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Published Online: 26 January 2017
Accepted: January 2017

Coherent state transforms and the Weyl equation in Clifford analysis

Journal of Mathematical Physics 58, 013503 (2017); https://doi.org/10.1063/1.4974449
José Mourão1, João P. Nunes1, and Tao Qian2
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  • 1Department of Mathematics and Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, University of Lisbon, Lisbon, Portugal
  • 2Department of Mathematics, Faculty of Science and Technology, University of Macau, Macau, China
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, dμ) of ℂm-valued monogenic functions on ℝm+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 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, dμ) of solutions of the Weyl equation on the space-time ℝ × M.

Acknowledgments

The authors would like to thank Frank Sommen for useful comments. J.M. and J.P.N. thank also Pedro Girão and Jorge Silva for helpful discussions.
The authors were partially supported by Macau Government FDCT through the Project No. 099/2014/A2, Two related topics in Clifford analysis, and by the University of Macau Research Grant No. MYRG115(Y1-L4)-FST13-QT. J.M. and J.P.N. were also partly supported by FCT/Portugal through the Project Nos. UID/MAT/04459/2013, EXCL/MAT-GEO/0222/2012, and PTDC/MAT-GEO/3319/2014 and by the (European Cooperation in Science and Technology) COST Action No. MP1405 QSPACE.

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  27. Published by AIP Publishing.

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