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Praiseforthefirstedition“Quantumfieldtheoryisanextraordinarilybeautifulsubject,butitcanbeanintimidatingone.Theprofoundanddeeplyphysicalconceptsitembodiescangetlost,tothebeginner,amidstitstechnicalities.Inthisbook,Zeeimpartsthewisdomofanexperiencedandremarkablycreativepractitionerinauser-friendlystyle.IwishsomethinglikeithadbeenavailablewhenIwasastudent.”—FrankWilczek,MassachusettsInstituteofTechnology“Finally!Zeehaswrittenaground-breakingquantumfieldtheorytextbasedonthecourseImadehimteachwhenIchairedthePrincetonphysicsdepartment.Withutmostclarityhegivestheeagerstudentalight-heartedandeasy-goingintroductiontothemultifacetedwondersofquantumfieldtheory.IwishIhadthisbookwhenItaughtthesubject.”—MarvinL.Goldberger,President,Emeritus,CaliforniaInstituteofTechnology“Thisbookisfilledwithcharmingexplanationsthatstudentswillfindbeneficial.”—EdWitten,InstituteforAdvancedStudy“Thisbookisperhapsthemostuser-friendlyintroductorytexttotheessentialsofquantumfieldtheoryanditsmanymodernapplications.Withhisphysicallyintuitiveapproach,ProfessorZeemakesaserioustopicmorereachableforbeginners,reducingtheconceptualbarrierwhilepreservingenoughmathematicaldetailsnecessaryforafirmgraspofthesubject.”—BeiLokHu,UniversityofMaryland“LikethefamousFeynmanLecturesonPhysics,thisbookhastheflavorofagoodblackboardlecture.Zeepresentstechnicaldetails,butonlyinsofarastheyservethelargerpurposeofgivinginsightintoquantumfieldtheoryandbringingoutitsbeauty.”—StephenM.Barr,UniversityofDelaware“Thisisafantasticbook—exciting,amusing,unique,andveryvaluable.”—CliffordV.Johnson,UniversityofDurham“TonyZeeexplainsquantumfieldtheorywithaclearandengagingstyle.Forbuddingorseasonedcondensedmatterphysicistsalike,heshowsusthatfieldtheoryisanourishingnuttobecrackedandsavored.”—MatthewP.A.Fisher,KavliInstituteforTheoreticalPhysics“IwassoengrossedthatIspentallofSaturdayandSundayoneweekendabsorbinghalfthebook,tomywife’sdismay.Zeehasatalentforexplainingthemostabstruseandarcaneconceptsinanelegantway,usingtheminimumnumberofequations(thejokesandanecdoteshelp)....IwishthiswereavailablewhenIwasagraduatestudent.Buythebook,keepitbyyourbed,andrelishtheinsightsdeliveredwithsuchflairandgrace.”—N.P.Ong,PrincetonUniversityWhatreadersaresaying“Funny,chatty,physical:QFTeducationtransformed!!Thistextstandsapartfromothersinsomanywaysthatit’sdifficulttolistthemall....Theexpositionisbreezyandchatty.Thetextisneverboringtoread,andisattimesvery,veryfunny.Punsandjokesabound,asdoanecdotes....Abookwhichismucheasier,andmorefun,toreadthananyoftheothers.Zee’sskillsasapopularphysicswriterhavebeenusedtoexcellenteffectinwritingthistextbook....Wholeheartedlyrecommended.”—M.Haque“Areadable,andrereadableinstantclassiconQFT....Atanintroductorylevel,thistypeofbook—withitspedagogical(andoftenveryfunny)narrative—ispriceless.[It]isfulloffantasticinsightsakintoreadingtheFeynmanlectures.IhavesinceusedQFTinaNutshellasareviewfor[my]year-longcoursecoveringallofPeskinandSchroder,andhavebeenpleasantlysurprisedathowZeeisabletopreemptivelyanswermanyoftheopenquestionsthateludedmeduringmycourse....IvalueQFTinaNutshellthesamewayIdotheFeynmanlectures....It’satexttoteachanunderstandingofphysics.”