Duality symmetry for star products

TL;DR

The paper explores a duality symmetry in star product schemes, revisiting known formalisms and introducing a new algebra of operator symbols with an explicit kernel, enhancing the understanding of noncommutative structures in quantum mechanics.

Contribution

It introduces a duality symmetry for star products, revisits existing formalisms, and establishes a new star product with an explicit kernel, expanding the mathematical framework of quantum operator symbols.

Findings

Revisits Weyl-Wigner-Moyal, Husimi, Glauber-Sudarshan maps

Identifies dual partners for known star-product schemes

Derives a new star product with explicit kernel and examples

Abstract

A duality property for star products is exhibited. In view of it, known star-product schemes, like the Weyl-Wigner-Moyal formalism, the Husimi and the Glauber-Sudarshan maps are revisited and their dual partners elucidated. The tomographic map, which has been recently described as yet another star product scheme, is considered. It yields a noncommutative algebra of operator symbols which are positive definite probability distributions. Through the duality symmetry a new noncommutative algebra of operator symbols is found, equipped with a new star product. The kernel of the new star product is established in explicit form and examples are considered.