Path integral representation for inverse third order wave operator within the Duffin-Kemmer-Petiau formalism. Part I

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Within the framework of the Duffin-Kemmer-Petiau (DKP) formalism with a deformation, an approach to the construction of the path integral representation in parasuperspace for the Green’s function of a spin-1 massive particle in external Maxwell’s field is developed. For this purpose a connection between the deformed DKP-algebra and an extended system of the parafermion trilinear commutation relations for the creation and annihilation operators {a}_k^{pm } and for an additional operator a0 obeying para-Fermi statistics of order 2 based on the Lie algebra mathfrak{so} (2M + 2) is established. The representation for the operator a0 in terms of generators of the orthogonal group SO(2M correctly reproducing action of this operator on the state vectors of Fock space is obtained. An appropriate system of the parafermion coherent states as functions of para-Grassmann numbers is introduced. The procedure of the construction of finite-multiplicity approximation for determination of the path integral in the relevant phase space is defined through insertion in the kernel of the evolution operator with respect to para-supertime of resolutions of the identity. In the basis of parafermion coherent states a matrix element of the contribution linear in covariant derivative {hat{D}}_{mu } to the time-dependent Hamilton operator hat{mathrm{mathscr{H}}}left(tau right) , is calculated in an explicit form. For this purpose the matrix elements of the operators a0, {a}_0^2 , the commutators [a0, {a}_n^{pm } ], [ {a}_0^2,{a}_n^{pm } ], and the product hat{A} [a0, {a}_n^{pm } ] with hat{A} ≡ exp( -ifrac{2pi }{3}{a}_0 ), were preliminary defined.