Su57524arez M Dorato and M R57524edei eds EPSA 2007 Launch of the European Philosophy of Science Association Springer 2009 1 Introduction The idea of duality is at the core of the most relevant developments in recent fundamental physics During the ID: 80372
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DualitiesandintertheoreticrelationsElenaCastellaniInM.Suarez,M.DoratoandM.Redei(eds),EPSA2007:LaunchoftheEuropeanPhilosophyofScienceAssociation,Springer20091IntroductionTheideaofdualityisatthecoreofthemostrelevantdevelopmentsinrecentfundamentalphysics.Duringthelastfortyyearstheoreticalphysicshasusedthenotionofdualityindi erentwaysandframeworks:intheso-calleddualresonancemodelofthelatesixties,whichgavebirthtoearlystringtheory;inthecontextofquantumeldtheory,whereagroundbreakinggeneralizationofelectromagneticdualitywasconjecturedbyClausMontonenandDavidOlivein1977;insupersymmetricstringtheory,wherevarioussortsofdualitiesareplayingakeyroleinthetheoreticalelaboration.Thispaperisconcernedwiththesignicanceofphysicaldualitiesfromtheviewpointofphilosophyofscience.Theideaisthat,foritspeculiarity,this`new'ingredientintheoryconstructioncanopenunexpectedperspectivesforthecurrentphilosophicalre ectiononcontemporaryphysics.1Inparticular,dualitiesrepresentanunusualtypeofintertheoryrelation,themeaningofwhichdeservestobeinvestigated.Itistheaimofthepapertoshowhowdiscussingthispointbringsintoplay,atthesametime,whatisintendedbya`theory'andinwhichsensedualitiesaretobeconsidered`symmetries'(iftheyare). DepartmentofPhilosophy,UniversityofFlorence,viaBolognese52,50139,Firenze,Italy.E-mail:elena.castellani@uni.it1Thephilosophicalliteratureonphysicaldualitiesisstillverymeagre.Thephilosophersofphysicsarejuststartingtoturntheirattentiontostringtheoryanditsformsofdualities.OneofthefewcontributionsinthisdirectionisDawid(2007).1 Consideringtheroleandmeaningofphysicaldualitiesingeneralposesimmediatelyaproblem.Thedualitiesappliedinrecentfundamentalphysicsareofdi erentformsandstatus.Whilesomeofthemseemtohaveasoundbasis,othersarejusttheoreticalconjecturesandagoodpartofthelastdevel-opmentsgroundedondualitiesarestillatawork-in-progressstage.Nonethe-less,inmostofthecaseswheredualitiesareappliedinaquantumframeworkitispossibletoindividuatesomecommonrelevantcharacteristicfeatures.Adualitytypethatresultsparticularlyrepresentativefromthispointofviewistheso-calledelectromagneticduality(EMduality).EMdualityalsorepre-sentstherstformofdualityexplicitlyappliedintwentiethcenturyphysics:namely,inP.A.M.Dirac'sfamoustwopapers(published,respectively,in1931and1948)onhis`theoryofmagneticpoles'.Itthereforeo ersanap-propriate,howeverspecic,casestudytobeginwith.Startingtoinvestigatethesignicanceofphysicaldualitiesbyfocussingonthiscasestudyistheobjectofthepaper.2ThecaseofelectromagneticdualityElectromagneticdualityasformulatedbyDiracis,inasense,theprototypeoftoday'sphysicaldualities.InthisSectionwepresentabriefsurveyofthedevelopmentofthisdualityideafromtheclassicaltothequantumcontext.2.1EMduality(1):classicalelectrodynamicsEMdualityisgroundedontheideathatthereisasubstantialsymmetrybetweenelectricityandmagnetism.Thisisanoldidea,goingbacktoMichaelFaradayandrstmademoreprecisewiththeformulationbyJamesClerkMaxwellofhisfamousequationsregulatingthebehaviourofelectricandmagneticelds.Incurrentnotation(usingaunitsystemforwhichc=1),Maxwell'sequationsread:2 