邓克尔双光子/薛定谔代数及其应用

中文摘要 AI 辅助

本研究引入了新型的邓克尔双光子代数与邓克尔-薛定谔代数，它们是标准六维双光子代数与薛定谔代数经反射扩展后的对应物。这些扩展通过一个代表反射的对合生成元实现，在适当极限下，当该附加生成元不存在时，它们会退化为未变形的双光子代数与薛定谔代数。对于这两种被证明同构的代数结构，我们推导了相关的三次多项式卡西米尔不变量。此外，我们识别出若干具有结构重要性的子代数，并计算了它们对应的二次卡西米尔不变量。邓克尔双光子代数的表示在一维邓克尔谐振子的福克空间上构造，随后被转化为相关邓克尔-福克-巴格曼表示内的全纯实现。最后，对于邓克尔-薛定谔代数，我们给出了基于邓克尔空间导数的显式向量场实现，同时保持时间连续。将该实现应用于前述子代数的卡西米尔不变量时，得到了1+1维下的一组六个非平凡邓克尔微分-差分方程，其中尤其包括著名的热-薛定谔方程的邓克尔版本。

英文摘要

The work introduces the novel Dunkl two-photon and Dunkl-Schrödinger algebras as reflection-extended counterparts of the standard six-dimensional two-photon and Schrödinger algebras. These extensions are realized through the presence of an involutive generator representing a reflection, and they reduce to the undeformed two-photon and Schrödinger algebras in the absence of this additional generator in an appropriate limit. For both algebraic structures, which are shown to be isomorphic, we derive the associated cubic polynomial Casimir invariant. Furthermore, we identify several structurally significant subalgebras and compute their corresponding quadratic Casimir invariants. Representations of the Dunkl two-photon algebra are constructed on the Fock space of the one-dimensional Dunkl harmonic oscillator and subsequently translated into holomorphic realizations within the associated Dunkl-Fock-Bargmann representation. Finally, for the Dunkl-Schrödinger algebra, we present an explicit vector field realization in terms of the Dunkl spatial derivative, while keeping time continuous. When applied to the Casimir invariants of the aforementioned subalgebras, this realization yields a family of six nontrivial Dunkl differential-difference equations in $1+1$ dimensions, including, in particular, a Dunkl version of the well-known heat-Schrödinger equation.

发表机构

• Universidad de Burgos（布尔戈斯大学）
• Università degli Studi di Milano（米兰大学）
• INFN Sezione di Milano（意大利国家核物理研究所米兰分部）

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29 pages