The Relativistic String All lecture courses on string theory start with a discussion of the point particle Ours is no exception Well take a 64258ying tour through the physics o f the relativistic point particle ID: 26962
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whichdonotyethaveexperimentalevidencebutwhichareconsideredtobelikely SEH=1 1/GN# (#h)2+1Mplh ...Eachofthesetermsisschematic:ifyouweretodothisexplicitly,youwouldÞndamessofindicescontractedindi!erentways.Weseethattheinteractionsaresuppressed Thismakesthesituationmarginallyworse,withtheÞrstdiver-Figure1: Þeldtheories,validonlyuptosomeenergyscale.Onedealswiththedivergencesbysimplyadmittingignorancebeyondthisscaleandtreating#asa degreesoffreedomareneededtocompletegravity.SingularitiesOnlyaparticlephysicistwouldphraseallquestionsabouttheuniverseintermsofscatteringamplitudes.Ingeneralrelativitywetypicallythinkaboutthegeometryasawhole,ratherthanbastardizingtheEinstein-Hilbertactionanddiscussingperturba-tionsaroundßatspace.Inthislanguage,thequestionofhigh-energyphysicsturnsintooneofshortdistancephysics.ClassicalgeneralrelativityisnottobetrustedinregionswherethecurvatureofspacetimeapproachesthePlanckscaleandultimatelybecomessingular.Aquantumtheoryofgravityshouldresolvethesesingularities.Thequestionofspacetimesingularitiesismorallyequivalenttothatofhigh-energyscattering.Bothprobetheultra-violetnatureofgravity.Aspacetimegeometryismadeofacoherentcollectionofgravitons,justastheelectricandmagneticÞeldsinalaseraremadefromacollectionofphotons.TheshortdistancestructureofspacetimeisgovernedÐafterFouriertransformÐbyhighmomentumgravitons.Understandingspacetimesingularitiesandhigh-energyscatteringaredi!erentsidesofthesamecoin.Therearetwosituationsingeneralrelativitywheresingularitytheoremstellusthatthecurvatureofspacetimegetslarge:atthebigbangandinthecenterofablackhole.Theseprovidetwoofthebiggestchallengestoanyputativetheoryofquantumgravity.GravityisSubtleItisoftensaidthatgeneralrelativitycontainstheseedsofitsowndestruction.ThetheoryisunabletopredictphysicsatthePlanckscaleandfreelyadmitstoit.Problems hole,thenyouÕreonyourown:withoutatheoryofquantumgravity,noonecantellyouwhatfateliesinstoreatthesingularity.Yet,ifyouaresmartandstayoutsideoftheblackhole,youÕllbehardpushedtoseeanye!ectsofquantumgravity.ThisisbecauseNaturehasconspiredtohidePlanckscalecurvaturesfromourinquisitiveeyes.Inthecaseofblackholesthisisachievedthroughcosmiccensorshipwhichisaconjectureinclassicalgeneralrelativitythatsayssingularitiesarehiddenbehindhorizons.Inthecaseofthebigbang,itisachievedthroughinßation,washingawayanytracesfromtheveryearlyuniverse.Natureappearstoshieldusfromthee!ectsofquantumgravity,whetherinhigh-energyscatteringorinsingularities.IthinkitÕsfairtosaythatnooneknowsifthisconspiracyispointingatsomethingdeep,orismerelyinconvenientforscientiststryingtoprobethePlanckscale.Whilehorizonsmayprotectusfromtheworstexcessesofsingularities,theycomewithproblemsoftheirown.Thesearetheunknownunknowns:di$cultiesthatarisewhencurvaturesaresmallandgeneralrelativitysaysÒtrustmeÓ.Theentropyofblackholesandtheassociatedparadoxofinformationlossstronglysuggestthatlocalquan-tumÞeldtheorybreaksdownatmacroscopicdistancescales.AttemptstoformulatequantumgravityindeSitterspace,orinthepresenceofeternalinßation,hintatsimilardi$culties.Ideasofholography,blackholecomplimentarityandtheAdS/CFTcorre-spondenceallpointtowardsnon-locale!ectsandtheemergenceofspacetime.Thesearethedeeppuzzlesofquantumgravityandtheirrelationshiptotheultra-violetpropertiesofgravityisunclear.AsaÞnalthought,letmementiontheoneobservationthathasanoutsidechanceofbeingrelatedtoquantumgravity:thecosmologicalconstant.Withanenergyscaleof#%10!3eVitappearstohavelittletodowithultra-violetphysics.Ifitdoeshaveitsoriginsinatheoryofquantumgravity,itmusteitherbeduetosomesubtleÒunknownunknownÓ,orbecauseitisexplainedawayasanenvironmentalquantityasinstringtheory. andtheotherforces.Giventhatwehavethistheorysittinginourlaps,itwouldbe whereµ=0,...,D"1andúXµ=dXµ/d&.WeÕveintroducedanewparameter&whichlabelsthepositionalongtheworldlineoftheparticleasshownbythedashedlinesintheÞgure.Thisactionhasasimpleinterpretation:itisjustthepropertime$dsalongtheworldline.NaivelyitlooksasifwenowhaveDphysicaldegreesoffreedomratherthanD"1because,aspromised,thetimedirectionX0'tisamongourdynamicalvariables:X0=X0(&).However,thisisanillusion.Toseewhy,weneedtonotethattheaction &/d÷&|.Meanwhile,thevelocitieschangeas d÷&dX!d÷&"µ!.TheupshotofthisisthatnotallDdegreesoffreedomXµarephysical.Forexample,supposeyouÞndasolutiontothissystem,sothatyouknowhowX0changeswith& where#isaLorentztransformationsatisfying#µ!"!"##"="µ#,whilecµcorrespondstoaconstanttranslation.Wehavemadeallthesymmetriesmanifestatthepriceofintroducingagaugesymmetryintooursystem.Asimilargaugesymmetrywillarise )%=0.Usingtheusualrepresentationofthemomentumoperatorpµ="i#/ 2"em2) ThisistheNambu-Gotoactionforarelativisticstring.Action=Area:ACheckIfyouÕreunfamiliarwithdi!erentialgeometry,theargu-dldl12!"Figure6: X#&.Iftheanglebetweenthesetwovectorsis+,thentheareaisthengivenbyds2=|%dl1||%dl #'(""det**'(#( "gg'(#(Xµ)=0,(1.23)whichcoincideswiththeequationofmotion(1.21)fromtheNambu-Gotoaction,exceptthatg'(isnowanindependentvariablewhichisÞxedbyitsownequationofmotion.Todeterminethis,wevarytheaction(rememberingagainthat'!"g="12!"gg'('g'(=+12!"gg'('g'(),'S="T2!d2('g'(,! "12!"gg'(g"## g'(isnÕtquitethesameasthepull-backmetric*'(deÞnedinequation(1.12);thetwodi!erbytheconformalfactorf.However,thisdoesnÕtmatterbecause,ratherremarkably,fdropsoutoftheequationofmotion(1.23).Thisisbecausethe!"gtermscalesasf,whiletheinversemetricg'(scalesasf!1andthetwopiecescancel.WethereforeseethatNambu-GotoandthePolyakovactionsresultinthesameequationofmotionforX.Infact,wecanseemoredirectlythattheNambu-GotoandPolyakovactionscoincide.Wemayreplaceg'( d2(! g')g(*"g'*g()). 2"'(""## #=0andúX0=R,whereRisaconstantthatisneededondimensionalgrounds.Theinterpretationofthisconstantwillbecomeclearshortly.Then,writingXµ=(t,%x),theequationofmotionforspatialcomponentsisthefreewaveequation,¬%x"%x##=0whiletheconstraintsbecomeú%x ()(+2! thestring.Thiscanbechecked,forexample,bystudyingtheNoethercurrents RealityofXµrequiresthatthecoe n),,÷)µn=(÷) )µnand÷)µ m,p m,n )!n=4)#.�n0÷ ,wehavetwoexpressionsfortheinvariantmass:oneintermsofright-movingoscillators