The Fuzzy Sphere Star-Product and Spin Networks

TL;DR

This paper develops a graphical spin network technique to analyze the fuzzy sphere's non-commutative star-product, clarifying its relation to deformation quantization and introducing a novel approach in non-commutative geometry.

Contribution

It introduces a new graphical method using spin networks to analyze the fuzzy sphere's star-product, linking non-commutative geometry with Penrose's spin networks.

Findings

Graphical technique effectively analyzes the star-product expansion.

Clarifies the connection between fuzzy sphere product and deformation quantization.

Potentially broadens the application of spin networks in non-commutative geometry.

Abstract

We analyze the expansion of the fuzzy sphere non-commutative product in powers of the non-commutativity parameter. To analyze this expansion we develop a graphical technique that uses spin networks. This technique is potentially interesting in its own right as introducing spin networks of Penrose into non-commutative geometry. Our analysis leads to a clarification of the link between the fuzzy sphere non-commutative product and the usual deformation quantization of the sphere in terms of the star-product.