By Giovanni Giachetta, Luigi Mangiarotti, Gennadi Sardanashvily

Modern quantum box concept is principally 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 e-book goals to supply an entire mathematical origin of Lagrangian classical box concept and its BRST extension for the aim of quantization. according to the traditional geometric formula of concept of nonlinear differential operators, Lagrangian box conception is handled in a really common atmosphere. 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 such a lot bodily correct box theories - gauge idea on important bundles, gravitation thought on average bundles, conception of spinor fields and topological box thought - are awarded in a whole manner. This ebook is designed for theoreticians and mathematical physicists focusing on box idea. The authors have attempted all through to supply the required mathematical heritage, hence making the exposition self-contained.

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It follows that tangent spaces to leaves of a foliation F constitute an involutive distribution T F on Z, called the tangent bundle to the foliation F. The factor bundle V F = T Z/T F, called the normal bundle to F, has transition functions independent of coordinates z i . Let T F ∗ → Z denote the dual of T F → Z. 32) of vector bundles over Z. It should be emphasized that leaves of a foliation need not be closed or imbedded submanifolds. , if z ∈ U , then a leaf through z also belongs to U . A pair (Z, F) where F is a foliation of Z is called a foliated manifold.

CodimE. 1) Given a k-order differential equation E, one can always construct its r-order jet prolongation as follows. 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.

3. , this is a section of the affine bundle π11 : J 1 J 1 Y → J 1 Y. Every connection on a fibre bundle Y → X gives rise to the second order one by means of a world connection on X as follows. 11), but not of π11 . 40) on X, one can construct the affine morphism sK : J 1 J 1 Y → J 1 J 1 Y, i i (xλ , y i , yλi , yλi , yλµ ) ◦ sK = (xλ , y i , yλi , yλi , yµλ − Kλ ν µ (yνi − yνi )), such that π11 = J 1 π01 ◦ sK [53]. 50) + yλj ∂j Γiµ + ν Kλ µ (yνi − Γiν )]∂iµ ), which is an affine morphism Γ J 1 Y −→ J 1 J 1 Y π01 π11 ❄ ❄ Γ Y −→ J 1 Y over the connection Γ.

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