Slice Segal-Bargmann transform

The Segal-Bargmann transform is a unitary map between the Schrodinger and Fock space, which is used, for example, to show the integrability of quantum Rabi models. Slice monogenic functions provide the framework in which functional calculus for quaternionic quantum mechanics can be developed. In this paper, a generalisation of the Segal-Bargmann transform, to the context of slice monogenic functions, is constructed and studied in detail. It is shown to interact appropriately with the recently constructed slice Fourier transform. This leads furthermore to a construction of a slice Fock space, which is shown to be a reproducing kernel space.