From Geometry to Quantum Mechanics: In Honor of Hideki Omori

326 Pages·2007·2.06 MB·English

by  K. D. Elworthy (auth.)

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
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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.