EMdualitycanthenbeexpressedastheinvarianceofthesourcelessMaxwell'sequationsunder`rotations'oftheelectricandmagneticelds.Thiscanbebettervisualizedbyintroducingthecomplexvectoreld~E+i~B,intermsofwhichMaxwell'sequationscanbewritteninthefollowingconciseform:~r(~E+i~B)=0;(5)~r^(~E+i~B)=i@ @t(~E+i~B):Maxwell'sequationsintheaboveformremaininvariantunderthedualityrotations:~E+i~B!ei(~E+i~B):(6)Intheseterms,itiseasytoseethattheenergyandmomentumdensitiesoftheelectromagneticeld,representedrespectivelybythefollowingtwoexpressions,E=1 2j~E+i~Bj2=1 2(E2+B2);(7)P=1 2i(~E+i~B)^(~E+i~B)=~E^~B;areinvariantwithrespecttotheEMdualitytransformations.Tosumup:Whennosourcetermsarepresent,thedualityDexchangestherolesoftheelectricandmagneticeldswhileleavingthe`physics'{thatis,theMaxwell'sequationsandphysicalrelevantquantitiessuchastheenergyandmomentumdensitiesoftheelectromagneticeld{invariant.Whenelectricsourcetermsarepresent,theMaxwellequationsarenolongerinvariantunderthedualityDandEMsymmetryisbroken.2.1.2RestoringEMdualityinthepresenceofsourcesThereisawaytorestorethesymmetrybetweentheelectricandmagneticeldsinthepresenceofsources:thatis,byincludingmagneticsourceterms.Assumingtheexistenceofamagneticdensityofchargegandmagnetic4 current~jg,inadditiontotheusualelectricchargedensityeandelectriccurrent~je,theMaxwell'sequationstaketheform~r~E=e;(8)~r~B=g;�~r^~E=~jg+@~B @t;~r^~B=~je+@~E @t:Theseequationsareinvariantunderthefollowingdualitytransformation,interchangingtherolesoftheelectricandmagneticeldsand{atthesametime{therolesoftheelectricandmagneticchargesandcurrents:~E!~B;~B!�~E;(9)e;~je!g;~jg;g;~jg!�e;�~je:Intermsofthecomplexvectoreld~E+i~B,theaboveequationscanbewrittenconciselyas:~r(~E+i~B)=e+ig;(10)~r^(~E+i~B)=i~je+i~jg+@ @t~E+i~B:Theseequationsareinvariantunderthedualityrotations:~E+i~B!ei~E+i~B;(11)e+ig!ei(e+ig);~je+i~jg!ei~je+i~jg:Maxwell'sequationscanthusbemodiedtoaccomodatetheinclusionofmagneticchargesandcurrents.Theproblemisthatisolatedmagneticcharges,theso-calledmagneticmonopoles(or,inDirac'sterminology,magneticpoles),haveneverbeenobserved.Ifwebreakamagnetbarintwoparts,wealwaysobtaintwosmallermagnetsandneveranisolatedNorthpoleandanisolatedSouthpole.QuotingDirac(1948,p.817):\Theeldequationsofelectrody-namicsaresymmetricalbetweenelectricandmagneticforces.Thesymmetry5 Ontheotherhand,thevectorpotential~Aisintroducedinstandardelectro-magnetismbytakingadvantageoftheabsenceofmagneticsourceterms:~r~B=0!~B=~r^~A(12)(forall~A;~r(~r~^~A)=0).Thisseemstoimplythatquantummechanicsisinconsistentwiththepres-enceofmagneticcharge.Dirachadthustoaddressthefollowingconsistencyissue:whetheritwaspossibletoincludeparticlescarryingamagneticchargewithoutdisturbingtheconsistencyofthecouplingofelectromagnetismtoquantummechanics.Theargumentheproposedinhis1931paperforsolvingthisapparentinconsistencyisremarkableundermanyaspects.Inparticular,itrepresentsoneoftherstexampleofanexplicituseoftopologicalconsiderationsintheearlytwentiethcenturyphysics.Indevelopinghisargument,centeredontherelationbetweenthephasechangeofthewavefunctionsroundclosedcurvesandthe