Star product and ordered moments of photon creation and annihilation operators

TL;DR

This paper introduces a star-product scheme for symbols based on normally ordered powers of photon creation and annihilation operators, providing explicit kernels and new factorial sum relations, with applications to quantum dynamics.

Contribution

It develops a novel star-product framework for ordered moments of photon operators, including explicit kernels and analysis of its properties, advancing quantum phase space methods.

Findings

Explicit star-product kernel and intertwining kernels derived.

New sum relations for factorials discovered.

Time evolution equations in ordered moments formulated.

Abstract

We develop a star-product scheme of symbols defined by the normally ordered powers of the creation and annihilation photon operators, (a\dag)^m a^n. The corresponding phase space is a two-dimensional lattice with nodes (m,n) given by pairs of nonnegative integers. The star-product kernel of symbols on the lattice and intertwining kernels to other schemes are found in explicit form. Analysis of peculiar properties of the star-product kernel results in new sum relations for factorials. Advantages of the developed star-product scheme for describing dynamics of quantum systems are discussed and time evolution equations in terms of the ordered moments are derived.