Quantum Field Theory And Topology Kvantovai︠a︡ Teorii︠a︡ Poli︠a︡ I Topologii︠a︡. English

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. By Albert S. Schwarz. 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 1. The Simplest Lagrangians -- 2. Quadratic Lagrangians -- 3. Internal Symmetries -- 4. Gauge Fields -- 5. Particles Corresponding To Nonquadratic Lagrangians -- 6. Lagrangians Of Strong, Weak And Electromagnetic Interactions -- 7. Grand Unifications -- 8. Topologically Stable Defects -- 9. Topological Integrals Of Motion -- 10. A Two-dimensional Model. Abrikosov Vortices -- 11. 't Hooft-polyakov Monopoles -- 12. Topological Integrals Of Motion In Gauge Theory -- 13. Particles In Gauge Theories -- 14. The Magnetic Charge -- 15. Electromagnetic Field Strength And Magnetic Charge In Gauge Theories -- 16. Extrema Of Symmetric Functionals -- 17. Symmetric Gauge Fields -- 18. Estimates Of The Energy Of A Magnetic Monopole -- 19. Topologically Non-trivial Strings -- 20. Particles In The Presence Of Strings -- 21. Nonlinear Fields -- 22. Multivalued Action Integrals -- 23. Functional Integrals -- 24. Applications Of Functional Integrals To Quantum Theory -- 25. Quantization Of Gauge Theories -- 26. Elliptic Operators -- 27. The Index And Other Properties Of Elliptic Operators -- 28. Determinants Of Elliptic Operators -- 29. Quantum Anomalies -- 30. Instantons -- 31. The Number Of Instanton Parameters -- 32. Computation Of The Instanton Contribution -- 33. Functional Integrals For A Theory Containing Fermion Fields -- 34. Instantons In Quantum Chromodynamics -- 35. Topological Spaces -- 36. Groups -- 37. Gluings -- 38. Equivalence Relations And Quotient Spaces -- 39. Group Representations -- 40. Group Actions -- 41. The Adjoint Representation Of A Lie Group -- 42. Elements Of Homotopy Theory -- 43. Applications Of Topology To Physics. Albert Schwarz. Translation Of: Kvantovai︠a︡ Teorii︠a︡ Poli︠a︡ I Topologii︠a︡. Includes Bibliographical References (p. [263]-267 And Index.

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.