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String Theory In Nutshell

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2012-07-25 0人阅读 0 0 0 暂无简介 举报

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BookreviewStringTheoryinaNutshellEliasKiritsisPrinceton,NJ:PrincetonUniversityPresspp,UK£,US$ISBN:Thebook‘StringTheoryinaNutshell’byEliasKiritsisprovidesacomprehensiveintroductiontomodernstringtheoryStringtheoryistheleadingcandidateforatheorythatsuccessfullyunifiesallfundamentalforcesofnature,includinggravityThesubjecthasbeencontinuouslydevelopingsincetheearlys,withclassictextbooksonthesubjectbeingthoseofGreen,SchwarzandWitten()andPolchinski()Sincethelatterwaspublishedtherehavebeensubstantialdevelopments,inparticularinunderstandingblackholesandgravitygaugetheorydualitiesAtextbooktreatmentofthisimportantmaterialisclearlyneeded,bothbystudentsandresearchersinstringtheoryandbymathematiciansandphysicistsworkinginrelatedfieldsThisbookhasagoodselectionofmaterial,startingfrombasicsandmovingintoclassicandmoderntopicsInparticular,Kiritsis’presentationofthebasicmaterialiscomplementarytothatoftheearliertextbooksandheincludesanumberoftopicswhicharenoteasilyfoundorcoveredadequatelyelsewhere,forexample,loopcorrectionstostringeffectivecouplingsOverallthebooknicelycoversthemajoradvancesofthelasttenyears,including(nonperturbative)stringdualities,blackholephysics,AdSCFTandmatrixmodelsItprovidesaconcisebutfairlycompleteintroductiontothesesubjectswhichcanbeusedbothbystudentsandbyresearchersMoreovertheemphasisisonresultsthatarereasonablyestablished,asisappropriateforatextbookconcisesummariesaregivenforsubjectswhicharestillinflux,withreferencestorelevantreviewsandpapersApositivefeatureofthebookisthatthebibliographysectionsattheendofeachchapterprovideacomprehensiveguidetotheliteratureThebibliographiespointtoreviewsandpedagogicalpapersonsubjectscoveredinthisbookaswellasthosethatwereomittedItisrareforatextbooktocontainsuchaselfcontainedanddetailedguidetotheliteratureInall,thebookcontainsnearlyfivehundredexercisesforthegraduatelevelstudent,whichareusefulbothinteachingcoursesonstringtheoryandforthosewhoarestudyingbythemselvesAnicefeatureofthisbookisthatreferencesaremadetospecificpagesearlierinthebook,ratherthantochapters,whichishelpfulforstudentsworkingthroughthebookontheirownInsummary,‘StringTheoryinaNutshell’isavaluableadditiontotheexistingstringtheorytextbooksitiscomplementarytothepreviousbooksandgivesagoodtreatmentofsubsequentdevelopmentsItislikelytobecomeastaplereferenceonthesubject,usedbothbystudentsandresearchersKostasSkenderisInstituteofTheoreticalPhysics,UniversiteitvanAmsterdam,Valckenierstraat,NLXEAmsterdam,TheNetherlandsContentsPrefacexvAbbreviationsxviiIntroductionPrehistoryTheCaseforStringTheoryAStringyHistoricalPerspectiveConventionsBibliographyClassicalStringTheoryThePointParticleRelativisticStringsOscillatorExpansionsClosedstringsOpenstringsTheVirasoroconstraintsBibliographyExercisesQuantizationofBosonicStringsCovariantCanonicalQuantizationLightconeQuantizationSpectrumoftheBosonicStringUnorientedStringsOpenstringsandChanPatonfactorsPathIntegralQuantizationIntroductionPrehistoryThequestinphysicshasbeenhistoricallydominatedbyunravelingthesimplicityofthephysicallaws,movingmoreandmoretowardtheelementaryAlthoughthisisnotguaranteedtosucceedindefinitely,ithasbeenvindicatedsofarTheotherorganizingtendencyofthehumanmindistoward“unification”:findingauniqueframeworkfordescribingseeminglydisparatephenomenaThephysicsofthelatenineteenthandtwentiethcenturiesisaseriesofdiscoveriesandunificationsMaxwellunifiedelectricityandmagnetismEinsteindevelopedthegeneraltheoryofrelativitythatunifiedtheprincipleofrelativityandgravityInthelates,therewasaculminationoftwodecades’effortsintheunificationofelectromagnetismandquantummechanicsInthesands,thetheoryofweakandelectromagneticinteractionswasalsounifiedMoreover,aroundthesameperiodtherewasalsoawiderconceptualunificationThreeofthefourfundamentalforcesknownweredescribedbygaugetheoriesThefourth,gravity,isalsobasedonalocalinvariance,albeitofadifferenttype,andsofarstandsapartThecombinedtheory,containingthequantumfieldtheoriesoftheelectroweakandstronginteractionstogetherwiththeclassicaltheoryofgravity,formedtheStandardModeloffundamentalinteractionsItisbasedonthegaugegroupSU()×SU()×U()ItsspingaugebosonsmediatethestrongandelectroweakinteractionsThematterparticlesarequarksandleptonsofspininthreecopies(knownasgenerationsanddifferingwidelyinmass),andaspinparticle,theHiggsboson,stillexperimentallyelusive,thatisresponsibleforthespontaneousbreakingoftheelectroweakgaugesymmetryToday,wehavesomeintriguingevidencethatevengravitymaybeastrongcouplingfacetofanextraunderlyingfourdimensionalgaugetheory|ChapterTheStandardModelhasbeenexperimentallytestedandhassurvivedthirtyyearsofacceleratorexperimentsThishighlysuccessfultheory,however,isnotsatisfactory:•Aclassicaltheory,namely,gravity,describedbygeneralrelativity,mustbeaddedtotheStandardModelinordertoagreewithexperimentaldataThistheoryisnotrenormalizableatthequantumlevelInotherwords,newinputisneededinordertounderstanditshighenergybehaviorThishasbeenachallengetothephysicscommunitysincethesand(apartfromstringtheory)verylittlehasbeenlearnedonthissubjectsincethen•ThethreeSMinteractionsarenotcompletelyunifiedThegaugegroupissemisimpleGravityseemsevenfurtherfromunificationwiththegaugetheoriesArelatedproblemisthattheStandardModelcontainsmanyparametersthatlookaprioriarbitrary•Themodelisunstableasweincreasetheenergy(hierarchyproblemofmassscales)andthetheorylosespredictivityasonestartsmovingfarfromcurrentacceleratorenergiesandclosertothePlanckscaleGaugebosonsareprotectedfromdestabilizingcorrectionsbecauseofgaugeinvarianceThefermionsareequallyprotectedduetochiralsymmetriesTherealculpritistheHiggsbosonSeveralattemptshavebeenmadetoimproveontheproblemsaboveThefirstattemptsfocusedonimprovingonunificationTheygaverisetothegrandunifiedtheories(GUTs)AllinteractionswerecollectedinasimplegroupSU()inthebeginning,butalsoSO(),E,andothersThefermionsofagivengenerationwereorganizedinthe(larger)representationsoftheGUTgroupThereweresuccessesinthisendeavor,includingthepredictionofsinθWandthepredictionoflightrighthandedneutrinosinsomeGUTsHowever,therewasaneedforHiggsbosonstobreaktheGUTsymmetrytotheSMgroupandthehierarchyproblemtookitstollbymakingittechnicallyimpossibletoengineeralightelectroweakHiggsThephysicscommunityrealizedthatthefocusmustbeonbypassingthehierarchyproblemAfirstideaattackedtheproblematitsroot:itattemptedtobanishtheHiggsbosonasanelementarystateandtoreplaceitwithextrafermionicdegreesoffreedomItintroducedanewgaugeinteraction(termedtechnicolor)whichboundsthesefermionsstronglyoneofthetechnihadronsshouldhavetherightpropertiestoreplacetheelementaryHiggsbosonasresponsiblefortheelectroweaksymmetrybreakingThenegativesideofthislineofthoughtisthatitreliedonthenonperturbativephysicsofthetechnicolorinteractionRealisticmodelbuildingturnedouttobedifficultandeventuallythislineofthoughtwasmostlyabandonedAcompetingideareliedonanewtypeofsymmetry,supersymmetry,thatconnectsbosonstofermionsThispropertyturnedouttobeessentialsinceitcouldforcethebadmanneredspinbosonstobehaveaswellastheirspinpartnersThisworkswell,butsupersymmetrystipulatedthateachSMfermionmusthaveaspinsuperpartnerwithequalmassThisbeingobviouslyfalse,supersymmetrymustbespontaneouslybrokenatWiththeexceptionoftheneutrinosectorthatwassuspectedtobeincompleteandiscurrentlythesourceofinterestingdiscoveriesIntroduction|anenergyscalenotfarawayfromtoday’sacceleratorenergiesFurtheranalysisindicatedthatthebreakingofglobalsupersymmetryproducedsuperpartnerswhosemasseswerecorrelatedwiththoseofthealreadyknownparticles,inconflictwithexperimentaldataToavoidsuchconstraintsglobalsupersymmetryneededtobepromotedtoalocalsymmetryAsasupersymmetrytransformationisinasensethesquarerootofatranslation,thisentailedthatatheoryoflocalsupersymmetrymustalsoincorporategravityThistheorywasfirstconstructedinthelates,andwasfurthergeneralizedtomakemodelbuildingpossibleTheflipsideofthiswasthattheinclusionofgravityopenedthePandora’sboxofnonrenormalizabilityofthetheoryHopesthat(extended)supergravitymightberenormalizablesoonvanishedInparallelwiththedevelopmentsabove,apartofthecommunityresurrectedtheoldideaofKaluzaandKleinofunifyinggravitywiththeothergaugeinteractionsIfonestartsfromahigherdimensionaltheoryofgravityandcompactifiesthetheorytofourdimensions,oneendswithfourdimensionalgravityplusextragaugeinteractionsAlthoughgravityinhigherdimensionsismoresingularintheUV,physicistshopedthatatleastattheclassicallevelonewouldgetatheorythatisveryclosetotheStandardModelAlthoughprogresswasmade,astumblingblockturnedouttobeobtainingafourdimensionalchiralspectrumoffermionsasintheSMAlthoughnoneofthedirectionsaboveprovidedafinalandsuccessfultheory,theingredientswereveryinterestingideasthatmanyfeltwouldformapartoftheultimatetheoryTheCaseforStringTheoryStringtheoryhasbeentheleadingcandidateoverthepasttwodecadesforatheorythatconsistentlyunifiesallfundamentalforcesofnature,includinggravityItgainedpopularitybecauseitprovidesatheorythatisUVfiniteThebasiccharacteristicofthetheoryisthatitselementaryconstituentsareextendedstringsratherthanpointlikeparticlesasinquantumfieldtheoryThismakesthetheorymuchmorecomplicatedthanQFT,butatthesametimeitimpartssomeuniquepropertiesOneofthekeyingredientsofstringtheoryisthatitprovidesafinitetheoryofquantumgravity,atleastinperturbationtheoryToappreciatethedifficultieswiththequantizationofEinsteingravity,welookatasinglegravitonexchangebetweentwoparticles(Figa))Then,theamplitudeisproportionaltoEMP,whereEistheenergyoftheprocessandMPisthePlanckmass,MP∼GeVItisrelatedtotheNewtonconstantasMP=()πGNAlthoughthereisnorigorousprooftoallordersthatthetheoryisUVfinite,thereareseveralallordersargumentsaswellasrigorousresultsatlowlooporderInclosedstringtheory,amplitudesmustbecarefullydefinedviaanalyticcontinuation,standardinSmatrixtheoryWhenopenstringsarepresent,therearedivergencesHowever,theyareinterpretedasIRdivergences(duetotheexchangeofmasslessstates)inthedualclosedstringchannelTheyaresubtractedinthe“Wilsonian”Smatrixelements|Chaptera)b)FigureGravitationalinteractionbetweentwoparticlesviagravitonexchangeTherefore,thegravitationalinteractionisirrelevantintheIR(E�MP)butstronglyrelevantintheUVInparticular,thisimpliesthatthetwogravitonexchangediagram(Figb))isproportionaltothedimensionlessratioE∫��EdE˜E˜∼,()MPMPwhereEistheoutgoingparticleenergyandE˜istheinternalparticleenergyThisisstronglyUVdivergentItisknownthatEinsteingravitycoupledtomatterisnonrenormalizableinperturbationtheorySupersymmetrymakestheUVdivergencesofterbutthenonrenormalizabilitypersistsTherearetwowaysoutofthis:•ThereisanontrivialUVfixedpointthatgovernsthehighenergybehaviorofquantumgravityTodate,nocredibleexampleofthispossibilityhasbeenoffered•ThereisnewphysicsatE∼MP(orevenlower)andEinsteingravityistheIRlimitofamoregeneraltheory,validatandbeyondthePlanckscaleYoucouldconsidertheanalogoussituationwiththeFermitheoryofweakinteractionsThere,anonrenormalizablecurrentcurrentinteractionwithsimilarproblemsoccurred,buttodayweknowthatthisistheIRlimitofthestandardweakinteractionmediatedbytheW±andZgaugebosonsSofar,thereisnoconsistentfieldtheorythatcanmakesenseatenergiesbeyondMPandcontainsgravityGoodreviewsoftheultravioletproblemsofEinsteingravitycanbefoundin,StringsprovideatheorythatinducesnewphysicsatthestringscaleMswhichinperturbationtheory(thestringcouplinggsbeingweak)ismuchlowerthanthePlanckscaleMPItisstilltrue,however,thatthestringperturbationtheorybecomesuncontrollablewhentheenergiesapproachthePlanckscaleTherearetwoimportantreasonswhyclosedstringtheorydoesnothaveUVdivergencesOneisthefactthatstringdynamicsandinteractionsareinextricablylinkedtothegeometryoftwodimensionalsurfacesForexample,forclosedstrings,bydecomposingthestringIntroduction|FigureOpenandclosedRiemannsurfacesFeynmandiagrams,itisobviousthatthereisessentiallyauniversalthreepointinteractioninthetheoryThisinteractionisdictatedbythetwodimensionalgeometryofclosedRiemannsurfacesasisobviousfromfigureTheotherrelatedreasonisthepresenceofaninfinitetowerofexcitationswithmassesinmultiplesofthestringscaleTheirinteractionsarecarefullytunedtobecomesoftatdistanceslargerthanthestringlength�sbutstilllongerthanthePlancklength�PForopenstringsthesituationissubtlerThere,UVdivergencesarepresent,butareinterpretedasIRclosedstringdivergencesinthedualclosedstringchannelThisUVIRopenclosedstringdualityisattheheartofmanyoftherecentdevelopmentsinthefieldAnotherkeyingredientofstringtheoryisthatitunifiesgravitywithgaugeinteractionsItdoesthisinseveraldifferentwaysThesimplestisviathetraditionalKKapproachSuperstringtheorytypicallyisdefinedintendimensionsStandardfourdimensionalvacuacanbeobtainedviacompactificationonasixdimensionalcompactmanifoldHowever,gaugesymmetrycanalsoarisefromDbranesthatsometimesarepartofthevacuum(asinorientifolds)ThereisevengaugesymmetrycomingfromanongeometricalpartofthetheoryashappensintheheteroticstringTheunifiedoriginofgravityandgaugesymmetryextendsevenfurthertootherinteractionsForexample,theYukawainteractions,crucialforgivingmasstotheSMparticles,arealsointimatelyrelatedtothegaugeinteractionsUnlikeearlierKaluzaKleinapproachestounification,stringtheoryiscapableofproviding,uponappropriatecompactifications,chiralmatterinfourdimensionsThishappensviaasubtleinterplaybetweenanomalyrelatedinteractionsandtheprocessofcompactificationAnothercharacteristicingredientofstringtheoryisthatthepresenceofspacetimefermionsinthetheoryimpliestheappearanceofspacetimesupersymmetryatleastathighenergiesSupersymmetry,consequently,isanimportantingredientofthetheoryTherearegoodreasonstobelievethatthetheoryisunique,althoughtherearemanypossiblevacuathatcouldbestableRecentunderstandingofnonperturbativedualitieshasstrengthenedthebeliefintheuniquenessofthisstringtheorystructureIthasalsopointedoutnewcornersintheoveralltheory,whichmanycallMtheory,thatarestillunchartedDespiteallthis,stringtheory,afterthirtyfiveyearsofresearch,hasmanyimportantquestionsstillunansweredPhysicistsfeelthatthefundamentaldefinitionofthetheoryisnotknownHowitfitstherealworldindetailisalsonotknownpresently∑|ChapterItmaybethatstringtheorywillturnintoan“intellectualclassicalblackhole”ItmayalsobethatitisthecorrectdescriptionofphysicsatshortdistancesTimeandexperimentwillshowAStringyHistoricalPerspectiveInthes,physiciststriedtomakesenseofalargeamountofexperimentaldatarelevanttothestronginteractionTherewerelotsofparticles(or“resonances”)andthesituationcouldbestbedescribedaschaoticSomeregularitieswereobserved:AlmostlinearReggebehaviorItwasnoticedthatarelativelylargenumberofresonancescouldbenicelyputon(almost)straightlinesbyplottingtheirmassversustheirspinJm=α,()α′withα′∼GeV−thisrelationwascheckeduptoJ=stdualityIfweconsiderascatteringamplitudeoftwohadrons→twohadrons(,→,),thenitcanbedescribedbytheMandelstaminvariantss=−(pp),t=−(pp),u=−(pp),()withstu=imiWeareusingametricwithsignature(−)Suchanamplitudedependsontheflavorquantumnumbersofhadrons(forexampleSU())Considertheflavorpart,whichiscyclicallysymmetricinflavorspaceForthefullamplitudetobesymmetric,itmustalsobecyclicallysymmetricinthemomentapiThissymmetryamountstotheinterchanget↔sThus,theamplitudeshouldsatisfyA(s,t)=A(t,s)ConsideratchannelcontributionduetotheexchangeofaspinJparticleofmassMThen,athighenergy(−s)JAJ(s,t)∼()t−MThus,thispartialamplitudeincreaseswithsanditsbehaviorbecomesworseforlargevaluesofJIfonesewsamplitudesofthisformtogethertomakealoopamplitude,thenthereareuncontrollableUVdivergencesforJ>Anyfinitesumofamplitudesoftheform()hasthisbadUVbehaviorMoreover,suchafinitesumhasnoschannelpolesHowever,ifoneallowsaninfinitenumberoftermsthenitisconceivablethattheUVbehaviormightbedifferentAproposalforsuchadualamplitudewasmadebyVeneziano,�(−α(s))�(−α(t))A(s,t)=,()�(−α(s)−α(t))where�isthestandardEuler�functionandα(s)=α()α′s()Introduction|Byusingthestandardpropertiesofthe�function,itcanbecheckedthattheamplitude()hasaninfinitenumberofs,tchannelpoles:∞∑(α(s))···(α(s)n)A(s,t)=−()n!α(t)−nn=Inthisexpansion,thes↔tinterchangesymmetryof()isnotmanifestThepolesin()correspondtotheexchangeofaninfinitenumberofparticlesofmassM=(n−αα′())andhighspinsItcanalsobecheckedthatthehighenergybehavioroftheVenezianoamplitudeissofterthananylocalquantumfieldtheoryamplitude,andtheinfinitenumberofpolesiscrucialforthisItwassubsequentlyrealizedbyNambu,Goto,Nielsen,andSusskindthatsuchamplitudescameoutoftheoriesofrelativisticstringsHo

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