p-Adic Dirac Equations, Continuous-Time Quantum Walks, and Quantum Networks

Abstract

We introduce a new class of p-adic Dirac equations, formulated in the standard axiomatic framework of quantum mechanics, in which the ordinary spatial derivatives are replaced by non-local operators built from arbitrary integrable kernels. We diagonalize the resulting free Dirac Hamiltonian in momentum space, construct its plane-wave solutions, determine its spectrum, and establish a p-adic charge-conjugation symmetry relating particle and antiparticle sectors. We then discretize the free equation in two ways, both giving genuine continuous-time quantum walks rather than the discrete-time, coined walks that dominate the existing literature: a first construction on a countable covering of the underlying p-adic space, and a second, more explicit construction on a finite, tree-structured graph, for which we prove that the transition probabilities, once the internal (particle/antiparticle) components of the wavefunction are combined, form a genuine, properly normalized set of transition probabilities at every instant of time; in other words, ignoring the internal structure of the walk, it behaves exactly like an ordinary random walk on that graph. Building on this stochastic-matrix property, we discuss how the resulting construction can serve as the foundation of a quantum network with genuinely relativistic-type internal degrees of freedom, complementing earlier, non-relativistic p-adic quantum neural networks. To the best of our knowledge, this is the first continuous-time quantum walk whose free dynamics coincides exactly with a Dirac equation on a hierarchical graph. We close with a discussion of the open mathematical and computational problems raised by this construction.