Berezin quantization of the Schrodinger algebra

TL;DR

This paper explores the Berezin quantization of the Schrödinger algebra, revealing its structure by analyzing its components and their representations, with a focus on decoupling the sl(2) and Heisenberg parts.

Contribution

It introduces a novel realization of the Schrödinger algebra's sl(2) component from the Heisenberg algebra, facilitating a clearer understanding of its structure through Berezin representation.

Findings

Decoupling of sl(2) and Heisenberg components achieved

Berezin representation computed explicitly

Structural insights into the Schrödinger algebra obtained

Abstract

We examine the Schrodinger algebra in the framework of Berezin quantization. First, the Heisenberg-Weyl and sl(2) algebras are studied. Then the Berezin representation of the Schrodinger algebra is computed. In fact, the sl(2) piece of the Schrodinger algebra can be decoupled from the Heisenberg component. This is accomplished using a special realization of the sl(2) component that is built from the Heisenberg piece as the quadratic elements in the Heisenberg-Weyl enveloping algebra. The structure of the Schrodinger algebra is revealed in a lucid way by the form of the Berezin representation.