Functional Representations for Fock Superalgebras

TL;DR

This paper develops functional representations of Fock superalgebras using superspace functions, introduces Gaussian integration in infinite dimensions, and derives a Mehler formula for the Ornstein-Uhlenbeck semigroup.

Contribution

It constructs superanalogs of classical function spaces and representations, extending the mathematical framework of Fock spaces to superspaces with new integration and ordering techniques.

Findings

Defined Gaussian integration on infinite dimensional superspaces

Constructed superanalogs of classical function spaces with reproducing kernels

Derived a Mehler formula for the Ornstein-Uhlenbeck semigroup

Abstract

The Fock space of bosons and fermions and its underlying superalgebra are represented by algebras of functions on a superspace. We define Gaussian integration on infinite dimensional superspaces, and construct superanalogs of the classical function spaces with a reproducing kernel -- including the Bargmann-Fock representation -- and of the Wiener-Segal representation. The latter representation requires the investigation of Wick ordering on Z2-graded algebras. As application we derive a Mehler formula for the Ornstein-Uhlenbeck semigroup on the Fock space.