Discreteness of area in noncommutative space

TL;DR

This paper introduces an area operator in the Moyal noncommutative plane, revealing a discrete spectrum without a minimum-area limit, challenging previous assumptions and highlighting issues in quantum-gravity geometric measurements.

Contribution

It defines a new area operator for the Moyal plane and demonstrates its discrete spectrum, contradicting prior expectations of a minimum-area principle.

Findings

The spectrum of the area operator is discrete.

No minimum-area principle characterizes the spectrum.

Highlights issues in heuristic quantum-gravity analyses.

Abstract

We introduce an area operator for the Moyal noncommutative plane. We find that the spectrum is discrete, but, contrary to the expectation formulated by other authors, not characterized by a "minimum-area principle". We show that an intuitive analysis of the uncertainty relations obtained from Moyal-plane noncommutativity is fully consistent with our results for the spectrum, and we argue that our area operator should be generalizable to several other noncommutative spaces. We also observe that the properties of distances and areas in the Moyal plane expose some weaknesses in the line of reasoning adopted in some of the heuristic analyses of the measurability of geometric spacetime observables in the quantum-gravity realm.