—FlipTanedo“Oneofthosebooksapersoninterestedintheoreticalphysicssimplymustown!Arealscientificmasterpiece.IboughtitatthetimeIwasaphysicssophomoreandthatwasthebestchoiceIcouldhavemade.Itwasthisbookthattriggeredmyinterestinquantumfieldtheoryandcrystallizedmydreamsofbecomingatheoreticalphysicist....Themaingoalofthebookistomakethereadergainrealintuitioninthefield.Amazing...amusing...realfun.Whatalsodistinguishesthisbookfromothersdealingwithasimilarsubjectisthatitiswrittenlikeatale....Ifeelenormouslyfortunatetohavecomeacrossthisbookatthebeginningofmyadventurewiththeoreticalphysics....DefinitelythebestquantumfieldtheorybookIhaveeverread.”—Anonymous“IhaveusedQuantumFieldTheoryinaNutshellastheprimarytext....Iamimmenselypleasedwiththebook,andrecommendithighly....Don’tletthe‘damnthetorpedoes,fullsteamahead’approachscareyouoff.Onceyougetusedtoseeingthephysicsquickly,Ithinkyouwillfindtheexperienceverysatisfyingintellectually.”—JimNapolitano“ThisisundoubtedlythebestbookIhaveeverreadaboutthesubject.Zeedoesafantasticjobofexplainingquantumfieldtheory,inawayIhaveneverseenbefore,andIhavereadmostoftheotherbooksonthistopic.Ifyouarelookingforquantumfieldtheoryexplanationsthatareclear,precise,concise,intuitive,andfuntoread—thisisthebookforyou.”—Anonymous“Oneofthemostartisticanddeepestbookseverwrittenonquantumfieldtheory.Amazing...extremelypleasant...alotofverydeepandilluminatingremarks....IrecommendthebookbyZeetoeverybodywhowantstogetaclearideawhatgoodphysicsisabout.”—SlavaMukhanov“Perfectforlearningfieldtheoryonyourown—byfartheclearestandeasiesttofollowbookI’vefoundonthesubject.”—IanZ.Lovejoy“Abeautifullywrittenintroductiontothemodernviewoffields...breezyandenchanting,leadingtoexceptionalclaritywithoutsacrificingdepth,breadth,orrigorofcontent....[It]passesmytestoftruegreatness:IwishithadbeenthefirstbookonthistopicthatIhadfound.”—JeffreyD.Scargle“Abreezeoffreshair...arealliterarygemwhichwillbeusefulforstudentswhomaketheirfirststepsinthisdifficultsubjectandanenjoyabletreatforexperts,whowillfindnewanddeepinsights.Indeed,theNutshellislikeabrightlightsourceshiningamongtallandheavytrees—themanymoreformalbooksthatexist—andhelpsseeingtheforestasawhole!...IhavebeenpracticingQFTduringthepasttwodecadesandwithallmyexperienceIwasthrilledwithenjoymentwhenIreadsomeofthesections.”—JoshuaFeinberg“Thistextnotonlyteachesup-to-datequantumfieldtheory,butalsotellsreadershowresearchisactuallydoneandshowsthemhowtothinkaboutphysics.[Itteachesthingsthat]peopleusuallysay‘cannotbelearnedfrombooks.’[Itis]inthesamestyleasFearfulSymmetryandEinstein’sUniverse.Allthreebooks...areclassics.”—YuShi“Ibelongtothe[groupof]enthusiasticlaymenhavingenoughcuriosityandinsistence...butlackingthemasteryofadvancedmathandphysics....Ireallycouldnotseetheforestforthetrees.ButatlonglastIgotthisbook!”—MakayAttila“MorefunthananyotherQFTbookIhaveread.ThecomparisonstoFeynman’swritingsmadebyseveralofthereviewersseemquiteapt....Hisenthusiasmisquiteinfectious....Idoubtthatanyotherbookwillsparkyourinterestlikethisonedoes.”—StephenWandzura“I’mhavingablastreadingthisbook.It’sbothdeepandentertaining;thisisararebreed,indeed.Iusuallypreferthemoreformalstyle(bigLandaufan),butIhavetosaythatwhenZeehasthetalenttopresentthingshisway,it’sadefiniteplus.”