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 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.

23 pages. Misprints corrected, references updated, one remark added at the end of the section 3. To appear in Review in Mathematical Physics

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• Closed star product on noncommutative ${\mathbb{R}}^3$ and scalar field dynamics
• Slavnov-Taylor identities, non-commutative gauge theories and infrared divergences
• Spectral triplets, statistical mechanics and emergent geometry in non-commutative quantum mechanics
• Single Extra Dimension from $\kappa$-Poincaré and Gauge Invariance
• Spectral theorem in noncommutative field theories: Jacobi dynamics
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