uxofthemagneticeld~Bthroughclosedsurfaces,Diracinfactappliedideasinvolvingthestructureofthespaceinthelarge(whatisnowknownasglobaltopology).3Theresultheobtainedwasthefollowing:theintroductionofmagneticchargecanbeconsistentwiththequantumtheoryprovideditsvaluesare`quantized'.Inhisownwords(Dirac1931,p.68):\Ourtheorythusallowsisolatedmagneticpoles,butthestrengthofsuchpolesmustbequantised,thequantum0beingconnectedwiththeelectronicchargeebyhc/e0=2."Incurrentnotation(denotingmagneticchargebygandusingtheunitsystemh=c=1),Dirac'sresultwasthatamagneticchargegcanoccurinthepresenceofanelectricchargeeifthefollowingcondition,knownasDiracquantizationcondition,issatised:eg=2nn=0;1;2;::::(13)Thisconditionhasanimmediatestrikingconsequence:themereexistenceofamagneticchargegsomewhereintheuniverseimpliesthequantizationofelectriccharge,sinceanyelectricchargemustthenoccurinintegermultiplesoftheunit2=g.InDirac'swords(ibid.),\Thetheoryalsorequiresaquan-tisationofelectriccharge,sinceanychargedparticlemovingintheeldofa 3OnDirac'santicipationoftopologicalideasinphysicssee,forexample,Olive(2003).7 3ThemeaningofEMdualityInclassicalelectrodynamics(withmagneticsourcetermsincluded),wehaveseenthattheEMdualitytransformation~E!~B;~B!�~E;(14)e;~je!g;~jg;g;~jg!�e;�~je;exchanges,atthesametime,therolesoftheelectricandmagneticeldsandtherolesoftheelectricandmagneticchargesandcurrents,whileleavingthephysicsinvariant.`Thephysics'meanstheMaxwell'sequationsandtherelevantphysicalquantities(suchastheenergyandmomentumdensitiesoftheelectromagneticeld).EMdualityisthusasymmetryofthetheory,expressingtheequivalenceofthefollowingdualwaysofdescribingthesamephysics:1)Description1.Thephysicsisdescribedintermsof:theelectriceld~E1andthemagneticeld~B1;theelectricchargeandcurrentdensitiese1and~je1,andthemagneticchargeandcurrentdensitiesg1and~jg1.2)Description2.Thephysicsisdescribedintermsof:theelectriceld~E2=~B1andthemagneticeld~B2=�~E1;theelectricchargeandcurrentdensitiese2=g1andje2=jg1,andthemagneticchargeandcurrentdensitiesg2=�e1and~jg2=�~je1.Thismeans,inconcrete,thatacalculationofaphysicalquantityintheframeworkofdescription1canbeobtainedbymeansofanothercalculationinthedualframeworkofdescription2.Forexample,calculatingtheforceoftheelectriceld~E1onaparticlewithelectricchargee1intheframeworkofdescription1isthesameascalculatingtheforceofthemagneticeld~B2onaparticlewithmagneticchargeg2=�e1intheframeworkofdescription2.Forthedualityissueofconcernhere,thisdoesnotsaymuch.Theideaofasymmetrybetweenelectricityandmagnetismis,ofcourse,moreprofoundthenwhattheaboveconsiderationcanshow.Inparticular,ithasplayedaveryimportantheuristicroleinthehistoryofpre-quantumelectrodynamics{9 thinkaboutitsin uenceonFaraday'sdiscoveryofelectromagneticinductionorEinstein's1905workonspecialrelativity.Butitisonlyinthequantumcontextthatthefulltheoreticalsignicanceofphysicaldualitiesdoesactuallyemerge.InordertohaveacompletegraspontherealmeaningofEMdualityinquantumphysics,weshouldfollowthedevelopmentofthisideainquantumeldtheoryandstringtheory.Inthispaperwepursueamuchmoremodestscope.WeremainintheconceptualrangeoftheprecedingSection,andconsiderwhatcanbeextractedfromDirac'sseminalworkfortheissueatstake.InfactDiracanticipatedsomuchthat,onthebasisofhisresults,itispossibletogetanideaofsomegeneralfeaturesoftoday'sphysicaldualities.Herewefocusonthemoststrikingofthesefeatures:thatis,thefactthatdualitiestypicallyinterrelateweakandstrongcoupling.Thisisknown,inthephysicsliterature,asweak-strongduality.IntheframeworkofDirac'stheoryofmagneticmonopoles,itiseasytoseehowtheweak-stronginterchangenaturallyfollowsfromassumingEMdualityandthequantizationcondition.Aswehaveseen,EMdualityimpliesinterchangingelectricandmagneticcharges:EMduality:e!g;g!