AlejandroRivero∗
Dep.Economia,Univ.CarlosIIIMadrid
February2,2008
Abstract
Itissuggestedthatgenerationsarelinkedtotheneedofcalculatingcurvatureofspaceviaadeformedordiscretecalculus.Quantizationwouldlimitthedeformation,buildingthreegenerations,andnotfour,asotherinterpretationcouldimply.
1Introduction
Itisknown[8,11]thattheusualambiguityinthedefinitionof(partial)deriva-tivesofafunctionbecomesasourceofproblemswhenwegotoquantumtheory.ThesimplestexampleisFeynmanquantummechanics(or0+1dimensionalfieldtheory),wheredifferentelectionsofdiscretisationforthederivativedrivetodif-ferentorderingrulesinthequantizedtheory;Weylcorrespondstothesymmetricone,Born-Jordantotheforwardderivative,andsoon,andevenmoreexoticeffectscouldbegotbygaugingtheambiguity.
Mypaper[17]hasbeenunderstoodin[2]asifitweregivingtheoreticalsupporttotheexistenceofafourthgeneration.Indeed,asitispointedin[17]andalsopreviouslyin[16],Ibelievethattheambiguitiesofadiscretecurvatureonlypointtothreegenerations,theextantambiguitybeingabsorbedinascaleparameterwhichrelatestoPlankconstant(and,tothedeformationparameterofthecalculus).Itismyfaultthatthisprocessisjusthalfcooked,insparsereferencesattheendof[15]and,asappendix,in[14].IapologizeIcannothelpwithadetailedexampleyet,soinordertotrytocleartheconfusion,thispapercanonlytoexpandontheideasofourprevious[17].
So,pleaseconsiderthisnoteasatraditionalconferenceposter,tryingtoputillustrationstothepreviouswork.Theexamplesofthepreviouspaperswereformulatedinthecontextofnoncommutativegeometry.Herewewillkeepourselvesintheconceptualcloud.
2Pictorialimage
Themainobservation,figure(1),isthattodefineavectortangentatapointx0weneedtobuildtwoseriesofpointsapproachingx0,sothelimit(x2−x1)/ǫwillgivethetangentvector.Itseemsthattherearetwoambiguities,tochoose
x2andtochoosex1,butoneofthemcanbeabsorbedintothescaleparameter.Theextantambiguityistheoneweproposedtoconsideramass.
x2
x0
x1Figure1:q-tangentvectortothepointx0
Considernow,asinfigure(2),acurveofwhichwewanttoknowthecur-vatureatapoint.Thisimpliestogivefourpoints,sotheorthogonaltotwoq-tangentvectorswillcrossmarkingthepositionoftheradiusofcurvature,andgivingustheinverseofthecurvaturewhenthecontinuouslimitisapproached.
Againthescalecanabsorboneoftheambiguities,andwehavethreefreeparameters,thatwecanidentifywiththreemasses.
Whengoingtohigherdimensions,theplayisnotmoreambiguous,butitismorecomplicated(Figure3).Thecurvaturetensorisbuildfromthecurvaturesofthegeodesicsurfacetangenttoeach2-plane.Everydirectionmustbecon-sidered,anddependencesbetweentheambiguitiesoforthogonaldirectionsarenotclear.
Therearewaystosimplifythetask,forinstanceaskingforadditionalre-strictions(isotropy,homogeneity)tothespace-timemanifold.Inanycase,weshouldexpectnowtomultiplyourdegenerationtimesthenumberofdimensionofspacetime,tobeabletocopewitheverycurvature.Thuswewillgetfourparticles(fromspace-timedimension)andthreegenerations(fromambiguity).
Last,considerhowthescaleparametercouldcomeintoplay.Everytangentvectorisdefinedviatalestheorem,figure(4),relatingthetime(orparameterofthecurve)withthedistance.Classicallyeachtriangleiswelldefinedateachpointofthecurve.Now,doubtscanberaisedwhenwemakeanintegration(figure5)inthe”deformed”way,beforeanylimit.Itseemsthatweshouldintroduceanscaleparameterǫ,tobeabletoaddthequantitiesofeachtriangle,andthatthecontinuouslimitshouldcomewhenǫ→0.
TheprocessasIseeitisalittlemoreinvolved,asfirstoneneedstousescaleinvariancetogofromabareǫtoarenormalizedh,andtheclassicallimitish→0.ThismethodisneededbecausethegroupoidofpathsgivenbyConnesdoesnotadd(x,y,t)(y,z,t)to(x,z,2t)butto(x,z,t).
2
x4
x3x2x0
x1
Figure2:Curvatureofacurvethroughx0
3Interdisciplinarywork
Themostpopularofalltheregularisationproblemsistheoneoffermiondou-bling,whichhappenswhenwetrytoquantizefermionsinlatticequantumfieldtheory.Here,ifweuseonlythesymmetricdefinitionofderivativewefinishwithasetof2dfermionsforeachinitialfermioninthetheory.Wilsoncuresthisbyincorporatingtheambiguitytothetheory,buildingeachderivativeasacombinationofsymmetricandantisymmetricpart,andthengivingahighmasstotheantisymmetriccombination,whichcontrolstheunwanteddegreesoffree-dom.Recently[12,9]Wilson’approachhasbeenresuscitedandtheGinsparg-Wilsonconditionislookingforaplaceinnoncommutativegeometry.L¨uscher’approachisperhapsclosesttoDimakisorMajidones,butBalachandran[1]isalreadylookingforaroleforitneartotheaxiomaticofNCGmanifolds.
