Clifford Algebra-valued Segal-Bargmann Transform

Abstract

A classical Segal-Bargmann transform maps square-integrable functions on R to holomorphic square-integrable functions on C with respect to some Gaussian measure. In this work, we extend the classical Segal-Bargmann transform to functions taking values in a Clifford algebra. We establish that in this setting, the generalized Segal-Bargmann transform is a unitary isomorphism mapping Clifford algebra-valued square-integrable functions on Rⁿ with respect to some Gaussian measure to monogenic square-integrable functions on Rⁿ⁺¹ with respect to another Gaussian measure. We also discuss about the differential and multiplication operators in the monogenic functions space as well as certain properties of their domains.