Metric Properties of the Fuzzy Sphere

TL;DR

This paper investigates the metric properties of the fuzzy sphere, demonstrating that its spectral distances converge to those of the classical sphere in the high-spin limit, bridging quantum and classical geometries.

Contribution

It provides a detailed description of the metrics on the fuzzy sphere and proves convergence of the spectral distances to the classical sphere metrics.

Findings

Spectral distances on the fuzzy sphere form a converging sequence.

Bloch coherent states approximate the classical sphere in the high-spin limit.

The fuzzy sphere's metrics approach those of the classical sphere as spin increases.

Abstract

The fuzzy sphere, as a quantum metric space, carries a sequence of metrics which we describe in detail. We show that the Bloch coherent states, with these spectral distances, form a sequence of metric spaces that converge to the round sphere in the high-spin limit.