Advanced Classical Field Theory

Giachetta, G.; Mangiarotti, L.; Sardanashvily, G.

Авторы: Giachetta G., Mangiarotti L., Sardanashvily G.
Год издания: 2009
Издательство: WORLD SCIENTIFIC PUBL CO PTE LTD
Местоположение издательства: 5 TOH TUCK LINK, SINGAPORE,SINGAPORE, 596224
Объём: 382 страниц
ISBN: 10 981-283-895--3
Аннотация: Preface Contemporary quantum ﬁeld theory is mainly developed as quantization of classical ﬁelds. Classical ﬁeld theory thus is a necessary step towards quantum ﬁeld theory. This book provides an exhaust mathematical foundation of Lagrangian classical ﬁeld theory and its BRST extension for the purpose of quantization. Lagrangian theory of Grassmann-graded (even and odd) ﬁelds on ﬁbre bundles and graded manifolds is presented in the book in a very general setting. It is adequately formulated in geometric and algebraic topological terms of the jet manifolds and the variational bicomplex. The main ingredients in this formulation are cohomology of the variational bicomplex, the global ﬁrst variational formula, variational symmetries and supersymmetries, the ﬁrst Noether theorem, Noether identities, the direct and inverse second Noether theorems, and gauge symmetries. Degenerate Lagrangian ﬁeld theories are comprehensively investigated. The hierarchies of their non-trivial reducible Noether identities and gauge symmetries are described in homology terms. The relevant direct and inverse second Noether theorems are formulated in a very general setting. The study of degeneracy of Lagrangian ﬁeld theory straightforwardly leads to its BRST extension by Grassmann-graded antiﬁelds and ghosts which constitute the chain and cochain complexes of non-trivial Noether identities and gauge symmetries. In particular, a gauge operator is prolonged to a nilpotent BRST operator, and an original ﬁeld Lagrangian is extended to a non-trivial solution of the classical master equation of Lagrangian BRST theory. This is a preliminary step towards quantization of classical Lagrangian ﬁeld theory in terms of functional integrals. The basic ﬁeld theories, including gauge theory on principal bundles, gravitation theory on natural bundles, theory of spinor ﬁelds and topological ﬁeld theory, are presented in the book in a very complete way. Our book addresses to a wide audience of theoreticians and mathematical physicists, and aims to be a guide to advanced diﬀerential geometric and algebraic topological methods in ﬁeld theory. With respect to mathematical prerequisites, the reader is expected to be familiar with the basics of diﬀerential geometry of ﬁbre bundles. We have tried to give the necessary mathematical background, thus making the exposition self-contained. For the sake of convenience of the reader, several relevant mathematical topics are compiled in Appendixes. Contents. Preface. Introduction. 1. Diﬀerential calculus on ﬁbre bundles. 1.1 Geometry of ﬁbre bundles. 1.1.1 Manifold morphisms. 1.1.2 Fibred manifolds and ﬁbre bundles. 1.1.3 Vector and aﬃne bundles. 1.1.4 Vector ﬁelds, distributions and foliations. 1.1.5 Exterior and tangent-valued forms. 1.2 Jet manifolds. 1.3 Connections on ﬁbre bundles. 1.3.1 Connections as tangent-valued forms. 1.3.2 Connections as jet bundle sections. 1.3.3 Curvature and torsion. 1.3.4 Linear connections. 1.3.5 Aﬃne connections. 1.3.6 Flat connections. 1.3.7 Second order connections. 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. 2.4.1 Cartan and Hamilton–De Donder equations. 2.4.2 Lagrangian conservation laws. 2.4.3 Gauge conservation laws. Superpotential. 2.4.4 Non-regular quadratic Lagrangians. 2.4.5 Reduced second order Lagrangians. 2.4.6 Background ﬁelds. 2.4.7 Variation Euler–Lagrange equation. Jacobi ﬁelds. 2.5 Appendix. Cohomology of the variational bicomplex. 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. 3.6 Appendix. Cohomology of the Grassmann-graded variational bicomplex. 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. 4.5 Appendix. Noether identities of diﬀerential operators. 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. 5.8.1 Gauge ﬁeld Lagrangian. 5.8.2 Conservation laws. 5.8.3 BRST extension. 5.8.4 Matter ﬁeld Lagrangian. 5.9 Yang–Mills supergauge theory. 5.10 Reduced structure. Higgs ﬁelds. 5.10.1 Reduction of a structure group. 5.10.2 Reduced subbundles. 5.10.3 Reducible principal connections. 5.10.4 Associated bundles. Matter and Higgs ﬁelds. 5.10.5 Matter ﬁeld Lagrangian. 5.11 Appendix. Non-linear realization of Lie algebras. 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. 6.7 Appendix. Aﬃne world connections. 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. 8.1.1 Characteristic classes of principal connections. 8.1.2 Flat principal connections. 8.1.3 Chern classes of unitary principal connections. 8.1.4 Characteristic classes of world 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. Bibliography. Index
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