quantum field theory by hagen kleinert

PARTICLES AND QUANTUM FIELDS

Particles and Quantum Fields

Hagen KleinertProfessor of Physics

Freie Universitat Berlin

Preface

This book arose from lectures I gave at the Free University Berlin over the last40 years years. The lectures were intended to prepare graduate students for theirresearch work in either condensed matter many-body physics or in particle physics.As a general broadly-based introduction, the book will eventually require a moredetailed and up-to-date account of the exciting developments in nonabelian gaugetheories that have taken place over the last two decades. These have pretxyy muchestablished quantum chromodynamics (QCD) as a probably true theory of stronginteractions.

Another extension into the strong-coupling limit (or critical limit) of quantumfield theory has already been published as a monograph.1 with interesting applica-tions into the theory od very rare events 2

I thank Axel Pelster for carefully reading parts of these lecture notes and fortheir useful comments and corrections. A number of printing errors were correctedby my wife Dr. Annemarie Kleinert whom I thank for a her patience and sacrificeof many sunny weekends. Without her persistent ecouragements the book wouldnever have been finished.

H. Kleinert

Berlin, October 2013

1H. Kleinert and V. Schulte-Frohlinde, Critical Properties of 4-Theories , World Scientific,Singapore 2001, pp. 1489 (http://klnrt.de/b8).

2H. Kleinert, Quantum Field Theory of Black-Swan Events , EPL 100, 10001 (2013)(http://klnrt.de/399/399-TAI-PEH.pdf); a Effective Action and Field Equation for BEC fromWeak to Strong Couplings , (http://klnrt.de/403).

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Contents

1 Fundamentals 11.1 Classical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Relativistic Mechanics in Curved Spacetime . . . . . . . . . . . . . . 101.3 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3.1 Bragg Reflections and Interference . . . . . . . . . . . . . . 111.3.2 Matter Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3.3 Schrodinger Equation . . . . . . . . . . . . . . . . . . . . . 141.3.4 Particle Current Conservation . . . . . . . . . . . . . . . . . 18

1.4 Diracs Bra-Ket Formalism . . . . . . . . . . . . . . . . . . . . . . . 181.4.1 Basis Transformations . . . . . . . . . . . . . . . . . . . . . 191.4.2 Bracket Notation . . . . . . . . . . . . . . . . . . . . . . . . 211.4.3 Continuum Limit . . . . . . . . . . . . . . . . . . . . . . . . 221.4.4 Generalized Functions . . . . . . . . . . . . . . . . . . . . . 241.4.5 Schrodinger Equation in Dirac Notation . . . . . . . . . . . 251.4.6 Momentum States . . . . . . . . . . . . . . . . . . . . . . . 271.4.7 Incompleteness and Poissons Summation Formula . . . . . 29

1.5 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.5.1 Uncertainty Relation . . . . . . . . . . . . . . . . . . . . . . 321.5.2 Density Matrix and Wigner Function . . . . . . . . . . . . . 331.5.3 Generalization to Many Particles . . . . . . . . . . . . . . . 34

1.6 Time Evolution Operator . . . . . . . . . . . . . . . . . . . . . . . . 351.7 Properties of the Time Evolution Operator . . . . . . . . . . . . . . 381.8 Heisenberg Picture of Quantum Mechanics . . . . . . . . . . . . . . 401.9 Interaction Picture and Perturbation Expansion . . . . . . . . . . . 431.10 Time Evolution Amplitude . . . . . . . . . . . . . . . . . . . . . . . 441.11 Fixed-Energy Amplitude . . . . . . . . . . . . . . . . . . . . . . . . 471.12 Free-Particle Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . 491.13 Quantum Mechanics of General Lagrangian Systems . . . . . . . . . 531.14 Particle on the Surface of a Sphere . . . . . . . . . . . . . . . . . . . 581.15 Spinning Top . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611.16 Classical and Quantum Statistics . . . . . . . . . . . . . . . . . . . . 68

1.16.1 Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . 691.16.2 Grand-Canonical Ensemble . . . . . . . . . . . . . . . . . . 70

1.17 Density of States and Tracelog . . . . . . . . . . . . . . . . . . . . . 75Appendix 1A Simple Time Evolution Operator . . . . . . . . . . . . . . . 77

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Appendix 1B Convergence of the Fresnel Integral . . . . . . . . . . . . . . 77Appendix 1C The Asymmetric Top . . . . . . . . . . . . . . . . . . . . . . 78Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

2 Field Formulation of Many-Body Quantum Physics 822.1 Mechanics and Quantum Mechanics for n Nonrelativistic Particles . 822.2 Identical Particles Bosons and Fermions . . . . . . . . . . . . . . 85

2.3 Creation and Annihilation Operators for Bosons . . . . . . . . . . . 912.4 Schrodinger Equation for Noninteracting Bosons in Terms of Cre-

ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

2.5 Second Quantization and Symmetrized Product Representation . . . 972.6 Bosons with Two-Body Interactions . . . . . . . . . . . . . . . . . . 1012.7 Quantum Field Formulation of Many-Boson Schrodinger Equations . 1022.8 Canonical Formalism in Quantum Field Theory . . . . . . . . . . . . 104

2.9 More General Creation and Annihilation Operators . . . . . . . . . 1092.10 Quantum Field Formulation of Many-Fermion Schrodinger Equations 1112.11 Free Nonrelativistic Particles and Fields . . . . . . . . . . . . . . . . 113

2.12 Second-Quantized Current Conservation Law . . . . . . . . . . . . . 1162.13 Free-Particle Propagator . . . . . . . . . . . . . . . . . . . . . . . . 1172.14 Quantum Statistic of Free Nonrelativistic Fields . . . . . . . . . . . 120

2.14.1 Thermodynamic Quantities . . . . . . . . . . . . . . . . . . 120

2.14.2 The Degenerate Fermi Gas Near T = 0 . . . . . . . . . . . . 1262.14.3 Degenerate Bose Gas Near T = 0 . . . . . . . . . . . . . . . 1312.14.4 High Temperatures . . . . . . . . . . . . . . . . . . . . . . . 137

2.15 Noninteracting Bose Gas In Trap . . . . . . . . . . . . . . . . . . . . 138

2.15.1 Bose Gas in Finite Box . . . . . . . . . . . . . . . . . . . . 1382.15.2 Harmonic and General Power Trap . . . . . . . . . . . . . . 1402.15.3 Anharmonic Trap in Rotating Bose-Einstein Gas . . . . . . 141

2.16 Temperature Green Functions of Free Particles . . . . . . . . . . . . 1432.17 Calculating the Matsubara Sum via Poisson Formula . . . . . . . . . 1482.18 Nonequilibrium Quantum Statistics . . . . . . . . . . . . . . . . . . 1492.19 Linear Response and Time-Dependent Green Functions for T 6= 0 . . 150

2.20 Spectral Representations of Green Functions for T 6= 0 . . . . . . . 1532.21 Other Important Green Functions . . . . . . . . . . . . . . . . . . . 1552.22 Hermitian Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . 158

2.23 Harmonic Oscillator Green Functions for T 6= 0 . . . . . . . . . . . . 1592.23.1 Creation Annihilation Operators . . . . . . . . . . . . . . . 1602.23.2 Real Field Operators . . . . . . . . . . . . . . . . . . . . . . 162

Appendix 2A Permutation Group and Representations . . . . . . . . . . . 164

Appendix 2B Treatment of Singularities in Zeta-Function . . . . . . . . . . 1692B.1 Finite Box . . . . . . . . . . . . . . . . . . . . . . . . . . . 1692B.2 Harmonic Trap . . . . . . . . . . . . . . . . . . . . . . . . . 172

Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

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3 Interacting Non-Relativistic Particles 1763.1 Weakly Interacting Bose Gas . . . . . . . . . . . . . . . . . . . . . . 1773.2 Weakly Interacting Fermi Gas . . . . . . . . . . . . . . . . . . . . 187

3.2.1 Electrons in a Metal . . . . . . . . . . . . . . . . . . . . . . 1873.3 Superconducting Electrons . . . . . . . . . . . . . . . . . . . . . . . 194

3.3.1 Zero Temperature . . . . . . . . . . . . . . . . . . . . . . . 2003.4 Renormalized Theory at Strong Interactions . . . . . . . . . . . . . 204

3.4.1 Finite Temperature . . . . . . . . . . . . . . . . . . . . . . . 2063.5 Crossover to Strong Couplings . . . . . . . . . . . . . . . . . . . . . 210

3.5.1 Bogoliubov Bose Gas at Finite Temperature . . . . . . . . . 2113.6 Bose Gas at Strong Interactions . . . . . . . . . . . . . . . . . . . . 2143.7 Corrections Due to Omitted Interaction Hamiltonian . . . . . . . . . 231Appendix 3A Two-Loop Momentum Integrals . . . . . . . . . . . . . . . . 234Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

4 Relativistic Free Particles and Fields 2394.1 Relativistic Particles . . . . . . . . . . . . . . . . . . . . . . . . . . 2394.2 Differential Operators for Lorentz Transformations . . . . . . . . . . 2464.3 Space Inversion and Time Reversal . . . . . . . . . . . . . . . . . . 2554.4 Relativistic Free Scalar Fields . . . . . . . . . . . . . . . . . . . . . 2574.5 Other Symmetries of Scalar Action . . . . . . . . . . . . . . . . . . . 264

4.5.1 Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . 2644.5.2 Space Inversion . . . . . . . . . . . . . . . . . . . . . . . . . 2654.5.3 Time Reversal . . . . . . . . . . . . . . . . . . . . . . . . . 2674.5.4 Charge Conjugation . . . . . . . . . . . . . . . . . . . . . . 270

4.6 Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . 2714.6.1 Action and Field Equations . . . . . . . . . . . . . . . . . . 2714.6.2 Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . . 2734.6.3 Lorentz Transformation Properties of Electromagnetic Fields 276

4.7 Other Symmetries of Electromagnetic Action . . . . . . . . . . . . . 2784.7.1 Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . 2794.7.2 Space Inversion, Time Reversal, and Charge Conjugation . . 279

4.8 Plane-Wave Solutions of Maxwells Equations . . . . . . . . . . . . 2804.9 Massive Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 284

4.9.1 Action and Field Equations . . . . . . . . . . . . . . . . . . 2844.9.2 Plane Wave Solutions for Massive Vector Fields . . . . . . . 285

