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However, Bell has shown that a theory of this type could not be consistent with all of the predictions of quantum mechanics. It is reasonable to assume that fields in a theory of the Einstein type could be described as classical dynamical systems. We begin by formulating conditions for an operator representation in Hilbert space of the phase functions of a finite classical system. It is proved that such a representation cannot be commutative. The Weyl representation fulfills the required conditions: in Appendix A it is shown to map the quadratically integrable phase functions isometrically into the Hilbert-Schmidt operators. It follows that the image 77 of the phase probability density is a statistical operator in the sense of von Neumann, though not generally nonnegative definite. The Weyl representation is then formally extended to infinite classical systems and applied to interacting relativistic tensor wave fields (spinor fields, not being single valued, present difficulties). With the choiceħ =π\n−1, this representation satisfies the requirements of canonical quantization. The Hamiltonian operator and operators for the usual field quantities are the same as in quantum field theory. However, an additional field interaction term appears, modifying von Neumann’s equation for the time dependence ofU. Essential divergence difficulties should not arise, since no divergences occur in classical field theory. This representation of classical field theory is amenable, in just the same way as quantum field theory, to a configuration space representation and a corresponding particle interpretation. These aspects will be discussed in a later publication.",{"EN":493},"A Noncommutative representation of classical dynamics. Connections with field quantization",{"VOID":495},"[\"6076035628281674358\"]",{"VOID":497},"P. A. M. Dirac:Proc. Roy. Soc.,114, 243 a, 710 (1927).\nA. Einstein:Albert Einstein: Philosopher-Scientist, edited byP. A. Schilfp (New York, 1951), p. 675.\nJ. S. Bell:Physics,1, 195 (1964);Proc. S.I.F., Course IL (New York, 1972), p. 170.\nJ. F. Clauser, M. A. Horne, A. Shimony andR. A. Holt:Phys. Rev. Lett.,23, 880 (1969). Other possible violations are discussed byP. M. Pearle:Phys. Rev. D,2, 1418 (1970).\nJ. von Neumann :Mathematical Foundations of Quantum Mechanics (Princeton, 1955), p. 313.\nJ. von Neumann:Mathematical Foundations of Quantum Mechanics (Princeton, 1955), p. 316.\nN. Dunford andJ. T. Schwartz:Linear Operators (New York, 1963), p. 905.\nJ. E. Moyal:Proc. Cambridge Phil. Soc.,45, 99 (1949).\nG. A. Baker:Phys. Rev.,109, 2198 (1958).\nJ. C. T. Pool:Jour. Math. Phys.,7, 66 (1966).\nH. Weyl:The Theory of Groups and Quantum Mechanics (New York, 1931), p. 275.\nG. Wentzel:Quantum Theory of Fields (New York, 1949), p. 12.\nG. Wentzel:Quantum Theory of Fields (New York, 1949), p. 51, 123.\nG. Wentzel:Quantum Theory of Fields (New York, 1949), p. 4. This statement is true also for all of the usual field quantities.\nG. Wentzel:Quantum Theory of Fields (until 1947), Theoretical Physics in the Twentieth Century, edited byFierz, Marcus and Weisskopf (New York, 1960), p. 48.\nJ. G. Sinai:Dokl. Akad. Nauk SSSR,153, 1261 (1963).\nA. I. Khinchin:Mathematical Foundations of Statistical Mechanics (New York, 1949), p. 56.\nA. Blanc-Lapierre,P. Casal andA. Tortrat:Méthods mathématiques de la mécanique statistique (Paris, 1959), p. 61.\nE. Brout andI. Prigogine:Physica,22, 35 (1956).\nP. Bocchieri andA. Loinger:Lett. Nuovo Cimento,4, 310 (1970).\nM. H. Stone:Linear Transformations in Hilbert space (New York, 1964), p. 58.\nM. H. Stone:Linear Transformations in Hilbert Space (New York, 1964), p. 3.\nM. H. Stone:Linear Transformations in Hilbert Space (New York, 1964), p. 64.\nW. O. Kermack andW. H. McCrea:Proc. Edinburg Math. 