ProgressinMathematicaPlhysics Volume39 Editors-in-Chief AnneBoutetdeMonvel,Universite Paris VIIDenis Diderot GeraldKaiser,Centerfor Signals andWaves, Austin, TX Editorial Board C. Berenstein,University ofMaryland, College Park SirM.Berry,University ofBristol P.Blanchard,Universitdt Bielefeld M.Eastwood,University ofAdelaide A.S.Fokas,Imperial College ofScience, Technology andMedicine D.Stemheimer,UniversitedeBourgogne, Dijon C.Tracy,University ofCalifornia, Davis Fabrizio Colombo Irene Sabadini Franciscus Sommen Daniele C. Struppa Analysis of Dirac Systems and Computational Algebra Springer Science+Business Medi~ LLC Fabrizio Colombo lrene Sabadini Politecnico di Milano Politecnico di Milano Dipartimento di Matematica Dipartimento di Matematica 20133 Milano 20133 Milano ltaly ltaly Franciscus Sommen Daniele C. Stroppa Ghent University George Mason University Faculty of Engineering Department of Mathematical Sciences Department of Mathematical Analysis Fairfax, VA 22030 9000 Ghent USA Belgium AMS Subject Classifications: Primary: 30035; Secondary: 16E05, 33N05 Library of Congress CataIogIng-In-PubHcadon Data Analysis of Dirac systems and computational algebra / Fabrizio Colombo .. let al.]. p. cm. - (Progress in mathematical physics ; v. 39) Inc1udes bibliographical references and index. ISBN 978-1-4612-6469-9 ISBN 978-0-8176-8166-1 (eBook) DOI 10.1007/978-0-8176-8166-1 1. Mathematical physics. 2. Dirac equation. 3. Clifford algebras. 4. Differential n. equations, Partial. 5. Mathematical analysis. I. Colombo, Fabrizio. Progress in mathematical physics ; v. 39. QC20.AS3 2004 530.15'2'57-dc22 2004053657 ISBN 978-1-4612-6469-9 Printed on acid-free paper. ©2004 Springer Science+Business Media New York Originally published by Birkhlluser Boston in 2004 ~ GD Softcover reprint ofthe hardcover 1st edition 2004 ajpJ All rights reserved. 1ms work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. Tbe use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to property rights. 987654321 SPIN 10843874 www.birkhasuer-science. carn ToBiancaandClotilde-FC,IS Tomy sons-FS ForLisa-DCS Contents Preface be 1 BackgroundMaterial 1 1.1 Algebraictools. . . . . . . .. . . .. . . . 1 1.1.1 Commutativealgebra 1 1.1.2 Crobnerbases: a quickintroduction 11 1.1.3 Sheaftheory 23 1.2 Analyticaltools.. .. . . . . .. 34 1.2.1 Topologicallinear space.s 34 1.2.2 Distributions . . . .. . . 40 1.2.3 Fouriertransformandfundamentaslolutions 52 1.3 Elements ofhyperfunction theory. .. .. . . .. . . 60 1.3.1 Hyperfunctions in onevariable. . . . .. . . 60 1.3.2 Cohomologicaplropertiesofthe sheaf ofholomorphic functions 74 1.3.3 Hyperfunctions ofseveralvariables 78 1.3.4 Fouriertransform.. 85 1.4 Appendix:category theory. . .. . 88 2 ComputationalAlgebraicAnalysis 93 2.1 A primer ofalgebraic analysi.s. . .. . . . . . . . 93 2.2 The Ehrenprei-sPalamodovFundamentalPrinciple . 116 2.3 The FundamentalPrinciple for hyperfunctions . 126 2.4 Usingcomputationaallgebra software .131 viii Contents 3 TheCauchy-FueteSrystemanditsVariations 139 3.1 Regular functions ofonqeuaternionicvariable. . .139 3.2 Quaternionichyperfunctions in one variable... .148 3.3 Severalquaternionicvariables: ananalyticapproach .166 3.4 Severalquaternionicvariables: an algebraic approac.h .173 3.5 The Moisil-Theodorescu system. . . .. . . . . . . .198 4 SpecialFirstOrderSystemsin CliffordAnalysis 209 4.1 Introductionto Cliffordalgebras .. . . . .209 4.1.1 StandardCliffordalgebras . .. . . . . .209 4.1.2 Endomorphisms and spinor spaces. . .218 4.1.3 Classificationsofreal Cliffordalgebras . 222 4.2 Introductionto Cliffordanalysis .225 4.2.1 Diracoperators. . . . .225 4.2.2 Radial algebra .229 4.2.3 Fischer decomposition. . .231 4.3 The Dirac complexfortw,othreeand fouroperators . 