4.10 Gravitational Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 2894.10.1 Action and Field Equations . . . . . . . . . . . . . . . . . . 2894.10.2 Lorentz Transformation Properties of Gravitational Field . . 2934.10.3 Other Symmetries of Gravitational Action . . . . . . . . . . 2934.10.4 Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . 2934.10.5 Space Inversion, Time Reversal, and Charge Conjugation . . 2944.10.6 Gravitational Plane Waves . . . . . . . . . . . . . . . . . . 294

4.11 Relativistic Free Fermi Fields . . . . . . . . . . . . . . . . . . . . . . 299

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4.12 Spin-1/2 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3004.13 Other Symmetries of Dirac Action . . . . . . . . . . . . . . . . . . . 310

4.13.1 Translations and Poincare group . . . . . . . . . . . . . . . 3104.13.2 Space Inversion . . . . . . . . . . . . . . . . . . . . . . . . . 3104.13.3 Charge Conjugation . . . . . . . . . . . . . . . . . . . . . . 3214.13.4 Time Reversal . . . . . . . . . . . . . . . . . . . . . . . . . 3244.13.5 Transformation Properties of Currents . . . . . . . . . . . . 325

4.14 Majorana Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3264.14.1 Plane-Wave Solutions of Dirac Equation . . . . . . . . . . . 329

4.15 Lorentz Transformation of Spinors . . . . . . . . . . . . . . . . . . 3414.16 Precession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

4.16.1 Wigner Precession . . . . . . . . . . . . . . . . . . . . . . . 3434.16.2 Thomas Precession . . . . . . . . . . . . . . . . . . . . . . . 3454.16.3 Spin Four-Vector and Little Group . . . . . . . . . . . . . . 346

4.17 Weyl Spinor Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 3494.18 Higher Spin Representations . . . . . . . . . . . . . . . . . . . . . . 351

4.18.1 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3514.18.2 Extension to Lorentz Group . . . . . . . . . . . . . . . . . . 3544.18.3 Finite Representation Matrices . . . . . . . . . . . . . . . . 356

4.19 Higher Spin Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 3634.19.1 Plane-Wave Solutions . . . . . . . . . . . . . . . . . . . . . 365

4.20 Vector Field as Higher Spin Field . . . . . . . . . . . . . . . . . . . 3674.21 Rarita-Schwinger field for Spin 3/2 . . . . . . . . . . . . . . . . . . . 368Appendix 4A Derivation of Baker-Campbell-Hausdorff Formula . . . . . . 369Appendix 4B Wigner Rotations and Lobatschewksi Geometry of Rapidities 371

4B.1 Wigner Precession . . . . . . . . . . . . . . . . . . . . . . . 3724B.2 Thomas Precession . . . . . . . . . . . . . . . . . . . . . . . 3744B.3 Calculation in Four-Dimensional Representation . . . . . . 3744B.4 Hyperbolic Geometry . . . . . . . . . . . . . . . . . . . . . 375

Appendix 4C Clebsch-Gordan Coefficients . . . . . . . . . . . . . . . . . . 377Appendix 4D Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . 381Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

5 Classical Radiation 3855.1 Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . . . 3855.2 Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3855.3 Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

5.3.1 Gravitational Field of Matter Source . . . . . . . . . . . . . 3905.3.2 Quadrupole Moment . . . . . . . . . . . . . . . . . . . . . . 3935.3.3 Average Radiated Energy . . . . . . . . . . . . . . . . . . . 3965.3.4 Relation to Gravitational Interaction Energy . . . . . . . . 396

5.4 Simple Models for Sources of Gravitational Radiation . . . . . . . . 3985.4.1 Vibrating Quadrupole . . . . . . . . . . . . . . . . . . . . . 3985.4.2 Two Rotating Masses . . . . . . . . . . . . . . . . . . . . . 400

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5.4.3 Particle Falling into Star . . . . . . . . . . . . . . . . . . . . 4065.4.4 Cloud of Colliding Stars . . . . . . . . . . . . . . . . . . . . 407

5.5 Orders of Magnitude of Different Radiation Sources . . . . . . . . . 4105.6 Detection of Gravitational Waves . . . . . . . . . . . . . . . . . . . . 4125.7 Attractive Gravity versus Repulsive Electromagnetism between Like

Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4165.8 Nonlinear Gravitational Waves . . . . . . . . . . . . . . . . . . . . . 417Appendix 5A Nonexistence of Gravitational Waves for D = 3 and D = 2 . 418Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

6 Relativistic Particles and Fields in External Electromagnetic Po-tential 4666.1 Charged Point Particles . . . . . . . . . . . . . . . . . . . . . . . . . 466

6.1.1 Coupling to Electromagnetism . . . . . . . . . . . . . . . . 4676.1.2 Spin Precession in Atom . . . . . . . . . . . . . . . . . . . . 4696.1.3 Relativistic Equation of Motion for Spin Vector and Thomas

Precession . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4726.2 Charged Particle in Schrodinger Theory . . . . . . . . . . . . . . . . 4756.3 Charged Relativistic Fields . . . . . . . . . . . . . . . . . . . . . . . 478

6.3.1 Scalar Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 4786.3.2 Dirac Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 478

6.4 Pauli Equation from Dirac Theory . . . . . . . . . . . . . . . . . . . 4796.5 Relativistic Wave Equations in Coulomb Potential . . . . . . . . . . 481

6.5.1 Reminder of Schrodinger Equation with Coulomb Potential 4826.5.2 Klein-Gordon Field in Coulomb Potential . . . . . . . . . . 4846.5.3 Dirac Field in Coulomb Potential . . . . . . . . . . . . . . . 485

6.6 Green Function in External Electromagnetic Field . . . . . . . . . . 4916.6.1 Scalar Field in Constant Electromagnetic Field . . . . . . . 4916.6.2 Dirac Field in Constant Electromagnetic Field . . . . . . . . 4976.6.3 Dirac Field in Electromagnetic Plane-Wave Field . . . . . . 499

Appendix 6A Spinor Spherical Harmonics . . . . . . . . . . . . . . . . . . 502Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503

7 Quantization of Relativistic Free Fields 5047.1 Scalar Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

7.1.1 Real Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5057.1.2 Field Quantization . . . . . . . . . . . . . . . . . . . . . . . 5057.1.3 Propagator of Free Scalar Particles . . . . . . . . . . . . . . 5117.1.4 Complex Case . . . . . . . . . . . . . . . . . . . . . . . . . 5147.1.5 Energy of Free Charged Scalar Particles . . . . . . . . . . . 5167.1.6 Behavior under Discrete Symmetries . . . . . . . . . . . . . 518

7.2 Spacetime Behavior of Propagators . . . . . . . . . . . . . . . . . . 5237.2.1 Wick Rotation . . . . . . . . . . . . . . . . . . . . . . . . . 5237.2.2 Feynman Propagator in Minkowski Space . . . . . . . . . . 526

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7.2.3 Retarded and Advanced Propagators . . . . . . . . . . . . . 5287.2.4 Comparison of Singular Functions . . . . . . . . . . . . . . 533

7.3 Free Dirac Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5367.3.1 Field Quantization . . . . . . . . . . . . . . . . . . . . . . . 5377.3.2 Energy of Free Dirac Particles . . . . . . . . . . . . . . . . . 5407.3.3 Lorentz Transformation Properties of Particle States . . . . 5427.3.4 Behavior under Discrete Symmetries . . . . . . . . . . . . . 551

7.4 Free Photon Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5557.4.1 Field Quantization . . . . . . . . . . . . . . . . . . . . . . . 5567.4.2 Covariant Field Quantization . . . . . . . . . . . . . . . . . 5627.4.3 Behavior under Discrete Symmetries . . . . . . . . . . . . . 588

7.5 Massive Vector Bosons . . . . . . . . . . . . . . . . . . . . . . . . . 5887.5.1 Field Quantization . . . . . . . . . . . . . . . . . . . . . . . 5907.5.2 Energy of Massive Vector Particle . . . . . . . . . . . . . . 5917.5.3 Propagator of Massive Vector Particle . . . . . . . . . . . . 5937.5.4 Behavior under Discrete Symmetry Transformations . . . . 597

7.6 Spin-3/2 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5987.7 Gravitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6007.8 Spin-Statistics Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 6017.9 CPT-Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6047.10 Physical Consequences of Vacuum Fluctuations. Casimir Effect . . . 6057.11 Zeta Function Regularization . . . . . . . . . . . . . . . . . . . . . . 6117.12 Dimensional Regularization . . . . . . . . . . . . . . . . . . . . . . . 6157.13 Accelerated Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6187.14 Free Green Functions of n Fields . . . . . . . . . . . . . . . . . . . . 619

7.14.1 Wicks Theorem . . . . . . . . . . . . . . . . . . . . . . . . 6247.15 Functional Form of Wicks Theorem . . . . . . . . . . . . . . . . . . 627

7.15.1 Thermodynamic Version of Wicks Theorem . . . . . . . . . 632Appendix 7A Lienard-Wiechert Potential . . . . . . . . . . . . . . . . . . . 636Appendix 7B Equal-Time Commutator from Time-Ordered Product . . . 637Appendix 7C Euler-Maclaurin formula . . . . . . . . . . . . . . . . . . . . 639Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643

8 Continuous Symmetries and Conservation Laws; Noethers Theo-rem 6458.1 Point Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645

8.1.1 Continuous Symmetries and Conservation Law . . . . . . . 6458.1.2 Alternative Derivation . . . . . . . . . . . . . . . . . . . . . 647

8.2 Displacement and Energy Conservation . . . . . . . . . . . . . . . . 6488.3 Momentum and Angular Momentum . . . . . . . . . . . . . . . . . . 650

8.3.1 Translational Invariance in Space . . . . . . . . . . . . . . . 6508.3.2 Rotational Invariance . . . . . . . . . . . . . . . . . . . . . 6518.3.3 Center-of-Mass Theorem . . . . . . . . . . . . . . . . . . . . 6528.3.4 Conservation Laws resulting from Lorentz Invariance . . . . 654

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8.4 Generating the Symmetries . . . . . . . . . . . . . . . . . . . . . . . 6568.5 Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658

8.5.1 Continuous Symmetry and Conserved Currents . . . . . . . 6588.5.2 Alternative Derivation . . . . . . . . . . . . . . . . . . . . . 6598.5.3 Local Symmetries . . . . . . . . . . . . . . . . . . . . . . . 660

8.6 Canonical Energy Momentum Tensor . . . . . . . . . . . . . . . . . 6628.6.1 Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . 6638.6.2 Dirac Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 664

8.7 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . 6668.8 Four-Dimensional Angular Momentum . . . . . . . . . . . . . . . . . 6678.9 Spin Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669

