Advanced Classical Field TheoryContemporary 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 OCo gauge theory on principal bundles, gravitation theory on natural bundles, theory of spinor fields and topological field theory OCo 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
Introduction | 1 |
1 Differential calculus on fibre bundles | 5 |
2 Lagrangian field theory on fibre bundles | 61 |
3 Grassmanngraded Lagrangian field theory | 99 |
4 Lagrangian BRST theory | 129 |
5 Gauge theory on principal bundles | 165 |
6 Gravitation theory on natural bundles | 215 |
Other editions - View all
Advanced Classical Field Theory G. Giachetta,L. Mangiarotti,Gennadi? Aleksandrovich Sardanashvili Limited preview - 2009 |
Common terms and phrases
A-module affine bundle antifield associated automorphism BRST bundle atlas bundle coordinates bundle morphism called canonical cochain commutative ring composite bundle conservation law corresponding covariant defined differential bigraded algebra differential graded algebra differential operator elements exact sequence exterior forms fibre bundle finite G-bundle gauge symmetry gauge theory Given global section graded commutative graded commutative ring graded derivation graded manifold group G Hamiltonian Hamiltonian form holonomic isomorphic J¹Y jet bundle Lemma Let us consider Lie algebra linear world connection metric module monomorphism nilpotent Noether identities non-trivial order jet manifold Phys presheaf principal bundle principal connection pull-back Remark Rham cohomology sheaf sheaves spinor bundle splitting structure group subbundle submanifold tangent bundle tensor tetrad transition functions trivial typical fibre variational symmetry vector bundle vector field vertical virtue of Theorem world manifold