Fromthepointofviewofnoncommutativegeometry,itseems[10]thatnaivelatticefieldtheorydoesnotqualifyasadifferentialgeometry,asevenwhenmultiplecopiesofelementsaretaken(whichseemsneededtocopewithNCGfirstorderaxiom),itfailstofulfillPoincareduality.Togetoutofthistrap,thesuggestionshouldbetoactwiththeDiracoperatorindifferentpointsofthespace.
Wewilldothisbyintroducingsmallfinitedifferencesbetweenthefieldsat-tachedtoeachfermion,andrelatingthisdifferencetotheinverseofthemassreasonedinthepictorialshow,sothatintheverylowenergylimittheimple-menteddifferencebecomesequaltothederivativeitdiscretizes.OurgoalistoexpandNCGLagrangiantocontaininformationaboutquantizationambiguity,
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Figure3:Twocurvaturesofasurface
butwiththistechniquewealsogettointroducethespectrumofmasses.
TimeagoinBarcelona,Alain,inadualsessionwithAsthekar,suggestedthatthenewerversions[5,6]oftheConnes-Lotapproachshouldbeseenasanlowenergyapproximationtoacompletelynoncommutativespaceonlyvisibleathighenergy.Insomesense,herethemethodologyisreversed,bettingfirstofa”verynoncommutative”modelandtryingtoguessamethodtogodowntolowenergy.Atverylowenergy,onlygravityshouldbeseen,whileatintermediateenergy,Connes-ChamseddineorConnes-Lotshouldbesuitableapproximations.
JusttovisualizesuchapproximationwewanttokeepourselvesusingtheDiracoperatorformalism.Ontheotherhand,fromtheTangentGroupoidconstruction,weknowthatthesetoffunctionsoverthetangentspaceTMofamanifoldcanbepasted,viaWeylquantization[3,4],tothesetofkernelsk(x,y)ofoperatorsinaHilbertspace,andthisformalismisveryneartothefinitedifferenceschemeproposedabove.Itshouldbeniceifourcandidatesfordifferentialformshadsomedualitywiththisspaceofoperators.Alsobecausethetangentgroupoidseemsclosertoq-deformationsasmadebyMajidandothers,andtothenon-commutativeformalismusedbylatticetheoretistsintheabovereferredworks.
4Mass
Basicmass,aswehaveseen,shouldbefixedbythepositionofthevectoroffigure1respectiveofthepointwhich”differentiation”isassignedto.
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Figure4:Tangentvectorasalimitor”instantaneousvelocity”
Figure5:Summation(integration)acrossasectionofthetangentbundle.Shouldalengthscalebedefinedateverypoint?
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Foreachparticle,amassrelationshipcanbeimposedaskingtothesecondderivativestogivethesameresultasaiterationofthefirstderivatives.Sothetwovectorsofacurvaturewouldbegivenbythepositionsoftheextremepointsofthevectorgivingthefirstderivative.
Suchrelationshipcouldbenotneededifsomegeometricalconsistencycondi-tionswhereimposedtothemassmatrix.Forinstance,itisknowthatPoincareDualityforcestheConnes-Lottmodelgenerationstobedegeneratedinmass.Morerestringentconditionscouldappear.
Finally,themassrelationbetweendifferentparticlesisthetouchiestpoint.Mybetistolinkittoapreferredkindofmetrics,withapreferredsetofcoordinatesystems.Verymuchasithappensinasphericalsetofcoordinates:avariationacrossδrhasnoadditionalweight,variationsacrossδθ,δφcarryanadditionalweightr,andvariationacrossδφcarriesanadditionaltermrespecttoδθ.In[16],thiswasstatedwithanobscurecomparisonbetweenquarksandangles.
5Acknowledgements
TogaugetheambiguityisasuggestionofE.Follana,yettoexplore.Cheerfullyacknowledged,aswellasalotofsupporttodiscussotherideas.MomentumspacewasseenasconfigurationspacewithafieldofforcesinsomecoffeetalkswithJ.I.Martinez,accordingmyoldnotebooks.Again,yettoexplore.And,asnotedelsewhere,themainthesisofthispapersurfacedwhileabedroomtalkwithJ.GuerreroatVietri,whereIwasdriventothinkinanalogiesbetweenthefourfundamentalfermionsandafourdimensionalvolumeform.Moreaboutthiscanbeforthcomingin[18],wherewewonderabouttherelationbetweenjunkremovalandthePauliantisymmetrizationofNfermions.
TherelaxingambianceprovidedbythefolksofLaLatina,theMadridinnerneighborhoodwhereIamdoingahalfsabbatical,shouldalsobeacknowledged.
References
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fourthgeneration,hep-ph/9911203[3]JFCari˜nenaetal.,Connes’TangentGroupoidandDeformationQuanti-zation,J.ofGeom.andPhys,v32(1999).math/9802102[4]A.Connes,NonCommutativeGeometry,AcademicPress1994[5]A.Connes,NonCommutativeGeometryandReality
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