دانلود کتاب An Introduction to Quantum Field Theory

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AN INTRODUCTION TO QUANTUM FIELD THEORY......Page 1
Half-title......Page 2
About......Page 4
Title Page......Page 6
Copyright Page......Page 7
Dedication......Page 8
Contents......Page 10
Preface......Page 14
Acknowledgements and thanks......Page 17
Part I: Scalar fields......Page 20
Local field theory......Page 22
Lagrange equations......Page 23
1.2 Relativistic scalar fields......Page 24
Units......Page 25
Free field......Page 26
First quantization......Page 27
Transformed density......Page 29
Form invariance and symmetry transformations......Page 30
Noether's theorem......Page 31
Application to Lorentz transformations......Page 32
Phase invariance and global symmetry......Page 34
Groups and representations......Page 35
Lie groups and algebras......Page 36
The nonabelian groups U(2) and SU(2)......Page 37
Conserved currents......Page 38
Lorentz transformations......Page 39
The Lorentz group......Page 40
Lie algebra......Page 41
Finite and discrete transformations......Page 43
The Poincaré group and its generators......Page 44
Exercises......Page 45
Canonical quantization......Page 48
Poisson brackets for the scalar field......Page 49
Quantization of the scalar field......Page 50
2.2 Quantum symmetries......Page 51
States and the Poincaré algebra......Page 52
Transformations of states......Page 53
Causality......Page 54
Internal symmetries......Page 55
The Hamiltonian in momentum space......Page 57
Discrete system......Page 58
Fock space as a solution......Page 59
States of the continuum Fock space......Page 60
Bose–Einstein statistics......Page 61
Relativistic invariance......Page 62
Charged scalar field......Page 63
Wave packets and particle interpretation......Page 65
Field matrix elements......Page 66
Stueckelberg–Feynman Green function......Page 67
2.5 Interacting fields and scattering......Page 68
Spectral assumptions......Page 69
In- and out-states......Page 70
The S-matrix......Page 71
Reduction formulas......Page 72
Exercises......Page 74
3.1 The path integral......Page 77
The path integral......Page 78
Reduction to classical quantities......Page 79
Configuration space......Page 81
Wick rotation......Page 82
Semiclassical approximation......Page 83
Time-ordered products......Page 85
Generating functionals......Page 86
Path integral for scalar fields......Page 87
Free-field Green functions......Page 88
Perturbative expansion......Page 89
3.2 The path integral and coherent states......Page 90
Coherent states......Page 91
Path integral for coherent states......Page 92
Source dependence for the harmonic oscillator......Page 93
Path integral for the free field......Page 95
Path integral in terms of fields......Page 97
The S-matrix......Page 98
S-matrix for the free field......Page 99
Normalization......Page 100
Free-field and Feynman diagrams......Page 101
Example......Page 102
Time ordering......Page 104
Vacuum bubbles......Page 106
Feynman rules; symmetry factors......Page 107
Connected, truncated and one-particle irreducible diagrams......Page 108
Generating functionals......Page 109
Exercises......Page 111
Fourier transform......Page 113
Tree and loop diagrams......Page 115
4.2 The S-matrix......Page 117
Normalization......Page 118
4.3 Cross sections......Page 119
Total and differential cross sections......Page 123
Cross section in tree approximation......Page 125
Interactions......Page 126
Generating functionals......Page 127
S-matrix......Page 128
Feynman rules for the charged field......Page 129
Crossing......Page 131
The T* product......Page 132
Exercises......Page 133
Part II: Fields with spin......Page 136
Single-valued representations......Page 138
Rotational spinors and SU(2)......Page 139
Lorentz spinors and SL(2, C)......Page 140
Scalars and vectors......Page 141
Vectors......Page 142
General representations......Page 143
Weyl equation......Page 144
Dirac equation and four-component spinors......Page 145
Dirac matrices......Page 146
Form invariance......Page 147
γ[sub(5)] and projection......Page 148
