Length and distance on a quantum space

Abstract: This contribution is an introduction to the metric aspect of noncommutative geometry, with emphasize on the Moyal plane. 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 arXiv:0912.0906, as well as between coherent states arXiv:1110.6164. We also question the difference (which remains hidden in the commutative case) between the spectral distance and the notion of quantum length inherited from the length operator defined in various models of noncommutative space-time (DFR and \theta-Minkowski). We recall that a standard procedure in noncommutative geometry, consisting in doubling the spectral triple, allows to fruitfully confront the spectral distance with the quantum length. Finally we refine the idea of discrete vs. continuous geodesics in the Moyal plane, introduced in arXiv:1106.0261.