—PierreJouvelot“RequiredreadingforQFT:[it]heraldstheintroductionofabookonquantumfieldtheorythatyoucansitdownandread.Myprofessor’slecturesmademuchmoresenseasIfollowedalonginthisbook,becauseconceptswereactuallyEXPLAINED,notjustworkedout.”—AlexanderScott“Notyourfather’squantumfieldtheorytext:Iparticularlyappreciatethatthingsaremotivatedphysicallybeforetheirmathematicalarticulation....Mostespeciallythough,theauthor’s‘heuristic’descriptionsarethebestIhavereadanywhere.Fromthemalonetheessentialideasbecomecrystalclear.”—DanDillQuantumFieldTheoryinaNutshellThispageintentionallyleftblankQuantumFieldTheoryinaNutshellSECONDEDITIONA.ZeePRINCETONUNIVERSITYPRESS.PRINCETONANDOXFORDCopyright©2010byPrincetonUniversityPressPublishedbyPrincetonUniversityPress,41WilliamStreet,Princeton,NewJersey08540IntheUnitedKingdom:PrincetonUniversityPress,6OxfordStreet,Woodstock,OxfordshireOX201TWAllRightsReservedLibraryofCongressCataloging-in-PublicationDataZee,A.Quantumfieldtheoryinanutshell/A.Zee.—2nded.p.cm.Includesbibliographicalreferencesandindex.ISBN978-0-691-14034-6(hardcover:alk.paper)1.Quantumfieldtheory.I.Title.QC174.45.Z442010530.14′3—dc222009015469BritishLibraryCataloging-in-PublicationDataisavailableThisbookhasbeencomposedinScalaLFwithZzTEXbyPrincetonEditorialAssociates,Inc.,Scottsdale,ArizonaPrintedonacid-freepaper.press.princeton.eduPrintedintheUnitedStatesofAmerica10987654321Tomyparents,whovaluededucationaboveallelseThispageintentionallyleftblankContentsPrefacetotheFirstEditionxvPrefacetotheSecondEditionxixConvention,Notation,andUnitsxxvIPartI:MotivationandFoundationI.1WhoNeedsIt?3I.2PathIntegralFormulationofQuantumPhysics7I.3FromMattresstoField17I.4FromFieldtoParticletoForce26I.5CoulombandNewton:RepulsionandAttraction32I.6InverseSquareLawandtheFloating3-Brane40I.7FeynmanDiagrams43I.8QuantizingCanonically61I.9DisturbingtheVacuum70I.10Symmetry76I.11FieldTheoryinCurvedSpacetime81I.12FieldTheoryRedux88IIPartII:DiracandtheSpinorII.1TheDiracEquation93II.2QuantizingtheDiracField107II.3LorentzGroupandWeylSpinors114II.4Spin-StatisticsConnection120xii|ContentsII.5VacuumEnergy,GrassmannIntegrals,andFeynmanDiagramsforFermions123II.6ElectronScatteringandGaugeInvariance132II.7DiagrammaticProofofGaugeInvariance144II.8Photon-ElectronScatteringandCrossing152IIIPartIII:RenormalizationandGaugeInvarianceIII.1CuttingOffOurIgnorance161III.2RenormalizableversusNonrenormalizable169III.3CountertermsandPhysicalPerturbationTheory173III.4GaugeInvariance:APhotonCanFindNoRest182III.5FieldTheorywithoutRelativity190III.6TheMagneticMomentoftheElectron194III.7PolarizingtheVacuumandRenormalizingtheCharge200III.8BecomingImaginaryandConservingProbability207IVPartIV:SymmetryandSymmetryBreakingIV.1SymmetryBreaking223IV.2ThePionasaNambu-GoldstoneBoson231IV.3EffectivePotential237IV.4MagneticMonopole245IV.5NonabelianGaugeTheory253IV.6TheAnderson-HiggsMechanism263IV.7ChiralAnomaly270VPartV:FieldTheoryandCollectivePhenomenaV.1Superfluids283V.2Euclid,Boltzmann,Hawking,andFieldTheoryatFiniteTemperature287V.3Landau-GinzburgTheoryofCriticalPhenomena292V.4Superconductivity295V.5PeierlsInstability298V.6Solitons302V.7Vortices,Monopoles,andInstantons306VIPartVI:FieldTheoryandCondensedMatterVI.1FractionalStatistics,Chern-SimonsTerm,andTopologicalFieldTheory315VI.2QuantumHallFluids322Contents|xiiiVI.3Duality331VI.4TheσModelsasEffectiveFieldTheories340VI.5FerromagnetsandAntiferromagnets344VI.6SurfaceGrowthandFieldTheory347VI.7Disorder:ReplicasandGrassmannianSymmetry350VI.8RenormalizationGroupFlowasaNaturalConceptinHighEnergyandCondensedMatterPhysics356VIIPartVII:GrandUnificationVII.1QuantizingYang-MillsTheoryandLatticeGaugeTheory371VII.2ElectroweakUnification379VII.3QuantumChromodynamics385VII.4LargeNExpansion394VII.5GrandUnification407VII.6ProtonsAreNotForever413VII.7SO(10)Unification421VIIIPartVIII:GravityandBeyondVIII.1GravityasaFieldTheoryandtheKaluza-KleinPicture433VIII.2TheCosmologicalConstantProblemandtheCosmicCoincidenceProblems448VIII.3EffectiveFieldTheoryApproachtoUnderstandingNature452VIII.4Supersymmetry:AVeryBriefIntroduction461VIII.5AGlimpseofStringTheoryasa2-DimensionalFieldTheory469ClosingWords473NPartNN.1GravitationalWavesandEffectiveFieldTheory479N.2GluonScatteringinPureYang-MillsTheory483N.3SubterraneanConnectionsinGaugeTheories497N.4IsEinsteinGravitySecretlytheSquareofYang-MillsTheory?513MoreClosingWords521AppendixA:GaussianIntegrationandtheCentralIdentityofQuantumFieldTheory523AppendixB:ABriefReviewofGroupTheory525xiv|ContentsAppendixC:FeynmanRules534AppendixD:VariousIdentitiesandFeynmanIntegrals538AppendixE:DottedandUndottedIndicesandtheMajoranaSpinor541SolutionstoSelectedExercises545FurtherReading559Index563PrefacetotheFirstEditionAsastudent,Iwasrearingatthebit,afteracourseonquantummechanics,tolearnquantumfieldtheory,butthebooksonthesubjectallseemedsoformidable.Fortunately,IcameacrossalittlebookbyMandlonfieldtheory,whichgavemeatasteofthesubjectenablingmetogoonandtacklethemoresubstantivetexts.IhavesincelearnedthatotherphysicistsofmygenerationhadsimilargoodexperienceswithMandl.Inthelastthreedecadesorso,quantumfieldtheoryhasveritablyexplodedandMandlwouldbehopelesslyoutofdatetorecommendtoastudentnow.ThusIthoughtofwritingabookontheessentialsofmodernquantumfieldtheoryaddressedtothebrightandeagerstudentwhohasjustcompletedacourseonquantummechanicsandwhoisimpatienttostarttacklingquantumfieldtheory.Ienvisagedarelativelythinbook,thinatleastincomparisonwiththemanyweightytomesonthesubject.Ienvisagedthestyletobebreezyandcolloquial,andthechoiceoftopicstobeidiosyncratic,certainlynotencyclopedic.Ienvisagedhavingmanyshortchapters,keepingeachchapter“bite-sized.”Thechallengeinwritingthisbookistokeepitthinandaccessiblewhileatthesametimeintroducingasmanymoderntopicsaspossible.Atoughbalancingact!Intheend,IhadtobeunrepentantlyidiosyncraticinwhatIchosetocover.Notetotheprospectivebookreviewer:Youcanalwayscriticizethebookforleavingoutyourfavoritetopics.Idonotapologizeinanyway,shape,orform.Mymottointhisregard(andinlifeaswell),takenfromtheRickyNelsonsong“GardenParty,”is“Youcan’tpleaseeveryonesoyougottapleaseyourself.”Thisbookdiffersfromotherquantumfieldtheorybooksthathavecomeoutinrecentyearsinseveralrespects.Iwanttogetacrosstheimportantpointthattheusefulnessofquantumfieldtheoryisfarfromlimitedtohighenergyphysics,amisleadingimpressionmygenerationoftheoreticalphysicistswereinculcatedwithandwhichamazinglyenoughsomerecenttextbooksonxvi|PrefacetotheFirstEditionquantumfieldtheory(allwrittenbyhighenergyphysicists)continuetofoster.Forinstance,thestudyofdrivensurfacegrowthprovidesaparticularlyclear,transparent,andphysicalexampleoftheimportanceoftherenormalizationgroupinquantumfieldtheory.Insteadofbeingentangledinallsortsofconceptualirrelevanciessuchasdivergences,wehavetheobviouslyphysicalnotionofchangingtherulerusedtomeasurethefluctuatingsurface.OtherexamplesincluderandommatrixtheoryandChern-SimonsgaugetheoryinquantumHallfluids.Ihopethatcondensedmattertheorystudentswillfindthisbookhelpfulingettingafirsttasteofquantumfieldtheory.Thebookisdividedintoeightparts,1withtwodevotedmoreorlessexclusivelytocondensedmatterphysics.Itrytogivethereaderatleastabriefglimpseintocontemporarydevelopments,forexample,justenoughofatasteofstringtheorytowhettheappetite.Thisbookisperhapsalsoexceptionalinincorporatinggravityfromthebeginning.Sometopicsaretreatedquitedifferentlythanintraditionaltexts.IintroducetheFaddeev-PopovmethodtoquantizeelectromagnetismandthelanguageofdifferentialformstodevelopYang-Millstheory,forexample.Theemphasisisresoundinglyontheconceptualratherthanthecomputational.TheonlycalculationIcarryoutinallitsgorydetailsisthatofthemagneticmomentoftheelectron.Throughout,specificexamplesratherthanheavyabstractformalismwillbefavored.Insteadofdealingwiththemostgeneralcase,Ialwaysoptforthesimplest.Ihadtostruggleconstantlybetweenclarityandwordiness.Intryingtoanticipateandtominimizewhatwouldconfusethereader,IoftenfindthatIhavetobelaborcertainpointsmorethanwhatIwouldlike.Itriedtoavoidthedreadedphrase“Itcanbeshownthat...”asmuchaspossible.Otherwise,Icouldhavewrittenamuchthinnerbookthanthis!Thereareindeedthinnerbooksonquantumfieldtheory:Ilookedatacoupleanddiscoveredthattheyhardlyexplainanything.ImustconfessthatIhaveanalmostinsatiabledesiretoexplain.Asthemanuscriptgrew,thelistoftopicsthatIreluctantlyhadtodropalsokeptgrowing.Somanybeautifulresults,butsolittlespace!Italmostmakesmeilltothinkaboutallthestuff(bosonization,instanton,conformalfieldtheory,etc.,etc.)Ihadtoleaveout.Asonecolleagueremarked,thenutshellisturningintoacoconutshell!ShelleyGlashowoncedescribedthegenesisofphysicaltheories:“Tapestriesaremadebymanyartisansworkingtogether.Thecontributionsofseparateworkerscannotbediscernedinthecompletedwork,andthelooseandfalsethreadshavebeencoveredover.”IregretthatotherthangivingafewtidbitshereandthereIcouldnotgointothefascinatinghistoryofquantumfieldtheory,withallitsdefeatsandtriumphs.OnthoseoccasionswhenIrefertooriginalpapersIsufferfromthatdisconcertingquirkofhumanpsychologyoftendingtofavormyownmorethandecorummighthaveallowed.Icertainlydidnotattemptatruebibliography.1MurrayGell-MannusedtotalkabouttheeightfoldwaytowisdomandsalvationinBuddhism(M.Gell-MannandY.Ne’eman,TheEightfoldWay).ReadersfamiliarwithcontemporaryChineseliteraturewouldknowthatthecelestialdragonhaseightparts.PrefacetotheFirstEdition|xviiThegenesisofthisbookgoesbacktothequantumfieldtheorycourseItaughtasabeginningassistantprofessoratPrincetonUniversity.IhadtheenormousgoodfortuneofhavingEdWittenasmyteachingassistantandgrader.EdproducedlucidlywrittensolutionstothehomeworkproblemsIassigned,totheextentthatthenextyearIwenttothechairmantoask“WhatiswrongwiththeTAIhavethisyear?Heisnothalfasgoodastheguylastyear!