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 (Gayral et al. (2004) [17]) is not a spectral metric space in the sense of Bellissard et al. (2010)[19]. This motivates the study of truncations of the spectral triple, based on M-n (C) with arbitrary n is an element of N, which turn out to be compact quantum metric spaces in the sense of Rieffel. Finally the distance is explicitly computed for n = 2. (C) 2011 Elsevier B.V. All rights reserved.

The spectral distance in the moyal plane

Martinetti P.;
2011-01-01

Abstract

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 (Gayral et al. (2004) [17]) is not a spectral metric space in the sense of Bellissard et al. (2010)[19]. This motivates the study of truncations of the spectral triple, based on M-n (C) with arbitrary n is an element of N, which turn out to be compact quantum metric spaces in the sense of Rieffel. Finally the distance is explicitly computed for n = 2. (C) 2011 Elsevier B.V. All rights reserved.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11567/974896
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