Quantum deformations of the arc length metric

TL;DR

This paper introduces a q-deformation of the arc length metric on the circle, resulting in a family of quantum metric spaces that depend continuously on the deformation parameter and do not correspond to classical metrics.

Contribution

It constructs a new quantum metric space via q-deformation of the classical arc length metric, extending the concept of metrics in noncommutative geometry.

Findings

The q-deformation arises from the Dirac operator with q-analogues of integers.

The quantum metric space only makes sense at the level of quantum metric spaces, not classical.

The family of quantum metrics depends continuously on the deformation parameter q.

Abstract

We investigate a q-deformation of the arc length metric on the unit circle. This q-deformation arises naturally from the Dirac operator by replacing the standard integers by their q-deformed analogues. Nonetheless, we show that the corresponding metric structure only makes sense at the level of quantum metric spaces as introduced by Marc Rieffel. This means that the quantum metric we obtain on the continuous functions on the circle does not arise from a classical metric on the circle. In the special case where q equals one we recover the usual arc length metric and we show that our family of quantum metric spaces depend continuously on the deformation parameter with respect to David Kerr's complete Gromov-Hausdorff distance.