Global symmetries......Page 149
The Maxwell Lagrangian......Page 150
Gauge invariance and gauge conditions......Page 151
Plane waves......Page 152
5.4 Interactions and local gauge invariance......Page 153
Local gauge invariance......Page 154
Local nonabelian gauge invariance......Page 156
Infinitesimal transfrmations......Page 157
Higgs mechanism in SU(2) × U(1)......Page 159
Gauge conditions for nonabelian fields......Page 162
Exercises......Page 163
Conserved currents......Page 166
6.2 Unitary representations of the Poincaré group......Page 167
Method of Wigner......Page 168
Fields and solutions......Page 170
Klein–Gordon equation......Page 171
Massive Dirac equation......Page 172
Spin basis......Page 174
Massive vector solutions......Page 176
6.4 Massless solutions......Page 177
Dirac equation......Page 178
Helicity......Page 179
Free Dirac theory......Page 181
Anticommutation relations......Page 182
States and Fermi statistics......Page 183
Particles, antiparticles and spin......Page 184
Spin and statistics......Page 185
Vector fields......Page 186
Operator content......Page 189
Parity conservation......Page 190
Standard model of leptonic electroweak interactions......Page 191
Exercises......Page 193
7.1 Fermionic path integrals......Page 195
Anticommuting numbers......Page 196
Grassmann integrals......Page 197
Mixed integrals and transformations......Page 198
Gaussian integrals......Page 199
Fermionic field theories......Page 200
7.2 Fermions in an external field......Page 201
Feynman propagator......Page 203
Sign of the permutation......Page 204
Absolute sign......Page 205
Feynman rules......Page 206
Fermion loops and functional determinants......Page 207
7.3 Gauge vectors and ghosts......Page 208
Two-dimensional analogy......Page 209
Gauge theory......Page 211
Effective Lagrangian......Page 213
Covariant and physical gauges......Page 214
Generator......Page 216
Reduction for fermions and vectors......Page 218
Example......Page 220
Cross sections and averaging......Page 221
Exercises......Page 222
Feynman rules......Page 223
Green functions in momentum space......Page 224
The S-matrix for Bhabha scattering......Page 226
Unpolarized cross sections and traces......Page 227
Dirac matrix theorems......Page 229
Evaluation of the cross section......Page 231
Helicity conservation......Page 233
The Klein–Nishina formula......Page 236
Decoupling of unphysical photons......Page 239
Zero-mass cross sections......Page 240
Magnitude of the charge......Page 241
Lagrangian......Page 242
Muon decay and charged currents......Page 243
Neutrino–electron elastic scattering......Page 245
8.4 Quantum chromodynamics and quark–quark scattering......Page 246
Lagrangian in SU(N)......Page 247
S-matrix elements......Page 248
Cross section......Page 249
Feynman rules......Page 250
Physical ancl unphysical polarizations......Page 252
8.6 Parton-model interpretation of QCD cross sections......Page 256
Deeply inelastic scattering......Page 257
The Drell–Yan cross section......Page 259
Exercises......Page 261
Part III: Renormalization......Page 264
9.1 One-loop example......Page 266
9.2 Wick rotation in perturbation theory......Page 269
9.3 Dimensional regularization......Page 271
Ultraviolet divergences......Page 272
Angular integrals......Page 273
Special functions......Page 275
One-loop diagrams in n dimensions......Page 276
Dimensionally continued Lagrangians......Page 277
Functional modifications......Page 279
Summary of one-loop integrals......Page 280
Analytic continuation in the n-plane......Page 282
Two-loop example......Page 283
Weinberg's theorem......Page 284
9.5 Time-ordered perturbation theory......Page 285
9.6 Unitarity......Page 290
Cut graphs......Page 291
Proof of equation (9.80)......Page 293
Sum over final states......Page 295
Exercises......Page 297
10.1 ϕ[sup(3)][sub(4)] and mass renormalization......Page 299
The mass counterterm......Page 300
Physical parameters......Page 302
Generalized minimal subtraction......Page 304
The physical mass and the renormalized mass......Page 305
Tadpole diagram......Page 306
