Connes distance function on fuzzy sphere and the connection between geometry and statistics

TL;DR

This paper extends the computation of Connes spectral distance to the fuzzy sphere within non-commutative quantum mechanics, revealing a link between geometry and statistics through analysis of states and coherent states.

Contribution

It applies an existing spectral distance algorithm to the fuzzy sphere, demonstrating the geometric-statistical connection in this new non-commutative setting.

Findings

Computed Connes spectral distance on fuzzy sphere

Established a connection between geometry and statistics

Analyzed distances between mixed states and coherent states

Abstract

An algorithm to compute Connes spectral distance, adaptable to the Hilbert-Schmidt operatorial formulation of non-commutative quantum mechanics, was developed earlier by introducing the appropriate spectral triple and used to compute infinitesimal distances in the Moyal plane, revealing a deep connection between geometry and statistics. In this paper, using the same algorithm, the Connes spectral distance has been calculated in the Hilbert-Schmidt operatorial formulation for the fuzzy sphere whose spatial coordinates satisfy the $su(2)$ algebra. This has been computed for both the discrete, as well as for the Perelemov's $SU(2)$ coherent state. Here also, we get a connection between geometry and statistics which is shown by computing the infinitesimal distance between mixed states on the quantum Hilbert space of a particular fuzzy sphere, indexed by $n\in\mathbb{Z}/2$.