Fermion Quasi-Spherical Harmonics

TL;DR

This paper introduces Fermion Quasi-Spherical Harmonics, a novel scalar representation of fermion spin angular momentum with unique double-valued properties on the sphere, expanding the mathematical tools for quantum angular momentum analysis.

Contribution

It presents a new class of spherical harmonics for half-odd-integer values of angular momentum, with specific symmetry and domain properties, not previously documented.

Findings

Derived explicit forms of Fermion Quasi-Spherical Harmonics.

Showed these functions have 4π symmetry and double-valued nature.

Provided a table of these functions for practical use.

Abstract

Spherical Harmonics, $Y_\ell^m(\theta,\phi)$, are derived and presented (in a Table) for half-odd-integer values of $\ell$ and $m$. These functions are eigenfunctions of $L^2$ and $L_z$ written as differential operators in the spherical-polar angles, $\theta$ and $\phi$. The Fermion Spherical Harmonics are a new, scalar and angular-coordinate-dependent representation of fermion spin angular momentum. They have $4\pi$ symmetry in the angle $\phi$, and hence are not single-valued functions on the Euclidean unit sphere; they are double-valued functions on the sphere, or alternatively are interpreted as having a double-sphere as their domain.