Clifford Algebra-Valued Segal–Bargmann Transform and Taylor Isomorphism

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 $$\mathbb {R}^n$$ .