Quantum deformations of the arc length metric
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866908850865569792 |
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| author | Hansen, Rasmus Hauge Kaad, Jens |
| author_facet | Hansen, Rasmus Hauge Kaad, Jens |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_21085 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quantum deformations of the arc length metric Hansen, Rasmus Hauge Kaad, Jens Operator Algebras 46L87, 46L30 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. |
| title | Quantum deformations of the arc length metric |
| topic | Operator Algebras 46L87, 46L30 |
| url | https://arxiv.org/abs/2602.21085 |