The Jacobi group and the squeezed states - some comments

TL;DR

This paper explores the mathematical structure of squeezed states via the Jacobi group, deriving explicit formulas for scalar products, orthonormal bases, and reproducing kernels in a holomorphic representation.

Contribution

It provides a detailed analysis of the Jacobi group's role in realizing squeezed states and derives explicit formulas for scalar products and reproducing kernels.

Findings

Explicit form of the weight function for the scalar product

Orthogonality of base functions established

Reproducing kernel and holomorphic representation derived

Abstract

The generalized coherent states attached to the Jacobi group realize the squeezed states. Imposing hermitian conjugacy to the generators of the Jacobi algebra, we find out the form of the weight function appearing in the scalar product. We show effectively the orthonormality of the base functions with respect to the scalar product. From the explicit form of the reproducing kernel, we find out the expression of the multiplier in a holomorphic representation of the Jacobi group.