237 4.3.1 Caseoftwooperators. .237 4.3.2 Caseofthreeoperators . . .238 4.3.3 Case offouorperators. . . .248 4.4 Specialsystems in Cliffordanalysis . 250 4.4.1 Generalizedsystems. . . . .250 4.4.2 Systems using theWittbasis . 253 4.4.3 Combinatorialsystems .. . . .262 5 SomeFirstOrderLinearOperatorsin Physics 267 5.1 Physics and algebra ofMaxwellanPdrocafields .270 5.2 Variations on Maxwellsystem in the spaceboifquaternions . 277 5.3 PropertiesofVz-regularfunctions . . . . .. . . . . 284 5.4 The Diracequationand thelinearizationproblem. . 293 5.5 Octonionic Diracequation. . . . . . . . .. . .. . .296 6 OpenProblemsandAvenuesforFurtherResearch 307 6.1 The Cauchy-Fuetersystem . 307 6.2 The Dirac system. .308 6.3 Miscellanea. . . .. . . . . .309 Bibliography 313 Index 327 Preface Cliffordalgebras have been widelyusaesdamathematicatlool forthe descrip tion of physicalphenomen.aThe smallestnoncommutativeCliffordalgebrais the algebralHl ofquaternions,firstintroducedby Hamilton, in which it is pos sible to formulate several physical laws by means of some special first order differentiaolperators.For example,ifwewrite aquaternionqby means offour realcoordinatesas q = Xo+ iXl + jX2 + kX3, we canintroducethe so-called Moisil-Theodorescuoperatordefined by 'D .8 . 8 k 8 = ~8Xl +J8X2 + 8X3 and wecan apply it tqouaternionvalued functions ofthe for1m= 10+ih + j12 + kf3 to get the homogeneouesquation'DI = O. Fromthepoint of view of physicalapplicationsthis approachis particularlyinterestingbecause if we write1= 10+F, withF = ih +j12 +kh, then 'DI ='D(fo+F) = -'V.F +'V10+'V x F and henceanylHl-valuedfunctionI inthe kerneolf'D givesriseto apair(fo, F) satisfying the system 'V ·F=O { 'V10 +'V x F = 0. If10 == const, for.example10 == 0,thenwehavethatirrotationaalnd solenoidal vector fields (.ie.,the solutions of the previous system) are purely vector func tions regular inthe senseoftMheoisil-Theodorescuoperator'D.The book[111] x Preface is entirely devoted to showing hoqwuaternions(andquaternionvalued func tions) are wellsuited to thsteudyofanarrayofdifferent physical phenomena. In the samespiri,tImaeda[92]considered functions definedonreqaulaternions with values in the algebra ofcomplqeuxaternions,the so-calledbiquaternion.s He thenintroduceda newoperatorD whose kernel is the space of the so calledD-regularfunctions.Itcan be shownthata functionF: JH[ - JH[~ C, F = ao+ibo+a+ib isD-regularif and only ifit satisfies the system This system represents Maxwelle'squationsifthe vectors,ab are the magnetic and the electric field respectiv;ebloyis relatedto the electric density charge and to the electriccurrentdensity.The scalarao is supposed to beconstant to avoid the existence of magnetic monopoles. The notioDn-roefgularityis an interestingone because theoperatorD, at least formally, is well known tomathematicianass a simplevariationof theCauchy-Fueteroperato.rFrom this point ofview,the workIomfaedashowshowatransformationoftime into imaginarytime can be usedto derivethe Maxweeqlluationsfrom theCauchy Fuetersystem. The earlie(rthoughless known) work of Lanczosintroduced similar ideas (seehis collected wor[k1s14]). In thesettingofbiquaternionsit is also possible to consideortherkinds of complexifiedFueter-typeoperators(see[198])which,as shownin[59],describe spin1/2masslessfield.s Anotherfundamentalequationin physics is the Diracequation.As is well known,the classicalconservation energy lEaw= p2/2m+V and thequantum mechanicaloperatorsassociated to energy anmdomentumiOt, -iV',respec tively, give rise to the Schr6dingeerquationthatis first order in time and second order in spac.eThis deductionis not satisfactory, apsointedout by Dirac,since in thetheoryof