| Description |
1 online resource (x, 382 pages) |
| Contents |
1. Differential calculus on fibre bundles. 1.1. Geometry of fibre bundles. 1.2. Jet manifolds. 1.3. Connections on fibre bundles. 1.4. Composite bundles. 1.5. Higher order jet manifolds. 1.6. Differential operators and equations. 1.7. Infinite order jet formalism -- 2. Lagrangian field theory on fibre bundles. 2.1. Variational bicomplex. 2.2. Lagrangian symmetries. 2.3. Gauge symmetries. 2.4. First order Lagrangian field theory -- 3. Grassmann-graded Lagrangian field theory. 3.1. Grassmann-graded algebraic calculus. 3.2. Grassmann-graded differential calculus. 3.3. Geometry of graded manifolds. 3.4. Grassmann-graded variational bicomplex. 3.5. Lagrangian theory of even and odd fields -- 4. Lagrangian BRST theory. 4.1. Noether identities. The Koszul-Tate complex. 4.2. Second Noether theorems in a general setting. 4.3. BRST operator. 4.4. BRST extended Lagrangian field theory -- 5. Gauge theory on principal bundles. 5.1. Geometry of Lie groups. 5.2. Bundles with structure groups. 5.3. Principal bundles. 5.4. Principal connections. Gauge fields. 5.5. Canonical principal connection. 5.6. Gauge transformations. 5.7. Geometry of associated bundles. Matter fields. 5.8. Yang-Mills gauge theory. 5.9. Yang-Mills supergauge theory. 5.10. Reduced structure. Higgs fields -- 6. Gravitation theory on natural bundles. 6.1. Natural bundles. 6.2. Linear world connections. 6.3. Lorentz reduced structure. Gravitational field. 6.4. Space-time structure. 6.5. Gauge gravitation theory. 6.6. Energy-momentum conservation law -- 7. Spinor fields. 7.1. Clifford algebras and Dirac spinors. 7.2. Dirac spinor structure. 7.3. Universal spinor structure. 7.4. Dirac fermion fields -- 8. Topological field theories. 8.1. Topological characteristics of principal connections. 8.2. Chern-Simons topological field theory. 8.3. Topological BF theory. 8.4. Lagrangian theory of submanifolds -- 9. Covariant Hamiltonian field theory. 9.1. Polysymplectic Hamiltonian formalism. 9.2. Associated Hamiltonian and Lagrangian systems. 9.3. Hamiltonian conservation laws. 9.4. Quadratic Lagrangian and Hamiltonian systems. 9.5. Example. Yang-Mills gauge theory. 9.6. Variation Hamilton equations. Jacobi fields -- 10. Appendixes. 10.1. Commutative algebra. 10.2. Differential operators on modules. 10.3. Homology and cohomology of complexes. 10.4. Cohomology of groups. 10.5. Cohomology of Lie algebras. 10.6. Differential calculus over a commutative ring. 10.7. Sheaf cohomology. 10.8. Local-ringed spaces. 10.9. Cohomology of smooth manifolds. 10.10. Leafwise and fibrewise cohomology |
| Summary |
Contemporary quantum field theory is mainly developed as quantization of classical fields. Therefore, classical field theory and its BRST extension is the necessary step towards quantum field theory. This book aims to provide a complete mathematical foundation of Lagrangian classical field theory and its BRST extension for the purpose of quantization. Based on the standard geometric formulation of theory of nonlinear differential operators, Lagrangian field theory is treated in a very general setting. Reducible degenerate Lagrangian theories of even and odd fields on an arbitrary smooth manifold are considered. The second Noether theorems generalized to these theories and formulated in the homology terms provide the strict mathematical formulation of BRST extended classical field theory. The most physically relevant field theories - gauge theory on principal bundles, gravitation theory on natural bundles, theory of spinor fields and topological field theory - are presented in a complete way. This book is designed for theoreticians and mathematical physicists specializing in field theory. The authors have tried throughout to provide the necessary mathematical background, thus making the exposition self-contained |
| Bibliography |
Includes bibliographical references (pages 359-367) and index |
| Notes |
Print version record |
| Subject |
Field theory (Physics) -- Mathematics
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Lagrange equations.
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SCIENCE -- Waves & Wave Mechanics.
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Field theory (Physics) -- Mathematics
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Lagrange equations
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| Form |
Electronic book
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| Author |
Mangiarotti, L
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Sardanashvili, G. A. (Gennadiĭ Aleksandrovich)
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World Scientific (Firm)
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| ISBN |
9789812838964 |
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9812838961 |
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