Coherent State Transforms and the Weyl Equation in Clifford Analysis

Abstract

We study a transform, inspired by coherent state transforms, from the Hilbert space of Clifford algebra valued square integrable functions $L^2({\mathbb{R}}^m,dx)\otimes {\mathbb{C}}_m$ to a Hilbert space of solutions of the Weyl equation on ${\mathbb{R}}^{m+1}=\mathbb{R}\times {\mathbb{R}}^m$, namely to the Hilbert space $\mathcal{M}{L}^2({\mathbb{R}}^{m+1},d\mu )$ of ${\mathbb{C}}_m$-valued monogenic functions on ${\mathbb{R}}^{m+1}$ which are $L^2$ with respect to an appropriate measure $d\mu$. We prove that this transform is a unitary isomorphism of Hilbert spaces and that it is therefore an analog of the Segal-Bargmann transform for Clifford analysis. As a corollary we obtain an orthonormal basis of monogenic functions on ${\mathbb{R}}^{m+1}$. We also study the case when ${\mathbb{R}}^m$ is replaced by the $m$-torus ${\mathbb{T}}^m.$ Quantum mechanically, this extension establishes the unitary equivalence of the Schrödinger representation on $M$, for $M={\mathbb{R}}^m$ and $M={\mathbb{T}}^m$, with a representation on the Hilbert space $\mathcal{M}{L}^2(\mathbb{R}\times M,d\mu )$ of solutions of the Weyl equation on the space-time $\mathbb{R}\times M$.

Cited by in corpus (2)

• The Cholewinski-Fock space in the Slice Hyperholomorphic Setting
• Clifford Algebra-valued Segal-Bargmann Transform and Taylor Isomorphism