From Geometry to Quantum Mechanics: In Honor of Hideki Omori PDF
Preview From Geometry to Quantum Mechanics: In Honor of Hideki Omori
Progress in Mathematics Volume252 SeriesEditors HymanBass JosephOesterle´ AlanWeinstein From Geometry to Quantum Mechanics In Honor of Hideki Omori Yoshiaki Maeda Peter Michor Takushiro Ochiai Akira Yoshioka Editors Birkha¨user Boston • Basel • Berlin YoshiakiMaeda PeterMichor DepartmentofMathematics Universita¨tWein FacultyofScienceandTechnology Faculta¨tfu¨rMathematik KeioUniversity,Hiyoshi Nordbergstrasse15 Yokohama223-8522 A-1090Wein Japan Austria TakushiroOchiai AkiraYoshioka NipponSportsScienceUniversity DepartmentofMathematics DepartmentofNaturalScience TokyoUniversityofScience 7-1-1,Fukazawa,Setagaya-ku Kagurazaka Tokyo158-8508 Tokyo102-8601 Japan Japan MathematicsSubjectClassification(2000):22E30,53C21,53D05,00B30(Primary);22E65,53D17, 53D50(Secondary) LibraryofCongressControlNumber:2006934560 ISBN-10:0-8176-4512-8 eISBN-10:0-8176-4530-6 ISBN-13:978-0-8176-4512-0 eISBN-13:978-0-8176-4530-4 Printedonacid-freepaper. (cid:2)c2007Birkha¨userBoston Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permissionofthepublisher(SpringerScience+BusinessMediaLLC,RightsandPermissions,233 SpringStreet,NewYork,NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsor scholarlyanalysis.Useinconnectionwithanyformofinformationstorageandretrieval,electronic adaptation,computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterde- velopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,evenifthey arenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyare subjecttoproprietaryrights. 987654321 www.birkhauser.com (SB) HidekiOmori,2006 Contents Preface .......................................................... ix CurriculumVitae HidekiOmori ..................................................... xiii PartI GlobalAnalysisandInfinite-DimensionalLieGroups 1 AspectsofStochasticGlobalAnalysis K.D.Elworthy .................................................... 3 ALieGroupStructureforAutomorphismsofaContactWeylManifold NaoyaMiyazaki ................................................... 25 PartII RiemannianGeometry 45 ProjectiveStructuresofaCurveinaConformalSpace OsamuKobayashi.................................................. 47 DeformationsofSurfacesPreservingConformalorSimilarityInvariants AtsushiFujioka,Jun-ichiInoguchi..................................... 53 GlobalStructuresofCompactConformallyFlatSemi-SymmetricSpacesof Dimension3andofNon-ConstantCurvature MidoriS.Goto .................................................... 69 DifferentialGeometryofAnalyticSurfaceswithSingularities TakaoSasai....................................................... 85 viii Contents PartIII SymplecticGeometryandPoissonGeometry 91 TheIntegrationProblemforComplexLieAlgebroids AlanWeinstein .................................................... 93 Reduction,InductionandRicciFlatSymplecticConnections MichelCahen,SimoneGutt .......................................... 111 LocalLieAlgebraDeterminesBaseManifold JanuszGrabowski.................................................. 131 LieAlgebroidsAssociatedwithDeformedSchoutenBracketof2-VectorFields KentaroMikami,TadayoshiMizutani................................... 147 Parabolic Geometries Associated with Differential Equations of FiniteType KeizoYamaguchi,TomoakiYatsui ..................................... 161 PartIV QuantizationsandNoncommutativeGeometry 211 TowardGeometricQuantumTheory HidekiOmori ..................................................... 213 ResonanceGyronsandQuantumGeometry MikhailKarasev ................................................... 253 ASecondaryInvariantofFoliatedSpacesandTypeIIIλvonNeumannAlgebras HitoshiMoriyoshi.................................................. 277 TheGeometryofSpace-TimeandItsDeformations:APhysicalPerspective DanielSternheimer, ................................................ 287 GeometricObjectsinanApproachtoQuantumGeometry HidekiOmori,YoshiakiMaeda,NaoyaMiyazaki,AkiraYoshioka ............ 