10.2 Power counting and renormalizability......Page 307
Renormalizability......Page 308
Renormalization......Page 309
Mass and wave function renormalization......Page 310
Coupling-constant renormalization......Page 312
Momentum subtraction......Page 314
Generalized minimal subtraction......Page 315
Renormalized and unrenormalized Green functions......Page 316
The S-matrix......Page 317
Fixing the renormalized coupling and mass......Page 318
Two-loop renormalization of ϕ[sup(3)][sub(6)]......Page 319
Two-loop calculations......Page 321
Two-loop renormalization constants......Page 323
The nature of renormalization proofs......Page 324
Renormalization theorems......Page 326
BPHZ renormalization......Page 327
10.5 Introduction to the renormalization group......Page 328
Beta function with minimal subtraction......Page 329
Solution of the renormalization group equation......Page 330
Fixed points......Page 333
Green functions......Page 335
Exercises......Page 336
11.1 Gauge theories at one loop......Page 338
Fermion self-energy......Page 340
Vacuum polarization......Page 342
Vector wave function renormalization......Page 343
Charge renormalization in QED......Page 344
Furry's theorem......Page 345
Three-gluon vertex......Page 346
Renormalized Lagrangian for QED......Page 347
Gluon self-energy......Page 348
Vanishing of scaleless integrals......Page 349
Fermion–gluon vertex......Page 350
Renormalized Lagrangian in QCD......Page 351
Renormalization group for gauge theories......Page 352
11.2 Renormalization and unitarity in QED......Page 353
Field transformations......Page 355
Gauge transformations and Ward identities in QED......Page 356
Graphical proof of Ward identities and transversality......Page 358
Current conservation and nonrenormalization......Page 359
Renormalization-constant identities to all orders......Page 360
Transversality of the photon self-energy......Page 361
The case of QED: Z[sub(1)] = Z[sub(ψ)]......Page 362
Gauge invariance of the S-matrix......Page 363
Unitarity of the S-matrix......Page 366
Becchi–Rouet–Stora (BRS) invariance......Page 367
Invariance......Page 368
Ward identities......Page 369
Gauge invariance of the physical S-matrix......Page 371
Uncut and cut graphs......Page 372
Ward identity for cut diagrams......Page 373
Unitarity......Page 374
Renormalization and broken symmetry......Page 376
Identities from global transformations......Page 377
Chiral rotation and naive identity......Page 378
Triangle diagrams......Page 379
Tensor structure......Page 380
Finite integrals......Page 381
Higher orders and the Adler–Bardeen theorem......Page 382
Exercises......Page 383
Part IV: The nature of perturbative cross sections......Page 386
Tensor structure and form factors......Page 388
Vertex correction......Page 389
Reduction of Dirac structure......Page 390
Zero momentum transfer......Page 391
Infrared divergence......Page 392
Order-q expansion......Page 393
Infrared approximation......Page 394
High-energy behavior......Page 395
Fermion self-energy......Page 396
12.2 Order-α infrared bremsstrahlung......Page 397
Virtual corrections......Page 399
Photon emission......Page 400
Infrared cancellation......Page 401
Classical analogy and energy resolution......Page 402
On-shell renormalization to all loops......Page 403
Infrared power counting......Page 404
Grammer–Yennie decomposition......Page 406
Fermion lines as Green functions......Page 407
Change of variables......Page 408
Exponentiation of infrared divergences......Page 409
Real photons......Page 411
Physical parameters......Page 413
Born cross section for e[sup(+)]e[sup(–)] annihilation at high energy......Page 414
Virtual correction......Page 419
Determination of α[sub(s)]......Page 420
The running coupling and the QCD scale parameter......Page 421
Comparison of QED and QCD......Page 422
Angular resolutions and jet cross sections......Page 423
Two-jet cross section......Page 424
Differential jet cross sections......Page 426
Exercises......Page 427
13.1 Analytic structure of Feynman diagrams......Page 430
Poles, singular surfaces and contour integrals......Page 431
Landau equations......Page 434
