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Whatisthecorrectframework forQuantumFieldTheories? YujiTachikawa 1/29 • Myresearcharea theoreticalhighenergyphysics/stringtheory/mathematics • Myobsessionthesedays whatisthe correctframework for quantumfieldtheories? 3/29 Disclaimer: I don’tyethaveananswer! ButI’lltrytoexplainwhyit’sanimportantand/orinterestingquestion. 4/29 Disclaimer: I don’tyethaveananswer! ButI’lltrytoexplainwhyit’sanimportantand/orinterestingquestion. I don’thaveanycutefiguresintheslideseither. AndI don’thaveanyexcuseforit. I’msorry. 4/29 Whatisa quantum field theory? • Itdescribes quantum propertiesof fields, • whereafieldis anythingthatisextendedalongthespacetime, e.g. electromagneticfield, electronfield, … 5/29 A prototypical quantumfieldtheory (QFT) is the QuantumElectrodynamics • whichdescribesthequantizedelectromagneticfield interactingwithelectronsetc., and • wasestablishedaround1950. Sincethentherehasbeenasteadyprogress. 6/29 Bynowweknowthat ourworld, atthemostmicroscopiclevelthatis experimentallyaccessible, isdescribedbya quantumfieldtheory called the StandardModel, establishedtheoreticallyinthe1970s. Itsfinalpiece, theHiggsboson, wasconfirmedexperimentallyin2012, andbothhighenergytheoristsandexperimentalistsarelookingfor physicsbeyondtheStandardModel. 7/29 Quantumfieldtheories alsoappearubiquitouslyin condensedmatterphysics. Forexample, physicsatthesecond-orderphasetransitionisoften describedby conformalfieldtheories, whichformacertainsubclassofquantumfieldtheories. Thisubiquityisnotsurprising, since quantumfields arejust the quantumversionofanythingthatareextendedalongthespace. 8/29 Duetoitsubiquity, manypeoplehaveworkedonquantumfieldtheories overitshistoryofmorethanhalfacentury. Theoreticalpredictionsagreewellwithexperimentalresults. One extremeexampleisthe anomalousmagneticmomentofelectron: ae = 1 159 652 181.7... × 10−12 Inthiscasetheagreementisto 12decimalplaces. AgreementsinthecaseoftheStandardModelvs. theLargeHadron Colliderexperimentsarequiteimpressivetoo. 9/29 Evenwiththisimpressiveagreement, I say wedon’tyetknowwhat’sthe correctframeworktostudyquantumfieldtheories. Thisisincontrasttothesituationfor quantummechanicsorgeneral relativity, forwhich I say weknowthecorrectframeworks. WhydoI sayso? 10/29 Inthecaseofquantummechanicsorgeneralrelativity, I canexplainat leastoneframeworktomathematiciansinafewsentences. Quantummechanics It’sastudyofunitaryoperatorsactingonaHilbertspace. Generalrelativity It’sastudyofadifferentialequationsatisfiedby theRiemanntensorofaLorentzianmetriconamanifold. Notbad. 11/29 Butthen, whatisaquantumfieldtheory? QuantumFieldTheory ??? Asaphysicist, I thinkI know, butI don’thaveawaytoexplainittomathematicians. I believeit’snotjustaproblemaboutme. 12/29 Manypeopletriedtoformulatemathematicallysomesmallpartofwhat theyknewaboutquantumfieldtheories: topological QFT vertex operator algebras later 1980s late 1980s algebraic QFT 1960s axiomatic QFT 1950s QFT as known to physicists 13/29 However, innoneoftheframeworkswecanexpresse.g. thecomputationsthatledtotheimpressiveprediction ae = 1 159 652 181.7... × 10−12 fortheanomalousmagneticmomentoftheelectron. • Theframework axiomaticQFT is toobroadandgeneric tocarryout thisparticularcomputation • Theframeworks vertexoperatoralgebras and topologicalQFT are toonarrow andexcludetherealQuantumElectrodynamics 14/29 Thatsaid, themethodtocomputethisvalue ae = 1 159 652 181.7... × 10−12 usingquantumelectrodynamicsis explainedin quantumfieldtheorytextbooksforphysicists. Doesn’titsufficetomathematicallyformulatewhatarewrittenthere? I don’tthinkso. 