On the distance observable in the Moyal plane and in a novel two-dimensional space with string-theory pregeometry

TL;DR

This paper introduces a new geometric observable called the distance operator in the Moyal plane using a pregeometric phase-space approach, and explores a novel string-based spacetime with fuzzy coordinates inspired by string theory uncertainty principles.

Contribution

It develops a distance operator for the Moyal plane via pregeometric phase-space formulation and proposes a new string-based spacetime with extended points and fuzzy geometry.

Findings

Defined a distance operator for the Moyal plane.

Identified limitations in previous area descriptions in the Moyal plane.

Proposed a string-based spacetime with extended points and fuzzy coordinates.

Abstract

Motivation for the study of spacetime noncommutativity comes primarily from its possible use in investigations of (Planck-scale) spacetime fuzziness, but most work focuses on S-matrix/field-theory observables and still very little has been established for geometric observables. We argue that it might be useful to exploit the "pregeometric" formulation of spacetime noncommutativity, which in particular describes the coordinates of the Moyal plane in terms of the phase-space coordinates of a point particle "living" in a auxiliary/fictitious spacetime ($\{x_1,x_2\}_{Moyal} \equiv \{q,p\}_{particle}$). This leads us straightforwardly to a distance operator for the Moyal plane, and allows us to expose some limitations of a previous attempt to describe the "area of a disc" in the Moyal plane. We also observe that from our pregeometric perspective it is rather natural to contemplate a spacetime whose pregeometric picture is based on the phase-space coordinates (fields) of a string. The fact that such "stringspaces" essentially provide spacetime points with extendedness is relevant for the fuzziness of geometric spacetime observables in ways that we preliminarily characterize through an analysis of the distance observable on a two-dimensional stringspace and through the observation that by implementing the Amati-Ciafaloni-Veneziano/Gross-Mende uncertainty relation in the phase space of the pregeometric string one could have stringspaces with a novel type of fuzzy coordinates.