Integral representations of the star product corresponding to the $s$-ordering of the creation and annihilation operators

TL;DR

This paper derives a new integral representation for the star product associated with the s-ordering of creation and annihilation operators, enabling continuous variation between different operator orderings.

Contribution

It introduces a novel integral representation for the s-ordered star product using reproducing formulas, expanding the mathematical tools for quantum operator ordering.

Findings

Derived a new integral representation for the s-ordered star product.

Compared properties of different kernels for star products.

Connected the s-ordering with intermediate quantization schemes.

Abstract

A new integral representation is obtained for the star product corresponding to the $s$-ordering of the creation and annihilation operators. This parametric ordering convention introduced by Cahill and Glauber enables one to vary the type of ordering in a continuous way from normal order to antinormal order. Our derivation of the corresponding integral representation is based on using reproducing formulas for analytic and antianalytic functions. We also discuss a different representation whose kernel is a generalized function and compare the properties of this kernel with those of the kernels of another family of star products which are intermediate between the $qp-$ and $pq$-quantization.