Metrics and causality on Moyal planes

Abstract

Metrics structures stemming from the Connes distance promote Moyal planes to the status of quantum metric spaces. We discuss this aspect in the light of recent developments, emphasizing the role of Moyal planes as representative examples of a recently introduced notion of quantum (noncommutative) locally compact space. We move then to the framework of Lorentzian noncommutative geometry and we examine the possibility of defining a notion of causality on Moyal plane, which is somewhat controversial in the area of mathematical physics. We show the actual existence of causal relations between the elements of a particular class of pure (coherent) states on Moyal plane with related causal structure similar to the one of the usual Minkowski space, up to the notion of locality.

33 pages. Improved version; a summary added at the end of the introduction, misprints corrected. Version to appear in Contemporary Mathematics

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Cited by in corpus (9)

• The Lorentzian distance formula in noncommutative geometry
• Exact Partition Functions for Gauge Theories on ${\mathbb{R}}_{\lambda }^3$
• Single Extra Dimension from $\kappa$-Poincaré and Gauge Invariance
• Spectral Distance on Lorentzian Moyal Plane
• Quantum causality constraints on kappa-Minkowski space-time
• Connes spectral distance and nonlocality of generalized noncommutative phase spaces
• Quantum causality in $\kappa$-Minkowski and related constraints
• Connes distance of $2D$ harmonic oscillators in quantum phase space
• Connes spectral distances, quantum discord and coherence of qubits