## 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. |

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### 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

affine bundle antifield associated automorphism BRST bundle atlas bundle coordinates bundle morphism called canonical cochain composite bundle conservation law corresponding covariant deﬁned Deﬁnition diﬀerent diﬀerential differential bigraded algebra differential graded algebra differential operator elements Euler-Lagrange exact sequence exterior forms fibre bundle ﬁnite frame bundle LX G-bundle gauge symmetry gauge theory Given global section graded commutative ring graded derivation graded manifold group G Hamiltonian form holonomic horizontal isomorphic jet bundle Lagrangian Lemma Let us consider Lie algebra linear world connection metric module monomorphism nilpotent Noether identities non-trivial one-form order jet manifold presheaf principal bundle principal connection pull-back Remark Rham cohomology sheaf sheaves smooth manifold spinor bundle spinor structure splitting structure group subbundle submanifold takes the form tangent bundle tensor tetrad transition functions trivial typical ﬁbre variational symmetry vector bundle vector field vertical virtue of Theorem world manifold