Pierre Martinetti, Luca Tomassini
We study the metric aspect of the Moyal plane from Connes' noncommutative geometry point of view. First, we compute Connes' spectral distance associated with the natural isometric action of R^2 on the algebra of the Moyal plane A. We show that the distance between any state of A and any of its translated is precisely the amplitude of the translation. As a side result, we obtain the spectral distance between coherent states of the quantum harmonic oscillator as the Euclidean distance on the plane, multiplied by the Planck length. Second, we compute the spectral distance in the double Moyal plane, intended as the product - in the sense of spectral triples - of (the minimal unitization of) A by C^2. On the set of states obtained by translation of an arbitrary state of A, this distance is given by Pythagoras theorem. Applied to the Doplicher-Fredenhagen-Roberts model of quantum spacetime [DFR], these two theorems show that Connes' spectral distance and the DFR quantum length coincide on the set of states of optimal localization. On the way, we also prove some Pythagoras inequalities for the product of arbitrary unital & non-degenerate spectral triples. Some of the results of this paper can be thought as a continuation of arXiv:0912.0906, as well as a companion to arXiv:1106.0261.
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http://arxiv.org/abs/1110.6164
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