CONNES DISTANCE BY EXAMPLES: HOMOTHETIC SPECTRAL METRIC SPACES

Abstract

We study metric properties stemming from the Connes spectral distance on three types of non-compact non-commutative spaces which have received attention recently from various viewpoints in the physics literature. These are the non-commutative 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 the 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 in detail.