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.
Categories: hep-th math-ph math.MP