Missing the point in noncommutative geometry

TL;DR

This paper argues that in noncommutative geometries, the concept of points or arbitrarily small regions is not meaningful, challenging traditional notions of localized regions in space.

Contribution

It demonstrates that points cannot be defined or operationally realized in noncommutative geometries, and explores how smooth manifolds may emerge as approximations.

Findings

Small regions are not definable in Connes' spectral triple framework.

Small regions lack operational meaning in the Moyal-Weyl approach.

Points do not exist in noncommutative geometries.

Abstract

Noncommutative geometries generalize standard smooth geometries, parametrizing the noncommutativity of dimensions with a fundamental quantity with the dimensions of area. The question arises then of whether the concept of a region smaller than the scale - and ultimately the concept of a point - makes sense in such a theory. We argue that it does not, in two interrelated ways. In the context of Connes' spectral triple approach, we show that arbitrarily small regions are not definable in the formal sense. While in the scalar field Moyal-Weyl approach, we show that they cannot be given an operational definition. We conclude that points do not exist in such geometries. We therefore investigate (a) the metaphysics of such a geometry, and (b) how the appearance of smooth manifold might be recovered as an approximation to a fundamental noncommutative geometry.