Coherent State Transforms and the Weyl Equation in Clifford Analysis

TL;DR

This paper introduces a Clifford algebra-valued transform analogous to the Segal-Bargmann transform, establishing a unitary isomorphism between certain Hilbert spaces of functions and solutions to the Weyl equation, with implications for quantum representations.

Contribution

It constructs a Clifford algebra-based coherent state transform that is a unitary isomorphism, linking Hilbert spaces of square-integrable functions and monogenic solutions of the Weyl equation, extending to torus cases.

Findings

The transform is a unitary isomorphism between the Hilbert spaces.

It provides an orthonormal basis of monogenic functions.

Establishes quantum equivalence of Schrödinger representations on Euclidean space and torus.

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 Schr\"odinger 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 ${\mathbb R}\times M$.