# Length and distance on a quantum space

May 13, 2012

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.

open access link

Proceeding of Science (CORFU2011)040 (2012)

Keywords:

noncommutative geometry, DFR model, Moyal plane