�e;whileDirac'squantizationconditionimpliesthattheelectricandmagneticcharges(thatis,theelectricandmagneticcouplingconstants)aresorelated:Quantizationcondition::eg=2n:Puttingthetwotogether,weobtain:e!g=2n e;g!�e=�2n g:Thismeansthatifthechargeeissmall,thechargegintowhichitistrans-formedisstrongandviceversa.Thatis:inquantumphysics,EMdualityrelatesweakandstrongcoupling.Ingeneral,turningtothemoreappropriatecontextofquantumeldtheoryandstringtheory,whathappensisthatdualitiestypicallyrelateatheoreticaldescriptionconcerningastrong-couplingregimetoanotherde-scriptionconcerningaweak-couplingregime(whileleavingthe`physics'in-variant).Thatis,dualitiesexchangephysicalregimesthatareverydi erent,withtheremarkableconsequencethatcalculationsinvolvingstrongforcesinonetheoreticaldescriptioncanbeobtainedfromcalculationsinvolvingweak10 sortofsymmetryisrepresentedbydualities,iftheseareindeedsymmetries(asiscommonlyassumed).Itisusuallysaidthatdualtheories,ordualtheoreticaldescriptions,areconnectedwithoneanotherbytransformations`leavingthephysicsinvari-ant':dualitiesareinthissense`symmetries'.Thiscanbemademoreprecisebyspecifyingthemeaningoftheexpression`leavingthephysicsinvariant'.Ifbythisweintendthatthedynamicalequationsofthetheoryremaininvari-ant,asintheEMdualitycasediscussedinSection2(wheretheMaxwell'sequationsareinvariantunderthedualitytransformationD),thenthedualityisasymmetryofthetheoryintheprecisesensenormallyusedincontempo-raryphysics.Thatis,thesenseaccordingtowhichGisasymmetrygroupofatheoryifthedynamicalequations(orthe`action')ofthetheoryareinvari-antunderthetransformations(thataretheelements)ofthegroupG.Thesymmetriespostulatedthroughtheso-calledinvarianceprinciplesofphysics,suchasthespace-timesymmetriesandthegaugesymmetriesoftheStandardModelofparticlephysics,arepropertiesofphysicaltheoriesinthissense.But,ingeneral,thedualitiesusedintodayphysicsrelatetwodi erenttheoreticaldescriptionsthatconcerndi erentscalelevelsandpresentap-parentlydi erentontologicalscenarios.Thesedescriptionscaneveninvolvedi erentactions(orHamiltonians)anddi erentelds.7Inwhichsense,then,theyarejusttwodi erentformulationsofthesameunderlyingtheory,asiscommonlymaintained?Thisclearlydependsonthemeaningattributedtothenotionoftheory.Theclueisgivenbytheextendedsenseinwhichdu-alityisconsideredasymmetry.Thatis:the`theory'isidentiedonthebasisofwhatremainsinvariantunderthedualitytransformations,the`samephysics'thatisdi erentlydescribedbymeansofthedualformulations.Andthis`samephysics',accordingtothephysicistsworkingonthesubject,isgivenbythespectraandthetransitionamplitudes.8Wethusarriveatanapparently`phenomenological'understandingofthenotionofatheorythatmayseemparadoxicalinsuchahighlymathematizedandfarawayfromcommon(and,fornow,possible)experienceasisstringtheory.Notethatsuchanotionisnotnewinthehistoryofquantumphysics:thinkabouttheideologybehindHeisenberg'smatrixmechanicsinthe1920sortheS-matrixapproachdominatinginthe1960s(whichwas,itisworth 7Acompanionpaperinpreparationisdevotedtoexamininginsomedetailthispoint,byfocussingonthedevelopmentsofelectromagneticdualityinquantumeldtheoryandstringtheory8See,forexample,Polchinski(1998),Section4.12 J.Polchinski(1998),\QuantumgravityatthePlancklength",arXiv:hep-th/9812104.14