On the $\star$-product quantization and the Duflo map in three dimensions

TL;DR

This paper explores the star product on R^3 derived from the Moyal product on R^4, connecting it to the Duflo map, and applies it to compute the hydrogen atom spectrum in quantum mechanics.

Contribution

It introduces a specific star product on R^3 via reduction from R^4 and relates it to the Duflo map, providing a new framework for quantum mechanical applications.

Findings

Matrix basis for the algebra of functions on su(2) dual

Explicit realization of the algebra in this basis

Spectrum of the hydrogen atom computed using the star product

Abstract

We analyze the star product induced on the algebra of functions on R^3 by a suitable reduction of the Moyal product defined on F(R^4). This is obtained through the identification of R^3 with the dual of a three dimensional Lie algebra. We consider the su(2) case, exhibit a matrix basis and realize the algebra of functions on its dual in such a basis. The relation to the Duflo map is discussed. As an application to quantum mechanics we compute the spectrum of the hydrogen atom.