Fermion States on the Sphere $S^2$

TL;DR

This paper derives the spectrum and eigenfunctions of the Dirac operator on the sphere, revealing nonzero integer eigenvalues and unique spinor eigenfunctions classified by SU(2) representations, differing from standard spherical spinors.

Contribution

It provides explicit solutions for the Dirac operator on the sphere, including nonstandard eigenfunctions classified by SU(2) representations with half-integer angular momenta.

Findings

Eigenvalues are nonzero integers.

Eigenfunctions are special linear combinations of spherical spinors.

Eigenfunctions are classified by SU(2) representations with half-integer angular momenta.

Abstract

We solve for the spectrum and eigenfunctions of Dirac operator on the sphere. The eigenvalues are nonzero whole numbers. The eigenfunctions are two-component spinors which may be classified by representations of the SU(2) group with half-integer angular momenta. They are not the conventional spherical spinors but special linear combinations of those.