Projective Fourier Duality and Weyl Quantization

TL;DR

This paper reexamines the Weyl-Wigner correspondence through the lens of Kac algebras, extending the framework to include projective structures for a better understanding of Fourier duality in quantum mechanics.

Contribution

It introduces an extension of Kac algebras to incorporate physical requirements, establishing a new dual framework for the translation group on the plane using projective Kac algebras.

Findings

Weyl formula emerges as part of the duality mapping.

Extended Kac structures accommodate physical constraints.

New duality framework for translation groups on the plane.

Abstract

The Weyl-Wigner correspondence prescription, which makes large use of Fourier duality, is reexamined from the point of view of Kac algebras, the most general background for noncommutative Fourier analysis allowing for that property. It is shown how the standard Kac structure has to be extended in order to accommodate the physical requirements. An Abelian and a symmetric projective Kac algebras are shown to provide, in close parallel to the standard case, a new dual framework and a well-defined notion of projective Fourier duality for the group of translations on the plane. The Weyl formula arises naturally as an irreducible component of the duality mapping between these projective algebras.