Dirac Wave Functions of Positive Energy with Arbitrarily Small Position Uncertainty
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Quantum Physics arXiv:2603.04569 (quant-ph) [Submitted on 4 Mar 2026] Title:Dirac Wave Functions of Positive Energy with Arbitrarily Small Position Uncertainty Authors:Ilmar Bürck, Roderich Tumulka View a PDF of the paper titled Dirac Wave Functions of Positive Energy with Arbitrarily Small Position Uncertainty, by Ilmar B\"urck and Roderich Tumulka View PDF HTML (experimental) Abstract:We consider wave functions in the Hilbert space $\mathcal{H}=L^2(\mathbb{R}^3,\mathbb{C}^4)$ of a single Dirac particle, specifically from the positive-energy subspace $\mathcal{H}_+$ of the free Dirac Hamiltonian. Over the decades, various authors conjectured that for wave functions from $\mathcal{H}_+$, there is a positive lower bound to the position uncertainty $\sigma_x$; in other words, that such states cannot be arbitrarily narrow in $x$. Building on work by Bracken and Melloy, we show that this conjecture is false. (In fact, they already stated that this conjecture is false and already had a counter-example, but their proof that it is a counter-example had a gap.) Comments: Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2603.04569 [quant-ph] (or arXiv:2603.04569v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2603.04569 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Roderich Tumulka [view email] [v1] Wed, 4 Mar 2026 19:56:07 UTC (564 KB) Full-text links: Access Paper: View a PDF of the paper titled Dirac Wave Functions of Positive Energy with Arbitrarily Small Position Uncertainty, by Ilmar B\"urck and Roderich TumulkaView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-03 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?) Links to Code Toggle Papers with Code (What is Papers with Code?) 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?)