Clifford Algebra-valued Segal-Bargmann Transform and Taylor Isomorphism

TL;DR

This paper extends classical Segal-Bargmann theory to Clifford algebra-valued functions, establishing new unitary isomorphisms among various function spaces related to Clifford analysis.

Contribution

It introduces a Clifford algebra-valued Segal-Bargmann transform and Taylor isomorphism, generalizing classical concepts to Clifford algebra contexts.

Findings

Established unitary isomorphisms among Clifford-valued function spaces

Connected square-integrable functions with monogenic functions and tensor functionals

Extended wave-particle duality concepts to Clifford algebra framework

Abstract

Classical Segal-Bargmann theory studies three Hilbert space unitary isomorphisms that describe the wave-particle duality and the configuration space-phase space. In this work, we generalized these concepts to Clifford algebra-valued functions. We establish the unitary isomorphisms among the space of Clifford algebra-valued square-integrable functions on $\mathbb{R}^n$ with respect to a Gaussian measure, the space of monogenic square-integrable functions on $\mathbb{R}^{n+1}$ with respect to another Gaussian measure and the space of Clifford algebra-valued linear functionals on symmetric tensor elements of $\mathbb{R}^n$.