Introduction to quantum field theory

This text explains the features of quantum and statistical field systems that result from their field-theoretic nature and are common to different physical contexts. It supplies the practical tools for carrying out calculations and discusses the meaning of the results. The central concept is that of effective action (or free energy), and the main technical tool is the path integral, although other formalisms are also mentioned. The author emphasizes the simplest models first, then progresses to discussions of real systems before addressing more general and rigorous conclusions. The book is structured around carefully selected problems, which are solved in detail.

Author Notes

Valerij G. Kiselev is a scientist at the Section of Medical Physics, Department of Diagnostic Radiology, University of Freiburg (Germany).
Yakov Shnir is a leading scientist at the Institute for Theoretical Physics, University of Cologne (Germany)
Arthur Ya. Tregubovich is a senior scientist at the Institute of Physics, National Academy of Sciences of Belarus

Preface	p. xiii
I The Path Integral in Quantum Mechanics	p. 1
1 Action in Classical Mechanics	p. 3
1.1 The Variational Principle and Equations of Motion	p. 3
1.2 A Mathematical Note: The Notion of the Functional	p. 6
1.3 The Action as a Function of The Boundary Conditions	p. 9
1.4 Symmetries of the Action and Conservation Laws	p. 13
2 The Path Integral in Quantum Mechanics	p. 17
2.1 The Green Function of the Schrodinger Equation	p. 17
2.2 The Path Integral	p. 21
2.3 The Path Integral for Free Motion	p. 25
Free Motion: Straightforward Calculation of the Path Integral	p. 26
Free Motion: Path Integral Calculation by the Stationary Phase Method	p. 27
2.4 The Path Integral for the Harmonic Oscillator	p. 31
2.5 Imaginary Time and the Ground State Energy	p. 33
3 The Euclidean Path Integral	p. 39
3.1 The Symmetric Double Well	p. 39
Quantum Mechanical Instantons	p. 42
The Contribution from the Vicinity of the Instanton Trajectory	p. 49
Calculation of the Functional Determinant	p. 53
3.2 A Particle in a Periodic Potential. Band Structure	p. 61
3.3 A Particle on a Circle	p. 65
3.4 Conclusions	p. 67
II Introduction to Quantum Field Theory	p. 71
4 Classical and Quantum Fields	p. 73
4.1 From Large Number of Degrees of Freedom to Particles	p. 73
4.2 Energy-Momentum Tensor	p. 77
4.3 Field Quantization	p. 79
Canonical Quantization	p. 79
Quantization via Path Integrals	p. 81
4.4 The Equivalence of QFT and Statistical Physics	p. 83
4.5 Free Field Quantization: From Fields to Particles	p. 86
Momentum Space	p. 86
Normal Modes	p. 88
Zero-Point Energy	p. 89
Elementary Excitations of the Field	p. 91
5 Vacuum Energy in [phi superscript 4] Theory	p. 99
5.1 Casimir Effect	p. 100
Simple Calculation of Casimir Energy	p. 100
Casimir Energy: Calculation via Path Integral	p. 103
5.2 Effective Potential of [phi superscript 4] Theory	p. 107
Calculation of U[subscript eff] ([phi])	p. 109
The Explicit Form of U[subscript eff]	p. 111
Renormalization of Mass and Coupling Constant	p. 113
Running Coupling Constant, Dimensional Transmutation and Anomalous Dimensions	p. 117
Effective Potential of the Massive Theory	p. 124
6 The Effective Action in [phi superscript 4] Theory	p. 131
6.1 Correlation Functions and the Generating Functional	p. 132
6.2 Z[J], W[J] and Correlation Functions of the Free Field	p. 135
The Classical Green Function	p. 136
Correlation Functions	p. 138
6.3 Generating Functionals in [phi superscript 4] Theory	p. 141
[phi superscript 4] Theory	p. 141
Generating Functionals: Expansion in [lambda]	p. 141
Generating Functionals: the Loop Expansion	p. 146
6.4 Effective Action	p. 148
Expansion of the Functional Determinant	p. 151
7 Renormalization of the Effective Action	p. 159
7.1 Momentum Space	p. 159
Explicit Form of the Diagrams	p. 162
7.2 The Structure of Ultraviolet Divergencies	p. 165
7.3 Pauli-Villars Regularization	p. 168
Calculation of Integrals	p. 171
About Dimensional Regularization	p. 173
7.4 The Regularized Inverse Propagator	p. 174
Analytic Continuation to Minkowski Space	p. 176
7.5 Renormalization	p. 179
Renormalization of Mass	p. 179
Renormalization of the Coupling Constant	p. 181
Renormalization of the Wave Function	p. 182
7.6 Conclusion	p. 185
8 Renormalization Group	p. 189
8.1 Renormalization Group	p. 189
Renormalization Group Equation	p. 189
General Solution of RG Equation	p. 192
Explicit Example	p. 195
8.2 Scale Transformations	p. 197
