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

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 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 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 emphasize 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) A by C 2 . We show that on the set of states obtained by translation of an arbitrary state of A, this distance is given by Pythagoras theorem. On the way, we prove some Pythagoras inequalities for the product of arbitrary unital & 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.

Minimal length in quantum space and integration of the line element in noncommutative geometry

“…[28]] The coherent state ω c κ is the translated of the ground state of the quantum harmonic oscillator, with translation √ 2λ P κ. That is to say…”

“…In this section, we list the results on the quantum length and the spectral distance between generalized coherent states obtained in [27] for the former, in [8] and [28] for the latter.…”

“…In the following, we review several recent results on the spectral distance and the length operator, mainly from [8,27,28,15]. We begin by recalling in section 2 how to retrieve, in the commutative case Q µν = 0, the Euclidean distance from either the length operator L or Connes' formula (1.8).…”

“…We also recall the results on the spectral distance d D for various classes of states of the Moyal algebra, including the eigenstates of the Hamiltonian of the quantum harmonic oscillator [8] and the coherent states [28]. Section 6 presents the strategy developed in [27] in order to compare the quantum length with the spectral distance, despite an obvious discrepancy (the latter vanishes between a state and itself, the former does not as soon as l P = 0).…”

“…Starting by questioning "how to define a standard meter in a space whose coordinates no longer commute ? ", we list several recent results regarding Connes's spectral distance calculated between eigenstates of the quantum harmonic oscillator [8], as well as between coherent states [28]. We also question the difference (which remains hidden in the commutative case) between the spelral distance and the notion of quantum length inherited from the length operator defined in various models of noncommutative space-time (DFR and θ -Minkowski).…”