Complex Weyl symbols of metaplectic operators: an elementary approach

TL;DR

This paper provides explicit formulas for the Berezin and complex Weyl symbols of metaplectic operators using holomorphic representations, connecting classical and quantum symbol calculus.

Contribution

It introduces an elementary approach to derive explicit formulas for the symbols of metaplectic operators, bridging holomorphic representations and classical Weyl calculus.

Findings

Explicit formulas for Berezin and Weyl symbols of metaplectic operators

Recovery of classical Weyl symbol formulas for quadratic forms

Connections established between holomorphic and classical symbol representations

Abstract

We give explicit formulas for the Berezin symbols and the complex Weyl symbols of the metaplectic representation operators by using the holomorphic representations of the Jacobi group. Then we recover some known formulas for the symbols of the metaplectic operators in the classical Weyl calculus, in particular for the classical Weyl symbol of the exponential of an operator whose Weyl symbol is a quadratic form.