8.9.1 Electromagnetic Fields . . . . . . . . . . . . . . . . . . . . . 6698.9.2 Dirac Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 672

8.10 Symmetric Energy-Momentum Tensor . . . . . . . . . . . . . . . . . 6748.10.1 Gravitational Field . . . . . . . . . . . . . . . . . . . . . . . 676

8.11 Internal Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . 6778.11.1 U(1)-Symmetry and Charge Conservation . . . . . . . . . . 6778.11.2 SU(N)-Symmetry . . . . . . . . . . . . . . . . . . . . . . . 6788.11.3 Broken Internal Symmetries . . . . . . . . . . . . . . . . . . 678

8.12 Generating the Symmetry Transformations on Quantum Fields . . . 6788.13 Energy Momentum Tensor of a Relativistic Massive Point Particle . 6808.14 Energy Momentum Tensor of Massive Charged Particles in an Elec-

tromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684

9 Scattering and Decay of Particles 6869.1 Quantum-Mechanical Description . . . . . . . . . . . . . . . . . . . 686

9.1.1 Schrodinger Picture . . . . . . . . . . . . . . . . . . . . . . 6869.1.2 Heisenberg Picture . . . . . . . . . . . . . . . . . . . . . . . 6879.1.3 Interaction Picture . . . . . . . . . . . . . . . . . . . . . . . 6889.1.4 Neumann-Liouville expansion . . . . . . . . . . . . . . . . . 6889.1.5 Mller Operators . . . . . . . . . . . . . . . . . . . . . . . . 6909.1.6 Lippmann-Schwinger equation . . . . . . . . . . . . . . . . . 6939.1.7 Discrete States . . . . . . . . . . . . . . . . . . . . . . . . . 6959.1.8 Gell-Mann --Low Formulas . . . . . . . . . . . . . . . . . . . 696

9.2 Scattering in External Potential . . . . . . . . . . . . . . . . . . . . 7019.2.1 The T -Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 7019.2.2 Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . 7059.2.3 Partial Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 7079.2.4 Off Shell T -Matrix . . . . . . . . . . . . . . . . . . . . . . . 7139.2.5 Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . 7169.2.6 Partial Wave Decomposition of Cross Section . . . . . . . . 7209.2.7 Dirac -Function Potential . . . . . . . . . . . . . . . . . . . 7209.2.8 Spherical Square-Well Potential . . . . . . . . . . . . . . . . 723

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9.3 Two-Particle Scattering . . . . . . . . . . . . . . . . . . . . . . . . . 725

9.3.1 Center-of-Mass Scattering Cross Section . . . . . . . . . . . 727

9.3.2 Laboratory Scattering Cross Section . . . . . . . . . . . . . 7289.4 Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732

9.5 Optical Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732

9.6 Initial- and Final-State Interactions . . . . . . . . . . . . . . . . . . 733

9.7 Tests of Time-Reversal Violations . . . . . . . . . . . . . . . . . . . 734

9.7.1 Strong and Electromagnetic Interactions . . . . . . . . . . . 735

9.7.2 Selection Rules in Weak Interactions . . . . . . . . . . . . . 736

9.7.3 Phase of Weak Amplitudes from Time-Reversal Invariance 737

Appendix 9A Green Function in Arbitrary Dimensions . . . . . . . . . . . 738Appendix 9B Partial Waves in Arbitrary Dimensions . . . . . . . . . . . . 740

Appendix 9C Spherical Square-Well Potential in D Dimensions . . . . . . 745


10 Quantum Field Theoretic Perturbation Theory 748

10.1 The Interacting n-Point Function . . . . . . . . . . . . . . . . . . . 74810.2 Perturbation Expansion for Green Functions . . . . . . . . . . . . . 751

10.3 Feynman Rules for 4-theory . . . . . . . . . . . . . . . . . . . . . . 752

10.3.1 The Vacuum Graphs . . . . . . . . . . . . . . . . . . . . . . 754

10.4 The Two-Point Function . . . . . . . . . . . . . . . . . . . . . . . . 757

10.5 The Four-Point Function . . . . . . . . . . . . . . . . . . . . . . . . 759

10.6 Connected Green Function . . . . . . . . . . . . . . . . . . . . . . . 761

10.6.1 One-Particle Irreducible Graphs . . . . . . . . . . . . . . . . 765

10.6.2 Momentum Space Version of Diagrams . . . . . . . . . . . . 76710.7 Green Functions and Scattering Amplitudes . . . . . . . . . . . . . . 769

10.8 Wick Rules for Scattering Amplitudes . . . . . . . . . . . . . . . . . 776

10.9 Thermal Perturbation Theory . . . . . . . . . . . . . . . . . . . . . 777


11 Extracting Finite Results from Perturbation Series. Regularization,Renormalization 781

11.1 Vacuum Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781

11.2 The Two- and Four-Point Functions . . . . . . . . . . . . . . . . . . 78411.3 Divergences, Cutoff, and Counterterms . . . . . . . . . . . . . . . . 786

11.4 Bare Theory and Multiplicative Renormalization . . . . . . . . . . . 793

11.5 Dimensional Regularization of Integrals . . . . . . . . . . . . . . . . 797

11.6 Renormalization of Amplitudes . . . . . . . . . . . . . . . . . . . . . 811

11.7 Additive Renormalization of Vacuum Energy . . . . . . . . . . . . . 814

11.8 Generalization to O(N)-Symmetric Model . . . . . . . . . . . . . . . 814

11.9 Finite S-Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . . 820

Appendix 11AAlternative Proof of Veltman Integral Rule . . . . . . . . . . 822


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12 Quantum Electrodynamics 825

12.1 Gauge Invariant Interacting Theory . . . . . . . . . . . . . . . . . . 825

12.1.1 Reminder of Classical Electrodynamics of Point Particles . . 826

12.1.2 Electrodynamics and Quantum Mechanics . . . . . . . . . . 828

12.1.3 Principle of Nonholonomic Gauge Invariance . . . . . . . . . 830

12.1.4 Electrodynamics and Relativistic Quantum Mechanics . . . 831

12.2 Noethers Theorem and Gauge Fields . . . . . . . . . . . . . . . . . 832

12.3 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834

12.4 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 837

12.5 Ward-Takahashi identity . . . . . . . . . . . . . . . . . . . . . . . . 842

12.6 Magnetic Moment of Electron . . . . . . . . . . . . . . . . . . . . . 843

12.7 Decay of Atomic State . . . . . . . . . . . . . . . . . . . . . . . . . 847

12.8 Rutherford Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 851

12.8.1 Classical Cross Section . . . . . . . . . . . . . . . . . . . . . 851

12.8.2 Quantum-Mechanical Born Approximation . . . . . . . . . . 853

12.8.3 Relativistic Born Approximation . . . . . . . . . . . . . . . 853

12.9 Compton Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 857

12.9.1 Classical Result . . . . . . . . . . . . . . . . . . . . . . . . . 857

12.9.2 Klein-Nishina Formula . . . . . . . . . . . . . . . . . . . . . 859

12.10 Electron-Positron Annihilation . . . . . . . . . . . . . . . . . . . . . 863

12.11 Positronium Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 869

12.12 Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 870

12.12.1 Classical Bremsstrahlung . . . . . . . . . . . . . . . . . . . 870

12.12.2 Bremsstrahlung in Mott Scattering . . . . . . . . . . . . . . 873

12.13 Electron-Electron Scattering . . . . . . . . . . . . . . . . . . . . . . 876

12.14 Electron-Positron Scattering . . . . . . . . . . . . . . . . . . . . . . 878

12.15 Anomalous Magnetic Moment of Electron and Muon . . . . . . . . 881

12.15.1 Form Factors . . . . . . . . . . . . . . . . . . . . . . . . . . 886

12.15.2 Charge Radius . . . . . . . . . . . . . . . . . . . . . . . . . 887

12.15.3 Anomalous Magnetic Moment . . . . . . . . . . . . . . . . . 888

12.16 Vacuum Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . 892

12.17 Dimensional Regularization . . . . . . . . . . . . . . . . . . . . . . . 896

12.18 Two-Dimensional QED . . . . . . . . . . . . . . . . . . . . . . . . . 897

12.19 Self-Energy of Electron . . . . . . . . . . . . . . . . . . . . . . . . . 898

12.20 Ward-Takahashi identity . . . . . . . . . . . . . . . . . . . . . . . . 901

12.21 Lamb Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903

12.21.1 Rough Estimate of Effect of Vacuum Fluctuations . . . . . . 903

12.21.2 Relativistic Estimate . . . . . . . . . . . . . . . . . . . . . . 906

12.21.3 Effect of Wave Functions . . . . . . . . . . . . . . . . . . . 906

12.21.4 Effect of the Anomalous Magnetic Moment . . . . . . . . . 915

Appendix 12ACalculation of Dirac Trace for Klein-Nishina formula . . . . 917


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13 Functional Integral Representation of Quantum Field Theory 92313.1 Functional Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 92313.2 Gaussian Functional Integral . . . . . . . . . . . . . . . . . . . . . . 92413.3 Functional Formulation for Free Quantum Fields . . . . . . . . . . . 92613.4 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92913.5 Euclidean Quantum Field Theory . . . . . . . . . . . . . . . . . . . 93213.6 Functional Integral Representation for Fermions . . . . . . . . . . . 93313.7 Relation Between Z[j] and Partition Function . . . . . . . . . . . . 93713.8 Bosons and Fermions in a Single State . . . . . . . . . . . . . . . . . 94213.9 Free Energy for Free Scalar Fields . . . . . . . . . . . . . . . . . . . 94413.10 Interacting Nonrelativistic Fields . . . . . . . . . . . . . . . . . . . . 945

13.10.1 Functional Formulation . . . . . . . . . . . . . . . . . . . . 94713.10.2 Grand-Canonical Ensembles at Zero Temperature . . . . . . 948

13.11 Interacting Relativistic Fields . . . . . . . . . . . . . . . . . . . . . . 95413.12 Plasma Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 956

13.12.1 General Formalism . . . . . . . . . . . . . . . . . . . . . . . 95613.12.2 Physical Consequences . . . . . . . . . . . . . . . . . . . . . 960

13.13 Gauge Fields and Gauge Fixing . . . . . . . . . . . . . . . . . . . . 96313.14 Nontrivial Gauge and Fadeev-Popov Ghosts . . . . . . . . . . . . . . 97013.15 Functional Formulation of Quantum Electrodynamics . . . . . . . . 973

