[PDF]  Advanced Quantum mechanics by Ashok Das  book free download

Advanced Quantum mechanics book PDF

Contents

Preface . . . . . . . . . . . . . . . . . .  . . . . . . . . . vii

1 Relativistic equations . . . . . . . . . . . . . . 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . 1

1.2 Notations. . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Klein-Gordon equation . . .  10

1.3.1 Klein paradox . . . 14

1.4 Dirac equation . . . 19

1.5 References . . . 26

2 Solutions of the Dirac equation . . . 27

2.1 Plane wave solutions . . . 27

2.2 Normalization of the wave function . . . 34

2.3 Spin of the Dirac particle. . . 40

2.4 Continuity equation . . . 44

2.5 Dirac’s hole theory  . . . 47

2.6 Properties of the Dirac matrices  . . . 49

2.6.1 Fierz rearrangement . . . 58

2.7 References . . . 62

3 Properties of the Dirac equation . . . 65

3.1 Lorentz transformations . . . 65

3.2 Covariance of the Dirac equation . .  . 72

3.3 Transformation of bilinears . . . 82

3.4 Projection operators, completeness relation . . . 84

3.5 Helicity .  . . 92

3.6 Massless Dirac particle . . . 94

3.7 Chirality .  . . 99

3.8 Non-relativistic limit of the Dirac equation. . . 105

3.9 Electron in an external magnetic field . . . 107

3.10 Foldy-Wouthuysen transformation. . . 111

3.11 Zitterbewegung . . . 117

3.12 References . . . 122

4 Representations of Lorentz and Poincar´e groups . . . 125

4.1 Symmetry algebras . . . 125

4.1.1 Rotation  . . . 125

4.1.2 Translation  . . . 129

4.1.3 Lorentz transformation . .  . 130

4.1.4 Poincar´e transformation . . . 133

4.2 Representations of the Lorentz group . . . 135

4.2.1 Similarity transformations and representations . . . 140

4.3 Unitary representations of the Poincar´e group . . . 147

4.3.1 Massive representation . .  . 151

4.3.2 Massless representation . . . 155

4.4 References . . . 160

5 Free Klein-Gordon field theory . .  . 161

5.1 Introduction .  . . 161

5.2 Lagrangian density . . . 163

5.3 Quantization.  . . 167

5.4 Field decomposition . . . 171

5.5 Creation and annihilation operators. . . 175

5.6 Energy eigenstates .  . . 186

5.7 Physical meaning of energy eigenstates . . .  190

5.8 Green’s functions . . . 194

5.9 Covariant commutation relations . . . 205

5.10 References . . . 209

6 Self-interacting scalar field theory . . . 211

6.1 N¨other’s theorem . . . 211

6.1.1 Space-time translation . . . 215

6.2 Self-interacting φ

4

theory. . . . 219

6.3 Interaction picture and time evolution operator . . . 223

6.4 S-matrix .  . . 229

6.5 Normal ordered product and Wick’s theorem . . . . 233

6.6 Time ordered products and Wick’s theorem . . . . . 241

6.7 Spectral representation and dispersion relation . . . 246

6.8 References  . . . 254

7 Complex scalar field theory . . . 257

7.1 Quantization.  . . . 257

7.2 Field decomposition.  . . 260

7.3 Charge operator . . . . 263

7.4 Green’s functions .  . . 268

7.5 Spontaneous symmetry breaking and the Goldstone

theorem . .. . . . 270

7.6 Electromagnetic coupling. . . . . 281

7.7 References .  . . 283

8 Dirac field theory. .  . . 285

8.1 Pauli exclusion principle . . . . 285

8.2 Quantization of the Dirac field . . . . 286

8.3 Field decomposition. . . . . 291

8.4 Charge operator . . . . 297

8.5 Green’s function  . . . 300

8.6 Covariant anti-commutation relations. . .  303

8.7 Normal ordered and time ordered products . . . . . 305

8.8 Massless Dirac fields . . . . 308

8.9 Yukawa interaction  . . . 312

8.10 Feynman diagrams . . . . . 318

8.11 References . . . . . 325

9 Maxwell field theory . . . . . 327

9.1 Maxwell’s equations . . . . . 327

9.2 Canonical quantization . . . . 330

9.3 Field decomposition . . . . 335

