{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T10:55:35Z","timestamp":1753872935755,"version":"3.41.2"},"reference-count":30,"publisher":"AIP Publishing","issue":"1","content-domain":{"domain":["pubs.aip.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2011,1,1]]},"abstract":"<jats:p>The Klein\u2013Gordon equation satisfied by the wave function in general relativity is solved for the metric of the closed and open universe corresponding to Einstein\u2013De Sitter\u2013Friedmann isotropic cosmological model. The angular dependences are specified by spherical harmonics for the longitude and latitude, and for the hyperlatitude by modified spherical harmonics having as variable circular functions for the closed universe and hyperbolic functions for the open universes. The time dependence of the probabilistic wave function is similar for the closed and open universes and is obtained in the following three cases: (I) constant Hubble parameter, (II) constant decceleration parameter, and (III) uniform matter and energy distribution, which corresponds to the Hubble parameter a linear function of time. Thus six solutions are obtained, namely, the three cases I\u2013III each for closed and open isotropic universes. For each of these six solutions is considered: (i) the existence of singularities in space\u2013time including asymptotic time in the future or past, (ii) the square integrability of the wave function over the full extent of the four-dimensional space\u2013time, and (iii) the existence or otherwise of a positive probability density associated with the wave function.<\/jats:p>","DOI":"10.1063\/1.3512995","type":"journal-article","created":{"date-parts":[[2011,1,10]],"date-time":"2011-01-10T23:54:36Z","timestamp":1294703676000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":1,"title":["On a wave-particle in closed and open isotropic universes"],"prefix":"10.1063","volume":"52","author":[{"given":"L. M. B. C.","family":"Campos","sequence":"first","affiliation":[{"name":"Instituto Superior T\u00e9cnico Centro de Ci\u00eancias e Tecnologias Aeron\u00e1uticas e Espaciais (CCTAE), and \u00c1rea Cient\u00edfica de Mec\u00e2nica Aplicada e Aeroespacial (ACMAA), , 1049-001 Lisboa, Portugal"}]}],"member":"317","published-online":{"date-parts":[[2011,1,10]]},"reference":[{"volume-title":"Handbook of Mathematical Function","year":"1965","key":"2023080303553631300_c1"},{"volume-title":"Quantum Fields in Curved Space-Time","year":"1982","key":"2023080303553631300_c2"},{"volume-title":"Mathematical Theory of Black Holes","year":"1992","key":"2023080303553631300_c3"},{"volume-title":"Relativistic Astrophysics","year":"1985","key":"2023080303553631300_c4"},{"volume-title":"Quantum Mechanics","year":"1930","key":"2023080303553631300_c5"},{"volume-title":"General Theory of Relativity","year":"1975","key":"2023080303553631300_c6"},{"volume-title":"Mathematical Theory of Relativity","year":"1922","key":"2023080303553631300_c7"},{"key":"2023080303553631300_c8","first-page":"891","article-title":"Zur Elektrodynamik bewegter K\u00f6rper","volume":"17","year":"1905","journal-title":"Ann. Phys. Chem."},{"key":"2023080303553631300_c9","first-page":"799","article-title":"Die Grundlage der allgemeinen relativitatstheorie","year":"1915","journal-title":"Sitzber. Preuss. Akad. Wiss. Berlin"},{"volume-title":"The Principle of Relativity","year":"1952","key":"2023080303553631300_c10"},{"volume-title":"Theory of Differential Equations","year":"1890\u20131906","key":"2023080303553631300_c11"},{"volume-title":"Treatise of Differential Equations","year":"1929","key":"2023080303553631300_c12"},{"key":"2023080303553631300_c13","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1007\/BF01332580","article-title":"Ueber die Krmmung des Raumes","volume":"10","year":"1922","journal-title":"Zeits. Phys."},{"volume-title":"The Large-Scale Structure of Space-Time","year":"1973","key":"2023080303553631300_c14"},{"volume-title":"The Physical Principles of Quantum Theory","year":"1949","key":"2023080303553631300_c15"},{"volume-title":"Spherical and Ellipsoidal Harminics","year":"1955","key":"2023080303553631300_c16"},{"key":"2023080303553631300_c17","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1073\/pnas.15.3.168","article-title":"The relation between distance and radial velocity amang extragalactic nebulare","volume":"15","year":"1929","journal-title":"Proc. Natl. Acad. Sci. U.S.A."},{"volume-title":"Ordinary Differential Equations","year":"1956","key":"2023080303553631300_c18"},{"volume-title":"Differentialgleichungen","year":"1972","key":"2023080303553631300_c19"},{"volume-title":"Field Theory","year":"1969","key":"2023080303553631300_c20"},{"volume-title":"Relativistic Quantum Theory","year":"1972","key":"2023080303553631300_c21"},{"volume-title":"Quantum Mechanics","year":"1973","key":"2023080303553631300_c22"},{"volume-title":"Quantum Mechanics","year":"1999","key":"2023080303553631300_c23"},{"volume-title":"Gravitation","year":"1972","key":"2023080303553631300_c24"},{"volume-title":"Expanding Universes","year":"1956","key":"2023080303553631300_c25"},{"volume-title":"Relativistic Quantum Field Theory","year":"2005","key":"2023080303553631300_c26"},{"key":"2023080303553631300_c27","first-page":"229","article-title":"On the curvature of space","volume":"20","year":"1917","journal-title":"Proc. K. Ned. Akad. Wet."},{"volume-title":"Bessel Functions","year":"1944","key":"2023080303553631300_c28"},{"volume-title":"Quantum Theory of Fields","year":"1949","key":"2023080303553631300_c29"},{"key":"2023080303553631300_c30","first-page":"1143","article-title":"Quantum theory of gravity","volume":"160","year":"1967","journal-title":"Phys. Rev."}],"container-title":["Journal of Mathematical Physics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pubs.aip.org\/aip\/jmp\/article-pdf\/doi\/10.1063\/1.3512995\/15961411\/013508_1_online.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/pubs.aip.org\/aip\/jmp\/article-pdf\/doi\/10.1063\/1.3512995\/15961411\/013508_1_online.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,8,3]],"date-time":"2023-08-03T03:55:56Z","timestamp":1691034956000},"score":1,"resource":{"primary":{"URL":"https:\/\/pubs.aip.org\/jmp\/article\/52\/1\/013508\/955872\/On-a-wave-particle-in-closed-and-open-isotropic"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,1,1]]},"references-count":30,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2011,1,1]]}},"URL":"https:\/\/doi.org\/10.1063\/1.3512995","relation":{},"ISSN":["0022-2488","1089-7658"],"issn-type":[{"type":"print","value":"0022-2488"},{"type":"electronic","value":"1089-7658"}],"subject":[],"published-other":{"date-parts":[[2011,1]]},"published":{"date-parts":[[2011,1,1]]},"article-number":"013508"}}