13.15.1 Decay Rate of Dirac Vacuum in Constant Electric Field . . 97413.15.2 Constant Electric and Magnetic Background Fields . . . . . 97813.15.3 Decay Rate in Constant Electromagnetic Field . . . . . . . 98113.15.4 Effective Action in Purely Magnetic Field . . . . . . . . . . 98213.15.5 Heisenberg-Euler Lagrangian . . . . . . . . . . . . . . . . . 98313.15.6 Alternative Derivation for Constant Magnetic Field . . . . . 986

Appendix 13APropagator of the Bilocal Pair Field . . . . . . . . . . . . . . 990Appendix 13BFluctuations around the Composite Field . . . . . . . . . . . 992Appendix 13CTwo-Loop Heisenberg-Euler Effective Action . . . . . . . . . 994Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995

14 Formal Properties of Perturbation Theory 100414.1 Connectedness Structure of Feynman Diagrams . . . . . . . . . . . . 100414.2 Functional Differential Equations . . . . . . . . . . . . . . . . . . . . 100514.3 Decomposition of Green Functions into Connected Green Functions 100714.4 Functional Differential Equation for W [j[ . . . . . . . . . . . . . . . 100914.5 Iterative Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100914.6 Vertex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101114.7 The Generating Functional for Vertex Functions . . . . . . . . . . . 101114.8 Functional Differential Equation for [] . . . . . . . . . . . . . . . 101714.9 Effective Action as a Basis for Variational Calculations . . . . . . . 102014.10 Effective Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102014.11 Higher Effective Actions . . . . . . . . . . . . . . . . . . . . . . . . . 102114.12 High Orders in Simple Model . . . . . . . . . . . . . . . . . . . . . . 1026

xviii


15 Path Integral Calculation of Effective Action. Loop Expansion 103015.1 General Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103015.2 Quadratic Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . 103315.3 Massless Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103815.4 Effective Action to Second Order in h . . . . . . . . . . . . . . . . . 104215.5 Effective Action to all Orders in h . . . . . . . . . . . . . . . . . . . 1046Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1048

16 Systematic Graphical Construction of Feynman Diagrams . . . 105016.1 Generalized Scalar 4-Theory . . . . . . . . . . . . . . . . . . . . . . 105116.2 Basic Graphical Operations . . . . . . . . . . . . . . . . . . . . . . . 1053

16.2.1 Cutting Lines . . . . . . . . . . . . . . . . . . . . . . . . . . 105316.2.2 Removing Lines . . . . . . . . . . . . . . . . . . . . . . . . . 1056

16.3 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 105716.4 Functional Differential Equation for W = lnZ . . . . . . . . . . . . 105816.5 Recursion Relation and Graphical Solution . . . . . . . . . . . . . . 106016.6 Scalar 2A-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106216.7 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 106416.8 Functional Differential Equation for W = lnZ . . . . . . . . . . . . 106416.9 Recursion Relation and Graphical Solution . . . . . . . . . . . . . . 106516.10 Computer Generation of Diagrams . . . . . . . . . . . . . . . . . . . 106716.11 Matrix Representation of Diagrams . . . . . . . . . . . . . . . . . . 106716.12 Practical Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069

16.12.1 Connected Vacuum Diagrams . . . . . . . . . . . . . . . . . 106916.12.2 Two- and Four-Point Functions from Cutting Lines . . . . . 107216.12.3 Two- and Four-Point Function from Removing Lines . . . . 1073

17 Spontaneous Symmetry Breakdow 108417.1 Scalar O(N)-Symmetric 4-Theory . . . . . . . . . . . . . . . . . . . 108417.2 Nambu-Goldstone Particles . . . . . . . . . . . . . . . . . . . . . . . 1090

17.2.1 The Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 109017.2.2 General Considerations . . . . . . . . . . . . . . . . . . . . 109217.2.3 Experimental Consequences . . . . . . . . . . . . . . . . . . 1094

17.3 Domain Walls in O(1)-Symmetric Theory . . . . . . . . . . . . . . . 109517.4 Vortex Lines in O(2)-Symmetric Theory . . . . . . . . . . . . . . . . 1100Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106

18 Scalar Quantum Electrodynamics 110718.1 Action and Generating Functional . . . . . . . . . . . . . . . . . . . 110718.2 Meissner-Higgs Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 111018.3 Spatially Varying Ground States . . . . . . . . . . . . . . . . . . . . 111618.4 Two Natural Length Scales . . . . . . . . . . . . . . . . . . . . . . . 1117

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18.5 Planar Domain Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . 111918.6 Surface Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112518.7 Single Vortex Line and Critical Field Hc1 . . . . . . . . . . . . . . . 112618.8 Critical Field Hc2 where Superconductivity is Destroyed . . . . . . . 113218.9 Order of the Superconducting Phase Transition . . . . . . . . . . . . 1135Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136

19 Exactly Solvable O(N)-Symmetric 4-Theory for Large N 113719.1 Introduction of a Collective Field . . . . . . . . . . . . . . . . . . . . 113719.2 The Limit of Large N . . . . . . . . . . . . . . . . . . . . . . . . . . 114019.3 Variational Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 1147

19.3.1 Non-trivial Ground States . . . . . . . . . . . . . . . . . . . 114819.4 Special Features of Two Dimensions . . . . . . . . . . . . . . . . . . 115219.5 Experimental Consequences . . . . . . . . . . . . . . . . . . . . . . . 115419.6 Correlation Functions for Large N . . . . . . . . . . . . . . . . . . . 1157Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1160

20 Non-Linear -Model 116120.1 Definition of Classical Heisenberg Model . . . . . . . . . . . . . . . . 116120.2 Spherical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116420.3 Free Energy and Gap Equation in D > 2 Dimensions . . . . . . . . . 1165

20.3.1 High-Temperature Phase . . . . . . . . . . . . . . . . . . . 116720.3.2 Low-Temperature Phase . . . . . . . . . . . . . . . . . . . . 1168

20.4 Approaching the Critical Point . . . . . . . . . . . . . . . . . . . . . 117020.5 Physical Property of Bare Temperature . . . . . . . . . . . . . . . . 117120.6 Spherical Model on Lattice . . . . . . . . . . . . . . . . . . . . . . . 117320.7 Background Field Treatment of Cold Phase . . . . . . . . . . . . . . 117720.8 Quantum Statistics at Nonzero Temperature Nonlinear -Model in

D Spacetime Dimensions . . . . . . . . . . . . . . . . . . . . . . . . 117920.8.1 Two-Dimensional Model . . . . . . . . . . . . . . . . . . . . 117920.8.2 Four-Dimensional Model . . . . . . . . . . . . . . . . . . . . 118420.8.3 Any Dimension . . . . . . . . . . . . . . . . . . . . . . . . . 1185

20.9 Criteria for Onset of Fluctuations in Ginzburg-Landau Theories . . . 119020.9.1 Ginzburgs Criterion . . . . . . . . . . . . . . . . . . . . . . 119120.9.2 Kleinerts Criterion . . . . . . . . . . . . . . . . . . . . . . . 119220.9.3 Experimental Consequences . . . . . . . . . . . . . . . . . . 1194

21 The Renormalization Group 119721.1 Example for Redundancy in Parametrization of Renormalized Theory119821.2 Renormalization Scheme . . . . . . . . . . . . . . . . . . . . . . . . 120021.3 The Renormalization Group Equation . . . . . . . . . . . . . . . . . 120221.4 Calculation of Coefficient Functions from Counter Terms . . . . . . 120321.5 Solution of Renormalization Group Equations for Vertex Functions . 120721.6 Renormalization Group for Effective Action and Potential . . . . . . 1211

xx

21.7 Approach to Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . 1213

21.8 Explicit Solution of the RGE close to D = 4 Dimensions . . . . . . . 1215

21.9 Further Critical Relations . . . . . . . . . . . . . . . . . . . . . . . . 1219

21.9.1 Scaling Relations Above Tc . . . . . . . . . . . . . . . . . . 1219

21.9.2 Scaling Relations Below Tc . . . . . . . . . . . . . . . . . . 1221

21.10 Comparison of Scaling Relations with Experiment . . . . . . . . . . 1223

21.11 Higher-Order Expansion . . . . . . . . . . . . . . . . . . . . . . . . 1225

21.12 Mean-Field Results for Critical Indices . . . . . . . . . . . . . . . . 1226

21.13 Effective Potential in the Critical Regime to Order . . . . . . . . . 1229

21.14 O(N)-Symmetric Theory . . . . . . . . . . . . . . . . . . . . . . . . 1234

21.15 Direct Scaling Form in the Limit of Large N . . . . . . . . . . 1236

22 Critical Properties of Non-linear -Model 1238

22.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 1238

22.2 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 1240

22.3 Symmetry Properties of Renormalized Effective Action . . . . . . . 1245

22.4 Critical Behavior in 2 + Dimensions . . . . . . . . . . . . . . . . . 1248

22.5 Critical Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1249

22.6 Two- and Three-Loop Results . . . . . . . . . . . . . . . . . . . . . 1256

22.7 Variational Resummation of -Expansions . . . . . . . . . . . . . . . 1259

22.7.1 Strong-Coupling Theory . . . . . . . . . . . . . . . . . . . . 1261

22.7.2 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . 1263

22.8 Relation of -Model to Quantum Mechanics of a Point Particle ona Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1268

22.9 Generalization of the Model . . . . . . . . . . . . . . . . . . . . . . . 1274


23 Exactly Solvable O(N)-Symmetric Four-Fermion Theory in 2 + Dimensions 1280

23.1 Scalar Self-Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 1280

23.2 Spontaneous Symmetry Breakdown . . . . . . . . . . . . . . . . . . 1285

23.3 Dimensionally Transmuted Coupling Constant . . . . . . . . . . . . 1286

23.4 Scattering Amplitude for Fermions . . . . . . . . . . . . . . . . . . . 1287

23.5 Nonzero Bare Fermion Mass . . . . . . . . . . . . . . . . . . . . . . 1294

23.6 Pairing Model and Dynamically Generated Goldstone Bosons . . . . 1295

23.7 Spontaneously Broken Symmetry . . . . . . . . . . . . . . . . . . . . 1302

23.8 Relation between Pairing and Gross-Neveu Model . . . . . . . . . . 1306

23.9 Comparison with O(N)-Symmetric 4-Theory . . . . . . . . . . . . . 1307

23.10 Two Phase Transitions in Chiral Gross-Neveu Model . . . . . . . . . 1312

23.11 Finite-Temperature Properties . . . . . . . . . . . . . . . . . . . . . 1315


xxi

24 Internal Symmetries of Strong Interactions 132824.1 Classification of Elementary Particles . . . . . . . . . . . . . . . . . 132824.2 Isospin in Nuclear Physics . . . . . . . . . . . . . . . . . . . . . . . 133124.3 Isospin in Pion Physics . . . . . . . . . . . . . . . . . . . . . . . . . 133624.4 SU(3)-Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133924.5 New Quarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135824.6 Tensor Representations and Young Tableaux . . . . . . . . . . . . . 135924.7 Effective Interactions among Hadrons . . . . . . . . . . . . . . . . . 1364

