Quantum Spacetime: a Disambiguation

Abstract

We review an approach to non-commutative geometry, where models are constructed by quantisation of the coordinates. In particular we focus on the full DFR model and its irreducible components; the (arbitrary) restriction to a particular irreducible component is often referred to as the "canonical quantum spacetime". The aim is to distinguish and compare the approaches under various points of view, including motivations, prescriptions for quantisation, the choice of mathematical objects and concepts, approaches to dynamics and to covariance. Some incorrect statements as "universality of Planck scale conflicts with Lorentz-Fitzgerald contraction and requires a modification of covariance", or "stability of the geometric background requires an absolute lower bound of (\Delta x^\mu)", or "violations of unitarity are due to time/space non-commutativity" are put in context, and discussed.