Canonical integral operators on the Fock space

TL;DR

This paper introduces a two-parameter family of integral operators on the Fock space, characterizes their boundedness and unitarity, relates them to classical transforms via the Bargmann transform, and constructs a new representation of SL(2,R).

Contribution

It defines and analyzes a novel class of integral operators on the Fock space, connecting them to classical transforms and group representations.

Findings

Operators include classical linear canonical transforms as special cases.

Precisely characterized when operators are bounded and unitary.

Established a new unitary projective representation of SL(2,R).

Abstract

In this paper we introduce and study a two-parameter family of integral operators on the Fock space $F^2(C)$. We determine exactly when these operators are bounded and when they are unitary. We show that, under the Bargmann transform, these operators include the classical linear canonical transforms as special cases. As an application, we obtain a new unitary projective representation for the special linear group $SL(2,R)$ on the Fock space.