Geometric structure of the relativistic quantum phase space
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Quantum Physics arXiv:2603.28836 (quant-ph) [Submitted on 30 Mar 2026] Title:Geometric structure of the relativistic quantum phase space Authors:Philippe Manjakasoa Randriantsoa, Ravo Tokiniaina Ranaivoson, Raoelina Andriambololona, Roland Raboanary, Wilfrid Chrysante Solofoarisina, Anjary Feno Hasina Rasamimanana View a PDF of the paper titled Geometric structure of the relativistic quantum phase space, by Philippe Manjakasoa Randriantsoa and 5 other authors View PDF HTML (experimental) Abstract:The relativistic quantum phase space (QPS) formalism extends classical phase space by incorporating both mean values and variance-covariance matrices of quantum states, thereby providing a unified setting where the uncertainty principle and relativistic covariance coexist. In this work we explore the basic geometric structure of the QPS for the signature \((1,4)\). We construct a scalar invariant built from the mean values and the inverse variance-covariance matrix, and prove its invariance under linear canonical transformations. For quantum states that saturate the uncertainty relations, and define the QPS itself, the invariant takes a value that encodes two fundamental length scales: a large scale characterising maximal coordinate uncertainties and a small scale characterising minimal coordinate uncertainties. From this invariance we derive a geometric equation that unifies the mean values and the quantum fluctuations. Analysing two asymptotic regimes reveals two physically significant limits: one leads to a curved spacetime geometry, consistent with current cosmological observations; the other yields a curved momenta space structure. These limits suggest a direct connection between quantum phase space geometry, cosmology, and quantum gravity, offering new perspectives on the origin of the quantum structure of spacetime. The results also resonate with the principle of Born reciprocity, which posits a fundamental duality between coordinates and momenta, and align with recent works on the relation between QPS and neutrino physics. Comments: Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2603.28836 [quant-ph] (or arXiv:2603.28836v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2603.28836 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Philippe Manjakasoa Randriantsoa [view email] [v1] Mon, 30 Mar 2026 10:24:19 UTC (12 KB) Full-text links: Access Paper: View a PDF of the paper titled Geometric structure of the relativistic quantum phase space, by Philippe Manjakasoa Randriantsoa and 5 other authorsView 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?)