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Noncommutative Geometry of the Moyal Plane: Translation Isometries, Connes’ Distance on Coherent States, Pythagoras Equality

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Abstract

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 \({\mathbb{R}^2}\) on the algebra of the Moyal plane \({\mathcal{A}}\). We show that the distance between any state of \({\mathcal{A}}\) and any of its translated states is precisely the amplitude of the translation. As a consequence, we obtain the spectral distance between coherent states of the quantum harmonic oscillator as the Euclidean distance on the plane. We investigate the classical limit, showing that the set of coherent states equipped with Connes’ spectral distance tends towards the Euclidean plane as the parameter of deformation goes to zero. The extension of these results to the action of the symplectic group is also discussed, with particular emphasis on the orbits of coherent states under rotations. Second, we compute the spectral distance in the double Moyal plane, intended as the product of (the minimal unitization of) \({\mathcal{A}}\) by \({\mathbb{C}^2}\). We show that on the set of states obtained by translation of an arbitrary state of \({\mathcal{A}}\), this distance is given by the Pythagoras theorem. On the way, we prove some Pythagoras inequalities for the product of arbitrary unital and non-degenerate spectral triples. 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.

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Authors and Affiliations

  1. CMTP & Dipartimento di Matematica, Università di Roma, Tor Vergata, 00133, Rome, Italy

    Pierre Martinetti & Luca Tomassini

  2. Dipartimento di Fisica, Università di Roma “Sapienza”, 00185, Rome, Italy

    Pierre Martinetti

  3. Dipartimento di Fisica, Università di Napoli Federico II, 00185, Rome, Italy

    Pierre Martinetti

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  1. Pierre Martinetti
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  2. Luca Tomassini
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Correspondence to Pierre Martinetti.

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Communicated by A. Connes

Work supported by the ERC Advanced Grant 227458 OACFT Operator Algebras & Conformal Field Theory and the ERG-Marie Curie fellowship 237927 Noncommutative geometry & quantum gravity.

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Martinetti, P., Tomassini, L. Noncommutative Geometry of the Moyal Plane: Translation Isometries, Connes’ Distance on Coherent States, Pythagoras Equality. Commun. Math. Phys. 323, 107–141 (2013). https://doi.org/10.1007/s00220-013-1760-8

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