer spin values,whereas boson spins have integer values.This connection between spin and symmetry (statistics) is proved within relativistic quantum field theory (the spin-statistics theo- rem).An important consequence in many-particle physics is the existence of Fermi-Dirac statistics and Bose-Einstein statistics.We shall begin in Sect. 1.1 with arks which follow 13 of Qu Mechanic For the later the first part,Sect.1. 1.1 Identical Particles,Many-Particle States, and Permutation Symmetry 1.1.1 States and Observables of Identical Particles We consider Nidentical particles (e.g.,electrons,mesons).The Hamiltonian H=H(1,2,·,N) (1.1.1) is symmetric in the variables 1,2,...,N.Here 1 =x1,o1 denotes the position and spin degrees of freedom of particle 1 and correspondingly for the other particles.Similarly,we write a wave function in the form =(1,2,.,N) (1.1.2) The permutation operator Pj,which interchanges i and j,has the following effect on an arbitrary N-particle wave function 1F.Schwabl.Quantum Mechanics,3 ed.,Springer,Berlin Heidelberg,2002;in subsequent citations this book will be referred to as QMI
1. Second Quantization In this first part, we shall consider nonrelativistic systems consisting of a large number of identical particles. In order to treat these, we will introduce a particularly efficient formalism, namely, the method of second quantization. Nature has given us two types of particle, bosons and fermions. These have states that are, respectively, completely symmetric and completely antisymmetric. Fermions possess half-integer spin values, whereas boson spins have integer values. This connection between spin and symmetry (statistics) is proved within relativistic quantum field theory (the spin-statistics theorem). An important consequence in many-particle physics is the existence of Fermi–Dirac statistics and Bose–Einstein statistics. We shall begin in Sect. 1.1 with some preliminary remarks which follow on from Chap. 13 of Quantum Mechanics1. For the later sections, only the first part, Sect. 1.1.1, is essential. 1.1 Identical Particles, Many-Particle States, and Permutation Symmetry 1.1.1 States and Observables of Identical Particles We consider N identical particles (e.g., electrons, π mesons). The Hamiltonian H = H(1, 2,... ,N) (1.1.1) is symmetric in the variables 1, 2,... ,N. Here 1 ≡ x1, σ1 denotes the position and spin degrees of freedom of particle 1 and correspondingly for the other particles. Similarly, we write a wave function in the form ψ = ψ(1, 2,... ,N). (1.1.2) The permutation operator Pij , which interchanges i and j, has the following effect on an arbitrary N-particle wave function 1 F. Schwabl, Quantum Mechanics, 3rd ed., Springer, Berlin Heidelberg, 2002; in subsequent citations this book will be referred to as QM I