Geometric structure of the relativistic quantum phase space

Quantum Physics [Submitted on 30 Mar 2026] Geometric structure of the relativistic quantum phase space Philippe Manjakasoa Randriantsoa, Ravo Tokiniaina Ranaivoson, Raoelina Andriambololona, Roland Raboanary, Wilfrid Chrysante Solofoarisina, Anjary Feno Hasina Rasamimanana 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: 12 pages 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 Submission history From: Philippe Manjakasoa Randriantsoa [view email] [v1] Mon, 30 Mar 2026 10:24:19 UTC (12 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-03 References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation 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 Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)