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The spectral distance in the Moyal plane. (English) Zbl 1226.81095

Summary: 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 \(\mathcal A\) of the algebra of Schwartz functions on \(\mathbb R^{2}\), we explicitly compute Connes’ spectral distance between the pure states of \(\mathcal A\) 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 [V. Gayral et al., Commun. Math. Phys. 246, No. 3, 569–623 (2004; Zbl 1084.58008)] is not a spectral metric space in the sense of J. V. Bellissard et al. [“Dynamical systems on spectral metric spaces”, arXiv:1008.4617]. This motivates the study of truncations of the spectral triple, based on \(M_{n}(\mathbb C)\) with arbitrary \(n\in \mathbb N\), which turn out to be compact quantum metric spaces in the sense of Rieffel. Finally the distance is explicitly computed for \(n=2\).

MSC:

81R60 Noncommutative geometry in quantum theory
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53D55 Deformation quantization, star products
81S10 Geometry and quantization, symplectic methods

Citations:

Zbl 1084.58008
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References:

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