论文标题
Duffin-Kemmer-Petiau形式主义中逆三阶波算子的路径积分表示。 ii
Path integral representation for inverse third order wave operator within the Duffin-Kemmer-Petiau formalism. II
论文作者
论文摘要
本文是我们论文第一部分的立即延续[1]。在这里，在帕拉格拉斯曼（Para-Grassmann）代数中，我们引入了一种非交流性的，关联的星星产品$*$（Moyal产品），这是对格拉斯曼数字代数中星级产品的直接概括。配备了星星产品的订单2的代数与创建和an灭操作员的代数$ a_ {n}^{\ pm} $遵守相同顺序的para-fermi统计数据之间的同构。考虑了两种独立的计算Moyal产品$*$的方法。结果表明，在基于派屈的相干状态来计算各种表达式的相干状态时，应不断考虑，我们在所谓的Ohnuki和Kamefuchi的全面状态 - 媒介空间$ \ mathfrak {u} _ {u} _ {\; g} $中，其定义$ para-grassmann norking para-grassmann norkity norking norking para-grassmann标准状态矢量空间$ \ Mathfrak {U} $（Fock Space）。否则，会出现各种各样的矛盾。使用扩展的状态矢量空间$ \ mathfrak {u} _ {\; g} $的直接后果是考虑二次casimir操作员$ \ hat {c} _ {2} _ {2} $和$ \ hat and $ \ hat {c} {c} {c} _ {2} _ {2}^{\ prime and $ ym + oorth $ so + ok y ym + ok ym of osthon of soy + 1）$，相应地。 Casimir操作员在状态向量上的操作规则，其矩阵元素的明确形式被定义，并获得Harish-Chandra操作员$ \hatΩ^2 $与Geyer Operator $ a_ {0}^{2} $之间的更一般的连接。引入了三星级产品，恒星指数和Moyal支架的概念。
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.
