Geometry, particles and fields

GEOMETRY, PARTICLES, AND FIELDS......Page 1
Title Page......Page 4
Preface......Page 8
Acknowledgements......Page 9
Contents......Page 10
Part I: Basic Properties of Particles and Fields......Page 22
General References to Part I......Page 23
1.1 The Electromagnetic Field......Page 24
1.2 The Introduction of Gauge Potentials in Electromagnetism......Page 28
1.3 Magnetic Flux......Page 32
1.4 Illustrative Example: The Gauge Potential of a Solenoid......Page 36
1.5 Relativistic Formulation of the Theory of Electromagnetism......Page 40
1.6 The Energy–Momentum Tensor......Page 43
Solutions of Worked Exercises......Page 49
2.1 Introduction......Page 52
2.2 Lagrangian Formalism for Particles: The Non-Relativistic Case......Page 53
2.3 Basic Principles of Quantum Mechanics......Page 60
2.4 Path Integrals — The Feynman Propagator......Page 63
2.5 Illustrative Example: The Free Particle Propagator......Page 67
2.6 Bohm–Aharonov Effect — Lorentz Force......Page 70
2.7 Gauge Transformation of the Schrödinger Wavefunction......Page 76
2.8 Quantum Mechanics of a Charged Particle as a Gauge Theory......Page 78
2.9 The Schrödinger Equation in the Path Integral Formalism......Page 83
2.10 The Hamiltonian Formalism......Page 86
2.11 Canonical Quantization and the Schrödinger Equation......Page 89
2.12 Illustrative Example: Superconductors and Flux Quantization......Page 93
Solutions of Worked Exercises......Page 100
3.1 Illustrative Example: Lagrangian Formalism for a String......Page 104
3.2 Lagrangian Formalism for Relativistic Fields......Page 107
3.3 Hamiltonian Formalism for Relativistic Fields......Page 111
3.4 The Klein–Gordon Field......Page 116
3.5 The Maxwell Field......Page 119
3.6 Spin of the Photon — Polarization of Electromagnetic Waves......Page 124
3.7 The Massive Vector Field......Page 128
3.8 The Cauchy Problem......Page 130
3.9 The Complex Klein–Gordon Field......Page 134
3.10 The Theory of Electrically Charged Fields as a Gauge Theory......Page 137
3.11 Charge Conservation as a Consequence of Gauge Symmetry......Page 140
3.12 The Equivalence of Real and Complex Field Theories......Page 143
Solutions of Worked Exercises......Page 145
4.1 Non-Linear Field Theories with a Degenerate Vacuum......Page 148
4.2 Topological Charges......Page 152
4.3 Solitary Waves......Page 156
4.4 Ground States for the Non-Perturbative Sectors......Page 159
4.5 Solitons......Page 165
4.6 The Bäcklund Transformation......Page 170
4.7 Dynamical Stability of Solitons......Page 174
4.8 The Particle Spectrum in Non-Linear Field Theories......Page 181
Solutions of Worked Exercises......Page 185
5.1 The Feynman Propagator Revisited......Page 188
5.2 Illustrative Example: The Harmonic Oscillator......Page 193
5.3 The Path Integral Revisited......Page 202
5.4 Illustrative Example: The Time-Dependent Oscillator......Page 209
5.5 Path Integrals and Determinants......Page 215
5.6 The Bohr–Sommerfeld Quantization Rule......Page 219
5.7 Instantons and Euclidean Field Theory......Page 233
5.8 Instantons and the Tunnel Effect......Page 240
5.9 Instanton Calculation of the Low Lying Energy Levels......Page 246
5.10 Illustrative Example: Calculation of the Parameter Δ......Page 257
Solutions of the Worked Exercises......Page 266
Part II: Basic Principles and Applications of Differential Geometry......Page 268
General References to Part II......Page 269
6.1 Coordinate Systems......Page 270
6.2 Differentiable Manifolds......Page 277
6.3 Productmanifolds and Manifolds Defined by Constraints......Page 285
6.4 Tangent Vectors......Page 290
6.5 Metrics......Page 297
6.6 The Minkowski Space......Page 305
6.7 The Action Principle for a Relativistic Particle......Page 312
6.8 Covectors......Page 321
6.9 Tensors......Page 329
6.10 Tensor Fields in Physics......Page 339
Solutions of Worked Exercises......Page 343
7.1 Introduction......Page 346
7.2 k-Forms — The Wedge Product......Page 348
7.3 The Exterior Derivative......Page 357
7.4 The Volume Form......Page 364
7.5 The Dual Map......Page 373
7.6 The Co-Differential and the Laplacian......Page 381
7.7 Exterior Calculus in 3 and 4 Dimensions......Page 387
7.8 Electromagnetism and the Exterior Calculus......Page 397
Solutions of Worked Exercises......Page 406
8.1 Introduction......Page 414
8.2 Submanifolds — Regular Domains......Page 416
8.3 The Integral of Differential Forms......Page 424
8.4 Elementary Properties of the Integral......Page 432
8.5 The Hilbert Product of Two Differential Forms......Page 441
8.6 The Lagrangian Formalism and the Exterior Calculus......Page 445
8.7 Integral Calculus and Electromagnetism......Page 450
8.8 The Nambu String and the Nielsen–Olesen Vortex......Page 463
8.9 Singular Forms......Page 475
Solutions of Worked Exercises......Page 482
9.1 Magnetic Charges and Currents......Page 490
9.2 The Dirac String......Page 497
9.3 Dirac\'s Lagrangian Principle for Magnetic Monopoles......Page 503
9.4 The Angular Momentum Due to a Monopole Field......Page 507
9.5 Quantization of the Angular Momentum......Page 512
9.6 The Gauge Transformation as a Unitary Transformation......Page 519
9.7 Quantizaton of the Magnetic Charge......Page 521
Solutions of Worked Exercises......Page 524
10.1 Local Properties of Smooth Maps......Page 532
10.2 Pull Backs of Co-Tensors......Page 541
10.3 Isometries and Conformal Maps......Page 551
10.4 The Conformal Group......Page 561
10.5 The Dual Map......Page 572
10.6 The Self-Duality Equation......Page 575
10.7 Winding Numbers......Page 583
10.8 The Heisenberg Ferromagnet......Page 591
10.9 The Exceptional ϕ^4-Model......Page 599
Solutions of Worked Exercises......Page 604
11.1 Conservation Laws......Page 608
11.2 Symmetries and Conservation Laws in Quantum Mechanics......Page 613
11.3 Conservation of Energy, Momentum and Angular Momentum in Quantum Mechanics......Page 617
11.4 Symmetries and Conservation Laws in Classical Field Theory......Page 621
11.5 Isometries as Symmetry Transformations......Page 627
11.6 The True Energy–Momentum Tensor for Vector Fields......Page 632
11.7 Energy–Momentum Conservation as a Consequence of Covariance......Page 635
11.8 Scale Invariance in Classical Field Theories......Page 639
11.9 Conformal Transformations as Symmetry Transformations......Page 646
Solutions of Worked Exercises......Page 653
Index of Subjects......Page 658