Interacting and gauge-covariant p-adic Dirac quantum walks

Determine whether coupling the free p-adic Dirac Hamiltonian to a genuinely p-adic electromagnetic or more general gauge potential yields a genuine continuous-time quantum walk with a stochastic-matrix structure analogous to that of the free finite-graph construction.

Background

The results in the paper concern only the free p-adic Dirac equation. Extending the construction to electromagnetic or gauge interactions would require defining an appropriate p-adic gauge-covariant coupling and then analyzing its discretization on the hierarchical graph.

A central unresolved issue is whether the stochastic-matrix property proved for the free walk survives in the presence of such interactions. This property is what permits the spinor walk, after summing over internal components, to induce an ordinary random walk on the graph.

References

Coupling \boldsymbol{H}_{0} to a genuinely p-adic electromagnetic (or more general gauge) potential, in the spirit of the gauge-covariant discrete-time walks of Arnault and Debbasch and of in the Archimedean setting, has not been carried out; nor is it known whether the resulting equation would still discretize into a genuine CTQW with a stochastic-matrix structure analogous to the one established in Section \ref{Section_CTQW_II}.

— 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 (2) (Interactions and gauge fields)