Extending coherent state transforms to Clifford analysis

Segal-Bargmann coherent state transforms can be viewed as unitary maps from L2 spaces of functions (or sections of an appropriate line bundle) on a manifoldX to spaces of square integrable holomorphic functions (or sections) onXC. It is natural to consider higher dimensional extensions of X based on Clifford algebras as they could be useful in studying quantum systems with internal, discrete, degrees of freedom corresponding to nonzero spins. Notice that extensions of X based on the Grassman algebra appear naturally in the study of supersymmetric quantum mechanics. In Clifford analysis the zero mass Dirac equation provides a natural generalization of the Cauchy-Riemann conditions of complex analysis and leads to monogenic functions. For the simplest but already quite interesting case of X = R we introduce two extensions of the Segal-Bargmann coherent state transform from L2(R, dx) ⊗ Rm to Hilbert spaces of slice monogenic and axial monogenic functions and study their properties. These two transforms are related by the dual Radon transform. Representation theoretic and quantum mechanical aspects of the new representations are studied.