The spectral distance on the Moyal plane

Details The spectral distance on the Moyal plane The spectral distance on the Moyal plane

2009-12-04

arxiv.org/abs/0912.0906v3

J.Geom.Phys.61:1881-1897,2011

Physics

High Energy Physics

High Energy Physics - Theory

Published version. Misprints corrected and references updated; Journal of Geometry and Physics (2011)

Scientific paper

10.1016/j.geomphys.2011.04.021

We study the noncommutative geometry of the Moyal plane from a metric point of view. Starting from a non compact spectral triple based on the Moyal deformation A of the algebra of Schwartz functions on R^2, we explicitly compute Connes' spectral distance between the pure states of A corresponding to eigenfunctions of the quantum harmonic oscillator. For other pure states, we provide a lower bound to the spectral distance, and show that the latest is not always finite. As a consequence, we show that the spectral triple [20] is not a spectral metric space in the sense of [5]. This motivates the study of truncations of the spectral triple, based on M_n(C) with arbitrary integer n, which turn out to be compact quantum metric spaces in the sense of Rieffel. Finally the distance is explicitly computed for n=2.

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