Star products: a group-theoretical point of view

TL;DR

This paper explores the construction of star products using group theory, providing explicit formulas and examining examples like phase space translations and the affine group, linking to quantum mechanics and wavelet analysis.

Contribution

It introduces a group-theoretical framework for star products, deriving explicit formulas and connecting to well-known structures like the Groenewold-Moyal product and wavelet analysis.

Findings

Explicit formulas for star products associated with group representations

Connection between star products and quantum phase space formulations

Link between affine group star products and wavelet analysis

Abstract

Adopting a purely group-theoretical point of view, we consider the star product of functions which is associated, in a natural way, with a square integrable (in general, projective) representation of a locally compact group. Next, we show that for this (implicitly defined) star product explicit formulae can be provided. Two significant examples are studied in detail: the group of translations on phase space and the one-dimensional affine group. The study of the first example leads to the Groenewold-Moyal star product. In the second example, the link with wavelet analysis is clarified.