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

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 $a^{\pm}_{k}$ and for an additional operator $a_{0}$ obeying para-Fermi statistics of order 2 based on the Lie algebra $\mathfrak{so}(2M+2)$ is established. The representation for the operator $a_{0}$ 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 $\hat{\cal H}(\tau)$, is calculated in an explicit form. For this purpose the matrix elements of the operators $a^{\phantom{2}}_0$, $a_{0}^{2}$, the commutators $[\hspace{0.03cm}a^{\phantom{\pm}\!}_{0}, a^{\pm}_{n}\hspace{0.02cm}]$, $[\hspace{0.03cm}a^{2}_{0}, a^{\pm}_{n}\hspace{0.02cm}]$, and the product $\hat{A}\hspace{0.03cm}[\hspace{0.03cm}a^{\phantom{\pm}\!}_{0}, a^{\pm}_{n}\hspace{0.02cm}]$ with $\hat{A} \equiv\exp\hspace{0.02cm}\bigl(-i\frac{2\pi}{3}\,a_{0}\bigr)$, were preliminary defined.