24.7.1 The Pion-Nucleon Interaction . . . . . . . . . . . . . . . . . 136424.7.2 The Decay (1232) N . . . . . . . . . . . . . . . . . . 136724.7.3 Vector Meson Decay (770) . . . . . . . . . . . . . . 137224.7.4 Vector Meson Decays (783) and (783) . . . 137324.7.5 Vector Meson Decays K(892) K . . . . . . . . . . . . . 137324.7.6 Axial Vector Meson Decay a1(1270) . . . . . . . . . . 137424.7.7 Coupling of (770)-Meson to Nucleons . . . . . . . . . . . . 1375

Appendix 24AUseful SU(3) Formulas . . . . . . . . . . . . . . . . . . . . . 1376Appendix 24BDecay rate for a1 . . . . . . . . . . . . . . . . . . . . . 1377Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1379

25 Symmetries Linking Internal and Spacetime Structure 138025.1 Approximate SU(4)-Symmetry of Nuclear Forces . . . . . . . . . . . 138025.2 Approximate SU(6)-Symmetry in Strong Interactions . . . . . . . . 138825.3 From SU(6) to Current Algebra . . . . . . . . . . . . . . . . . . . . 139625.4 Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1401

26 Weak Interactions 140426.0.1 Fermi Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 140426.0.2 Lepton Conservation . . . . . . . . . . . . . . . . . . . . . . 140826.0.3 Cabibbo Angle . . . . . . . . . . . . . . . . . . . . . . . . . 140826.0.4 Cabibbo Mass Matrix . . . . . . . . . . . . . . . . . . . . . 141026.0.5 Heavy Vector Bosons . . . . . . . . . . . . . . . . . . . . . . 1410

26.1 Standard Model of Electroweak Interactions . . . . . . . . . . . . . . 141226.2 Quantum Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . 1417

26.2.1 Kaon Oscillations . . . . . . . . . . . . . . . . . . . . . . . . 141726.2.2 General Flavor Mixing . . . . . . . . . . . . . . . . . . . . . 1420


27 Nonabelian Gauge Theory of Strong Interactions 142227.0.3 Local Color Symmetry . . . . . . . . . . . . . . . . . . . . . 142227.0.4 Gluon Action . . . . . . . . . . . . . . . . . . . . . . . . . . 1424

27.1 Quantization in the Coulomb gauge . . . . . . . . . . . . . . . . . . 142527.2 Direct Functional Quantization of Gauge Fields . . . . . . . . . . . . 143227.3 Equivalence of Landau and Coulomb gauges . . . . . . . . . . . . . 1438

27.3.1 Approximate Chiral Symmetry . . . . . . . . . . . . . . . . 1441

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Index 1443

List of Figures

1.1 Probability distribution of particle behind a double slit . . . . . . . 121.2 Relevant function

Nn=N e

2in in Poissons summation formula . . 301.3 Illustration of time-ordering procedure . . . . . . . . . . . . . . . . . 371.4 Triangular closed contour for Cauchy integral . . . . . . . . . . . . . 78

2.1 Average Bose occupation number . . . . . . . . . . . . . . . . . . . 1222.2 Average Fermi occupation number . . . . . . . . . . . . . . . . . . . 1232.3 Temperature behavior of specific heat of a free Fermi gas . . . . . . 1312.4 Temperature behavior of the chemical potential of a free Bose gas . 1322.5 Temperature behavior of fraction of degenerate bosons in the zero-

momentum state of a free Bose gas . . . . . . . . . . . . . . . . . . . 1332.6 Temperature behavior of the specific heat of a free boson gas. For

comparison we show the specific heat of the strongly interacting Boseliquid 4He, scaled down by a factor 2 to match the Dulong-Petit limitof the free Bose gas . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

2.7 Rotating trap potential for 2 > 0 and 2 < 0 . . . . . . . . . . . . 1422.8 Contour C in the complex z-plane . . . . . . . . . . . . . . . . . . . 1472.9 Finite-size corrections to critical temperature for N = 300 to infinity 172

3.1 Plot of the quasiparticle energies as a function of the momenta inan interacting Bose gas . . . . . . . . . . . . . . . . . . . . . . . . . 186

3.2 Common volume of two spheres at a distance q in momentum space 1923.3 Energy density in units Ry of electron gas in uniform background of

positive charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1943.4 Historical evolution of critical temperatures of onset of superconduc-

tivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1963.5 Approximate energy of a free electron near the Fermi surface in a

grand canonical ensemble . . . . . . . . . . . . . . . . . . . . . . . . 1983.6 Gap in the energy spectrum caused by attraction of pairs of electrons

with opposite spin and momenta . . . . . . . . . . . . . . . . . . . 1993.7 Spectrum of quasiparticles and holes . . . . . . . . . . . . . . . . . 2003.8 Gap in energy spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 2003.9 Solution of gap equation for weak attraction between electrons . . . 2103.10 Plot of the gap function and of the chemical potential against the

inverse s-wave scattering length . . . . . . . . . . . . . . . . . . . . 2113.11 Temperature dependence of the normal fraction u/ in Bose gas . . 214

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xxiv

3.12 Reduced gap s /a as a function of the reduced s-wave scatteringlength as = 8as/a = 8as

1/3 . . . . . . . . . . . . . . . . . . . . . 2223.13 Reduced energy per particle we1 = W1/Na as a function of the

reduced s-wave scattering length, compared with Bogoliubovs weak-coupling result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

3.14 Temperature dependence of the normal particle density . . . . . . . 2303.15 Diagrams picturing the Wick contractions . . . . . . . . . . . . . . 232

4.1 Six leptons and quarks . . . . . . . . . . . . . . . . . . . . . . . . . 3134.2 Asymmetry observed in the distribution of electrons from the -

decay of polarized 6027Co . . . . . . . . . . . . . . . . . . . . . . . . . 3144.3 of raising and lowering operators L+ and L upon the states |s,m 3544.4 Triangle formed by rapidities in a hyperbolic space. The sum of

angles is smaller than 1800. The angular defect yields the angle ofThomas precession . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376

5.1 Two equal masses M oscillating at the ends of a spring as a sourceof gravitational radiation . . . . . . . . . . . . . . . . . . . . . . . . 399

5.2 Two spherical masses in circular orbits about their center of mass. . 4015.3 Gravitational Amplitudes arriving at the earth from possible sources 4035.4 Two pulsars orbiting around each other . . . . . . . . . . . . . . . . 4045.5 Shift of time of periastron passage of PSR 1913+16 . . . . . . . . . 4055.6 Two masses in a Keplerian orbit around the common center-of-mass 4055.7 Spectrum of the gravitational radiation emitted by a particle of mass

m falling radially into a black hole of mass M . . . . . . . . . . . . . 4085.8 Particle falling radially toward a mass. . . . . . . . . . . . . . . . . . 4085.9 Distortions of a circular array of mass points by the passage of a

gravitational quandrupole wave . . . . . . . . . . . . . . . . . . . . 4145.10 Field lines of tidal forces of a gravitational wave . . . . . . . . . . . 415

6.1 Hydrogen spectrum according to Diracs theory. The fine-structuresplitting of the 2P -levels is about 10 times as big as the hyperfinesplitting and Lamb shift . . . . . . . . . . . . . . . . . . . . . . . . 488

7.1 Pole positions in the complex p0-plane in the integral representationsof Feynman propagators . . . . . . . . . . . . . . . . . . . . . . . . . 524

7.2 Wick rotation of the contour of integration in the complex p0-plane 5247.3 Integration contours in the complex p0-plane of the Fourier integral

for the various propagators . . . . . . . . . . . . . . . . . . . . . . . 5367.4 Different coupling schemes for two-particle states of total angular

momentum j and helicity m. . . . . . . . . . . . . . . . . . . . . . 5477.5 Geometry of the silver plates for the calculation of the Casimir effect 605

9.1 Behavior of wave function for different positions of a bound statenear the continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . 721

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9.2 Behavior of binding energy and scattering length in attractivesquare-well potential . . . . . . . . . . . . . . . . . . . . . . . . . . 724

9.3 Geometry of particle beams in a collider . . . . . . . . . . . . . . . . . 731

11.1 Singularities in complex q0-plane of Feynman propagator. There arepoles and cuts due to three- and more-particle intermediate statesin the diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 821

12.1 An electron on the mass shell absorbing several photons and leavingagain on the mass shell . . . . . . . . . . . . . . . . . . . . . . . . . 839

12.2 An electron on the mass shell absorbing several photons, plus anadditional photon, and leaving again on the mass shell . . . . . . . . 840

12.3 An internal electron loop absorbing several photons, plus an addi-tional photon, and leaving again on the mass shell . . . . . . . . . . 841

12.4 Transition of an atomic state from a state n with energy En to alower state n with energy En, thereby emitting a photon with afrequency = (En En)/h . . . . . . . . . . . . . . . . . . . . . . 847

12.5 Kinematics of Rutherford cross section . . . . . . . . . . . . . . . . . 852

12.6 Lowest-order Feynman Diagrams contributing to Compton Scatter-ing and giving rise to the Klein-Nishina formula . . . . . . . . . . . 857

12.7 Illustration of the photon polarization sum in Compton scattering . 861

12.8 Ratio between total relativistic Compton cross section and nonrela-tivistic Thomson cross section . . . . . . . . . . . . . . . . . . . . . 864

12.9 Lowest-Order Feynman Diagrams contributing to Electron-PositronAnnihilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864

12.10 Illustration of the photon polarization sum in electron-positron an-nihilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 867

12.11 Electron-positron annihilation cross section . . . . . . . . . . . . . . 867

12.12 Lowest-Order Feynman Diagrams contributing to decay of para-positronium decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 869

12.13 Lowest-Order Feynman Diagrams contributing to decay of or-thopositronium decay . . . . . . . . . . . . . . . . . . . . . . . . . . 870

