Path integrals and quantum processes / Mark S. Swanson, Department of Physics, University of Connecticut at Stamford.

Format:

Book

Author/Creator:

Swanson, Mark S., 1947- author.

Language:

English

Subjects (All):

Path integrals.

Quantum theory.

Quantum field theory.

Physical Description:

xii, 444 pages ; 24 cm

Place of Publication:

Mineola, New York : Dover Publications, Inc., [2014]

Summary:

This text offers a systematic presentation of the path integral approach to calculating transition elements, partition functions, and source functionals. The treatment contains a set of logical connections and applications which represent the scope of problems that the path integral can solve. Readers receive sufficient background to advance to related books and articles at the frontiers of theoretical physics. An introductory section covers mathematical preliminaries, progressing to examinations of quantum mechanical path integrals, an evaluation of the path integral, and an exploration of further applications. Subsequent chapters cover Grassmann variables, field theory and gauge field theory, perturbation theory, and nonperturbative results. Suitable for advanced undergraduates and graduate students of physics, the text requires only some familiarity with quantum mechanics. Exercises and lists of references throughout the book make it ideal for supplementary reading and self-study. Book jacket.

Contents:

Chapter 1 Mathematical Preliminaries 1

1.1 The Dirac Delta 2

1.2 Completeness 3

1.3 Functionals 9

1.4 Matrices 15

1.5 Gaussian Integrals 22

References 28

Chapter 2 Quantum Mechanical Path Integrals 31

2.1 Quantum Mechanics 31

2.2 The Path Integral Derived 35

2.3 The Sum over Histories 42

References 50

Chapter 3 Evaluating the Path Integral 51

3.1 The Free Particle 52

3.2 Motion with a Source 54

3.3 Continuum Techniques 62

3.4 Topological Measure 67

References 72

Chapter 4 Further Applications 75

4.1 Natural Units 75

4.2 Statistical Mechanics 77

4.3 Symmetry and Generating Functionals 83

4.4 Harmonic Oscillator Coherent States 92

4.5 Spontaneously Broken Symmetry 96

4.6 Constraints 105

References 115

Chapter 5 Grassmann Variables 117

5.1 Basic Definitions 118

5.2 Gaussian Grassmann Integrals 125

5.3 Classical Grassman Mechanics 128

5.4 Grassmann Quantum Mechanics 133

5.4.1 The Free Particle 138

5.4.2 The Harmonic Oscillator 139

5.5 Grassmann Path Integrals 141

5.6 Supersymmetric Quantum Mechanics 146

References 148

Chapter 6 Field Theory 151

6.1 A Mechanical Model 153

6.2 Relativity and Group Theory 158

6.3 Classical Free Fields 170

6.3.1 The Scalar Field 171

6.3.2 Spinor Fields 173

6.4 Symmetry and Noether's Theorem 182

6.4.1 Translational Invariance 186

6.4.2 Lorentz Invariance and Angular Momentum 187

6.4.3 Phase and Chiral Invariance 189

6.4.4 Charges as Symmetry Generators 190

6.5 Canonical Quantization 191

6.5.1 Scalar Field Quantization 191

6.5.2 Dirac Field Quantization 194

6.6 The S-Matrix 197

6.7 The Interaction Picture 204

6.8 The Path Integral for Field Theory 207

6.8.1 The Scalar Field Case 207

6.8.2 The Dirac Field Case 214

6.8.3 Euclidean Measure 218

6.8.4 Configuration Measure 222

References 229

Chapter 7 Gauge Field Theory 231

7.1 The Maxwell Field 232

7.1.1 The Classical Action 232

7.1.2 Gauge Invariance and Gauge Fixing 235

7.1.3 Quantization of the Free Gauge Field 239

7.2 QED as a Path Integral 249

7.3 Lie Algebras 263

7.4 Classical Yang-Mills Fields 270

7.5 Quantized Yang-Mills Fields 276

7.6 Topological Aspects of Gauge Fields 286

References 306

Chapter 8 Perturbation Theory 311

8.1 Generating Functionals 312

8.2 Ward-Takahashi Identities 319

8.3 Deriving the Feynman Rules 325

8.4 Renormalization 341

References 355

Chapter 9 Nonperturbative Results 357

9.1 The Goldstone Theorem 358

9.2 The Effective Potential 360

9.3 The Higgs-Kibble Mechanism 372

9.4 The SU(2)_L × U(1) Electroweak Model 383

9.5 Chiral Anomalies 389

9.6 Classical Solutions 403

9.6.1 The Kink Solution 403

9.6.2 Vacuum Tunnelling 410

9.6.3 Yang-Mills Instantons 416

9.6.4 The Abelian Magnetic Monopole 420

9.7 Applications of the Effective Potential 423

9.7.1 Finite Temperature and Symmetry Restoration 423

9.7.2 The Coleman-Weinberg Mechanism 428

9.7.3 The Gross-Neveu Model 431

References 436.

Notes:

Originally published: New York : Academic Press, 1992.

Includes bibliographical references and index.

ISBN:

0486493067

9780486493060

OCLC:

843532820

Publisher Number:

99958837198