Introduction to Quantum Field Theory

By: V. G. Kiselev, Y. M. Shnir, A. Ya Tregubovich

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Hardcover | 17 November 2000 | Edition Number 1

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456 Pages

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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.

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Preface	p. xiii
The Path Integral in Quantum Mechanics	p. 1
Action in Classical Mechanics	p. 3
The Variational Principle and Equations of Motion	p. 3
A Mathematical Note: The Notion of the Functional	p. 6
The Action as a Function of The Boundary Conditions	p. 9
Symmetries of the Action and Conservation Laws	p. 13
The Path Integral in Quantum Mechanics	p. 17
The Green Function of the Schrodinger Equation	p. 17
The Path Integral	p. 21
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
The Path Integral for the Harmonic Oscillator	p. 31
Imaginary Time and the Ground State Energy	p. 33
The Euclidean Path Integral	p. 39
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
A Particle in a Periodic Potential. Band Structure	p. 61
A Particle on a Circle	p. 65
Conclusions	p. 67
Introduction to Quantum Field Theory	p. 71
Classical and Quantum Fields	p. 73
From Large Number of Degrees of Freedom to Particles	p. 73
Energy-Momentum Tensor	p. 77
Field Quantization	p. 79
Canonical Quantization	p. 79
Quantization via Path Integrals	p. 81
The Equivalence of QFT and Statistical Physics	p. 83
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
Vacuum Energy in [phi superscript 4] Theory	p. 99
Casimir Effect	p. 100
Simple Calculation of Casimir Energy	p. 100
Casimir Energy: Calculation via Path Integral	p. 103
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
The Effective Action in [phi superscript 4] Theory	p. 131
Correlation Functions and the Generating Functional	p. 132
Z[J], W[J] and Correlation Functions of the Free Field	p. 135
The Classical Green Function	p. 136
Correlation Functions	p. 138
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
Effective Action	p. 148
Expansion of the Functional Determinant	p. 151
Renormalization of the Effective Action	p. 159
Momentum Space	p. 159
Explicit Form of the Diagrams	p. 162
The Structure of Ultraviolet Divergencies	p. 165
Pauli-Villars Regularization	p. 168
Calculation of Integrals	p. 171
About Dimensional Regularization	p. 173
The Regularized Inverse Propagator	p. 174
Analytic Continuation to Minkowski Space	p. 176
Renormalization	p. 179
Renormalization of Mass	p. 179
Renormalization of the Coupling Constant	p. 181
Renormalization of the Wave Function	p. 182
Conclusion	p. 185
Renormalization Group	p. 189
Renormalization Group	p. 189
Renormalization Group Equation	p. 189
General Solution of RG Equation	p. 192
Explicit Example	p. 195
Scale Transformations	p. 197
Scale Transformations at the Tree Level	p. 197
Gell-Mann--Low Equation	p. 199
Asymptotic Regimes	p. 200
Concluding Remarks	p. 207
Correlators in Terms of [Gamma phi]	p. 207
On the Properties of Perturbation Series	p. 210
On the Loop Expansion Parameter	p. 210
On the Asymptotic Nature of Perturbation Series	p. 214
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
Conclusion	p. 221
More Complex Fields and Objects	p. 225
Second Quantisation: From Particles to Fields	p. 227
Identical Particles and Symmetry of Wave Functions	p. 227
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
Fermions	p. 240
One-Particle Hamiltonian	p. 240
Creation and Annihilation Operators	p. 241
Many-Particle Hamiltonian	p. 242
Field Operator	p. 242
Path Integral For Fermions	p. 247
On the Formal Classical Limit for Fermions	p. 247
Grassmann Algebras: A Short Introduction	p. 249
Path Integral For Non-Relativistic Fermions	p. 258
Classical Pseudomechanics	p. 259
Path Integral Quantisation	p. 263
Generating Functional For Fermionic Fields	p. 267
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
Fermion Contribution to the Effective Potential	p. 281
Gauge Fields	p. 289
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
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
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
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
Hamiltonian Quantization of Yang-Mills Fields	p. 322
Quantization of Gauge Fields: Faddeev-Popov Method	p. 330
Coleman-Weinberg Effect	p. 333
Topological Objects in Field Theory	p. 343
Kink in 1 + 1 Dimensions	p. 344
A Few Words about Solitons	p. 347
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
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
SU(2) Instanton	p. 375
Nontrivial Solution	p. 375
On the Vacuum Structure of Yang-Mills Theory	p. 379
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
Some Integrals and Products	p. 405
Gaussian integrals	p. 405
Calculation of [Pi subscript n] (1 - x[superscript 2]/n[superscript 2]-[pi superscript 2]	p. 406
Calculation of [characters not reproducible] dx/x ln (1 - x)	p. 408
Calculation of [characters not reproducible] dx/x[superscript 2]+a[superscript 2] ln(1 + x[superscript 2])	p. 409
Feynman Parametrization	p. 411
Splitting of Energy Levels in Double-Well Potential	p. 413
Lie Algebras	p. 417
Elementary Definitions	p. 417
Examples of Lie Algebras	p. 419
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
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
Table of Contents provided by Syndetics. All Rights Reserved.

Introduction to Quantum Field Theory

At a Glance

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