Coherent state transforms and the Weyl equation in Clifford analysis

We study a transform, inspired by coherent state transforms, from the Hilbert space of Clifford algebra valued square integrable functions L²(ℝ^m, dx) ⊗ ℂ_m to a Hilbert space of solutions of the Weyl equation on ℝ^m+1 = ℝ × ℝ^m, namely, to the Hilbert space ℳL²(ℝ^m+1, dμ) of ℂ_m-valued monogenic functions on ℝ^m+1 which are L² with respect to an appropriate measure dμ. 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 ℝ^m+1. We also study the case when ℝ^m is replaced by the m-torus 𝕋^m. Quantum mechanically, this extension establishes the unitary equivalence of the Schrödinger representation on M, for M = ℝ^m and M = 𝕋^m, with a representation on the Hilbert space ℳL²(ℝ × M, dμ) of solutions of the Weyl equation on the space-time ℝ × M.