Clifford Coherent State Transforms on Spheres

Abstract

We introduce a one-parameter family of transforms, $U^t_{(m)}$, $t>0$, 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, $U_{(1)} : L^2(S^{1},d\sigma_{1}) \longrightarrow {\mathcal H}L^2({\mathbb{C} \setminus \{0\}},d\mu)$, 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 $S^m_{\mathbb{C}}$ 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.