Connes distance by examples: Homothetic spectral metric spaces

TL;DR

This paper investigates the metric properties of the Connes spectral distance on various noncommutative spaces, revealing their homothetic spectral metric space structure and providing explicit distance formulas.

Contribution

It demonstrates that certain noncommutative spaces are homothetic spectral metric spaces with explicit distance formulas, expanding understanding of their geometric properties.

Findings

Spaces have infinite connected components

Homothetic factors relate to determinants of Clifford metrics

New explicit spectral distance formulas obtained

Abstract

We study metric properties stemming from the Connes spectral distance on three types of non compact noncommutative spaces which have received attention recently from various viewpoints in the physics literature. These are the noncommutative Moyal plane, a family of harmonic Moyal spectral triples for which the Dirac operator squares to the harmonic oscillator Hamiltonian and a family of spectral triples with Dirac operator related to the Landau operator. We show that these triples are homothetic spectral metric spaces, having an infinite number of distinct pathwise connected components. The homothetic factors linking the distances are related to determinants of effective Clifford metrics. We obtain as a by product new examples of explicit spectral distance formulas. The results are discussed.