Mourao et al: Coherent state transforms and the Weyl equation in Clifford analysis


José Mourão, João P. Nunes, and Tao Qian, Coherent state transforms and the Weyl equation in Clifford analysis, Journal of Mathematical Physics, Volume 58, Issue 1, 10.1063/1.4974449, DOI: http://dx.doi.org/10.1063/1.4974449, Download: http://aip.scitation.org/doi/pdf/10.1063/1.4974449

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

Source: Email from journals_AT_aip-info.org, 7th Feb. 2017

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