Clifford Coherent State Transforms on Spheres

Abstract

We introduce a one-parameter family of transforms, $U_{(m)}^t$, $t>0$, from the Hilbert space of Clifford algebra valued square integrable functions on the $m$--dimensional sphere, $L^2(S^m,d{\sigma }_m)\otimes {\mathbb{C}}_{m+1}$, to the Hilbert spaces, $\mathcal{M}{L}^2({\mathbb{R}}^{m+1}\setminus \{0\},d{\mu }_t)$, of monogenic functions on ${\mathbb{R}}^{m+1}\setminus \{0\}$ which are square integrable with respect to appropriate measures, $d{\mu }_t$. We prove that these transforms are unitary isomorphisms of the Hilbert spaces and are extensions of the Segal-Bargman coherent state transform, $U_{(1)}:L^2(S^1,d{\sigma }_1)⟶\mathcal{H}{L}^2(\mathbb{C}\setminus \{0\},d\mu )$, to higher dimensional spheres in the context of Clifford analysis. In Clifford analysis it is natural to replace the analytic continuation from $S^m$ to $S_{\mathbb{C}}^m$ as in \cite{Ha1, St, HM} by the Cauchy--Kowalewski extension from $S^m$ to ${\mathbb{R}}^{m+1}\setminus \{0\}$. One then obtains a unitary isomorphism from an $L^2$--Hilbert space to an Hilbert space of solutions of the Dirac equation, that is to a Hilbert space of monogenic functions.

13 pages