Advanced Classical Field Theory

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. Contents......Page 8 Preface......Page 6 Introduction......Page 12 1.1 Geometry of fibre bundles......Page 16 1.1.1 Manifold morphisms......Page 17 1.1.2 Fibred manifolds and fibre bundles......Page 18 1.1.3 Vector and affine bundles......Page 23 1.1.4 Vector fields, distributions and foliations......Page 29 1.1.5 Exterior and tangent-valued forms......Page 32 1.2 Jet manifolds......Page 37 1.3 Connections on fibre bundles......Page 40 1.3.1 Connections as tangent-valued forms......Page 41 1.3.2 Connections as jet bundle sections......Page 43 1.3.3 Curvature and torsion......Page 45 1.3.4 Linear connections......Page 47 1.3.5 Affine connections......Page 49 1.3.6 Flat connections......Page 50 1.3.7 Second order connections......Page 52 1.4 Composite bundles......Page 53 1.5 Higher order jet manifolds......Page 57 1.6 Differential operators and equations......Page 62 1.7 Infinite order jet formalism......Page 65 2.1 Variational bicomplex......Page 72 2.2 Lagrangian symmetries......Page 77 2.3 Gauge symmetries......Page 81 2.4 First order Lagrangian field theory......Page 84 2.4.1 Cartan and Hamilton–De Donder equations......Page 86 2.4.2 Lagrangian conservation laws......Page 89 2.4.3 Gauge conservation laws. Superpotential......Page 91 2.4.4 Non-regular quadratic Lagrangians......Page 94 2.4.5 Reduced second order Lagrangians......Page 98 2.4.6 Background fields......Page 99 2.4.7 Variation Euler–Lagrange equation. Jacobi fields......Page 101 2.5 Appendix. Cohomology of the variational bicomplex......Page 103 3.1 Grassmann-graded algebraic calculus......Page 110 3.2 Grassmann-graded differential calculus......Page 115 3.3 Geometry of graded manifolds......Page 118 3.4 Grassmann-graded variational bicomplex......Page 126 3.5 Lagrangian theory of even and odd fields......Page 131 3.6 Appendix. Cohomology of the Grassmann-graded variational bicomplex......Page 136 4. Lagrangian BRST theory......Page 140 4.1 Noether identities. The Koszul–Tate complex......Page 141 4.2 Second Noether theorems in a general setting......Page 151 4.3 BRST operator......Page 158 4.4 BRST extended Lagrangian field theory......Page 161 4.5 Appendix. Noether identities of di erential operators......Page 165 5.1 Geometry of Lie groups......Page 176 5.2 Bundles with structure groups......Page 180 5.3 Principal bundles......Page 182 5.4 Principal connections. Gauge fields......Page 186 5.5 Canonical principal connection......Page 190 5.6 Gauge transformations......Page 192 5.7 Geometry of associated bundles. Matter fields......Page 195 5.8.1 Gauge field Lagrangian......Page 199 5.8.2 Conservation laws......Page 201 5.8.3 BRST extension......Page 203 5.8.4 Matter field Lagrangian......Page 205 5.9 Yang–Mills supergauge theory......Page 207 5.10.1 Reduction of a structure group......Page 209 5.10.2 Reduced subbundles......Page 211 5.10.3 Reducible principal connections......Page 213 5.10.4 Associated bundles. Matter and Higgs fields......Page 214 5.10.5 Matter field Lagrangian......Page 218 5.11 Appendix. Non-linear realization of Lie algebras......Page 222 6.1 Natural bundles......Page 226 6.2 Linear world connections......Page 230 6.3 Lorentz reduced structure. Gravitational field......Page 234 6.4 Space-time structure......Page 239 6.5 Gauge gravitation theory......Page 243 6.6 Energy-momentum conservation law......Page 247 6.7 Appendix. A ne world connections......Page 249 7.1 Clifford algebras and Dirac spinors......Page 254 7.2 Dirac spinor structure......Page 257 7.3 Universal spinor structure......Page 263 7.4 Dirac fermion fields......Page 269 8.1 Topological characteristics of principal connections......Page 274 8.1.1 Characteristic classes of principal connections......Page 275 8.1.2 Flat principal connections......Page 277 8.1.3 Chern classes of unitary principal connections......Page 281 8.1.4 Characteristic classes of world connections......Page 285 8.2 Chern–Simons topological field theory......Page 289 8.3 Topological BF theory......Page 294 8.4 Lagrangian theory of submanifolds......Page 297 9.1 Polysymplectic Hamiltonian formalism......Page 304 9.2 Associated Hamiltonian and Lagrangian systems......Page 309 9.3 Hamiltonian conservation laws......Page 315 9.4 Quadratic Lagrangian and Hamiltonian systems......Page 317 9.5 Example. Yang–Mills gauge theory......Page 324 9.6 Variation Hamilton equations. Jacobi fields......Page 327 10.1 Commutative algebra......Page 330 10.2 Differential operators on modules......Page 335 10.3 Homology and cohomology of complexes......Page 338 10.4 Cohomology of groups......Page 341 10.5 Cohomology of Lie algebras......Page 344 10.6 Differential calculus over a commutative ring......Page 345 10.7 Sheaf cohomology......Page 348 10.8 Local-ringed spaces......Page 357 10.9 Cohomology of smooth manifolds......Page 359 10.10 Leafwise and fibrewise cohomology......Page 365 Bibliography......Page 370 Index......Page 380 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. 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

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.

"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."--Jacket Contemporary quantum field theory is mainly developed as quantization of classical fields. This book aims to provide a mathematical foundation of Lagrangian classical field theory and its BRST extension for the purpose of quantization. It considers reducible degenerate Lagrangian theories of even and odd fields on an arbitrary smooth manifold.