Dirac operator on the Riemann sphere

TL;DR

This work provides an explicit solution for the Dirac operator on the Riemann sphere S^2, yielding a complete, orthonormal set of two-component spinors whose eigenvalues are nonzero integers and whose spinor structure forms SU(2) multiplets with half-integer angular momentum l = |λ| − 1/2. The eigenfunctions, constructed via Jacobi polynomials, reveal a clear SU(2) representation theory: λ determines l, and the corresponding Υ spinors come in 2l+1 member multiplets with well-defined ladder operators. A detailed comparison with conventional Ω spherical spinors shows Υ spinors diagonalize the Dirac operator on S^2 while Ω spinors diagonalize total angular momentum, establishing precise relations between the two bases through cartesian rotations. Time-reversal and complex-conjugation properties are fixed to respect SU(2) symmetry, and the paper provides explicit Cartesian realizations and transformation rules, enabling practical applications in problems with fermions on spherical geometries and in contexts such as spectral boundary conditions and fullerenes.

Abstract

We solve for spectrum, obtain explicitly and study group properties of eigenfunctions of Dirac operator on the Riemann sphere $S^2$. The eigenvalues $λ$ are nonzero integers. The eigenfunctions are two-component spinors that belong to representations of SU(2)-group with half-integer angular momenta $l = |λ| - \half$. 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.