15/29 WhatareexplainedinQFT textbooksforphysicistsare howtocompute things inthe quantumelectrodynamics oritsextension, the Standard Model. Buttheyarebothrather mundane fromamoremodernpointofviewof quantumfieldtheories. Afterall, althoughexperimentallyverifiedonlya fewyearsago, theStandardModelwastheoreticallyestablishedinthe early70s. Inasense, theproblemisthatquantumfieldtheoriesasknownto physicists alwaysmoveon, andthatthepreviousattemptstoformulate themwerealwaystooearly. 16/29 And themodernviewitselfisinflux inthelastfewyears. I’dliketoillustratethispointintherestofmytalk, usingmyownrecentresearchasanexample. 17/29 A bigtopicinourareais supersymmetric quantumfieldtheories. Supersymmetryallowsustorelate the bosons (photons, etc.) andthe fermions (electrons, etc.): a boson supersymmetry a fermion Ifitisphysicallyrealized, itcansolvevariousunsettlingquestions inthecurrentStandardModel. Thereare(theoretical)condensed-matterrealizationstoo. 18/29 ButI’mmoreinterestedinitstheoreticalstructure. Thentherecanbequantumfieldtheoriesthatcanhave multiplesupersymmetries: supersymmetry1 a fermion a boson supersymmetry2 supersymmetry2 another boson another fermion supersymmetry1 Iftherearetwo, itiscalledan N =2 supersymmetry, forhistoricalreasons. 19/29 Howmanysupersymmetries canaquantumfieldtheoryhave? That’satopiccoveredofteninthe firstchapter ofatextbookon supersymmetry, andthestandardanswer hasbeen asfollows: N = 1, N = 2, N =4 andthat’sit. I don’thavetimetoexplainwhytherecan’tbemorethanfour. Youmightask, whynot N = 3 ? 20/29 General Quantum Field Theories quite a lot N=1 still a lot N=2 a fair number N=3 nothing here N=4 a few Well, ifaquantumfieldtheoryhas N =3, youcaneasilycheckithasinfact N =4, orsopeoplesaid. 21/29 I myselfexplainedthusinalecturenote publishedtwoyearsagofromSpringer: Lecture Notes in Physics 890 Yuji Tachikawa N=2 Supersymmetric Dynamics for Pedestrians 22/29 Andstill, thispaperappearedon December20, 2015 ! where genuinely N =3 theorieswerefound, usingstringtheory. 23/29 General Quantum Field Theories quite a lot N=1 still a lot N=2 a fair number N=3 a few N=4 a few So, whywereallthetextbooks(includingmine)wrong? 24/29 Inthecaseofanobjectattachedtoamechanicalspring: Equationofmotion: m d2 dt2 x = −kx 25/29 Inthecaseofanobjectattachedtoamechanicalspring: Hamiltonian: Ht = ↓ Equationofmotion: m d2 dt2 1 2m p2 + k 2 x2 x = −kx 25/29 Inthecaseofanobjectattachedtoamechanicalspring: Lagrangian: S= ↕ Hamiltonian: Ht = ↓ Equationofmotion: ! m d2 dt2 dt( 1 2m m dx 2 k ( ) − x2 ) 2 dt 2 p2 + k 2 x2 x = −kx Wearethentaughtthat HamiltnianandLagrangianframeworksare equivalent. 25/29 Intextbooksonquantumfieldtheories, it’softensaid: Equations ofmotion: complicated. 26/29 Intextbooksonquantumfieldtheories, it’softensaid: Hamiltonians: Equations ofmotion: Ht , Hx , Hy , Hz , stillcomplicated. ↓ complicated. 26/29 Intextbooksonquantumfieldtheories, it’softensaid: Lagrangian: Hamiltonians: Equations ofmotion: S= ↓ ↓ ! (somethingsimple)d4 x Ht , Hx , Hy , Hz , stillcomplicated. complicated. RichardFeynmanevengotaNobelPrizeforcomingupwitha Lagrangianframeworkforquantumfieldtheories. 26/29 Butinthelastseveralyears, welearnedthat (Lagrangian: notavailable) Hamiltonians: Equations ofmotion: Ht , Hx , Hy , Hz , complicated. ↓ complicated. formanyquantumfieldtheories. 27/29 So, whydidallthetextbooks(includingmine)mistakenlystatethatany N =3 quantumfieldtheoryautomaticallyhas N =4 supersymmetry? Well, ifyouassumetheexistenceofa Lagrangian, • youwritedownallpossible N =2 supersymmetric Lagrangians • lookforanytheorythathasmorethan N =2 inthelist • youonlyfindthosewith N =4, neverwith N =3. Buttherecanbe N =3 theories withoutLagrangians, andindeedthereare. 28/29 Letmesummarize: Basically, allthetextbooksonquantumfieldtheoriesoutthere usean oldframeworkthatis simplytoonarrow, inthat itassumestheexistence ofaLagrangian. Thisisaseriousissue, becausewhenyoutrytocomeupe.g. witha theorybeyondtheStandardModel, peoplehabituallystartbywritinga Lagrangian…but thatmightbeputtingtoostronganassumption. Weneedtodosomething. 29/29