Paper detail

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

This paper is an immediate continuation of the first part of our paper [1]. Here, in a para-Grassmann algebra we introduce a noncommutative, associative star product $*$ (the Moyal product), which is a direct generalization of the star product in the algebra of Grassmann numbers. Isomorphism between the algebra of para-Grassmann numbers of order 2 equipped with the star product and with the algebra of creation and annihilation operators $a_{n}^{\pm}$ obeying the para-Fermi statistics of the same order is established. Two independent approaches to the calculation of the Moyal product $*$ are considered. It is shown that in calculating the matrix elements in the basis of parafermion coherent states of various expressions it should be taken into account constantly that we work in the so-called Ohnuki and Kamefuchi's generalized state-vector space $\mathfrak{U}_{\;G}$, whose state vectors include para-Grassmann numbers $ξ_{k}$ in their definition, instead of the standard state-vector space $\mathfrak{U}$ (the Fock space). Otherwise, the wide array of contradictions arises. An immediate consequence of using the extended state-vector space $\mathfrak{U}_{\;G}$ is a necessity to consider the quadratic Casimir operators $\hat{C}_{2}$ and $\hat{C}_{2}^{\prime}$ of the orthogonal groups $SO(2M)$ and $SO(2M + 1)$, correspondingly. The action rules of the Casimir operators on the state vectors, an explicit form of their matrix elements are defined and a more general connection between the Harish-Chandra operator $\hatω^2$ and the Geyer operator $a_{0}^{2}$ is obtained. The notions of the triple star product, the star exponent and the Moyal bracket are introduced.