Table Of ContentProgressinMathematicaPlhysics
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
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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
