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

TL;DR

This paper demonstrates that spherical harmonics can simultaneously realize both su(2) and su(1,1) Lie algebra representations, linking quantum numbers to discrete irreducible representations of these algebras.

Contribution

It introduces a novel approach to represent su(1,1) algebra using spherical harmonics, extending their application beyond the traditional su(2) symmetry.

Findings

Spherical harmonics encode su(2) and su(1,1) symmetries.

Positive and negative integer representations of su(1,1) are realized.

New representation of noncompact Lie algebra via spherical harmonics.

Abstract

The azimuthal and magnetic quantum numbers of spherical harmonics $Y_{l}^{m}(\theta,\phi)$ 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 $Y_{l}^{m}(\theta,\phi)$'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 $l+m$.