Curvature and curved-background p-adic Dirac walks

Determine whether a meaningful notion of curvature exists for Q_p^3, identify an appropriate non-Archimedean analogue of a Riemannian metric compatible with its ultrametric structure, and determine whether the p-adic continuous-time quantum-walk construction can be adapted to such a curved background.

Background

The paper contrasts its construction with Archimedean discrete-time quantum walks that converge to Dirac equations on curved background metrics. A corresponding geometric framework for p-adic space is not established here.

The unresolved problem has two linked components: developing suitable non-Archimedean geometric notions of curvature and metric, and adapting the hierarchical continuous-time quantum walk and p-adic Dirac dynamics to that geometry.

References

Whether a meaningful notion of curvature exists for \mathbb{Q}{p}{3}, and whether the CTQW construction of this paper can be adapted to such a background, is entirely open; even the correct non-Archimedean analogue of a Riemannian metric compatible with the ultrametric structure of \mathbb{Q}{p}{3} is not settled.

— p-Adic Dirac Equations, Continuous-Time Quantum Walks, and Quantum Networks  (2609.04358 - Zúñiga-Galindo, 3 Sep 2026) in Section 6, subsection “Open problems,” item (3) (Curved p-adic space-time)