Rigid Particle and its Spin Revisited

TL;DR

This paper revisits the representation of spinors using double valued spherical harmonics, clarifying previous objections and exploring the implications for rigid particles with extrinsic curvature in velocity space.

Contribution

It develops and justifies Pandres' arguments on spinor representations, correcting misconceptions and analyzing angular momentum in velocity space for rigid particles.

Findings

Double valued spherical harmonics can form a basis for spinor representations.

Proper choice of functions and inner product resolves hermiticity and orthogonality issues.

Orbital angular momentum in configuration space has integer eigenvalues, but differs in velocity space.

Abstract

The arguments by Pandres that the double valued spherical harmonics provide a basis for the irreducible spinor representation of the three dimensional rotation group are further developed and justified. The usual arguments against the inadmissibility of such functions, concerning hermiticity, orthogonality, behavior under rotations, etc., are all shown to be related to the unsuitable choice of functions representing the states with opposite projections of angular momentum. By a correct choice of functions and definition of inner product those difficulties do not occur. And yet the orbital angular momentum in the ordinary configuration space can have integer eigenvalues only, for the reason which have roots in the nature of quantum mechanics in such space. The situation is different in the velocity space of the rigid particle, whose action contains a term with the extrinsic curvature.