12.14 Trajectories in the simplest classical Bremsstrahlung process: Anelectron changing abruptly its momentum . . . . . . . . . . . . . . 871

12.15 Lowest-Order Feynman diagrams contributing to Bremsstrahlung.The vertical photon line indicates the nuclear Coulomb potential. . 874

12.16 The angles , , in the Bethe-Heitler cross section formula . . . . 875

12.17 Lowest-Order Feynman Diagrams contributing to Electron-ElectronScattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876

12.18 Kinematics of electron-electron scattering in the center of mass frame877

12.19 General form of the diagrams contributing to electron-positron scat-tering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 878

12.20 Lowest-order contributions to electron-positron scattering . . . . . . 879

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12.21 Experimental data for electron-electron and electron-positron scat-tering at = 900 as a function of the incident electron energy . . . 880

12.22 Cross section for Bhabha scattering at high energy . . . . . . . . . . 88112.23 The vertex correction responsible for the anomalous magnetic moment88212.24 Leading hadronic vacuum polarization corrections to a. . . . . . . . 89012.25 One-loop electroweak radiative corrections to a. The wiggly lines

are gluons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89112.26 Measured values of a and prediction of the standard model (SM) . 89212.27 Lowest-order Feynman diagram for the vacuum polarization . . . . . 89212.28 Lowest-order Feynman diagram for the vacuum polarization . . . . . 89812.29 Diagrammatic content in the calculation of the energy shift with the

help of Schrodinger wave functions. A hydrogen atom is representedby the fat line on the left which results from an infinite sum of photonexchanges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 910

13.1 The pure current piece of the collective action . . . . . . . . . . . . 95813.2 The non-polynomial self-interaction terms of the plasmons . . . . . 95913.3 Free plasmon propagator . . . . . . . . . . . . . . . . . . . . . . . . 959

14.1 Graphical solution of recursion relation (14.30) for the generatingfunctional W133j135 of all connected Green functions . . . . . . . . 1010

14.2 Tree decompositon of connected Green functions into one-particleirreducible perts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015

14.3 Graphical solution of functional differential equation (14.64) for theinteracting part int133135 of the effective action . . . . . . . . . . 1018

14.4 Recursion relation for two-paticle-irreducible graphs in the effectiveaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024

14.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102714.6 Approximations to F obtained from the extrema of the higher ef-

fective action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1028

17.1 Effective potential of the 4-theory for N = 2 in mean-field approx-mation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086

17.2 Magnetization 0 in mean-field approximation as a function of thetemperature ratio T/TMFc in mean-field approximation . . . . . . . . 1088

17.3 Magnetization 0 as a function of external source j in mean-fieldapproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1090

17.4 Plot of the symmetric double-well potential . . . . . . . . . . . . . 109617.5 Classical kink solution in double-well potential connecting the two

degenerate maxima in the reversed potential . . . . . . . . . . . . . 109717.6 Reversed double-well potential governing the motion of the position

as a function of the imaginary time x . . . . . . . . . . . . . . . . 109817.7 The order parameter = ||/|0| around a vortex line of strength

n = 1, 2, 3, . . . as a function of the reduced distance r = r/ where ris the distance from the axis and the healing length. . . . . . . . . 1102

xxvii

18.1 Dependence of order parameter and magnetic field H on the re-duced distance z for a planar domain wall between the normal andsuperconductive phases . . . . . . . . . . . . . . . . . . . . . . . . . 1122

18.2 Order parameter , and the magnetic field h for an n = 1 vortexline in a deep type-II superconductor with K = 10. . . . . . . . . . . 1131

18.3 Critical field hc1 at which a vortex line of strength n forms when itfirst invades a type-II superconductor, as a function of the parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1131

18.4 Spatial distribution magnetization of the order parameter (x) ina typical mixed state in which the vortex lines form a hexagonallattice (from W.M. Kleiner et al., See References.) . . . . . . . . . . 1132

20.1 Free energy as a function of for D = 2. The gap equation deter-mines the position of the maximum. . . . . . . . . . . . . . . . . . 1166

20.2 Free energy as a function of for D > 2. The gap equation deter-mines the position of the maximum. . . . . . . . . . . . . . . . . . 1167

20.3 Solution of the gap equation (20.48) for = 1 and large volume LD 1170

20.4 Temperature behavior of the correlation length . . . . . . . . . . . 1171

21.1 Curves in the (, g)-plane corresponding to the same physicalfermion mass Mf = , . . . , 5. These curves are the renormaliza-tion group trajectories of the O(N)-symmetric four-fermion modelin the limit N . . . . . . . . . . . . . . . . . . . . . . . . . . . 1200

21.2 Flow of the coupling constant g() as the scale parameter ap-proaches zero (infrared limit) . . . . . . . . . . . . . . . . . . . . . . 1216

22.1 Two-loop diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . 1257

22.2 Three-loop diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . 1257

22.3 Integrands of the Pade-Borel transform for the Pade approximants . 1267

22.4 The inverse of the critical exponent for the classical Heisenbergmodel in the O(3)-universality class is plotted as a function of = 4D1268

22.5 The inverse of the critical exponent for the O(3)-universality classis plotted as a function of = 4D . . . . . . . . . . . . . . . . . . 1269

22.6 The inverse of the critical exponent for the O(5)-universality classis plotted as a function of = 4D . . . . . . . . . . . . . . . . . . 1269

22.7 The inverse of the critical exponent for the O(1)-universality class(of the Ising model) is plotted as a function of = 4D . . . . . . 1270

22.8 The three successive approximations for at n = 3 (Heisenbergmodel) plotted as a function of x = M2 . . . . . . . . . . . . . . . 1270

22.9 The three successive approximations for at n = 4 plotted as afunction of x = M2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1271

22.10 The three successive approximations for at n = 5 plotted as afunction of x = M1 . . . . . . . . . . . . . . . . . . . . . . . . . . 1271

xxviii

22.11 The three successive approximations(1, 2, 3, 4) = (0.862357, 0.665451, 2.08686, 0.802416) for n = 1plotted as a function of x = M2 . . . . . . . . . . . . . . . . . . . 1271

22.12 The highest approximations (M = 4) for n = 3, 4, 5, and the 1/n-expansions to order 1/n2 . . . . . . . . . . . . . . . . . . . . . . . . 1272

23.1 One-loop Feynman diagram in the inverse propagator of the -field 1288

23.2 The function J(z)+2 in the denominator pf the -propagator (23.60)1290

23.3 The two transition lines in the Ng-plane of the chiral Gross-Neveumodel in 2 + dimensions . . . . . . . . . . . . . . . . . . . . . . . . 1314

23.4 Solution of the temperature dependent gap equation, showing thedecrease of the fermion mass M(T ) = (T ) with increasing temper-ature T/Tc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1319

24.1 The total and elastic + -proton cross section showing . . . . . . . 1329

24.2 The total and elastic -proton cross section . . . . . . . . . . . . . 1330

24.3 The photon-proton and photon-deuteron total cross sections . . . . . 1332

24.4 Mirror nuclei 5B11 and 6C

11 with their excited states . . . . . . . . . 1333

24.5 Singlets and triplets of isospin in the nuclei 6C14, 7N

14, 8O14 . . . . . 1334

24.6 Pseudoscalar meson octet states associated with the pions. The samepicture holds for the vector meson octet states with the replacement(24.58) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1341

24.7 Baryon octet states associated with the nucleons. . . . . . . . . . . 1341

24.8 Baryon decuplet states associated with the first resonance of thenucleons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1342

24.9 Quark content of the pseudoscalar meson octet fields. The particleand quark symbols denote the annihilation parts of the correspond-ing fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1344

24.10 Effect of raising and lowering operators on quark and antiquark states1348

24.11 Adding the fundamental weights in product representation space of3 and 3 vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1349

24.12 The states of the 3-representation with de Swart phases. . . . . . . 1350

24.13 The quark-antiquark content of the meson octet meson states withde Swart phase convention. . . . . . . . . . . . . . . . . . . . . . . 1350

24.14 Combination of indices a in the pseudoscalar octet field M a . . . . 1351

24.15 The quark content in the reduction of the product 3 3 = 6 + 3. . 1353

24.16 The octet and singlet states obtained from 3 3 in the product spaceof tree-quarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1353

24.17 The irreducible three-quark states 10 and 8 in the product 36 (thesymbol (. . .)s denotes complete symmetrization). . . . . . . . . . . 1355

24.18 The four quarks u, d, s, c and their position in the three-dimensionalweight space with the new quantum number charm . . . . . . . . 1359

xxix

25.1 The would-be SU(4) partner of the deuteron, with spin-1 andisospin-0, hiding in the complex energy plane just below the two-particle cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1385

25.2 The pseudoscalar and vector mesons of the 35 representation of SU(6)138925.3 The SU(3) content of the particles in the 56 representation of SU(6)

containing the JP = 1/2+ nucleons and the JP = 3/2+ resonances. 139025.4 The nucleon resonance of negative parity in the 70 representation of

SU(6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1390

25.5 Octet of spin-parity 12

+baryons . . . . . . . . . . . . . . . . . . . . 1391

26.1 Quark diagrams for K+ and K0 decays involving strangeness chang-ing neutral currents . . . . . . . . . . . . . . . . . . . . . . . . . . . 1416

26.2 Diagrams for theK0 + decay with compensating strangeness-changing neutral currents . . . . . . . . . . . . . . . . . . . . . . . . 1416

26.3 Oscillation in decay rate into + of K0 beam . . . . . . . . . . . 141926.4 Asymmetry of the number of mesons as a function of time . . . . . . 1419

27.1 Propagators in the Yang-Mills theory . . . . . . . . . . . . . . . . . 143827.2 Vertices in the Yang-Mills theory . . . . . . . . . . . . . . . . . . . . 1438

List of Tables

4.1 The transformation properties of various composite fields . . . . . . 326

4.2 The lowest Clebsch-Gordan coefficients. . . . . . . . . . . . . . . . . . 379

5.1 Binary systems as sources of gravitational radiation . . . . . . . . . . 402

5.2 Some observed parameters of PSR 1913+16 . . . . . . . . . . . . . . 403

5.3 Typical Astrophysical Sources of Gravitational Radiation . . . . . . . 409

16.1 Connected vacuum diagrams and their multiplicities of the 4-theoryup to five loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075

16.2 Connected diagrams of the two-point function and their multiplicitiesof the 4-theory up to four loops . . . . . . . . . . . . . . . . . . . . 1076

