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Quantum deformations of the arc length metric

Rasmus Hauge Hansen, Jens Kaad

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.

Quantum deformations of the arc length metric

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.
Paper Structure (10 sections, 25 theorems, 50 equations)

This paper contains 10 sections, 25 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. A $q$-deformed differential calculus on the unit circle
  4. Extending the $q$-derivative
  5. $q$-integration
  6. Quantum metrics derived from the $q$-derivative
  7. Some background on compact quantum metric spaces
  8. Quantum metrics on the circle
  9. Non-existence of certain metrics on the circle
  10. Continuity of the twist

Key Result

Proposition 3.3

The pair $(\mathcal{X}_q,\mathrm{d}_q)$ is a $\ast$-calculus over $\mathcal{O}(S^1)$.

Theorems & Definitions (49)

  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Definition 4.1
  • Lemma 4.2
  • Lemma 4.3
  • proof
  • Definition 4.4
  • Proposition 4.5
  • ...and 39 more