Quantum Dynamics from Lax Pair Theory: A Reconstruction from Spectrum Preservation
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Quantum Physics arXiv:2606.19664 (quant-ph) [Submitted on 18 Jun 2026] Title:Quantum Dynamics from Lax Pair Theory: A Reconstruction from Spectrum Preservation Authors:Péter Szabó View a PDF of the paper titled Quantum Dynamics from Lax Pair Theory: A Reconstruction from Spectrum Preservation, by P\'eter Szab\'o View PDF HTML (experimental) Abstract:We reconstruct unitary quantum dynamics from a minimal axiomatic foundation built on Hilbert-space observables and isospectral evolution. The only dynamical assumption is that physical time evolution is a continuous one-parameter flow of Hermitian observables that preserves their spectra, i.e. the possible outcomes of measurement. We show that this assumption is already sufficient to force the Lax form of quantum dynamics. The Heisenberg equation, the time-dependent and time-independent Schrödinger equations, conservation laws, and good quantum numbers then follow as theorems rather than postulates. In this formulation, Lax pair theory supplies the missing dynamical bridge between the measurement structure of a Hilbert space and standard quantum evolution: the Hamiltonian is not assumed, but emerges as the generator required for an isospectral observable flow. Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph); Chemical Physics (physics.chem-ph); History and Philosophy of Physics (physics.hist-ph) Cite as: arXiv:2606.19664 [quant-ph] (or arXiv:2606.19664v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2606.19664 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Péter Szabó [view email] [v1] Thu, 18 Jun 2026 00:20:45 UTC (11 KB) Full-text links: Access Paper: View a PDF of the paper titled Quantum Dynamics from Lax Pair Theory: A Reconstruction from Spectrum Preservation, by P\'eter Szab\'oView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-06 Change to browse by: math math-ph math.MP physics physics.chem-ph physics.hist-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?) 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?)