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球谐函数$Y_{l}^{m}(θ,ϕ)$：su（1，1）对l-m和l+m的正、负双曲表示

Spherical Harmonics $Y_{l}^{m}(θ,ϕ)$: Positive and Negative Integer Representations of su(1,1) for l-m and l+m

H. Fakhri

备注：
11 pages, no figures, version to appear in Adv. High Energy Phys
发表信息：
Adv. High Energy Phys. Volume 2016 (2016) 3732657
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摘要：

球谐函数$Y_{l}^{m}(θ,ϕ)$的方位角量子数和磁量子数描述了在$su(2)$李代数对称性实现的框架下，角动量算符的幅值和$z$分量对应的量子化.方位角量子数$l$通过用$l$表示的算符给它自己分配了一个附加的梯形对称性。在这里，它表明，同时实现的两个对称继承了积极和消极的$(l-m)$ -和$(l+m)$ -整数离散不可约表示$su(1,1)$李代数通过球面调和作为一个紧凑的流形。因此，除了对给定的$l$通过$Y_{l}^{m}(θ,ϕ)$实现$su(2)$紧李代数的酉不可约表示外，还可以对给定的$l-m$和$l+m$用球谐函数表示$su(1,1)$非紧李代数.

英文摘要：

The azimuthal and magnetic quantum numbers of spherical harmonics $Y_{l}^{m}(θ,ϕ)$ describe quantization corresponding to the magnitude and $z$-component of angular momentum operator in the framework of realization of $su(2)$ Lie algebra symmetry. The azimuthal quantum number $l$ allocates to itself an additional ladder symmetry by the operators which are written in terms of $l$. Here, it is shown that simultaneous realization of the both symmetries inherits the positive and negative $(l-m)$- and $(l+m)$-integer discrete irreducible representations for $su(1,1)$ Lie algebra via the spherical harmonics on the sphere as a compact manifold. So, in addition to realizing the unitary irreducible representation of $su(2)$ compact Lie algebra via the $Y_{l}^{m}(θ,ϕ)$'s for a given $l$, we can also represent $su(1,1)$ noncompact Lie algebra by spherical harmonics for given values of $l-m$ and $l+m$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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