Line element in quantum gravity: the examples of DSR and noncommutative geometry

TL;DR

This paper examines the concept of line elements in quantum spaces relevant to quantum gravity, focusing on noncommutative geometries and their implications for symmetries and distance measures.

Contribution

It critically analyzes the notion of line elements in noncommutative spaces and reinterprets existing results within the framework of noncommutative geometry.

Findings

Certain deformed Poincare transformations cannot serve as Noether symmetries due to Leibniz rule constraints.

Reinterpretation of Connes' distance formula in the context of quantum spaces.

Insights into the fundamental nature of line elements in noncommutative geometry.

Abstract

We question the notion of line element in some quantum spaces that are expected to play a role in quantum gravity, namely non-commutative deformations of Minkowski spaces. We recall how the implementation of the Leibniz rule forbids to see some of the infinitesimal deformed Poincare transformations as good candidates for Noether symmetries. Then we recall the more fundamental view on the line element proposed in noncommutative geometry, and re-interprete at this light some previous results on Connes' distance formula.