The Gelfand-Tsetlin bases for spherical monogenics in dimension 3

TL;DR

This paper constructs explicit orthogonal Gelfand-Tsetlin bases for spherical monogenics in three dimensions, providing new tools for solving Dirac equations with applications in mathematical physics.

Contribution

It introduces a Gelfand-Tsetlin basis approach to explicitly build orthogonal bases for spherical monogenics in dimension 3, enhancing previous analytic methods.

Findings

Explicit orthogonal bases with Appell property obtained

Comparison with previous analytic bases shows consistency

Gelfand-Tsetlin construction simplifies basis derivation

Abstract

The main aim of this paper is to recall the notion of the Gelfand-Tsetlin bases (GT bases for short) and to use it for an explicit construction of orthogonal bases for the spaces of spherical monogenics (i.e., homogeneous solutions of the Dirac or the generalized Cauchy-Riemann equation, respectively) in dimension 3. In the paper, using the GT construction, we obtain explicit orthogonal bases for spherical monogenics in dimension 3 having the Appell property and we compare them with those constructed by the first and the second author recently (by a direct analytic approach).