The Fuzzy Sphere: Star Product Induced From Generalized Squeezed States

TL;DR

This paper constructs a star product on the fuzzy sphere using generalized squeezed states, linking it to known formulations and providing new physical insights into the structure of noncommutative geometry.

Contribution

It introduces a new star product derived from generalized squeezed states on the fuzzy sphere, connecting it to existing models and elucidating the emergence of spherical harmonics.

Findings

Star product coincides with Gross and Presnajder's in simple cases

Squeezed states reproduce stereographic projection at large j

Spherical harmonics naturally emerge in this framework

Abstract

A family of states built from the uncertainty principle on the fuzzy sphere has been shown to reproduce the stereographic projection in the large $j$ limit. These generalized squeezed states are used to construct an associative star product which involves a finite number of derivatives on its primary functional space. It is written in terms of a variable on the complex plane. We show that it actually coincides with the one found by H.Gross and P.Presnajder in the simplest cases, endowing the later with a supplementary physical interpretation. We also show how the spherical harmonics emerge in this setting.