Orthogonal basis for spherical monogenics by step two branching

TL;DR

This paper develops an orthogonal basis for spherical monogenics in Euclidean space by analyzing their decomposition under Spin(p) x Spin(q), providing new tools for harmonic analysis of the Dirac operator.

Contribution

It introduces a Spin(p) x Spin(q)-invariant orthonormal basis for spherical monogenics, including a new basis constructed via an inductive approach with p=2.

Findings

Decomposition of spherical monogenics under Spin(p) x Spin(q)

Construction of a Spin(p) x Spin(q)-invariant orthonormal basis

New orthonormal basis for spherical monogenics using inductive methods

Abstract

Spherical monogenics can be regarded as a basic tool for the study of harmonic analysis of the Dirac operator in Euclidean space R^m. They play a similar role as spherical harmonics do in case of harmonic analysis of the Laplace operator on R^m. Fix the direct sum R^m = R^p x R^q. In this paper we will study the decomposition of the space M_n(R^m;C_m) of spherical monogenics of order n under the action of Spin(p) x Spin(q). As a result we obtain a Spin(p) x Spin(q)-invariant orthonormal basis for M_n(R^m;C_m). In particular, using the construction with p = 2 inductively, this yields a new orthonormal basis for the space M_n(R^m;C_m).