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