Conformal Groups in Geometry and Spin Structures

307 Pages·2008·2.478 MB·English

by  Pierre Anglè (auth.), Pierre Anglès (eds.)

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
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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