{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,2]],"date-time":"2026-04-02T20:20:12Z","timestamp":1775161212123,"version":"3.50.1"},"reference-count":76,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,1,4]],"date-time":"2024-01-04T00:00:00Z","timestamp":1704326400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>What is striking about de Broglie\u2019s foundational work on wave\u2013particle dualism is the role played by pseudo-Riemannian geometry in his early thinking. While exploring a fully covariant description of the Klein\u2013Gordon equation, he was led to the revolutionary idea that a variable rest mass was essential. DeWitt later explained that in order to obtain a covariant quantum Hamiltonian, one must supplement the classical Hamiltonian with an additional energy \u210f2Q from which the quantum potential emerges, a potential that Berry has recently shown also arises in classical wave optics. In this paper, we show how these ideas emerge from an essentially geometric structure in which the information normally carried by the wave function is contained within the algebraic description of the geometry itself, within an element of a minimal left ideal. We establish the fundamental importance of conformal symmetry, in which rescaling of the rest mass plays a vital role. Thus, we have the basis for a radically new theory of quantum phenomena based on the process of mass-energy flow.<\/jats:p>","DOI":"10.3390\/sym16010067","type":"journal-article","created":{"date-parts":[[2024,1,4]],"date-time":"2024-01-04T05:23:30Z","timestamp":1704345810000},"page":"67","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["de Broglie, General Covariance and a Geometric Background to Quantum Mechanics"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3616-1696","authenticated-orcid":false,"given":"Basil","family":"Hiley","sequence":"first","affiliation":[{"name":"Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Glen","family":"Dennis","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,4]]},"reference":[{"key":"ref_1","first-page":"225","article-title":"La m\u00e9canique ondulatoire et la structure atomique de la mati\u00e8re et du rayonnement","volume":"VIII","year":"1927","journal-title":"J. de Physique et Le Radium"},{"key":"ref_2","unstructured":"de Broglie, L., and Brillouin, L. 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