Fields

Warren S.

This text is meant to cover all the field theory every high energy theorist should know, but not all that any particular theorist might need to know. It is not meant as an introduction to research, but as a preliminary to such courses: We try to fill in the cracks that often lie between standard field theory courses and advanced specialized courses. For example, we have some discussion of string theory, but it is more oriented toward the strong interactions, where it has some experimental justification, rather than quantum gravity and unification, where its usefulness is still under investigation. We do not mention statistical mechanics, although many of the field theory methods we discuss are useful there. Also, we do not discuss any experimental results in detail; phenomenology and analysis of experiments deserve their own text. We give and apply the methods of calculation and discuss the qualitative features of the results, but do not make a numerical comparison to nature: We concentrate more on the “forest” than the “trees”.

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Table of contents :
PART ONE: SYMMETRY……Page 2
PART TWO: QUANTA……Page 3
PART THREE: HIGHER SPIN……Page 4
Scientific method……Page 5
Highlights……Page 8
Notes for instructors……Page 11
Notes for students……Page 13
Outline……Page 14
Acknowledgments……Page 18
SOME FIELD THEORY TEXTS……Page 19
1. Nonrelativity……Page 23
2. Fermions……Page 27
3. Lie algebra……Page 31
4. Relativity……Page 35
5. Discrete: C, P, T……Page 40
6. Conformal……Page 43
1. Matrices……Page 48
2. Representations……Page 50
3. Determinants……Page 55
4. Classical groups……Page 58
5. Tensor notation……Page 60
1. More coordinates……Page 65
2. Coordinate tensors……Page 67
3. Young tableaux……Page 71
4. Color and flavor……Page 73
5. Covering groups……Page 78
1. 3-vectors……Page 81
2. Rotations……Page 84
3. Spinors……Page 86
4. Indices……Page 87
5. Lorentz……Page 90
6. Dirac……Page 96
7. Chirality/duality……Page 98
1. Field equations……Page 100
2. Examples……Page 103
3. Solution……Page 104
4. Mass……Page 109
5. Foldy-Wouthuysen……Page 112
6. Twistors……Page 116
7. Helicity……Page 118
1. Algebra……Page 123
2. Supercoordinates……Page 124
3. Supergroups……Page 126
4. Superconformal……Page 129
5. Supertwistors……Page 130
1. General……Page 135
2. Fermions……Page 139
3. Fields……Page 140
4. Relativity……Page 142
5. Constrained systems……Page 148
1. Free……Page 152
2. Gauges……Page 156
3. Coupling……Page 157
4. Conservation……Page 158
5. Pair creation……Page 161
1. Nonabelian……Page 164
2. Lightcone……Page 168
3. Plane waves……Page 172
4. Self-duality……Page 173
5. Twistors……Page 176
6. Instantons……Page 179
7. ADHM……Page 183
8. Monopoles……Page 185
A. HIDDEN SYMMETRY……Page 190
1. Spontaneous breakdown……Page 191
2. Sigma models……Page 193
3. Coset space……Page 196
4. Chiral symmetry……Page 197
5. Stueckelberg……Page 200
6. Higgs……Page 202
1. Chromodynamics……Page 205
2. Electroweak……Page 209
3. Families……Page 213
4. Grand Unified Theories……Page 215
1. Chiral……Page 220
2. Actions……Page 222
3. Covariant derivatives……Page 224
4. Prepotential……Page 227
5. Gauge actions……Page 229
6. Breaking……Page 231
7. Extended……Page 234
V. QUANTIZATION……Page 240
1. Path integrals……Page 241
2. Semiclassical expansion……Page 245
3. Propagators……Page 249
4. S-matrices……Page 251
5. Wick rotation……Page 255
1. Particles……Page 259
2. Properties……Page 262
3. Generalizations……Page 265
4. Wick rotation……Page 268
1. Path integrals……Page 273
2. Graphs……Page 277
3. Semiclassical expansion……Page 282
4. Feynman rules……Page 286
