Gaussian transform of the Weil representation

TL;DR

This paper explores the Gaussian transform of the Weil representation, detailing its effects on Schwartz spaces and distributions, and discusses extending these ideas to infinite dimensions for quantum field applications.

Contribution

It provides a detailed description of the Gaussian transform of the Weil representation and proposes a framework for its infinite-dimensional generalization for quantum field quantization.

Findings

Explicit description of the Weil representation's image under Gaussian transform

Discussion on infinite-dimensional generalization for scalar field quantization

Potential applications in quantum field theory

Abstract

A description is given of the image of the Weil representation of the symplectic group in the Schwartz space and in the space of tempered distributions under the Gaussian integral transform. We also discuss the problem of infinite dimensional generalization of the Weil representation in the Schwartz space, in order to construct appropriate quantization of free scalar field.