Quantization on the sphere

Abstract

When phase-space is a sphere, then conventional quantization is not available and the Heisenberg algebra is most naturally replaced by so(3). Quantization may be carried out in terms of invariant star-products. This leads to the study of an interesting family of polynomials that are defined in a natural and intrinsic way on the enveloping algebra of sl(2). These polynomials are related to Legendre polynomials by a reordering rule that resembles the relation between Hermite polynomials and normal ordered monomials; they are identified as Pasternack polynomials and related to Jacobi polynomials. New orthogonality properties are found and interpreted in terms of unitary representations of SL(2,R) and SO(3).