{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,12]],"date-time":"2026-03-12T19:59:49Z","timestamp":1773345589587,"version":"3.50.1"},"reference-count":39,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,6,16]],"date-time":"2022-06-16T00:00:00Z","timestamp":1655337600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"l\u2019Ecole doctorale du Burundi"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>As an extension of Gabor signal processing, the covariant Weyl-Heisenberg integral quantization is implemented to transform functions on the eight-dimensional phase space x,k into Hilbertian operators. The x=x\u03bc values are space-time variables, and the k=k\u03bc values are their conjugate frequency-wave vector variables. The procedure is first applied to the variables x,k and produces essentially canonically conjugate self-adjoint operators. It is next applied to the metric field g\u03bc\u03bd(x) of general relativity and yields regularized semi-classical phase space portraits g\u02c7\u03bc\u03bd(x). The latter give rise to modified tensor energy density. Examples are given with the uniformly accelerated reference system and the Schwarzschild metric. Interesting probabilistic aspects are discussed.<\/jats:p>","DOI":"10.3390\/e24060835","type":"journal-article","created":{"date-parts":[[2022,6,17]],"date-time":"2022-06-17T01:48:12Z","timestamp":1655430492000},"page":"835","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Quantum Models \u00e0 la Gabor for the Space-Time Metric"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7346-7291","authenticated-orcid":false,"given":"Gilles","family":"Cohen-Tannoudji","sequence":"first","affiliation":[{"name":"Laboratoire de Recherche sur les Sciences de la Mati\u00e8re, LARSIM CEA, Universit\u00e9 Paris-Saclay, F-91190 Saint-Aubin, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7681-7672","authenticated-orcid":false,"given":"Jean-Pierre","family":"Gazeau","sequence":"additional","affiliation":[{"name":"CNRS, Astroparticule et Cosmologie, Universit\u00e9 Paris Cit\u00e9, F-75013 Paris, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9224-4739","authenticated-orcid":false,"given":"C\u00e9lestin","family":"Habonimana","sequence":"additional","affiliation":[{"name":"Ecole Normale Sup\u00e9rieure, Universit\u00e9 du Burundi, Bujumbura 1550, Burundi"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3388-3738","authenticated-orcid":false,"given":"Juma","family":"Shabani","sequence":"additional","affiliation":[{"name":"Ecole Doctorale, Universit\u00e9 du Burundi, Bujumbura 1550, Burundi"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"821","DOI":"10.1090\/S0273-0979-1980-14825-9","article-title":"On the role of the Heisenberg group in harmonic analysis","volume":"3","author":"Howe","year":"1980","journal-title":"Bull. 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