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The detector consists of a hemispheric electrostatic deflector for energy selection and a Faraday cup for flux measurements. The energy distribution of the positive component from 200 to 15 000 eV is thus obtained. Protons and α-particles in the interplanetary region are easily detected. Bulk velocity, density and most probable thermal velocity of the proton component outside the shock front are determined for the period 9 December through 31 December 1968. 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Carnap:Philos. Sci.,12, 72 (1945).",{},{"id":18,"text":985,"url":18,"identifiers":986},"See,e.g.,R. C. Tolman:The Principles of Statistical Mechanics (Oxford, 1938).",{},{"id":18,"text":988,"url":18,"identifiers":989},"SeeR. Carnap:The Basic System of Inductive Logic, Part I, inStudies in Inductive Logic and Probability, edited byR. Carnap andR. C. Jeffrey, Vol. 1 (Los Angeles, Cal., 1971), p. 33.",{},{"id":18,"text":991,"url":18,"identifiers":992},"SeeR. Carnap:The Basic System of Inductive Logic, Part II, inStudies in Inductive Logic and Probability, edited byR. C. Jeffrey, Vol.2 (Los Angeles, Cal., 1980), p. 7.",{},{"id":18,"text":994,"url":18,"identifiers":995},"D. 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The source of the geometry of these solutions is a viscous fluid which has not been thermalized. The solutions reported here are nonstatic generalizations of Gödel’s rotating cosmos. The solution discussed by Novello and Reboucas has been derived as a particular case.",{"EN":1006},"Rotating universes with viscous fluid and heat flux",{"VOID":1008},"[\"3893324631594083482\"]",{"VOID":1010},"M. Novello andM. J. Reboucas:Astrophys. J.,225, 719 (1978).\nD. Ray:J. Math. Phys. (N.Y.),21, 2797 (1980).\nM. J. Reboucas andJ. A. S. DeLimma:J. Math. Phys. (N.Y.),22, 2699 (1981).\nM. J. Reboucas:Nuovo Cimento, B,67, 120 (1982).\nJ. M. Bradely andE. Sviestins:Gen. Rel. Grav.,16, 1119 (1984).\nE. Sviestins:Gen Rel. Grav.,17, 521 (1985).\nG. L. Murphy:Phys. Rev. D,8, 4231 (1973).\nA. Banerjee andN. O. Santos:J. Math. Phys. (N.Y.),26, 878 (1985).\nM. B. Ribeiro andA. K. Sanyal:J. Math. Phys. (N.Y.),28, 657 (1987).\nA. Banerjee, M. B. Ribeiro andN. O. Santos:Astrophys. Space Sci.,136, 33 (1987).\nK. Gödel:Rev. Mod. Phys.,21, 447 (1949).",{"VOID":1012},"10.1007\u002FBF02728512","2024-08-30T18:01:24.559+00:00","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002FBF02728512",[1016,1031],{"id":1017,"sortIndex":19,"researcher":18,"roles":1018,"affiliations":1019,"properties":1028,"displayName":1030,"givenName":18,"familyName":18},"8e860dc8-6b3e-4225-9c69-01225473ed5b",[160],[1020],{"id":1021,"sortIndex":19,"affiliation":1022,"properties":18},"c9db722a-ab27-4c1c-822f-16caa995ac81",{"id":1021,"createTime":18,"updateTime":18,"relativeEntities":1023,"slug":18,"properties":1024,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":1027,"statistic":18},[],{"title":1025},{"VI":1026},"Department of Mathematics, The Ohio State University, Columbus, USA",[],{"title":1029},{"VI":1030},"S. S. 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Patel",{"url":1014,"publisher":1047,"properties":1066},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":1048,"slug":10,"properties":1049,"entityType":16,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":19,"subjectFields":1052,"manageAffiliations":1053,"indexDatabases":1054,"url":18,"thumbnailPath":18,"statistic":1061,"gsStatistic":18,"type":18,"analyzePriority":18},[],{"title":1050,"eissn":1051},{"EN":13},{"VOID":15},[],[],[1055],{"id":24,"indexDatabase":1056,"url":35,"indexYears":36,"academicFieldIds":18,"indexDatabaseRanking":37},{"id":26,"createTime":18,"updateTime":18,"relativeEntities":1057,"label":1058,"description":1059,"key":32,"publicationTags":1060,"standard":18},[],{"EN":29,"VI":29},{"EN":29,"VI":31},[34],{"impactFactor":19,"impactFactorByYear":1062,"i10Index":40,"i10IndexLast5Year":19,"totalPublication":41,"totalPublicationByYear":1063,"totalCitation":69,"totalCitationByYear":1064,"totalCitationPerPublication":99,"totalCitationPerPublicationByYear":1065,"hindexLast5Year":131,"hindex":131},{},{"1965":43,"1966":44,"1967":45,"1968":46,"1969":47,"1970":48,"1971":49,"1972":50,"1973":51,"1974":52,"1975":52,"1976":53,"1977":54,"1978":51,"1979":55,"1980":56,"1981":57,"1982":51,"1983":58,"1984":59,"1985":60,"1986":61,"1987":62,"1988":51,"1989":63,"1990":56,"1991":51,"1992":64,"1993":59,"1994":64,"1995":65,"1996":64,"2007":66,"2008":67,"2015":68},{"1965":71,"1966":72,"1967":73,"1968":74,"1969":75,"1970":76,"1971":77,"1972":78,"1973":79,"1974":80,"1975":81,"1976":82,"1977":83,"1978":84,"1979":85,"1980":86,"1981":87,"1982":87,"1983":88,"1984":89,"1985":90,"1986":91,"1987":55,"1988":92,"1989":93,"1990":94,"1991":46,"1992":95,"1993":96,"1994":96,"1995":97,"1996":98},{"1965":101,"1966":102,"1967":103,"1968":104,"1969":105,"1970":106,"1971":107,"1972":108,"1973":109,"1974":110,"1975":111,"1976":112,"1977":113,"1978":114,"1979":115,"1980":116,"1981":117,"1982":118,"1983":119,"1984":120,"1985":121,"1986":122,"1987":123,"1988":124,"1989":43,"1990":125,"1991":126,"1992":127,"1993":43,"1994":128,"1995":129,"1996":130},{"pages":1067,"volume":1069},{"VOID":1068},"419-424",{"VOID":418},{"total":19,"publishYear":420,"statisticByYear":1071},{},"2007-10-24",[37],{"id":1075,"createTime":1076,"updateTime":1077,"relativeEntities":1078,"slug":1079,"properties":1080,"entityType":151,"verifyStatus":152,"verifyTime":1091,"verifyNote":154,"languages":18,"translateLanguages":18,"viewCount":19,"primaryUrl":1092,"fullTextUrl":18,"authors":1093,"publicationType":173,"publisherRelationship":1109,"citationCount":18,"citationInfo":18,"publishDate":479,"publishYear":201,"citationAnalyzeStatus":480,"lastCitationAnalyze":1134,"indexDatabases":1135,"openAccess":18,"references":18,"isForceReanalyzing":207},"3ad2920e-6963-4449-8158-fad91b9d35f1","2024-02-06T16:26:26.803+00:00","2026-07-24T08:27:59.645+00:00",[],"Collective-quadrupole-states-of-even-nuclei",{"abstract":1081,"title":1083,"gsPaper":1085,"references":1087,"doi":1089},{"EN":1082},"Surface vibrations of even nuclei with quadrupole deformation are taken into account to compute the contribution of rotations to nuclear spectra. The complete range 0°⩽γ\n0⩽30° is covered with the same solution for the Hamiltonian equation. The agreement with the « small vibrations » hypothesis is better than that obtained with previous models.",{"EN":1084},"Collective quadrupole states of even nuclei",{"VOID":1086},"[\"1032377704800751291\"]",{"VOID":1088},"A. Bohr:Kgl. Danske Videnskab. Selskab, Mat.-fys. Medd.,26, No. 14 (1952).\nL. Wilets andM. Jean:Phys. Rev.,102, 788 (1956).\nA. S. Davydov:Nucl. Phys.,24, 682 (1961).\nA. S. Davydov andG. F. Filippov:Nucl. Phys.,8, 237 (1959).\nR. B. Moore andW. White:Can. Journ. Phys.,38, 1149 (1960).\nC. A. Mallmann:Phys. Rev. Lett.,2, 507 (1959).\nC. A. Mallmann:Nucl. Phys.,24, 535 (1961).\nJ. F. Suárez andE. Y. de Aisenberg:Nucl. Phys., A90, 449 (1967).\nA. S. Davydov andA. A. Chaban:Nucl. Phys.,20, 499 (1960).\nE. Y. de Aisenberg andJ. F. Suárez:Nucl. Phys., A97, 529 (1967).\nC. H. Townes andA. L. Schawlow:Microwave Spectroscopy (New York, 1955).",{"VOID":1090},"10.1007\u002FBF02712470","2024-06-25T02:05:06.621+00:00","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002FBF02712470",[1094],{"id":1095,"sortIndex":19,"researcher":18,"roles":1096,"affiliations":1097,"properties":1106,"displayName":1108,"givenName":18,"familyName":18},"14835daa-d1c0-4475-9851-b9043e9c0b5d",[160],[1098],{"id":1099,"sortIndex":19,"affiliation":1100,"properties":18},"c73d825c-00f5-47ff-8908-357f944eae85",{"id":1099,"createTime":18,"updateTime":18,"relativeEntities":1101,"slug":18,"properties":1102,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":1105,"statistic":18},[],{"title":1103},{"VI":1104},"Espectroscopía Nuclear, Comisión Nacional de Energía Atómica, Buenos Aires",[],{"title":1107},{"VI":1108},"J. F. 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