Clifford Coherent State Transforms on Spheres

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, $U_{(1)} : L^2(S^{1},d\sigma_{1}) \longrightarrow {\mathcal H}L^2({$\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 $S_{\mathbb{C}}^m$ 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.