16.3 Connected diagrams of the four-point function and their multiplicitiesof the 4-theory up to three loops . . . . . . . . . . . . . . . . . . . . 1077

16.4 Connected vacuum diagrams and their multiplicities of the 2A-theory up to four loops . . . . . . . . . . . . . . . . . . . . . . . . . . 1079

16.5 Unique matrix representation of all connected vacuum diagrams of4-theory up to the order p = 4. . . . . . . . . . . . . . . . . . . . . . 1080

16.6 Unique matrix representation of all connected two-point function of4-theory up to the order p = 4 . . . . . . . . . . . . . . . . . . . . . 1081

16.7 Unique matrix representation of all connected two-point function of4-theory up to the order p = 4 . . . . . . . . . . . . . . . . . . . . . 1082

18.1 The different critical magnetic fields in units of gauss for various im-purities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1131

20.1 Values of the lattice Yukawa potential vDl2 (0) of mass l2 at the origin

for different dimensions and l2 . . . . . . . . . . . . . . . . . . . . . . 1174

22.1 Equations determining the coefficients bn(g0) in the strong-couplingexpansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1266

22.2 Coefficients of the successive extension of the expansion coefficientsfor n = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1266



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24.1 Masses and lifetimes of the octet states associated with the nucleons . 134224.2 Structure Constants of SU(3) . . . . . . . . . . . . . . . . . . . . . . 134624.3 The symmetric couplings dabc . . . . . . . . . . . . . . . . . . . . . . 134724.4 Isoscalar Factors of SU(3) . . . . . . . . . . . . . . . . . . . . . . . . 1378

25.1 Action of the different interchange operators . . . . . . . . . . . . . . 138125.2 Action of spin and isospin operators in the expansion (25.7) . . . . . 138225.3 Eigenvalues of charge and other operators on quark states . . . . . . 1392

Any wide piece of ground is the potential site of a palace,

but theres no palace till its built.

Fernando Pessoa (1888-1935), The Book of Disquiet

1

Fundamentals

Before turning to the actual subject of this text it is useful to recall some basictheoretical background underlying the quantum field theory to be developed.1

1.1 Classical Mechanics

The orbits of a classical-mechanical system are described by a set of time-dependentgeneralized coordinates q1(t), . . . , qN(t). A Lagrangian

L(qi, qi, t) (1.1)

depending on q1, . . . , qN and the associated velocities q1, . . . , qN governs the dynam-ics of the system. The dots denote the time derivative d/dt. The Lagrangian is atmost a quadratic function of qi. The time integral

A[qi] = tb

tadt L(qi(t), qi(t), t) (1.2)

of the Lagrangian along an arbitrary path qi(t) is called the action of this path. Thepath being actually chosen by the system as a function of time is called the classicalpath or the classical orbit qcli (t). It has the property of extremizing the action incomparison with all neighboring paths

qi(t) = qcli (t) + qi(t) (1.3)

having the same endpoints q(tb), q(ta). To express this property formally, oneintroduces the variation of the action as the linear term in the Taylor expansion ofA[qi] in powers of qi(t):

A[qi] {A[qi + qi]A[qi]}lin term in qi . (1.4)

The extremal principle for the classical path is then

A[qi]

qi(t)=qcli (t)= 0 (1.5)

1This chapter is a small excerpt of the introductory sections to the textbook [1].

1

2 1 Fundamentals

for all variations of the path around the classical path qi(t) qi(t) qcli (t), whichvanish at the endpoints, i.e., which satisfy

qi(ta) = qi(tb) = 0. (1.6)

Since the action is a time integral of a Lagrangian, the extremality property canbe phrased in terms of differential equations. Let us calculate the variation of A[qi]explicitly:

A[qi] = {A[qi + qi]A[qi]}lin

= tb

tadt {L (qi(t) + qi(t), qi(t) + qi(t), t) L (qi(t), qi(t), t)}lin

= tb

tadt

{

L

qiqi(t) +

L

qiqi(t)

}

= tb

tadt

{

L

qi d

dt

L

qi

}

qi(t) +L

qiqi(t)

tb

ta

. (1.7)

The last expression arises from a partial integration of the qi term. Here, as in theentire text, repeated indices are understood to be summed (Einsteins summationconvention). The endpoint terms (surface or boundary terms) with the time t equalto ta and tb may be dropped, due to (1.6). Thus we find for the classical orbit q

cli (t)

the Euler-Lagrange equations :d

dt

L

qi=

L

qi. (1.8)

There is an alternative formulation of classical dynamics which is based on aLegendre-transformed function of the Lagrangian called the Hamiltonian

H Lqi

qi L(qi, qi, t). (1.9)

Its value at any time is equal to the energy of the system. According to the generaltheory of Legendre transformations [2], the natural variables which H depends onare no longer qi and qi, but qi and the generalized momenta pi, the latter beingdefined by the N equations

pi

qiL(qi, qi, t). (1.10)

In order to express the Hamiltonian H (pi, qi, t) in terms of its proper variables pi, qi,the equations (1.10) have to be solved for qi by a velocity function

qi = vi(pi, qi, t). (1.11)

This is possible provided the Hessian metric

hij(qi, qi, t) 2

qiqjL(qi, qi, t) (1.12)

1.1 Classical Mechanics 3

is nonsingular. The result is inserted into (1.9), leading to the Hamiltonian as afunction of pi and qi:

H (pi, qi, t) = pivi(pi, qi, t) L (qi, vi (pi, qi, t) , t) . (1.13)In terms of this Hamiltonian, the action is the following functional of pi(t) and qi(t):

A[pi, qi] = tb

tadt

[

pi(t)qi(t)H(pi(t), qi(t), t)]

. (1.14)

This is the so-called canonical form of the action. The classical orbits are now spec-ified by pcli (t), q

cli (t). They extremize the action in comparison with all neighboring

orbits in which the coordinates qi(t) are varied at fixed endpoints [see (1.3), (1.6)]whereas the momenta pi(t) are varied without restriction:

qi(t) = qcli (t) + qi(t), qi(ta) = qi(tb) = 0,

pi(t) = pcli (t) + pi(t).

(1.15)

In general, the variation is

A[pi, qi] = tb

tadt

[

pi(t)qi(t) + pi(t)qi(t)H

pipi

H

qiqi

]

= tb

tadt

{[

qi(t)H

pi

]

pi [

pi(t) +H

qi

]

qi

}

+ pi(t)qi(t)

tb

ta.

(1.16)

Since this variation has to vanish for the classical orbits, we find that pcli (t), qcli (t)

must be solutions of the Hamilton equations of motion

pi = H

qi,

qi =H

pi.

(1.17)

These agree with the Euler-Lagrange equations (1.8) via (1.9) and (1.10), as caneasily be verified. The 2N -dimensional space of all pi and qi is called the phasespace.

An arbitrary function O(pi(t), qi(t), t) changes along an arbitrary path as follows:

d

dtO (pi(t), qi(t), t) =

O

pipi +

O

qiqi +

O

t. (1.18)

If the path coincides with a classical orbit, we may insert (1.17) and find

dO

dt=

H

pi

O

qi O

pi

H

qi+

O

t

{H,O}+ Ot

.

(1.19)

4 1 Fundamentals

Here we have introduced the symbol {. . . , . . .} called Poisson brackets :

{A,B} Api

B

qi B

pi

A

qi, (1.20)

again with the Einstein summation convention for the repeated index i. The Poissonbrackets have the obvious properties

{A,B} = {B,A} antisymmetry, (1.21)

{A, {B,C}}+ {B, {C,A}}+ {C, {A,B}} = 0 Jacobi identity. (1.22)

If two quantities have vanishing Poisson brackets, they are said to commute.

The original Hamilton equations are a special case of (21.179):

d

dtpi = {H, pi} =

H

pj

piqj

pipj

H

qj= H

qi,

d

dtqi = {H, qi} =

H

pj

qiqj

qipj

H

qj=

H

pi.

(1.23)

By definition, the phase space variables pi, qi satisfy the Poisson brackets

{pi, qj} = ij ,{pi, pj} = 0,{qi, qj} = 0.

(1.24)

A function O(pi, qi) which has no explicit dependence on time and which, more-over, commutes with H (i.e., {O,H} = 0), is a constant of motion along the classicalpath, due to (21.179). In particular, H itself is often time-independent, i.e., of theform

H = H(pi, qi). (1.25)

Then, since H commutes with itself, the energy is a constant of motion.

The Lagrangian formalism has the virtue of being independent of the particularchoice of the coordinates qi. Let Qi be any other set of coordinates describing thesystem which is connected with qi by what is called a local

2 or point transformation

qi = fi(Qj , t). (1.26)

Certainly, to be of use, this relation must be invertible, at least in some neighborhoodof the classical path,

Qi = f1

i(qj , t). (1.27)

2The word local means here at a specific time. This terminology is of common use in fieldtheory where local means, more generally, at a specific spacetime point .


Otherwise Qi and qi could not both parametrize the same system. Therefore, fimust have a nonvanishing Jacobi determinant:

det

(

fiQj

)

6= 0. (1.28)

In terms of Qi, the initial Lagrangian takes the form

L(

Qj , Qj, t)

L(

fi (Qj , t) , fi (Qj, t) , t)

(1.29)

and the action reads

A = tb

tadt L

(

Qj(t), Qj(t), t)

= tb

tadt L

(

fi (Qj(t), t) , fi (Qj(t), t) , t)

.

(1.30)

By performing variations Qj(t), Qj(t) in the first expression while keepingQj(ta) = Qj(tb) = 0, we find the equations of motion

d

dt

L

Qj L

Qj= 0. (1.31)

The variation of the lower expression, on the other hand, gives

A = tb

tadt

(

L

qifi +

L

qifi

)

= tb

tadt

(

L

qi d

dt

L

qi

)

fi +L

qifi

tb

ta.

(1.32)

If qi is arbitrary, then so is fi. Moreover, with qi(ta) = qi(tb) = 0, also fivanishes at the endpoints. Hence the extremum of the action is determined equallywell by the Euler-Lagrange equations for Qj(t) [as it was by those for qi(t)].