”Somecolleaguesaskedmetowriteupmynotesforamuchneededtext(thoseweretheexcitingtimeswhengaugetheories,asymptoticfreedom,andscoresoftopicsnottobefoundinanytextsallhadtobelearnedsomehow)butawiserseniorcolleagueconvincedmethatitmightspelldisasterformyresearchcareer.Decadeslater,thetimehascome.IparticularlythankMurphGoldbergerforurgingmetoturnwhatexpositorytalentsIhavefromwritingpopularbookstowritingtextbooks.ItisalsoapleasuretosayawordinmemoryofthelateSamTreiman,teacher,colleague,andcollaborator,whoasamemberoftheeditorialboardofPrincetonUniversityPresspersuadedmetocommittothisproject.Iregretthatmyslowpaceinfinishingthebookdeprivedhimofseeingthefinishedproduct.OvertheyearsIhaverefinedmyknowledgeofquantumfieldtheoryindiscussionswithnumerouscolleaguesandcollaborators.Asastudent,Iattendedcoursesonquan-tumfieldtheoryofferedbyArthurWightman,JulianSchwinger,andSidneyColeman.Iwasfortunatethatthesethreeeminentphysicistseachhashisowndistinctivestyleandapproach.Thebookhasbeentested“inthefield”incoursesItaught.IuseditinmyfieldtheorycourseattheUniversityofCaliforniaatSantaBarbara,andIamgratefultosomeofthestudents,inparticularTedErler,AndrewFrey,SeanRoy,andDeanTownsley,forcomments.Ibenefittedfromthecommentsofvariousdistinguishedphysicistswhoreadallorpartsofthemanuscript,includingSteveBarr,DougEardley,MattFisher,MurphGoldberger,VictorGurarie,SteveHsu,Bei-lokHu,CliffordJohnson,MehranKardar,IanLow,JoePolchinski,ArkadyVainshtein,FrankWilczek,EdWitten,andespeciallyJoshuaFeinberg.Joshuaalsodidmanyoftheexercises.Talkingaboutexercises:Youdidn’tgetthisfarinphysicswithoutrealizingtheabsoluteimportanceofdoingexercisesinlearningasubject.Itisespeciallyimportantthatyoudomostoftheexercisesinthisbook,becausetocompensateforitsrelativeslimnessIhavetodevelopintheexercisesanumberofimportantpointssomeofwhichIneedforlaterchapters.Solutionstosomeselectedproblemsaregiven.Iwillmaintainawebpagehttp://theory.kitp.ucsb.edu/~zee/nuts.htmllistingalltheerrors,typographicalandotherwise,andpointsofconfusionthatwillundoubtedlycometomyattention.Ithankmyeditors,TrevorLipscombe,SarahGreen,andthestaffofPrincetonEditorialAssociates(particularlyCydWestmorelandandEvelynGrossberg)fortheiradviceandforseeingthisprojectthrough.Finally,IthankPeterZeeforsuggestingthecoverpainting.ThispageintentionallyleftblankPrefacetotheSecondEditionWhatonefoolcouldunderstand,anothercan.—R.P.Feynman1AppreciatingtheappreciatorsIthasbeennearlysixyearssincethisbookwaspublishedonMarch10,2003.Sinceauthorsoftenthinkofbooksastheirchildren,Imaylikenthefloodofappreciationfromreaders,students,andphysiciststothegloriousreportcardsabrightchildbringshomefromschool.Knowingthattherearepeoplewhoappreciatethecareandclaritycraftedintothepedagogyisamostgratifyingfeeling.Inworkingonthisnewedition,merelylookingatthetitlesofthecustomerreviewsonAmazon.comwouldlightenmytaskandquickenmypace:“Funny,chatty,physical.QFTeducationtransformed!,”“Areadable,andre-readableinstantclassiconQFT,”“AmustreadbookifyouwanttounderstandessentialsinQFT,”“Oneofthemostartisticanddeepestbookseverwrittenonquantumfieldtheory,”“Perfectforlearningfieldtheoryonyourown,”“Bothdeepandentertaining,”“Oneofthosebooksapersoninterestedintheoreticalphysicssimplymustown,”andsoon.InaPhysicsTod