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