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

Quantum Physics arXiv:2609.04358 (quant-ph) [Submitted on 3 Sep 2026] Title:p-Adic Dirac Equations, Continuous-Time Quantum Walks, and Quantum Networks Authors:W. A. Zúñiga-Galindo View a PDF of the paper titled p-Adic Dirac Equations, Continuous-Time Quantum Walks, and Quantum Networks, by W. A. Z\'u\~niga-Galindo View PDF HTML (experimental) 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. Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2609.04358 [quant-ph] (or arXiv:2609.04358v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2609.04358 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: W. A. Zuniga-Galindo [view email] [v1] Thu, 3 Sep 2026 18:20:40 UTC (29 KB) Full-text links: Access Paper: View a PDF of the paper titled p-Adic Dirac Equations, Continuous-Time Quantum Walks, and Quantum Networks, by W. A. Z\'u\~niga-GalindoView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-09 References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)