{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,11]],"date-time":"2026-09-11T01:39:20Z","timestamp":1789090760898,"version":"build-2803163510"},"reference-count":53,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2024,1,26]],"date-time":"2024-01-26T00:00:00Z","timestamp":1706227200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CMMI-2154151"],"award-info":[{"award-number":["CMMI-2154151"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The relation between de Broglie\u2019s double-solution approach to quantum dynamics and the hydrodynamic pilot-wave system has motivated a number of recent revisitations and extensions of de Broglie\u2019s theory. Building upon these recent developments, we here introduce a rich family of pilot-wave systems, with a view to reformulating and studying de Broglie\u2019s double-solution program in the modern language of classical field theory. Notably, the entire family is local and Lorentz-invariant, follows from a variational principle, and exhibits time-invariant, two-way coupling between particle and pilot-wave field. We first introduce a variational framework for generic pilot-wave systems, including a derivation of particle-wave exchange of Noether currents. We then focus on a particular limit of our system, in which the particle is propelled by the local gradient of its pilot wave. In this case, we see that the Compton-scale oscillations proposed by de Broglie emerge naturally in the form of particle vibrations, and that the vibration modes dynamically adjust to match the Compton frequency in the rest frame of the particle. The underlying field dynamically changes its radiation patterns in order to satisfy the de Broglie relation p=\u210fk at the particle\u2019s position, even as the particle momentum p changes. The wave form and frequency thus evolve so as to conform to de Broglie\u2019s harmony of phases, even for unsteady particle motion. We show that the particle is always dressed with a Compton-scale Yukawa wavepacket, independent of its trajectory, and that the associated energy imparts a constant increase to the particle\u2019s inertial mass. Finally, we see that the particle\u2019s wave-induced Compton-scale oscillation gives rise to a classical version of the Heisenberg uncertainty principle.<\/jats:p>","DOI":"10.3390\/sym16020149","type":"journal-article","created":{"date-parts":[[2024,1,29]],"date-time":"2024-01-29T04:57:07Z","timestamp":1706504227000},"page":"149","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Revisiting de Broglie\u2019s Double-Solution Pilot-Wave Theory with a Lorentz-Covariant Lagrangian Framework"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2787-961X","authenticated-orcid":false,"given":"David","family":"Darrow","sequence":"first","affiliation":[{"name":"Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7936-7256","authenticated-orcid":false,"given":"John W. M.","family":"Bush","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,26]]},"reference":[{"key":"ref_1","first-page":"507","article-title":"Ondes et quanta","volume":"177","year":"1923","journal-title":"Comptes Rendus"},{"key":"ref_2","unstructured":"de Broglie, L. (1930). An Introduction to the Study of Wave Mechanics, Methuen & Co."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"de Broglie, L. (1956). 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