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Advances in High Energy Physics
Volume 2016 (2016), Article ID 3732657, 7 pages
http://dx.doi.org/10.1155/2016/3732657
Research Article

Spherical Harmonics : Positive and Negative Integer Representations of for and

Department of Theoretical Physics and Astrophysics, Faculty of Physics, University of Tabriz, Tabriz 51666-16471, Iran

Received 26 November 2015; Accepted 15 February 2016

Academic Editor: Andrea Coccaro

Copyright © 2016 H. Fakhri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

Abstract

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