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Connections in Classical and Quantum Field Theoryкнига
- Авторы: Mangiarotti L., Sardanashvily G.
- Год издания: 2000
- Издательство: WORLD SCIENTIFIC PUBL CO PTE LTD
- Местоположение издательства: 5 TOH TUCK LINK, SINGAPORE,SINGAPORE, 596224
- Объём: 504 страниц
- ISBN: 981-02-2013-8
- Аннотация: Preface The present book is based on the graduate and post graduate courses of lectures given at the Department of Theoretical Physics of Moscow State University and the Department of Mathematics and Physics of Camerino University. It is addressed to a wide audience of physicists and mathematicians, and aims at showing in a unified way the role that the concept of a connection plays both in classical and quantum field theory. To our knowledge, this is the first attempt to present connections as a main geometrical object which underlies many relevant physical ideas. The concept of a connection in quantum field theory. is rather new. It is phrased in algebraic terms, in comparison with the purely geometric one used in classical field theory. This concept is based on modern development of quantum mechanics, SUSY models, BRST formalism, and non-commutative field theory. In our opinion, connections provide a new link between classical and quantum physics. In classical field theory, we follow the general notion of a connection on fibre bundles, formulated in terms of jet manifolds. We emphasize the role of connections as the main ingredient in dynamic equation theory, and Lagrangian and Hamiltonian formalisms. As is well-known, connections on principal bundles have a role of gauge potentials of fundamental interactions while, in gravitation theory, connections characterize the space-time geometry. Classical mechanics is also presented as a particular field theory, involving the concept of a connection in many aspects. The reader will find connections in quantum mechanics, too. In fact, we collect together the basic methods concerning different types of connections in quantum field theory. These are superconnections, connections in BRST formalism, topological field theory, theory of anomalies, and non-commutative geometry. Mathematics is not the primary scope of our book, but provides the powerful methods of studying contemporary field models. Contents. Preface. Introduction. 1 Geometric interlude. 1.1 Fibre bundles. 1.2 Differential forms and multivector fields. 1.3 Jet manifolds. 2 Connections. 2.1 Connections as tangent-valued forms. 2.2 Connections as jet bundle sections. 2.3 Curvature and torsion. 2.4 Linear connections. 2.5 Affine connections. 2.6 Flat connections. 2.7 Composite connections. 3 Connections in Lagrangian field theory. 3.1 Connections and dynamic equations. 3.2 The first variational formula. 3.3 Quadratic degenerate Lagrangians. 3.4 Connections and Lagrangian conservation laws. 4 Connections in Hamiltonian field theory. 4.1 Hamiltonian connections and Hamiltonian forms. 4.2 Lagrangian and Hamiltonian degenerate systems. 4.3 Quadratic and affine degenerate systems. 4.4 Connections and Hamiltonian conservation laws. 4.5 The vertical extension of Hamiltonian formalism. 5 Connections in classical mechanics. 5.1 Fibre bundles over R. 5.2 Connections in conservative mechanics. 5.3 Dynamic connections in time-dependent mechanics. 5.4 Non-relativistic geodesic equations. 5.5 Connections and reference frames. 5.6 The free motion equation. 5.7 The relative acceleration. 5.8 Lagrangian and Newtonian systems. 5.9 Non-relativistic Jacobi fields. 5.10 Hamiltonian time-dependent mechanics. 5.11 Connections and energy conservation laws. 5.12 Systems with time-dependent parameters. 6 Gauge theory of principal connections. 6.1 Principal connections. 6.2 The canonical principal connection. 6.3 Gauge conservation laws. 6.4 Hamiltonian gauge theory. 6.5 Geometry of symmetry breaking. 6.6 Effects of flat principal connections. 6.7 Characteristic classes. 6.8 Appendix. Homotopy, homology and cohomology. 6.9 Appendix. Cech cohomology. 7 Space-time connections. 7.1 Linear world connections. 7.2 Lorentz connections. 7.3 Relativistic mechanics. 7.4 Metric-affine gravitation theory. 7.5 Spin connections. 7.6 Affine world connections. 8 Algebraic connections. 8.1 Jets of modules. 8.2 Connections on modules. 8.3 Connections on sheaves. 9 Superconnections. 9.1 Graded tensor calculus. 9.2 Connections on graded manifolds. 9.3 Connections on supervector bundles. 9.4 Principal superconnections. 9.5 Graded principal bundles. 9.6 SUSY-extended field theory. 9.7 The Ne'eman-Quillen superconnection. 9.8 Appendix. K-Theory. 10 Connections in quantum mechanics. 10.1 Kahler manifolds modelled on Hilbert spaces. 10.2 Geometric quantization. 10.3 Deformation quantization. 10.4 Quantum time-dependent evolution. 10.5 Berry connections. 11 Connections in BRST formalism. 11.1 The canonical connection on infinite order jets. 11.2 The variational bicomplex. 11.3 Jets of ghosts and antifields. 11.4 The BRST connection. 12 Topological field theories. 12.1 The space of principle connections. 12.2 Connections on the space of connections. 12.3 Donaldson invariants. 13 Anomalies. 13.1 Gauge anomalies. 13.2 Global anomalies. 13.3 BRST anomalies. 14 Connections in non-commutative geometry. 14.1 Non-commutative algebraic calculus. 14.2 Non-commutative differential calculus. 14.3 Universal connections. 14.4 The Dubois-Violette connection. 14.5 Matrix geometry. 14.6 Connes' differential calculus. Bibliography. Index
- Добавил в систему: Сарданашвили Геннадий Александрович
Прикрепленные файлы
| № | Имя | Описание | Имя файла | Размер | Добавлен |
|---|---|---|---|---|---|
| 1. | Полный текст | book00.pdf | 5,8 МБ | 31 мая 2012 [sardanashvily] |