Dirac operator on the Riemann sphere

TL;DR

This paper explicitly solves for the spectrum and eigenfunctions of the Dirac operator on the Riemann sphere, revealing their group properties and providing a complete orthonormal set of spinor functions.

Contribution

It introduces a new set of eigenfunctions for the Dirac operator on the sphere, characterized by integer eigenvalues and SU(2) representations, expanding the understanding of spherical spinors.

Findings

Eigenvalues are nonzero integers.

Eigenfunctions form a complete orthonormal set.

Eigenfunctions correspond to SU(2) representations with half-integer angular momentum.

Abstract

We solve for spectrum, obtain explicitly and study group properties of eigenfunctions of Dirac operator on the Riemann sphere $S^2$. The eigenvalues $\lambda$ are nonzero integers. The eigenfunctions are two-component spinors that belong to representations of SU(2)-group with half-integer angular momenta $l = |\lambda| - \half$. They form on the sphere a complete orthonormal functional set alternative to conventional spherical spinors. The difference and relationship between the spherical spinors in question and the standard ones are explained.