303 Preface Hideki Omori is widely recognized as one of the world’s most creative and original mathematicians. This volume is dedicated to Hideki Omori on the occasion of his retirement from Tokyo University of Science. His retirement was also celebrated in April 2004 with an influential conference at the Morito Hall of Tokyo University of Science. Hideki Omori was born in Nishionmiya, Hyogo prefecture, in 1938 and was an undergraduate and graduate student at Tokyo University, where he was awarded his Ph.D degree in 1966 on the study of transformation groups on manifolds [3], which became one of his major research interests. He started his first research position at Tokyo Metropolitan University. In 1980, he moved to Okayama University, and then became a professor of Tokyo University of Science in 1982, where he continues to worktoday. Hideki Omori was invited to many of the top international research institutions, including the Institute for Advanced Studies at Princeton in 1967, the Mathematics Institute at the University of Warwick in 1970, and Bonn University in 1972. Omori received the Geometry Prize of the Mathematical Society of Japan in 1996 for his outstandingcontributionstothetheoryofinfinite-dimensionalLiegroups. ProfessorOmori’scontributionsaredeepandcoverawiderangeoftopicsasillus- tratedbythenumerouspapersandbooksinhislistofpublications.Hismajorresearch interestscoverthreetopics:Riemanniangeometry,thetheoryofinfinite-dimensional Lie groups, and quantization problems. He worked on isometric immersions of Rie- mannianmanifolds,wherehedevelopedamaximumprinciplefornonlinearPDEs[4]. Thismaximumprinciplehasbeenwidelyappliedtovariousproblemsingeometryas indicatedinChen–Xin[1].HidekiOmori’slastingcontributiontomathematicswasthe creationofthetheoryofinfinite-dimensionalLiegroups.Hisapproachtothistheory wasfoundedintheinvestigationofconcreteexamplesofgroupsofdiffeomorphisms with added geometric data such as differential structures, symplectic structures, con- tact structures, etc. Through this concrete investigation, Omori produced a theory of infinite-dimensional Lie groups going beyond the categories of Hilbert and Banach spacestothecategoryofinductivelimitsofHilbertandBanachspaces.Inparticular, thenotionandnamingofILH(orILB)LiegroupsisduetoOmori[O2].Furthermore, x Preface he extended his theory of infinite-dimensional Lie groups to the category of Fre´chet spacesinordertoanalyzethegroupofinvertiblezerothorderFourierintegralopera- tors on a closed manifold. In this joint work with Kobayashi, Maeda, and Yoshioka, thenotionofaregularFre´chetLiegroupwasformulated.Omoridevelopedandunified theseideasinhisbook[6]ongeneralizedLiegroups. Beginning in 1999, Omori focused on the problem of deformation quantization, whichhecontinuestostudytothisday.Heorganizedaprojectteam,calledOMMY aftertheinitialsoftheprojectmembers:Omori,Maeda,MiyazakiandYoshioka.Their firstworkshowedtheexistenceofdeformationquantizationforanysymplecticman- ifold. This result was produced more or less simultaneously by three different ap- proaches, due to Lecomte–DeWilde, Fedosov and Omori–Maeda–Yoshioka. The ap- proachoftheOmoriteamwastorealizedeformationquantizationasthealgebraofa “noncommutativemanifold.”Afterthisinitialsuccess,theOMMYteamhascontinued todeveloptheirresearchbeyondformaldeformationquantizationtotheconvergence problemfordeformationquantization,whichmayleadtonewgeometricproblemsand insights. Hideki Omori is not only an excellent researcher, but also a dedicated educator whohasnurturedseveralexcellentmathematicians.Omorihasaverycharmingsense ofhumorthatevenmakesitswayintohispapersfromtimetotime.Hehasafriendly personalityandlikestotalkmathematicsevenwithnon-specialists.Hismathematical ideas have directly influenced several researchers. In particular, he offered original ideasappearingintheworkofShiohamaandSugimoto[2],hiscolleagueandstudent, respectively,onpinchingproblems.DuringOmori’svisittotheUniversityofWarwick, hedevelopedagreatinterestintheworkofK.D.Elworthyonstochasticanalysis,and theyenjoyedmanydiscussionsonthistopic.ItisfairtosaythatOmoriwasthefirst persontointroduceElworthy’sworkonstochasticanalysisinJapan.Throughouttheir careers,ElworthyhasremainedoneofOmori’sbestresearchfriends. Inconclusion,HidekiOmoriisapioneerinJapaninthefieldofglobalanalysisfo- cusingonmathematicalphysics.Omoriiswellknownnotonlyforhisbrilliantpapers and books, but also for his general philosophy of physics. He always remembers the long history of fruitful interactions between physics and mathematics, going back to Newton’sclassicaldynamicsanddifferentiation,andEinstein’sgeneralrelativityand Riemanniangeometry.Fromthispointofview,Omorithinksthenextfruitfulinterac- tionwillbeageometricaldescriptionofquantummechanics.Hewillnodoubtbean activeparticipantinthedevelopmentofhisideaof“quantumgeometry.” The intended audience for this volume includes active researchers in the broad areas of differential geometry, global analysis, and quantization problems, as well as aspiringgraduatestudents,andmathematicianswhowishtolearnbothcurrenttopics intheseareasanddirectionsforfutureresearch. WefinallywishtothankAnnKostantforexperteditorialguidancethroughoutthe publicationofthisvolume.Wealsothankalltheauthorsfortheircontributionsaswell astheirhelpfulguidanceandadvice.Therefereesarealsothankedfortheirvaluable commentsandsuggestions.
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