relativity, space and time asrterictlyconnected, therefore forthreelativisticwaveequationweexpect first derivatives alsowith respect to space variab.les Considering therelativisticenergy conservation lawE2= m2 +p2 (without potentialV) and replacing thequantummechanicaloperator,soneobtainsa secondorderequationin space and time whosesolutiocnasnnotbeinterpreted as aprobabilitydensity. To overcome this difficul,tyone has to linearize the Jm equationas E = 2 +p2. This was done by Dirac byintroducingsuitable 4 x 4 matrices, called thgeammamatrices,and thenreplacing thequantum mechanicaolperatorsobtainingtheequation(incovariant form()i"(/1o/1-m)'ljJ = owhere0/1 = ox!" forfL = 1,2,3,00 = -ioxo' and"(/1are thegammamatrice.s The Diracequationis acentralone because its four solutions represent the ±! spin for a particle and its an-ptiarticle.The studyof systems of quarks deals withparticleshaving isospins. In this cas,ethe Diracequationdoes not Preface xi haveenoughcomponentsto describe this phenomenon, and the first idea which seems useful is to increase the sizeof the matrices. For ex,aamwplaeyto ex tendthe(3+1)space-timeframeworkto 8dimensionsand to take into account the increasingquantumnumbers andinternalsymmetriesassigned to elemen taryparticle,sis the use of octonion.sThe octonions form a non-associative algebra and this seems to prevent the usemofaatrixtranslationofthe eight dimensionalDiracequation,althoughin[45Jwehaveshownthatusingasuitable matrixproduct,definedoncomplexifiedoctonio,nitsispossibleto write an8x8 matrixthatrepresents the Dirac equation. Tmhiastrixapproachallowsone to linearizetherelativisticenergyconservation lawE2 = m2+p2. This generalized Diracequationgivesrise in anaturalwayto the Klei-nGordonequationas in the original Diracequation. But a more complexsituationcan be considered if we move to the Clif ford algebrasetting,and to thestudyof the so-called Diracoperatorin this framework.Ifa CliffordalgebraCn has basiseo= 1,ell'..,en with defining relationseiej + ejei = -2oij , i,j = 1,...,n, the Diracoperatoris defined as 8 = '£1=0e 8 (sometimes the sumstartsfromj = 1). In the casen = 3 the j xj Diracoperatorhas aphysicalmeaning (se[e35])sinceC4 can beembedded into C in astandardwayand wemaythinkofC4 asthe complexMinkowskispace 3 time.Thenthe complexifiedDiracoperatorcoincides with the Weyelquation formasslessfieldsofspin1/2. In this book wepropose a unifortmreatmenotfthese (and related) systems. Thepointofviewweadoptisbased onthe theory ofalgebraic analysis. Specifi cally,wetakeadvantageofthe relatively simpleformofthese systems (they are all systems oflineaprartialdifferentiaelquationswithconstantcoefficients)to studythecorrespondingmodulesoverasuitableringRofcomplexpolynomials. One may in fact consider a system oflinear differeentqiuaaltionsas anTl x TO matrixof differentiaolperatorsP(D), and ifS is a space of (generalized) functions,one seesthatP(D) definesanaturalmap whosekernel isthe objectionfteresttoanalysts. As has alreadyhappenedin geometry, however,it soon became clear among analystsas wellthatthe best way totreatsituationsof this kind was to in troducethe notion of sheaf, stohatwe will not consider simply spacebs,ut rathersheaves of (generalized) functions, to which the entire algethberaoircy will be applied.Partialdifferentialoperatorshave theimportantpropertyof acting onsuitablesheavesassheaf homomorphisms, and soSiifssuch ashea,f wenowhavethatP(D) isa sheaf homomorphism whosekernel (again a sheaf) wecustomarilydenote bySp. In what follows we will consider several different sheaves of (generalized) functions. Thetheorywe will describe works for a large class of shea,vfeosr example the sheaAf of realanalyticfunctions, E of infinitely differentiable