Note that the locality property is quite restrictive for the transformation of thegeneralized velocities qi(t). They will necessarily be linear in Qj :

qi = fi(Qj , t) =fiQj

Qj +fit

. (1.33)

In phase space, there exists also the possibility of performing local changes ofthe canonical coordinates pi, qi to new ones Pj, Qj . Let them be related by

pi = pi(Pj , Qj, t),

qi = qi(Pj, Qj, t),(1.34)

with the inverse relationsPj = Pj(pi, qi, t),

Qj = Qj(pi, qi, t).(1.35)

6 1 Fundamentals

However, while the Euler-Lagrange equations maintain their form under any localchange of coordinates, the Hamilton equations do not hold, in general, for any trans-formed coordinates Pj(t), Qj(t). The local transformations pi(t), qi(t) Pj(t), Qj(t)for which they hold, are referred to as canonical . They are characterized by the forminvariance of the action, up to an arbitrary surface term,

tb

tadt [piqi H(pi, qi, t)] =

tb

tadt

[

PjQj H (Pj, Qj, t)]

+ F (Pj, Qj, t)

tb

ta,

(1.36)

where H (Pj, Qj , t) is some new Hamiltonian. Its relation with H(pi, qi, t) must bechosen in such a way that the equality of the action holds for any path pi(t), qi(t)connecting the same endpoints (at least any in some neighborhood of the classicalorbits). If such an invariance exists then a variation of this action yields for Pj(t)and Qj(t) the Hamilton equations of motion governed by H

:

Pi = H

Qi,

Qi =H

Pi.

(1.37)

The invariance (1.36) can be expressed differently by rewriting the integral on theleft-hand side in terms of the new variables Pj(t), Qj(t),

tb

tadt

{

pi

(

qiPj

Pj +qiQj

Qj +qit

)

H(pi(Pj, Qj, t), qi(Pj , Qj, t), t)}

, (1.38)

and subtracting it from the right-hand side, leading to tb

ta

{(

Pj piqiQj

)

dQj piqiPj

dPj

(

H + piqit

H)

dt

}

= F (Pj, Qj , t)

tb

ta.

(1.39)

The integral is now a line integral along a curve in the (2N + 1)-dimensional space,consisting of the 2N -dimensional phase space variables pi, qi and of the time t.The right-hand side depends only on the endpoints. Thus we conclude that theintegrand on the left-hand side must be a total differential. As such it has to satisfythe standard Schwarz integrability conditions [3], according to which all secondderivatives have to be independent of the sequence of differentiation. Explicitly,these conditions are

piPk

qiQl

qiPk

piQl

= kl,

piPk

qiPl

qiPk

piPl

= 0, (1.40)

piQk

qiQl

qiQk

piQl

= 0,


andpit

qiPl

qit

piPl

=(H H)

Pl,

pit

qiQl

qit

piQl

=(H H)

Ql.

(1.41)

The first three equations define the so-called Lagrange brackets in terms of whichthey are written as

(Pk, Ql) = kl,

(Pk, Pl) = 0,

(Qk, Ql) = 0. (1.42)

Time-dependent coordinate transformations satisfying these equations are calledsymplectic. After a little algebra involving the matrix of derivatives

J =

Pi/pj Pi/qj

Qi/pj Qi/qj

, (1.43)

its inverse

J1 =

pi/Pj pi/Qj

qi/Pj qi/Qj

, (1.44)

and the symplectic unit matrix

E =

(

0 ijij 0

)

, (1.45)

we find that the Lagrange brackets (1.42) are equivalent to the Poisson brackets

{Pk, Ql} = kl,{Pk, Pl} = 0, (1.46){Qk, Ql} = 0.

This follows from the fact that the 2N 2N matrix formed from the Lagrangebrackets

L

(Qi, Pj) (Qi, Qj)(Pi, Pj) (Pi, Qj)

(1.47)

can be written as (E1J1E)TJ1, while an analogous matrix formed from thePoisson brackets

P

{Pi, Qj} {Pi, Pj}{Qi, Qj} {Qi, Pj}

(1.48)

is equal to J(E1JE)T . Hence L = P1, so that (1.42) and (1.46) are equivalent toeach other. Note that the Lagrange brackets (1.42) [and thus the Poisson brackets

8 1 Fundamentals

(1.46)] ensure piqi PjQj to be a total differential of some function of Pj and Qj inthe 2N -dimensional phase space:

piqi PjQj =d

dtG(Pj , Qj, t). (1.49)

The Poisson brackets (1.46) for Pi, Qi have the same form as those in Eqs. (21.182)for the original phase space variables pi, qi.

The other two equations (1.41) relate the new Hamiltonian to the old one. Theycan always be used to construct H (Pj, Qj , t) from H(pi, qi, t). The Lagrange brack-ets (1.42) or Poisson brackets (1.46) are therefore both necessary and sufficient forthe transformation pi, qi Pj, Qj to be canonical.

A canonical transformation preserves the volume in phase space. This followsfrom the fact that the matrix product J(E1JE)T is equal to the 2N 2N unitmatrix (1.48). Hence det (J) = 1 and

i

[dpi dqi] =

j

[dPj dQj ] . (1.50)

It is obvious that the process of canonical transformations is reflexive. It maybe viewed just as well from the opposite side, with the roles of pi, qi and Pj, Qjexchanged [we could just as well have considered the integrand (1.39) as a completedifferential in Pj, Qj , t space].

Once a system is described in terms of new canonical coordinates Pj , Qj, weintroduce the new Poisson brackets

{A,B} APj

B

Qj B

Pj

A

Qj, (1.51)

and the equation of motion for an arbitrary observable quantity O (Pj(t), Qj(t), t)becomes with (21.183)

dO

dt= {H , O} + O

t, (1.52)

by complete analogy with (21.179). The new Poisson brackets automatically guar-antee the canonical commutation rules

{Pi, Qj} = ij ,{Pi, Pj} = 0,{Qi, Qj} = 0.

(1.53)

A standard class of canonical transformations can be constructed by introducinga generating function F satisfying a relation of the type (1.36), but dependingexplicitly on half an old and half a new set of canonical coordinates, for instance

F = F (qi, Qj , t). (1.54)


One now considers the equation

tb

tadt [piqi H(pi, qi, t)] =

tb

tadt

[

PjQj H (Pj , Qj, t) +d

dtF (qi, Qj , t)

]

, (1.55)

replaces PjQj by PjQj + ddtPjQj , defines

F (qi, Pj, t) F (qi, Qj, t) + PjQj,

and works out the derivatives. This yields

tb

tadt

{

piqi + PjQj [H(pi, qi, t)H (Pj , Qj, t)]}

= tb

tadt

{

F

qi(qi, Pj, t)qi +

F

Pj(qi, Pj, t)Pj +

F

t(qi, Pj, t)

}

.

(1.56)

A comparison of the two sides yields the equations for the canonical transformation

pi =

qiF (qi, Pj, t),

Qj =

PjF (qi, Pj, t).

(1.57)

The second equation shows that the above relation between F (qi, Pj, t) andF (qi, Qj , t) amounts to a Legendre transformation.

The new Hamiltonian is

H (Pj, Qj, t) = H(pi, qi, t) +

tF (qi, Pj , t). (1.58)

Instead of (1.54) we could, of course, also have chosen functions with other mixturesof arguments such as F (qi, Pj, t), F (pi, Qj, t), F (pi, Pj, t) to generate simple canonicaltransformations.

A particularly important canonical transformation arises by choosing a gener-ating function F (qi, Pj) in such a way that it leads to time-independent momentaPj j. Coordinates Qj with this property are called cyclic. To find cyclic co-ordinates we must search for a generating function F (qj , Pj, t) which makes thetransformed H in (1.58) vanish identically. Then all derivatives with respect to thecoordinates vanish and the new momenta Pj are trivially constant. Thus we seek asolution of the equation

tF (qi, Pj, t) = H(pi, qi, t), (1.59)

where the momentum variables in the Hamiltonian obey the first equation of (1.57).This leads to the following partial differential equation for F (qi, Pj, t):

tF (qi, Pj, t) = H(qiF (qi, Pj, t), qi, t), (1.60)

10 1 Fundamentals

called the Hamilton-Jacobi equation. Here and in the sequel we shall often use theshort notations for partial derivatives t /t, qi /qi .

A generating function which achieves this goal is supplied by the action functional(1.14). When following the classical solutions starting from a fixed initial point andrunning to all possible final points qi at a time t, the associated actions of thesesolutions form a function A(qi, t). Expression (1.14) show that if a particle movesalong a classical trajectory, and the path is varied without keeping the endpointsfixed, the action changes as a function of the end positions (1.16) by

A[pi, qi] = pi(tb)qi(tb) pi(ta)qi(ta). (1.61)

From this we deduce immediately the first of the equations (1.57), now for thegenerating function A(qi, t):

pi =

qiA(qi, t). (1.62)

Moreover, the function A(qi, t) has the time derivative

d

dtA(qi(t), t) = pi(t)qi(t)H(pi(t), qi(t), t). (1.63)

Together with (1.62) this implies

tA(qi, t) = H(pi, qi, t). (1.64)

If the momenta pi on the right-hand side are replaced according to (1.62), A(qi, t)is indeed seen to be a solution of the Hamilton-Jacobi differential equation:

tA(qi, t) = H(qiA(qi, t), qi, t). (1.65)

1.2 Relativistic Mechanics in Curved Spacetime

The classical action of a relativistic spinless point particle in a curved four-dimensional spacetime is usually written as an integral

A = Mc2

dL(q, q) = Mc2

d

g q()q(), (1.66)

where is an arbitrary parameter of the trajectory. It can be chosen in the finaltrajectory to make L(q, q) 1, in which case it coincides with the proper time ofthe particle. For arbitrary , the Euler-Lagrange equation (1.8) reads

d

dt

[

1

L(q, q)g q

]

=1

2L(q, q)(g) q

q. (1.67)

If is the proper time where L(q, q) 1, this simplifies tod

dt(g q

) =1

2(g) q

q, (1.68)

1.3 Quantum Mechanics 11

or

g q =

(

1

2g g

)

qq. (1.69)

For brevity, we have denoted partial derivatives /q by . This partial derivativeis supposed to apply only to the quantity right behind it. At this point one introducesthe Christoffel symbol

1

2(g + g g), (1.70)

and the Christoffel symbol of the second kind [6]:

g. (1.71)

Then (1.69) can be written as

q + qq = 0. (1.72)

Since the solutions of this equation minimize the length of a curve in spacetime,they are called geodesics .

1.3 Quantum Mechanics

Historically, the extension of classical mechanics to quantum mechanics becamenecessary in order to understand the stability of atomic orbits and the discretenature of atomic spectra. It soon became clear that these phenomena reflect thefact that at a sufficiently short length scale, small material par

quantum field theory by hagen kleinert

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