Physical pictures and reduced diagrams......Page 435
The two-point function and normal thresholds......Page 436
One-loop example and light-cone variables......Page 437
Dispersion relations......Page 438
Infrared finiteness of σ[sub(tot)]......Page 440
Jet and soft subdiagrams......Page 441
Pinch surfaces with massless lines......Page 442
Application: jets in decay processes......Page 443
Application: physical pictures in leptoproduction......Page 444
Variables for pinch surfaces......Page 445
Homogeneous integral......Page 446
Pinch surfaces for nonexceptional Euclidean momenta......Page 447
Euclidean infrared power counting......Page 448
Threshold singularities and enhancements......Page 449
Anomalous thresholds......Page 450
Power counting for one-loop diagrams......Page 451
Jet power counting beyond one loop......Page 453
The electromagnetic vertex in ϕ[sup(4)]......Page 456
Power counting for cross sections and cut diagrams......Page 457
Finiteness of jet cross sections......Page 459
Initial-state interactions and the KLN theorem......Page 461
Interpretation......Page 465
Exercises......Page 466
14.1 Deeply inelastic scattering......Page 468
The hadronic tensor......Page 469
Born cross section and scaling......Page 473
Quark structure functions at one loop......Page 474
Parton-model structure functions......Page 478
Factorization theorem......Page 479
Factorization in perturbation theory......Page 480
Final-state interactions and factorization......Page 481
Parton interpretation of a pinch surface......Page 482
Factorization of spin matrices......Page 483
Minimal subtraction distributions......Page 484
One loop density......Page 486
Coefficient functions......Page 488
Factorization in the Drell–Yan process and universality......Page 489
One-loop corrections in the Drell–Yan process......Page 491
Other processes......Page 493
Nonsinglet structure functions......Page 494
Renormalization group......Page 496
One-loop anomalous dimensions......Page 497
Evolution equation......Page 498
Physical basis of scale breaking......Page 499
Generalized ladder structure......Page 500
Operator product expansion for T[sub(2)]......Page 503
Fourier transform......Page 504
Light-cone singularities and twist......Page 505
Moments and the operator product expansion......Page 506
Vacuum expectation of the operator product......Page 508
Exercises......Page 509
The Bethe–Salpeter equation and wave functions......Page 511
Amplitudes for bound states......Page 514
Convergence of the perturbative series......Page 516
Time evolution......Page 521
States and time ordering......Page 522
In- and out-fields......Page 524
Vacuum states......Page 525
Green functions......Page 526
Symmetry factors......Page 528
Generating functional of connected diagrams......Page 529
One-particle irreducible diagrams......Page 530
Higgs density (Section 5.4)......Page 533
Leptonic density......Page 534
Quark density......Page 535
Spontaneous symmetry breaking (Section 5.4)......Page 537
Parameters......Page 538
R[sub(ξ)]-gauges......Page 539
Time reversal......Page 542
Bosonic fields......Page 544
T invariance and noninvariance......Page 545
Charge conjugation......Page 546
C invariance and violation......Page 548
CP and CPT......Page 549
The Goldstone theorem......Page 551
Partially conserved axial vector currents (PCAC) and chiral symmetry......Page 553
The process π[sup(0)] → 2γ......Page 554
Lorentz group (Sections 1.5 and 5.1)......Page 558
SU(3) (Section 8.4)......Page 559
Normalization for SU(N) (Section 8.4)......Page 560
Trace and summation theorems (Sections 8.1 and 9.3)......Page 561
Solutions to the massive Dirac equation (Section 6.3 and exercise 6.2)......Page 562
Solutions to massless Dirac equation (Section 6.4)......Page 563
Cross sections (Section 4.3)......Page 564
One-loop integrals in dimensional regularization (Sections 9.3 and 9.4)......Page 565
Feynman rules for scalars, QED and QCD (Sections 3.4, 8.1, 8.4 and 8.5)......Page 566
Lagrange density and vertices......Page 567
References......Page 570
Index......Page 581
Back Cover......Page 592