The Schwinger action principle and effective action / David J. Toms.

Format:

Book

Author/Creator:

Toms, David J., 1953-

Contributor:

Emma Louise McClellan Fund.

Series:

Cambridge monographs on mathematical physics

Language:

English

Subjects (All):

Schwinger action principle.

Quantum theory.

Mathematical physics.

Physical Description:

xi, 495 pages : illustrations ; 26 cm.

Place of Publication:

Cambridge, UK ; New York : Cambridge University Press, 2007.

Summary:

This book is an introduction to the Schwinger action principle in quantum mechanics and quantum field theory, with applications to a variety of different models, not only those of interest to particle physics. The book begins with a brief review of the action principle in classical mechanics and classical field theory. It then moves on to quantum field theory, focusing on the effective action method. This is introduced as simply as possible by using the zero-point energy of the simple harmonic oscillator as the starting point. This allows the utility of the method, and the process of regularization and renormalization of quantum field theory, to be demonstrated with the minimum of formal development. The book concludes with a more complete definition of the effective action, and demonstrates how the provisional definition used earlier is the first term in the systematic loop expansion.

Several applications of the Schwinger action principle are given, including Bose-Einstein condensation, the Casimir effect and trapped Fermi gases. The renormalization of interacting scalar field theory is presented to two-loop order. This book will interest graduate students and researchers in theoretical physics who are familiar with quantum mechanics.

Contents:

1 Action principle in classical mechanics 1

1.1 Euler-Lagrange equations 1

1.2 Hamilton's principle 5

1.3 Hamilton's equations 8

1.4 Canonical transformations 12

1.5 Conservation laws and symmetries 22

2 Action principle in classical field theory 40

2.1 Continuous systems 40

2.2 Lagrangian and Hamiltonian formulation for continuous systems 43

2.4 Functional differentiation and Poisson brackets for field theory 54

2.5 Noether's theorem 60

2.6 The stress-energy-momentum tensor 66

2.7 Gauge invariance 75

2.8 Fields of general spin 83

2.9 The Dirac equation 88

3 Action principle in quantum theory 100

3.1 States and observables 100

3.2 Schwinger action principle 111

3.3 Equations of motion and canonical commutation relations 113

3.4 Position and momentum eigenstates 120

3.5 Simple harmonic oscillator 124

3.6 Real scalar field 132

3.7 Complex scalar field 140

3.8 Schrodinger field 144

3.9 Dirac field 151

3.10 Electromagnetic field 157

4 The effective action 169

4.2 Free scalar field in Minkowski spacetime 174

4.3 Casimir effect 178

4.4 Constant gauge field background 181

4.5 Constant magnetic field 186

4.6 Self-interacting scalar field 196

4.7 Local Casimir effect 203

5 Quantum statistical mechanics 208

5.2 Simple harmonic oscillator 213

5.3 Real scalar field 217

5.4 Charged scalar field 221

5.5 Non-relativistic field 228

5.6 Dirac field 234

5.7 Electromagnetic field 235

6 Effective action at finite temperature 238

6.1 Condensate contribution 238

6.2 Free homogeneous non-relativistic Bose gas 241

6.3 Internal energy and specific heat 245

6.4 Bose gas in a harmonic oscillator confining potential 247

6.5 Density of states method 258

6.6 Charged non-relativistic Bose gas in a constant magnetic field 267

6.7 The interacting Bose gas 278

6.8 The relativistic non-interacting charged scalar field 289

6.9 The interacting relativistic field 293

6.10 Fermi gases at finite temperature in a magnetic field 298

6.11 Trapped Fermi gases 309

7 Further applications of the Schwinger action principle 323

7.1 Integration of the action principle 323

7.2 Application of the action principle to the free particle 325

7.3 Application to the simple harmonic oscillator 329

7.4 Application to the forced harmonic oscillator 332

7.5 Propagators and energy levels 337

7.6 General variation of the Lagrangian 344

7.7 The vacuum-to-vacuum transition amplitude 348

7.8 More general systems 352

8 General definition of the effective action 368

8.1 Generating functionals for free field theory 368

8.2 Interacting fields and perturbation theory 374

8.3 Feynman diagrams 383

8.4 One-loop effective potential for a real scalar field 388

8.5 Dimensional regularization and the derivative expansion 394

8.6 Renormalization of [lambda phi superscript 4] theory 403

8.7 Finite temperature 415

8.8 Generalized CJT effective action 425

8.9 CJT approach to Bose-Einstein condensation 433

Appendix 1 Mathematical appendices 447

Appendix 2 Review of special relativity 462

Appendix 3 Interaction picture 469.

Notes:

Includes bibliographical references (pages 479-485) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Emma Louise McClellan Fund.

ISBN:

0521876761

9780521876766

OCLC:

122338040

Online:

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