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Conformal Groups in Geometry and Spin Structures PDF

307 Pages·2008·2.84 MB·English
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Progress in Mathematical Physics Volume50 Editors-in-Chief AnneBoutetdeMonvel,Universite´ ParisVIIDenisDiderot GeraldKaiser,CenterforSignalsandWaves,Austin,TX Editorial Board SirM.Berry,UniversityofBristol C.Berenstein,UniversityofMaryland,CollegePark P.Blanchard,Universita¨tBielefeld M.Eastwood,UniversityofAdelaide A.S.Fokas,ImperialCollegeofScience,TechnologyandMedicine C.Tracy,UniversityofCalifornia,Davis Pierre Angle`s Conformal Groups in Geometry and Spin Structures Birkha¨user Boston • Basel • Berlin PierreAngle`s LaboratoireEmilePicard InstitutdeMathe´matiquesdeToulouse Universite´PaulSabatier 31062ToulouseCedex9 France [email protected] MathematicsSubjectClassifications:11E88,15A66,17B37,20C30,16W55 LibraryofCongressControlNumber:2007933205 ISBN 978-0-8176-3512-1 eISBN978-0-8176-4643-1 Printedonacid-freepaper. (cid:1)c2008Birkha¨userBoston Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewrittenpermis- sionofthepublisher(Birkha¨userBoston,c/oSpringerScience+BusinessMediaLLC,RightsandPermis- sions,233SpringStreet,NewYork,NY10013,USA),exceptforbriefexcerptsinconnectionwithreviews orscholarlyanalysis.Useinconnectionwithanyformofinformationstorageandretrieval,electronicadap- tation,computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterdevelopedis forbidden. Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,eveniftheyare notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubjectto proprietaryrights. 987654321 www.birkhauser.com (Lap/SB) Tothememoryofmygrandparentsandmyfather,Camille; tomymother,Juliette,mywife,Claudie, mychildren,Fabrice,CatherineandMagali, mygrand-daughtersNoémie,EliseandJeanne andtothememoryofmyfriendPerttiLounesto. WilliamK.Clifford(1845–1879),MathematicianandPhilosopher.PortraitbyJohnCollier (bykindpermissionoftheRoyalSociety). “TheAngelofGeometryandtheDevilofAlgebrafightforthesoulofany mathematicalbeing.” AttributedtoHermannWeyl (CommunicatedbyRenéDeheuvelshimself accordingtoaprivateconversationwithH.Weyl) “C’estl’étudedugroupedesrotations(àtroisdimensions)quiconduisitHamiltonà ladécouvertedesquaternions;cettedécouverteestgénéraliséeparW.Cliffordqui, en1876,introduitlesalgèbresquiportentsonnom,etprouvequecesontdesproduits tensorielsd’algèbresdequaternionsoud’algèbresdequaternionsetd’uneextension quadratique. Retrouvées quatre ans plus tard par Lipschitz qui les utilisa pour donner une représentation paramétrique des transformations à n variables ... ces algèbres et la notion de ‘spineur’qui en dérive, devaient aussi connaıtre une grande vogue à l’époquemoderneenvertudeleurutilisationdanslesthéoriesquantiques.” NicolasBourbaki Elémentsd’histoiredesMathématiques Hermann,1969,p.173. Foreword Itisnotveryoftenthecase thatatreatiseandtextbookiscalledtobecomeastandard reference and text on a subject. Generally a comprehensive treatment on a subject is devoted to the specialist and a didactical textbook is a newer version of a series of guiding monographs. This book by Pierre Anglès is all these things in one: a good reference on the subject of Clifford algebras and conformal groups and the subjacentspinstructures,atextbookwherestudentsandevenspecialistsofanyone ofthesubjectscanlearnthefullmatter,andabridgebetweenthebasicapproachof GrassmannandCliffordoffindingalinearformthatcorrespondstoagivenquadratic formandallthestructureswhichcanbebuiltfromthosealgebrasandinparticular thepseudounitaryconformalspinstructures. Thenumerousreferences,startingintheforeworditselfandwithineachchapter supply the necessary connection to the state of the art of the subject as viewed by numerousotherauthorsandthecreativecontributionsofProfessorAnglèshimself.A freshapproachtothesubjectisfoundanywayandthischaracteristicisthebasisfor thisbooktobecome,aswesaid,astandardtextandreference. Besidestherigorousalgebraicapproachaconsistentgeometricalpointofview,in thegenealogyofWessel,Argand,Grassmann,Hamilton,Clifford,etc.andofCartan and Chevalley is found throughout the book. In fact it would be desirable that this transparencyofpresentationwouldbecontinuedoneday,byProfessorAnglès,inthe fieldofmathematicalphysicsandperhapsevenintheoreticalphysicswhereaclear connectionbetweenalgebra,geometryandspinstructureswithphysicaltheoretical structuresarealwayswelcome.Thesameappliestothepossibilityofextending,in thefuture,thenumerouspresentexercises,whichareaguidanceforthestudyofthe subject,toapplicationsinotherbranchesofmathematicsandtheoreticalphysics. Wefinallywanttostressthattheeffortoftheauthortoclearlypresentthedevel- opment from Clifford algebras through conformal real pseudo-euclidean geometry, pseudounitaryconformalspinstructuresandmoreadvancedapplicationshasresulted infactinabundantnewconceptsandmaterial. JaimeKeller UniversityofMexico

Description:
Conformal groups play a key role in geometry and spin structures. This book provides a self-contained overview of this important area of mathematical physics, beginning with its origins in the works of Cartan and Chevalley and progressing to recent research in spinors and conformal geometry.Key topi
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