Hypermonogenic solutions and plane waves of the Dirac operator in Rp x Rq

TL;DR

This paper introduces hypermonogenic solutions of the Dirac operator in product spaces, explores their properties including a Cauchy integral formula, and develops explicit methods for hypermonogenic plane wave solutions.

Contribution

It defines hypermonogenic solutions in Rp x Rq, establishes their fundamental properties, and provides explicit computational methods for hypermonogenic plane waves.

Findings

Established a Cauchy integral formula for hypermonogenic solutions

Defined hypermonogenic plane wave solutions

Provided explicit methods for computing these solutions

Abstract

In this paper we first define hypermonogenic solutions of the Dirac operator in Rp x Rq and study some basic properties, e.g., obtaining a Cauchy integral formula in the unit hemisphere. Hypermonogenic solutions form a natural function class in classical Clifford analysis. After that, we define the corresponding hypermonogenic plane wave solutions and deduce explicit methods to compute these functions.