Scale Transformations at the Tree Level	p. 197
Gell-Mann--Low Equation	p. 199
8.3 Asymptotic Regimes	p. 200
9 Concluding Remarks	p. 207
9.1 Correlators in Terms of [Gamma phi]	p. 207
9.2 On the Properties of Perturbation Series	p. 210
On the Loop Expansion Parameter	p. 210
On the Asymptotic Nature of Perturbation Series	p. 214
9.3 On [phi superscript 4] Theory with Large Coupling Constant	p. 219
The Cases d = 2 and d = 3: Second-Order Phase transitions	p. 219
The Cases d = 4: Possible Triviality of [phi superscript 4] Theory	p. 220
9.4 Conclusion	p. 221
III More Complex Fields and Objects	p. 225
10 Second Quantisation: From Particles to Fields	p. 227
10.1 Identical Particles and Symmetry of Wave Functions	p. 227
10.2 Bosons	p. 230
One-Particle Hamiltonian	p. 231
Creation and Annihilation Operators	p. 233
Total Hamiltonian	p. 235
The Field Operator	p. 236
Result: Recipe for Quantisation	p. 238
10.3 Fermions	p. 240
One-Particle Hamiltonian	p. 240
Creation and Annihilation Operators	p. 241
Many-Particle Hamiltonian	p. 242
Field Operator	p. 242
11 Path Integral For Fermions	p. 247
11.1 On the Formal Classical Limit for Fermions	p. 247
11.2 Grassmann Algebras: A Short Introduction	p. 249
11.3 Path Integral For Non-Relativistic Fermions	p. 258
Classical Pseudomechanics	p. 259
Path Integral Quantisation	p. 263
11.4 Generating Functional For Fermionic Fields	p. 267
11.5 Coupling of the Dirac Spinor and the [phi superscript 4] Scalar Fields	p. 272
Loop Expansion and Diagram Techniques	p. 273
Analysis of Divergences	p. 278
11.6 Fermion Contribution to the Effective Potential	p. 281
12 Gauge Fields	p. 289
12.1 Gauge Invariance	p. 289
The Basic Idea	p. 289
Example of a Globally Invariant Lagrangian	p. 290
Example of a Locally Invariant Lagrangian	p. 292
Lagrangian of Gauge Fields	p. 293
12.2 Dynamics of Gauge Invariant Fields	p. 298
Equations of Motion	p. 298
The Yang-Mills Equations	p. 299
The Total Energy	p. 300
Gauge Freedom and Gauge Conditions	p. 301
12.3 Spontaneously Broken Symmetry	p. 304
Vacuum and its Structure	p. 304
Goldstone Modes and Higgs Mechanism	p. 305
Elimination of Goldstone Modes. Goldstone Theorem	p. 307
Examples	p. 308
12.4 Quantization of Systems With Constraints	p. 310
Primary Constraints	p. 310
On Constrained Mechanical Systems	p. 312
Secondary Constraints	p. 312
The Matrix of Poisson Brackets	p. 313
First and Second Order Constraints	p. 314
Quantization	p. 317
Examples	p. 319
12.5 Hamiltonian Quantization of Yang-Mills Fields	p. 322
12.6 Quantization of Gauge Fields: Faddeev-Popov Method	p. 330
12.7 Coleman-Weinberg Effect	p. 333
13 Topological Objects in Field Theory	p. 343
13.1 Kink in 1 + 1 Dimensions	p. 344
13.2 A Few Words about Solitons	p. 347
13.3 Abrikosov Vortex	p. 350
Ginzburg-Landau Model of Superconductivity	p. 351
Nontrivial Solution	p. 352
Aharonov-Bohm Effect	p. 356
A Few Words about Topology and an Exotic String	p. 357
Vortex Solution in Other Contexts	p. 363
13.4 The 't Hooft-Polyakov Monopole	p. 364
Magnetic Properties of the Solution	p. 366
Lower Boundary on the Monopole Mass	p. 368
Dyons	p. 370
A Few Words About the Topology	p. 371
Do Monopoles Exist?	p. 373
13.5 SU(2) Instanton	p. 375
Nontrivial Solution	p. 375
On the Vacuum Structure of Yang-Mills Theory	p. 379
13.6 Quantum Kink	p. 383
Quantum Correction to the Mass of the Kink	p. 385
Physical Contents of Fluctuations around the Kink	p. 389
Elimination of Zero Mode	p. 391
Generating Functional	p. 395
A Some Integrals and Products	p. 405
A.1 Gaussian integrals	p. 405
A.2 Calculation of [Pi subscript n] (1 - x[superscript 2]/n[superscript 2]-[pi superscript 2]	p. 406
A.3 Calculation of [characters not reproducible] dx/x ln (1 - x)	p. 408
A.4 Calculation of [characters not reproducible] dx/x[superscript 2]+a[superscript 2] ln(1 + x[superscript 2])	p. 409
A.5 Feynman Parametrization	p. 411
B Splitting of Energy Levels in Double-Well Potential	p. 413
C Lie Algebras	p. 417
C.1 Elementary Definitions	p. 417
C.2 Examples of Lie Algebras	p. 419
C.3 The Idea of Classification. Levi-Maltsev Decomposition	p. 420
The Adjoint Representation	p. 420
Solvable and Nilpotent Algebras	p. 421
Reductive and Semisimple Algebras	p. 422
3.4 Classification of Complex Semisimple Lie Algebras	p. 424
The Cartan Subalgebra. Roots	p. 424
Properties of Roots. Cartan-Weyl Basis	p. 425
Cartan Matrix. Dynkin Schemes	p. 427
Compact Algebras	p. 429
Index	p. 432

Available:*

On Order

Summary

Summary

Author Notes

Table of Contents