By Giovanni Giachetta, Luigi Mangiarotti, Gennadi Sardanashvily

Modern quantum box thought is especially constructed as quantization of classical fields. consequently, classical box thought and its BRST extension is the mandatory step in the direction of quantum box thought. This publication goals to supply a whole mathematical beginning of Lagrangian classical box concept and its BRST extension for the aim of quantization. in accordance with the normal geometric formula of thought of nonlinear differential operators, Lagrangian box concept is handled in a truly common surroundings. Reducible degenerate Lagrangian theories of even and extraordinary fields on an arbitrary soft manifold are thought of. the second one Noether theorems generalized to those theories and formulated within the homology phrases give you the strict mathematical formula of BRST prolonged classical box theory.The so much bodily appropriate box theories - gauge idea on important bundles, gravitation concept on typical bundles, idea of spinor fields and topological box idea - are awarded in an entire manner. This booklet is designed for theoreticians and mathematical physicists focusing on box conception. The authors have attempted all through to supply the required mathematical history, hence making the exposition self-contained.

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We have the relations f ∗ (φ ∧ σ) = f ∗ φ ∧ f ∗ σ, df ∗ φ = f ∗ (dφ). In particular, given a fibre bundle π : Y → X, the pull-back onto Y of exterior forms on X by π provides the monomorphism of graded commutative algebras O∗ (X) → O∗ (Y ). Elements of its range π ∗ O∗ (X) are called basic forms. λr dxλ1 ∧ · · · ∧ dxλr , r! on Y such that u φ = 0 for an arbitrary vertical vector field u on Y are said to be horizontal forms. Horizontal forms of degree n = dim X are called densities. n , n! µ dx n!

Let us consider a repeated jet manifold σkr : J r J k Y → J k Y. 2) The s-order jet prolongation of the differential equation E is defined as a subset E(r) = (σkr )−1 (E) J k+r Y. January 26, 2009 2:48 World Scientific Book - 9in x 6in 52 book08 Differential calculus on fibre bundles In particular, if E(r) is a smooth submanifold of J k+r Y , then the s-order jet prolongation (E(r) )(s) of E(r) coincides with the (r + s)-order jet prolongation E(s+r) of E. A differential equation E is called regular if all finite order jet prolongations E(r) of E also are differential equations.

2. For any global section s : X → Y , there always exists a connection Γ such that s is an integral section of Γ. January 26, 2009 2:48 World Scientific Book - 9in x 6in 34 Differential calculus on fibre bundles Proof. 4. Treating connections as jet bundle sections, one comes to the following two constructions. (i) Let Y and Y be fibre bundles over the same base X. Given a connection Γ on Y → X and a connection Γ on Y → X, the fibre bundle Y ×Y → X X is provided with the product connection Γ × Γ : Y × Y → J 1 (Y × Y ) = J 1 Y × J 1 Y , X X X ∂ ∂ Γ × Γ = dxλ ⊗ ∂λ + Γiλ i + Γλj .

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Advanced Classical Field Theory by Giovanni Giachetta, Luigi Mangiarotti, Gennadi Sardanashvily
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