The Fock space in the slice hyperholomorphic setting

TL;DR

This paper introduces the Fock space in the slice hyperholomorphic setting, exploring its properties for quaternionic and Clifford algebra-valued functions, and initiating infinite dimensional analysis in this context.

Contribution

It is the first to define and analyze the Fock space in the slice hyperholomorphic framework for quaternions and Clifford algebras.

Findings

Defined the Fock space in the slice hyperholomorphic setting

Analyzed properties of slice regular and slice monogenic functions

Introduced the full Fock space for quaternions

Abstract

In this paper we introduce and study some basic properties of the Fock space (also known as Segal-Bargmann space) in the slice hyperholomorphic setting. We discuss both the case of slice regular functions over quaternions and also the case of slice monogenic functions with values in a Clifford algebra. In the specific setting of quaternions, we also introduce the full Fock space. This paper can be seen as the beginning of the study of infinite dimensional analysis in the quaternionic setting.