From Special Relativity to Feynman Diagrams.pdf

The hamiltonian formalism 222 84 canonical

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The Hamiltonian Formalism . . . . . . . . . . . . . . . . . . . . . . . . 222 8.4 Canonical Transformations and Conserved Quantities . . . . . . . 226 8.4.1 Conservation Laws in the Hamiltonian Formalism . . . 229 8.5 Lagrangian and Hamiltonian Formalism in Field Theories . . . . 232 8.5.1 Functional Derivative . . . . . . . . . . . . . . . . . . . . . . . 232 8.5.2 The Hamilton Principle of Stationary Action . . . . . . . 236 8.6 The Action of the Electromagnetic Field . . . . . . . . . . . . . . . . 238 8.6.1 The Hamiltonian for an Interacting Charge . . . . . . . . 243 8.7 Symmetry and the Noether Theorem . . . . . . . . . . . . . . . . . . . 245 8.8 Space–Time Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 8.8.1 Internal Symmetries . . . . . . . . . . . . . . . . . . . . . . . . 255 8.9 Hamiltonian Formalism in Field Theory . . . . . . . . . . . . . . . . 257 8.9.1 Symmetry Generators in Field Theories . . . . . . . . . . 260 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 9 Quantum Mechanics Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 263 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 9.2 Wave Functions, Quantum States and Linear Operators . . . . . 263 9.3 Unitary Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 9.3.1 Application to Non-Relativistic Quantum Theory . . . . 276 9.3.2 The Time Evolution Operator . . . . . . . . . . . . . . . . . 283 9.4 Towards a Relativistically Covariant Description . . . . . . . . . . 287 9.4.1 The Momentum Representation . . . . . . . . . . . . . . . . 293 9.4.2 Particles and Irreducible Representations of the Poincaré Group . . . . . . . . . . . . . . . . . . . . . . . 297 9.5 A Note on Lorentz Invariant Normalizations . . . . . . . . . . . . . 299 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 Contents xiii
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10 Relativistic Wave Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 10.1 The Relativistic Wave Equation . . . . . . . . . . . . . . . . . . . . . . 303 10.2 The Klein–Gordon Equation . . . . . . . . . . . . . . . . . . . . . . . . 307 10.2.1 Coupling of the Complex Scalar Field / ( x ) to the Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . 314 10.3 The Hamiltonian Formalism for the Free Scalar Field . . . . . . . 317 10.4 The Dirac Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 10.4.1 The Wave Equation for Spin 1/2 Particles . . . . . . . . . 319 10.4.2 Conservation of Probability . . . . . . . . . . . . . . . . . . . 323 10.4.3 Covariance of the Dirac Equation . . . . . . . . . . . . . . . 324 10.4.4 Infinitesimal Generators and Angular Momentum . . . . 326 10.5 Lagrangian and Hamiltonian Formalism . . . . . . . . . . . . . . . . 331 10.6 Plane Wave Solutions to the Dirac Equation . . . . . . . . . . . . . 334 10.6.1 Useful Properties of the u ( p , r ) and v ( p , r ) Spinors. . . 340 10.6.2 Charge Conjugation . . . . . . . . . . . . . . . . . . . . . . . . 343 10.6.3 Spin Projectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 10.7 Dirac Equation in an External Electromagnetic Field . . . . . . . 348 10.8 Parity Transformation and Bilinear Forms . . . . . . . . . . . . . . . 352 10.8.1 Bilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 11 Quantization of Boson and Fermion Fields . . . . . . . . . . . . . . . . . 359 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 11.2 Quantization of the Klein–Gordon Field . . . . . . . . . . . . . . . . 360 11.2.1 Electric Charge and its Conservation . . . . . . . . . . . . 375 11.3 Transformation Under the Poincaré Group . . . . . . . . . . . . . . . 377 11.3.1 Discrete Transformations . . . . . . . . . . . . . . . . . . . . . 379 11.4 Invariant Commutation Rules and Causality . . . . . . . . . . . . . 385 11.4.1 Green’s Functions and the Feynman Propagator . . . . . 388 11.5 Quantization of the Dirac Field . . . . . . . . . . . . . . . . . . . . . . 394 11.6 Invariant Commutation Rules for the Dirac Field . . . . . . . . . . 402 11.6.1 The Feynman Propagator for Fermions . . . . . . . . . . . 404 11.6.2 Transformation Properties of the Dirac Quantum Field 405 11.6.3 Discrete Transformations . . . . . . . . . . . . . . . . . . . . . 408 11.7 Covariant Quantization of the Electromagnetic Field . . . . . . . 412 11.7.1 Indefinite Metric and Subsidiary Conditions . . . . . . . 419 11.7.2 Poincaré Transformations and Discrete Symmetries . . 425 11.8 Quantum Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 426 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 xiv Contents
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12 Fields in Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 12.1 Interaction Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 12.2 Kinematics of Interaction Processes . . . . . . . . . . . . . . . . . . . 435 12.2.1 Decay Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 12.2.2 Scattering Processes . . . . . . . . . . . . . . . . . . . . . . . . 440 12.3 Dynamics of Interaction Processes . . . . . . . . . . . . . . . . . . . . 443 12.3.1 Interaction Representation . . . . . . . . . . . . . . . . . . . . 444 12.3.2 The Scattering Matrix . . . . . . . . . . . . . . . . . . . . . . . 446 12.3.3 Two-Particle Phase-Space Element . . . . . . . . . . . . . . 456 12.3.4 The Optical Theorem . . . . . . . . . . . . . . . . . . . . . . . 458 12.3.5 Natural Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 12.3.6 The Wick’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 462 12.4 Quantum Electrodynamics and Feynman Rules . . . . . . . . . . . 467 12.4.1 External Electromagnetic Field . . . . . . . . . . . . . . . . . 473 12.5 Amplitudes in the Momentum Representation . . . . . . . . . . . . 475 12.5.1
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