[论文解读] Quantum deformed algebras : Coherent states and special functions

J. D. Bukweli-Kyemba, Mahouton Norbert Hounkonnou|arXiv (Cornell University)|Jan 1, 2013

Algebraic structures and combinatorial models参考文献 63被引用 14

一句话总结

本文引入了一种广义的 ${\cal R}(p,q)$-形变海森堡代数，统一并扩展了 $q$-和 $(p,q)$-形变，使得能够构造新的相干态、超几何级数以及特殊函数，如 ${\cal R}(p,q)$-形变罗杰斯-谢戈尔多项式和连续赫米特多项式。该框架推广了已知结果，提供了霍普夫代数结构，并在非交换设置下扩展了微分与积分演算。

ABSTRACT

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.

研究动机与目标

• 使用参数 $\{q, l, \lambda\}$ 推广海森堡代数，以构造一类新的广义相干态。
• 开发一种统一方法，从形变单模振子代数的对易关系中推导结构函数。
• 推广与亚纯函数相关的双参数形变海森堡代数，从而导出 ${\cal R}(p,q)$-形变特殊函数。
• 通过一种与霍普夫代数性质相容的新代数结构，将 $q$-和 $(p,q)$-演算推广至非交换框架。
• 定义并全面表征 ${\cal R}(p,q)$-形变罗杰斯-谢戈尔与赫米特多项式，包括递推关系与生成函数。

提出的方法

• 使用参数 $\{q, l, \lambda\}$ 对标准海森堡代数进行形变，以生成满足克劳德标准的广义相干态。
• 推导相关算子的矩阵元，并证明主要形变态具有次泊松统计特性，从而定义广义度量。
• 构建统一形式化方法，直接从形变振子代数的对易关系中确定结构函数。
• 将 ${\cal R}(p,q)$-形变代数引入为与亚纯函数相关的双参数形变海森堡代数的推广。
• 定义新的超几何级数及 ${\cal R}(p,q)$-二项定理，作为已知 $(p,q)$-级数的推广。
• 通过定义 $\partial_{p,q,h}^{\mu,\nu}$ 及相关算子，建立非交换微分与积分演算，并构造相容的霍普夫代数结构。

实验结果

研究问题

• RQ1 如何利用三个参数 $\{q, l, \lambda\}$ 对海森堡代数进行形变，以生成具有次泊松统计特性的广义相干态？
• RQ2 从形变单模振子代数的对易关系中推导结构函数的通用方法是什么？
• RQ3 如何构建 ${\cal R}(p,q)$-形变代数，以统一并扩展已知的 $q$-和 $(p,q)$-形变？
• RQ4 ${\cal R}(p,q)$-形变罗杰斯-谢戈尔与赫米特多项式的性质及递推关系是什么？
• RQ5 如何在 ${\cal R}(p,q)$-代数框架内推广包含微分与积分的非交换演算？与之相容的霍普夫代数结构是什么？

主要发现

• 由 $\{q,l,\lambda\}$-形变海森堡代数构造的广义相干态表现出次泊松统计特性，证实其非经典性质，并支持在系统几何上定义广义度量。
• 建立了一种统一方法，可直接从形变振子代数的对易关系中推导结构函数，推广了已知结果，并支持相干态的系统构造。
• ${\cal R}(p,q)$-形变超几何级数被引入，并显示其推广了已知的 $(p,q)$-超几何级数，其中新导出的 ${\cal R}(p,q)$-二项定理作为特例。
• ${\cal R}(p,q)$-形变罗杰斯-谢戈尔多项式得到全面表征，包括其三项递推关系与差分方程，推广了标准与 $q$-形变版本。
• 通过 $\mathbb{H}_n(\cos\theta; p,q,\mu,\nu,h)$ 定义了连续 ${\cal R}(p,q)$-形变赫米特多项式，其满足包含 $p$-与 $q$-形变参数的三项递推关系。
• 明确构造了与 ${\cal R}(p,q)$-代数相容的霍普夫代数结构，将框架扩展至非交换几何与量子群领域。