Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1051quantum field theory, we should assume that X = R n and Y → X is a trivialaffine bundle. It follows that both the original coordinates (x α , y i , p α i )and the adapted coordinates (x α , y i , p a , p A ) on the Legendre bundle Π areglobal. Passing to field theory on an Euclidean space R n , we also assumethat the matrix a in the Lagrangian L (5.312) is positive–definite, i.e.,a AA > 0.Let us start from a Lagrangian (5.316) without gauge symmetries. Sincethe Lagrangian constraint space N L can be equipped with the adapted coordinatesp A , the generating functional of Euclidean Green functions of theLagrangian system in question reads [Bashkirov and Sardanashvily (2004)]∫ ∫Z = N −1 exp{ (L N + 1 2 Tr(ln σ 0)+iJ i y i +iJ A p A )ω} ∏ [dp A (x)][dy(x)],x(6.83)where L N is given by the expression (5.322) and σ 0 is the square matrixσ AB0 = M −1αAiM −1µBjσ 0ijαµ = δ AB (a AA ) −1 .The generating functional (6.83) a Gaussian integral of variables p A (x). Itsintegration with respect to p A (x) under the condition J A = 0 restarts thegenerating functional∫ ∫Z = N −1 exp{ (L + iJ i y i )ω} ∏ [dy(x)], (6.84)xof the original Lagrangian field system on Y with the Lagrangian (5.312).However, the generating functional (6.83) cannot be rewritten with respectto the original variables p µ i , unless a is a nondegenerate matrix function.In order to overcome this difficulty, let us consider a Lagrangian systemon the whole Legendre manifold Π with the Lagrangian L Π (5.319). Sincethis Lagrangian is constant along the fibres of the vector bundle Π → N L ,an integration of the generating functional of this field model with respect tovariables p a (x) should be finite. One can choose the generating functionalin the form [Bashkirov and Sardanashvily (2004)]∫ ∫Z = N −1 exp{ (L Π − 1 2 σ 1 ijαµp α i p µ j (6.85)+ 1 2 Tr(ln σ) + iJ iy i + iJ i µp µ i )ω} ∏ x[dp(x)][dy(x)].Its integration with respect to momenta p α i (x) restarts the generating functional(6.84) of the original Lagrangian system on Y . In order to get the