Quantum Field Theory and Topology

Albert S. Schwarz (auth.)0387547533, 9780387547534, 3540547533

In recent years topology has firmly established itself as an important part of the physicist’s mathematical arsenal. It has many applications, first of all in quantum field theory, but increasingly also in other areas of physics. The main focus of this book is on the results of quantum field theory that are obtained by topological methods. Some aspects of the theory of condensed matter are also discussed. Part I is an introduction to quantum field theory: it discusses the basic Lagrangians used in the theory of elementary particles. Part II is devoted to the applications of topology to quantum field theory. Part III covers the necessary mathematical background in summary form. The book is aimed at physicists interested in applications of topology to physics and at mathematicians wishing to familiarize themselves with quantum field theory and the mathematical methods used in this field. It is accessible to graduate students in physics and mathematics.

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Table of contents :
Front Matter….Pages I-VIII
Introduction….Pages 1-5
Definitions and Notations….Pages 6-9
Front Matter….Pages 11-11
The Simplest Lagrangians….Pages 13-16
Quadratic Lagrangians….Pages 17-18
Internal Symmetries….Pages 19-23
Gauge Fields….Pages 24-27
Particles Corresponding to Nonquadratic Lagrangians….Pages 28-29
Lagrangians of Strong, Weak and Electromagnetic Interactions….Pages 30-36
Grand Unifications….Pages 37-39
Front Matter….Pages 41-41
Topologically Stable Defects….Pages 43-55
Topological Integrals of Motion….Pages 56-61
A Two-Dimensional Model. Abrikosov Vortices….Pages 62-67
’t Hooft—Polyakov Monopoles….Pages 68-73
Topological Integrals of Motion in Gauge Theory….Pages 74-79
Particles in Gauge Theories….Pages 80-82
The Magnetic Charge….Pages 83-88
Electromagnetic Field Strength and Magnetic Charge in Gauge Theories….Pages 89-93
Extrema of Symmetric Functionals….Pages 94-96
Symmetric Gauge Fields….Pages 97-103
Estimates of the Energy of a Magnetic Monopole….Pages 104-108
Front Matter….Pages 41-41
Topologically Non-Trivial Strings….Pages 109-114
Particles in the Presence of Strings….Pages 115-121
Nonlinear Fields….Pages 122-127
Multivalued Action Integrals….Pages 128-131
Functional Integrals….Pages 132-137
Applications of Functional Integrals to Quantum Theory….Pages 138-145
Quantization of Gauge Theories….Pages 146-157
Elliptic Operators….Pages 158-162
The Index and Other Properties of Elliptic Operators….Pages 163-168
Determinants of Elliptic Operators….Pages 169-172
Quantum Anomalies….Pages 173-182
Instantons….Pages 183-193
The Number of Instanton Parameters….Pages 194-198
Computation of the Instanton Contribution….Pages 199-206
Functional Integrals for a Theory Containing Fermion Fields….Pages 207-215
Instantons in Quantum Chromodynamics….Pages 216-220
Front Matter….Pages 221-221
Topological Spaces….Pages 223-224
Groups….Pages 225-228
Gluings….Pages 229-232
Equivalence Relations and Quotient Spaces….Pages 233-234
Front Matter….Pages 221-221
Group Representations….Pages 235-240
Group Actions….Pages 241-244
The Adjoint Representation of a Lie Group….Pages 245-246
Elements of Homotopy Theory….Pages 247-256
Applications of Topology to Physics….Pages 257-259
Back Matter….Pages 261-276