arXiv 1301.01162013-01-01math-phmath.MP

量子变形代数：相干态和特殊函数

Quantum deformed algebras : Coherent states and special functions

J. D. Bukweli-Kyemba, M. N. Hounkonnou

备注：: arXiv admin note: text overlap with arXiv:1204.4705, arXiv:1211.3312, arXiv:1211.3373

摘要：

首先用参数集${q, l, λ}$对海森堡代数进行变形，生成一族新的广义相干态。在这个框架中，相关的运营商的矩阵元素的精确计算。证明了主要变形态统计量的亚泊松性质。这个属性用于确定一个广义度量。然后给出了由变形单模振子代数的对易关系计算结构函数的统一方法。推导了一种构造与变形代数相关的相干态的自然方法。给出已知的变形代数作为例子。其次，我们推广了一类与亚纯函数相关的双参数变形Heisenberg代数，称之为${\cal R}(p,q)$ -变形代数。探讨了相干态映射的相关族，并计算了相应的超几何级数。后者推广了已知的超几何级数，并给出了二项式定理的一个推广。所涉及的微分和积分的概念推广了通常的$q$ -和$(p,q)$ -微分和积分。导出了一个与${\cal R}(p,q)$ -代数相容的Hopf代数结构.利用Rogers-Szegö多项式的三项递推关系和相应的产生算符和湮灭算符所构成的量子代数，成功地给出了Rogers-Szegö多项式的一个新的特征，称为${\cal R}(p,q)$ -变形Rogers-Szegö多项式.给出了连续的${\cal R}(p,q)$变形Hermite多项式及其递推关系。新的代数关系提供和讨论。整个形式主义是在一个统一的方式进行，推广已知的相关结果，直接推导出的特殊情况。

英文摘要：

The Heisenberg algebra is first deformed with the set of parameters ${q, l, λ}$ to generate a new family of generalized coherent states. In this framework, the matrix elements of relevant operators are exactly computed. A proof on sub-Poissonian character of the statistics of the main deformed states is provided. This property is used to determine a generalized metric. A unified method of calculating structure functions from commutation relations of deformed single-mode oscillator algebras is then presented. A natural approach to building coherent states associated to deformed algebras is deduced. Known deformed algebras are given as illustration. Futhermore, we generalize a class of two-parameter deformed Heisenberg algebras related to meromorphic functions, called ${\cal R}(p,q)$-deformed algebra. Relevant families of coherent states maps are probed and their corresponding hypergeometric series are computed. The latter generalizes known hypergeometric series and gives to a generalization of the binomial theorem. The involved notions of differentiation and integration generalize the usual $q$- and $(p,q)$-differentiation and integration. A Hopf algebra structure compatible with the ${\cal R}(p,q)$-algebra is deduced. We succeed in giving a new characterization of Rogers- Szegö polynomials, called ${\cal R}(p,q)$-deformed Rogers-Szegö polynomials, by their three-term recursion relations and the associated quantum algebra built with corresponding creation and annihilation operators. Continuous ${\cal R}(p,q)$-deformed Hermite polynomials and their recursion relation are also deduced. Novel algebraic relations are provided and discussed. The whole formalism is performed in a unified way, generalizing known relevant results which are straightforwardly derived as particular cases.

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摘要：

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