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, L2(Sm,dσm)⊗Cm+1, to the Hilbert spaces, ML2(Rm+1∖{0},dμt), of solutions of the Euclidean Dirac equation on Rm+1∖{0} which are square integrable with respect to appropriate measures, dμt. We prove that these transforms are unitary isomorphisms of the Hilbert spaces and are extensions of the Segal-Bargman coherent state transform, U(1):L2(S1,dσ1)⟶HL2(C∖{0},dμ), to higher dimensional spheres in the context of Clifford analysis. In Clifford analysis it is natural to replace the analytic continuation from Sm to SCm as in (Hall, 1994; Stenzel, 1999; Hall and Mitchell, 2002) by the Cauchy-Kowalewski extension from Sm to Rm+1∖{0}. One then obtains a unitary isomorphism from an L2-Hilbert space to a Hilbert space of solutions of the Dirac equation, that is to a Hilbert space of monogenic functions.