Solutions of Koopman-von Neumann equations, their superpositions, orthogonality and uncertainties
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Quantum Physics arXiv:2512.11148 (quant-ph) [Submitted on 11 Dec 2025] Title:Solutions of Koopman-von Neumann equations, their superpositions, orthogonality and uncertainties Authors:Mustafa Amin, Mark A. Walton View a PDF of the paper titled Solutions of Koopman-von Neumann equations, their superpositions, orthogonality and uncertainties, by Mustafa Amin and 1 other authors View PDF HTML (experimental) Abstract:The Koopman-von Neumann (KvN) formulation brings classical mechanics to Hilbert space, but many techniques familiar from quantum mechanics remain missing. One would hope to solve eigenvalue problems, obtain orthonormal eigenstates of Hermitian operators and ascribe meaning to a coherent superposition of states, among other things. Here we consider the general KvN equation for a classical probability amplitude and show that its so-called gauge freedom allows the separation of variables. The amenability to Hilbert-space methods of the resulting KvN solutions is investigated. We construct superpositions from differently-gauged Liouvillian eigenstates, and find an orthonormal set among them. We find that some separable solutions describe the canonical ensemble with temperature related to the separation constant. Classical uncertainty relations arise naturally in the KvN formalism. We discuss one between the dynamical time and the Liouvillian in terms of the statistical description of classical systems. Comments: Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph); Classical Physics (physics.class-ph) Cite as: arXiv:2512.11148 [quant-ph] (or arXiv:2512.11148v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2512.11148 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Mustafa Amin [view email] [v1] Thu, 11 Dec 2025 22:11:06 UTC (25 KB) Full-text links: Access Paper: View a PDF of the paper titled Solutions of Koopman-von Neumann equations, their superpositions, orthogonality and uncertainties, by Mustafa Amin and 1 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2025-12 Change to browse by: math math-ph math.MP physics physics.class-ph 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?)