| Description |
1 online resource |
| Contents |
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 [symbol]. 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. Čech cohomology |
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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. Kähler 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 |
| Summary |
Geometrical notions and methods play an important role in both classical and quantum field theory, and a connection is a deep structure which apparently underlies the gauge-theoretical models. This collection of basic mathematical facts about various types of connections provides a detailed description of the relevant physical applications. It discusses the modern issues concerning the gauge theories of fundamental interactions. This text presents several levels of complexity, from the elementary to the advanced, and provides a considerable number of exercises. The authors have tried to give all the necessary mathematical background, thus making the book self-contained. This book should be useful to graduate students, physicists and mathematicians who are interested in the issue of deep interrelations between theoretical physics and geometry |
| Bibliography |
Includes bibliographical references and index |
| Subject |
Field theory (Physics) -- Mathematics
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Quantum field theory -- Mathematics
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Connections (Mathematics)
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Field theory (Physics)
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SCIENCE -- Waves & Wave Mechanics.
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Connections (Mathematics)
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Field theory (Physics)
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Field theory (Physics) -- Mathematics
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Quantum field theory -- Mathematics
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| Form |
Electronic book
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| Author |
Sardanashvili, G. A. (Gennadiĭ Aleksandrovich)
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| ISBN |
9789812813749 |
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9812813748 |
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