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Besides, the previous trajectory definition such that a value xj is confronted with every time moment tj should be changed too. We shall define a trajectory giving a value of a field ψ(x) = ψ(r, t), which may be multicomponents, in every cell. The paths integral presently involves a summation over all possible field values in every cell. We confront objects that were introduced in NQM with the following ones: j=1 [dqj , dpj ] → [dψ(x), dπ(x)], Formalism of functional integration L(qj , q˙j ), H(qj , pj ) → 29 d3 xL(ψ(x), ∂µ ψ(x)), d3 xH(ψ(x), π(x)), where π(x) is a momentum canonically conjugate to a field.

57) where every summation index takes the values from 1 to n. 56) this expansion is broken off. As an example, we consider one-dimensional Grassmann algebra G1 {y, y} = 0, y 2 = 0. For any element F (y) ∈ G1 , we have F (y) = F0 + yF1 . 58) To be specific, let us hold F (y) to be an ordinary number. Then F0 and F1 are ordinary and Grassmann numbers, respectively. We introduce differentiation and integration operations into G1 . Two kind of derivatives, left and right ones, exist in the Grassmann algebra owing to the anticommutation relations.

46) (∆c (x − y) is a causal Green function of a free scalar field). The term −iǫ ensures the Feynman prescription of the propagator poles detour. 43) in the Minkowski space. 45) has the following meaning: in the point y the source J(y) produces a free scalar particle that propagates from y to x and the source J(x) annihilates this particle in the point x. Expanding the exponential gives a series of terms corresponding to 0, 1, 2, . . free scalar particles. Now we switch on interaction (Lint = 0).