5. Semiclassical unitarity……Page 292
6. Cutting rules……Page 294
7. Cross sections……Page 297
8. Singularities……Page 300
9. Group theory……Page 302
1. Hamiltonian……Page 308
2. Lagrangian……Page 312
3. Particles……Page 315
4. Fields……Page 316
1. Radial……Page 320
2. Lorentz……Page 323
3. Massive……Page 325
4. Gervais-Neveu……Page 327
5. Super Gervais-Neveu……Page 330
6. Spacecone……Page 333
7. Superspacecone……Page 337
8. Background-field……Page 339
9. Nielsen-Kallosh……Page 345
10. Super background-field……Page 347
1. Yang-Mills……Page 351
2. Recursion……Page 355
3. Fermions……Page 357
4. Masses……Page 359
5. Supergraphs……Page 365
1. Dimensional renormalization……Page 370
2. Momentum integration……Page 373
3. Modified subtractions……Page 377
4. Optical theorem……Page 381
5. Power counting……Page 383
6. Infrared divergences……Page 387
1. Tadpoles……Page 391
2. Effective potential……Page 394
3. Dimensional transmutation……Page 397
4. Massless propagators……Page 398
5. Massive propagators……Page 401
6. Renormalization group……Page 405
7. Overlapping divergences……Page 408
1. Improved perturbation……Page 415
2. Renormalons……Page 420
3. Borel……Page 423
4. 1/N expansion……Page 426
1. Fermion……Page 432
2. Photon……Page 435
3. Gluon……Page 436
4. Grand Unified Theories……Page 442
5. Supermatter……Page 445
6. Supergluon……Page 447
7. Bosonization……Page 452
8. Schwinger model……Page 455
1. JWKB……Page 461
2. Axial anomaly……Page 464
3. Anomaly cancelation……Page 467
3. Pi-0 to 2 gamma……Page 470
5. Vertex……Page 472
6. Nonrelativistic JWKB……Page 475
1. Conformal anomaly……Page 480
2. e+e- to hadrons……Page 483
3. Parton model……Page 485
A. ACTIONS……Page 492
1. Gauge invariance……Page 493
2. Covariant derivatives……Page 496
3. Conditions……Page 501
4. Integration……Page 504
5. Gravity……Page 508
6. Energy-momentum……Page 511
7. Weyl scale……Page 514
1. Lorentz……Page 520
2. Geodesics……Page 522
3. Axial……Page 525
4. Radial……Page 527
5. Weyl scale……Page 532
1. Self-duality……Page 537
2. De Sitter……Page 538
3. Cosmology……Page 541
4. Red shift……Page 543
5. Schwarzschild……Page 545
6. Experiments……Page 551
7. Black holes……Page 554
1. Covariant derivatives……Page 558
2. Field strengths……Page 563
3. Compensators……Page 567
4. Scale gauges……Page 569
1. Integration……Page 575
2. Ectoplasm……Page 578
3. Component transformations……Page 581
4. Component approach……Page 582
5. Duality……Page 585
6. Superhiggs……Page 589
7. No-scale……Page 591
1. Dirac spinors……Page 594
2. Wick rotation……Page 597
3. Other spins……Page 600
4. Supersymmetry……Page 602
5. Theories……Page 605
6. Reduction to D=4……Page 608
XI. STRINGS……Page 613
1. Regge theory……Page 615
2. Classical mechanics……Page 618
3. Gauges……Page 621
4. Quantum mechanics……Page 626
5. Anomaly……Page 629
6. Tree amplitudes……Page 631
1. Massless spectrum……Page 638
2. Reality and orientation……Page 640
3. Supergravity……Page 641
4. T-duality……Page 642
5. Dilaton……Page 644
6. Superdilaton……Page 646
7. Conformal field theory……Page 648
8. Triality……Page 653
1. Spacetime lattice……Page 657
2. Worldsheet lattice……Page 661
3. QCD strings……Page 663
XII. MECHANICS……Page 668
1. Lightcone……Page 669
2. Algebra……Page 672
3. Action……Page 675
4. Spinors……Page 677
5. Examples……Page 679
1. Algebra……Page 684
2. Inner product……Page 685
3. Action……Page 687
4. Solution……Page 690
5. Spinors……Page 693
6. Masses……Page 694
7. Background fields……Page 695
8. Strings……Page 697
9. Relation to OSp(1,1/2)……Page 702
1. Antibracket……Page 705
2. ZJBV……Page 708
3. BRST……Page 712
AfterMath……Page 716
INDEX……Page 723
Comments on Warren Siegel’s Fields……Page 731