AI 中文总结
该研究探讨维格纳-杨量子力学系统的超对称结构，推广杨的表示并建立类库仑系统与维格纳-杨简谐振子的胡克-牛顿对偶性，揭示其隐藏超对称性。
AI 中文摘要
我们考虑实线上服从1950年维格纳提出的变形海森堡代数的量子系统，其显式坐标表示由杨于1951年给出，本质上与1989年邓克尔结合反射根系引入的差微分算子形式一致。在一定条件下，这类量子系统呈现超对称（SUSY）结构，其中反射算子作为 grading 算子。我们通过仅考虑相空间中两个运动方程中的一个，提出杨表示的推广，发现对应的非相互作用系统代表维滕的超对称量子力学模型。同时施加第二个运动方程，将维格纳和杨的讨论扩展到实线上的一般对称势，重新审视其原始结果。作为具体例子，我们讨论了简谐振子和吸引类库仑势V(x)=-γ/|x|，还建立了该类库仑系统与原始维格纳-杨简谐振子系统之间的胡克-牛顿对偶性。
英文摘要
We consider a quantum system on the real line obeying a deformed Heisenberg algebra originally proposed by Wigner in 1950. Its explicit coordinate representation was provided by Yang in 1951 and in essence is identical in form with Dunkl's difference-differential operator introduced in 1989 in connection with roots systems of refection groups. Under certain conditions such quantum systems exhibit a supersymmetric (SUSY) structure where the reflection operator acts as the grading operator. We present a generalisation of Yang's representation by first considering only of one the two equations of motion in phase space. The corresponding non-interacting system is found to represent Witten's model of SUSY quantum mechanics. Imposing also the second equation of motion the original result of Wigner and Yang is reconsidered by extending their discussion to general symmetric potentials on the real line. As explicit example we discuss the harmonic oscillator and an attractive Coulomb-like potential $V(x)=-γ/|x|$. We also establish a Hooke-Newton duality between this Coulomb-like system and the original Wigner-Yang harmonic oscillator system.
CommentsDedicated to Asim Orhan Barut (1926-1994) on the occasion of his $100^{\rm th}$ birthday. Talk presented at the GROUP36, Valladolid, Spain, July 13-17, 2026. 14 pages
