The spectral distance in the moyal plane
(2011) Journal of Geometry and Physics — Vol. 61, n° 10, p. 1881-1897 (2011)
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- 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 R2, 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 Mn(C) with arbitrary n∈N, which turn out to be compact quantum metric spaces in the sense of Rieffel. Finally the distance is explicitly computed for n=2. © 2011 Elsevier B.V.
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Cagnache, E., D’Andrea, F., Martinetti, P., & Wallet, J.-C. (2011). The spectral distance in the moyal plane. Journal of Geometry and Physics, 61(10), 1881-1897. https://doi.org/10.1016/j.geomphys.2011.04.021 (Original work published 2011)