arXiv 1906.117052019-06hep-thmath-phmath.MP

具有$sl(2|1)$动力学对称性的三维超共形量子力学

A three-dimensional superconformal quantum mechanics with $sl(2|1)$ dynamical symmetry

Ivan E. Cunha, Francesco Toppan

备注：: 30 pages, 5 figures; 4 references added
发表信息：: Phys. Rev. D 100, 125002 (2019)
DOI：: 10.1103/PhysRevD.100.125002

摘要：

我们构造了一个三维超共形量子力学（及其相应的de Alfaro-Fubini-Furlan变形振子），它具有$sl(2|1)$动力学对称性.在耦合参数$β\neq 0$的哈密顿量包含一个$\frac{1}{r^2}$势和一个自旋轨道（因此，一阶微分算子）相互作用项。在$β=0$中，恢复了未变形的三维振子的四个副本。哈密顿量在总$j$和轨道$l$角动量的每个扇区中对角化（系统的自旋为$\frac{1}{2}$）。变形振子的Hilbert空间由$sl(2|1)$最低权表示的直和给出。讨论了在给定的耦合常数$β$值下，可容许Hilbert空间的选择问题。计算了该模型的谱。真空能量（作为PDMATHAMT的函数）由递归之字形图案组成。能量本征值的简并性线性增长到$E\sim β$（以适当的单位），并且二次增长到$E>β$。正交能量本征态表示在相关的拉盖尔多项式和自旋球谐函数。将模型降维为$d=2$产生二维$sl(2|1)$变形振子的两个副本（分别为$β$和$-β$）。降维到$d=1$产生一维$D(2,1;α)$变形振子，$α$由$β$确定。

英文摘要：

We construct a three-dimensional superconformal quantum mechanics (and its associated de Alfaro-Fubini-Furlan deformed oscillator) possessing an $sl(2|1)$ dynamical symmetry. At a coupling parameter $β\neq 0$ the Hamiltonian contains a $\frac{1}{r^2}$ potential and a spin-orbit (hence, a first-order differential operator) interacting term. At $β=0$ four copies of undeformed three-dimensional oscillators are recovered. The Hamiltonian gets diagonalized in each sector of total $j$ and orbital $l$ angular momentum (the spin of the system is $\frac{1}{2}$). The Hilbert space of the deformed oscillator is given by a direct sum of $sl(2|1)$ lowest weight representations. The selection of the admissible Hilbert spaces at given values of the coupling constant $β$ is discussed. The spectrum of the model is computed. The vacuum energy (as a function of $β$) consists of a recursive zigzag pattern. The degeneracy of the energy eigenvalues grows linearly up to $E\sim β$ (in proper units) and quadratically for $E>β$. The orthonormal energy eigenstates are expressed in terms of the associated Laguerre polynomials and the spin spherical harmonics. The dimensional reduction of the model to $d=2$ produces two copies (for $β$ and $-β$, respectively) of the two-dimensional $sl(2|1)$ deformed oscillator. The dimensional reduction to $d=1$ produces the one-dimensional $D(2,1;α)$ deformed oscillator, with $α$ determined by $β$.

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