Coherent states on quaternion Slices and a measurable field of Hilbert spaces

TL;DR

This paper constructs and analyzes a family of reproducing kernel Hilbert spaces over quaternion slices using coherent states, demonstrating their structure as a measurable field and their relation to representations of quaternionic groups.

Contribution

It introduces a measurable field of Hilbert spaces over quaternion slices and links these to group representations, expanding the understanding of quaternionic Hilbert space structures.

Findings

The set of Hilbert spaces forms a measurable field.

Their direct integral is a reproducing kernel Hilbert space.

The spaces serve as representation spaces for irreducible unitary groups.

Abstract

A set of reproducing kernel Hilbert spaces are obtained on Hilbert spaces over quaternion slices with the aid of coherent states. It is proved that the so obtained set forms a measurable field of Hilbert spaces and their direct integral appears again as a reproducing kernel Hilbert space for a bigger Hilbert space over the whole quaternions. Hilbert spaces over quaternion slices are identified as representation spaces for a set of irreducible unitary group representations and their direct integral is shown to be a reducible representation for the Hilbert space over the whole quaternion field.