Basic aspects of symplectic Clifford analysis for the symplectic Dirac operator

TL;DR

This paper explores foundational elements of symplectic Clifford analysis related to the symplectic Dirac operator, focusing on symmetry operators, Fourier transform, kernels, and basis construction in low-dimensional spaces.

Contribution

It introduces new methods for analyzing symplectic Dirac operators, including symmetry operators, Fourier transforms, and basis constructions in symplectic Clifford analysis.

Findings

Analysis of first order symmetry operators

Development of symplectic Clifford-Fourier transform

Construction of bases for symplectic monogenics

Abstract

In the present article we study basic aspects of the symplectic version of Clifford analysis associated to the symplectic Dirac operator. Focusing mostly on the symplectic vector space of real dimension $2$, this involves the analysis of first order symmetry operators, symplectic Clifford-Fourier transform, reproducing kernel for the symplectic Fischer product and the construction of bases of symplectic monogenics for the symplectic Dirac operator.