{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,4]],"date-time":"2025-12-04T18:49:00Z","timestamp":1764874140514,"version":"build-2065373602"},"reference-count":102,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,3,6]],"date-time":"2024-03-06T00:00:00Z","timestamp":1709683200000},"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>We generalize Koopman\u2013von Neumann classical mechanics to poly symplectic fields and recover De Donder\u2013Weyl\u2019s theory. Compared with Dirac\u2019s Hamiltonian density, it inspires a new Hamiltonian formulation with a canonical momentum field that is Lorentz-covariant with symplectic geometry. We provide commutation relations for the classical and quantum fields that generalize the Koopman\u2013von Neumann and Heisenberg algebras. The classical algebra requires four fields that generalize spacetime, energy\u2013momentum, frequency\u2013wavenumber, and the Fourier conjugate of energy\u2013momentum. We clarify how first and second quantization can be found by simply mapping between operators in classical and quantum commutator algebras.<\/jats:p>","DOI":"10.3390\/sym16030316","type":"journal-article","created":{"date-parts":[[2024,3,7]],"date-time":"2024-03-07T04:19:02Z","timestamp":1709785142000},"page":"316","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Quantization of a New Canonical, Covariant, and Symplectic Hamiltonian Density"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4662-8592","authenticated-orcid":false,"given":"David","family":"Chester","sequence":"first","affiliation":[{"name":"Quantum Gravity Research, Topanga, CA 90290, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1485-1853","authenticated-orcid":false,"given":"Xerxes D.","family":"Arsiwalla","sequence":"additional","affiliation":[{"name":"Department of Information and Communication Technologies, Pompeu Fabra University, 08018 Barcelona, Spain"},{"name":"Wolfram Research, Champaign, IL 61820-7237, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4135-8685","authenticated-orcid":false,"given":"Louis H.","family":"Kauffman","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Illinois at Chicago, Chicago, IL 60607, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5739-546X","authenticated-orcid":false,"given":"Michel","family":"Planat","sequence":"additional","affiliation":[{"name":"Department of Micro NanoSciences and Systems, National Center for Scientific Research, FEMTO-ST Institute, University of Franche-Comt\u00e9, F-25044 Besan\u00e7on, France"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2938-3941","authenticated-orcid":false,"given":"Klee","family":"Irwin","sequence":"additional","affiliation":[{"name":"Quantum Gravity Research, Topanga, CA 90290, USA"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1073\/pnas.17.5.315","article-title":"Hamiltonian Systems and Transformation in Hilbert Space","volume":"17","author":"Koopman","year":"1931","journal-title":"Proc. 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