Fermion Quasi-Spherical Harmonics

Abstract

Spherical Harmonics, $Y_\ell^m(θ,ϕ)$, 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, $θ$ and $ϕ$. The Fermion Spherical Harmonics are a new, scalar and angular-coordinate-dependent representation of fermion spin angular momentum. They have $4π$ symmetry in the angle $ϕ$, 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.