The geometry of noncommutative space-time

TL;DR

This paper explores the geometric structure of noncommutative space-time arising from deformations of the Poincaré-Heisenberg algebra, linking it to gravitational effects and noncommutative geometry.

Contribution

It introduces a framework connecting algebraic deformation, noncommutative geometry, and gravity, emphasizing the role of noncommutative translations and tangent space structures.

Findings

Deformation requires a fundamental length scale.

Noncommutative translations relate to gravitational fields.

The approach connects with Finkelstein's work on space-time geometry.

Abstract

Stabilization, by deformation, of the Poincar\'{e}-Heisenberg algebra requires both the introduction of a fundamental lentgh and the noncommutativity of translations which is associated to the gravitational field. The noncommutative geometry structure that follows from the deformed algebra is studied both for the non-commutative tangent space and the full space with gravity. The contact points of this approach with the work of David Finkelstein are emphasized.