The higher spin $\Pi$-operator in Clifford analysis

TL;DR

This paper introduces a higher spin $\Pi$-operator in Clifford analysis, explores its properties, and applies it to establish existence and uniqueness results for a higher spin Beltrami equation.

Contribution

It defines the higher spin $\Pi$-operator, analyzes its properties, and applies it to solve a new higher spin Beltrami equation.

Findings

Established norm estimates and mapping properties of the higher spin $\Pi$-operator.

Proved existence and uniqueness of solutions to the higher spin Beltrami equation.

Extended Clifford analysis tools to higher spin field equations.

Abstract

Rarita-Schwinger fields are solutions to the relativistic field equation of spin-$3/2$ fermions in four dimensional flat spacetime, which are important in supergravity and superstring theories. Bure\v s et al. generalized it to arbitrary spin $k/2$ in 2002 in the context of Clifford algebras. In this article, we introduce the higher spin $\Pi$-operator related to the Rarita-Schwinger operator. Further, we investigate norm estimates, mapping properties and the adjoint operator of the higher spin $\Pi$-operator. As an application, a higher spin Beltrami equation is introduced, and existence and uniqueness of solutions to this higher spin Beltrami equation is established by the norm estimate of the higher spin $\Pi$-operator.