9.4 Photon propagator . . . . 342

9.5 Quantum electrodynamics . . . . 347

9.6 Physical processes . . . 350

9.7 Ward-Takahashi identity in QED . .  .  355

9.8 Covariant quantization of the Maxwell theory . . . . 360

9.9 References . . . . 376

10 Dirac method for constrained systems . . .  379

10.1 Constrained systems . . . . . 379

10.2 Dirac method and Dirac bracket. . . . 384

10.3 Particle moving on a sphere  . . . . 390

10.4 Relativistic particle . . . . 395

10.5 Dirac field theory . . . . 401

10.6 Maxwell field theory . . . . . 407

10.7 References . . . . 413

11 Discrete symmetries . . . . . 415

11.1 Parity. . .. . . . . . 415

11.1.1 Parity in quantum mechanics . . . . . . . . . . 417

11.1.2 Spin zero field . . . . . . . . . 424

11.1.3 Photon field . .. . . . . 428

11.1.4 Dirac field  . . . . 429

11.2 Charge conjugation . . . . . . . 436

11.2.1 Spin zero field . . . . . . . 437

11.2.2 Dirac field . . .. . . . . 441

11.2.3 Majorana fermions . . . . . .  . . . 449

11.2.4 Eigenstates of charge conjugation . . . . 453

11.3 Time reversal. . . . . 458

11.3.1 Spin zero field and Maxwell’s theory . . . . . . 464

11.3.2 Dirac fields . . . . 467

11.3.3 Consequences of T invariance . . . . . 473

11.3.4 Electric dipole moment of neutron . . . . . . . 477

11.4 CPT theorem . .  . 479

11.4.1 Equality of mass for particles and antiparticles 479

11.4.2 Electric charge for particles and antiparticles . 480

11.4.3 Equality of lifetimes for particles and antipar-ticles . .. . . . . . . 480

11.5 References .. . . . 482

12 Yang-Mills theory . .. . . . 485

12.1 Non-Abelian gauge theories  . . . . . 485

12.2 Canonical quantization of Yang-Mills theory . . . . . 502

12.3 Path integral quantization of gauge theories . . . . . 512

12.4 Path integral quantization of tensor fields . . . . . . 530

12.5 References . . . . . . 542

13 BRST invariance and its consequences . . .. . 545

13.1 BRST symmetry . . . . . . 545

13.2 Covariant quantization of Yang-Mills theory . . . . . 550

13.3 Unitarity . . . . . . 561

13.4 Slavnov-Taylor identity .. . . . 565

13.5 Feynman rules . . . . . 571

13.6 Ghost free gauges. . . . . 578

13.7 References . . . . . 581

14 Higgs phenomenon and the standard model . . . . . . . . . 583

14.1 St¨uckelberg formalism . . . . 583

14.2 Higgs phenomenon . . . . . 589

14.3 The standard model. . . . . . . 596

14.3.1 Field content . .  . . . . . 599

14.3.2 Lagrangian density . . . . . . 601

14.3.3 Spontaneous symmetry breaking . . . . 605

14.4 References . . . . . . . . 616

15 Regularization of Feynman diagrams. . . . . . . . . . . . . 619

15.1 Introduction . .. . . . . 619

15.2 Loop expansion . . . . . . . 621

15.3 Cut-off regularization . . . . . . . 623

15.3.1 Calculation in the Yukawa theory . . . . . . . . 631

15.4 Pauli-Villars regularization  . . . . 638

15.5 Dimensional regularization . . .  647

15.5.1 Calculations in QED . . . . . 656

15.6 References . . . . . . 666

16 Renormalization theory . .  . . . 669

16.1 Superficial degree of divergence . . . . 669

16.2 A brief history of renormalization . . . . . 679

16.3 Schwinger-Dyson equation . . . . . . 690

16.4 BPHZ renormalization .  . . . . . 692

16.5 Renormalization of gauge theories. . . . . 721

16.6 Anomalous Ward identity  . . . 724

16.7 References . . . .. . 732

17 Renormalization group and equation . . . . . . 733

17.1 Gell-Mann-Low equation . . . . 733

17.2 Renormalization group . . . . 739

17.3 Renormalization group equation . . . 744

17.4 Solving the renormalization group equation . . . . . 748

17.5 Callan-Symanzik equation . . . . . 759

17.6 References . . .. . . . 766

Index . .  . . . . . . 769

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