Whatisthecorrectframework

forQuantumFieldTheories?

YujiTachikawa

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• Myresearcharea

theoreticalhighenergyphysics/stringtheory/mathematics

• Myobsessionthesedays

whatisthe correctframework for quantumfieldtheories?

3/29

Disclaimer:

I don’tyethaveananswer!

ButI’lltrytoexplainwhyit’sanimportantand/orinterestingquestion.

I don’thaveanycutefiguresintheslideseither.

AndI don’thaveanyexcuseforit. I’msorry.

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 = 1159 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.

QuantummechanicsIt’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:

QFT as known to physicists

axiomatic QFT

algebraic QFT

vertex operatoralgebras

topological QFT

1950s

1960s

late 1980s

later 1980s

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However, innoneoftheframeworkswecanexpresse.g.

thecomputationsthatledtotheimpressiveprediction

ae = 1159 652 181.7... × 10−12

fortheanomalousmagneticmomentoftheelectron.

• Theframework axiomaticQFT is toobroadandgeneric tocarryout

thisparticularcomputation

• Theframeworks vertexoperatoralgebras and topologicalQFT are

toonarrow andexcludetherealQuantumElectrodynamics

14/29

Thatsaid, themethodtocomputethisvalue

ae = 1159 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 a fermionsupersymmetry

Ifitisphysicallyrealized, itcansolvevariousunsettlingquestions

inthecurrentStandardModel.

Thereare(theoretical)condensed-matterrealizationstoo.

18/29

ButI’mmoreinterestedinitstheoreticalstructure.

Thentherecanbequantumfieldtheoriesthatcanhave

multiplesupersymmetries:

a boson another boson

a fermion

another fermion

supersymmetry1

supersymmetry1supersymmetry2

supersymmetry2

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

N=1

N=2

N=3

N=4

nothing here

quite a lot

still a lot

a fair number

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

N=1

N=2

N=3

N=4

a few

quite a lot

still a lot

a fair number

a few

So, whywereallthetextbooks(includingmine)wrong?

24/29

Inthecaseofanobjectattachedtoamechanicalspring:

Lagrangian: S =

!dt(

m

2(dx

dt)2 −

k

2x2)

↕

Hamiltonian: Ht =1

2mp2 +

k

2x2

↓

Equationofmotion: md2

dt2x = −kx

Wearethentaughtthat HamiltnianandLagrangianframeworksare

equivalent.

25/29

Inthecaseofanobjectattachedtoamechanicalspring:

Lagrangian: S =

!dt(

m

2(dx

dt)2 −

k

2x2)

↕

Hamiltonian: Ht =1

2mp2 +

k

2x2

↓

Equationofmotion: md2

dt2x = −kx

Wearethentaughtthat HamiltnianandLagrangianframeworksare

equivalent.

25/29

Inthecaseofanobjectattachedtoamechanicalspring:

Lagrangian: S =

!dt(

m

2(dx

dt)2 −

k

2x2)

↕

Hamiltonian: Ht =1

2mp2 +

k

2x2

↓

Equationofmotion: md2

dt2x = −kx

Wearethentaughtthat HamiltnianandLagrangianframeworksare

equivalent.

25/29

Intextbooksonquantumfieldtheories, it’softensaid:

Lagrangian: S =

!(somethingsimple)d4x

↓Hamiltonians: Ht, Hx, Hy, Hz, stillcomplicated.

↓

Equations ofmotion: complicated.

RichardFeynmanevengotaNobelPrizeforcomingupwitha

Lagrangianframeworkforquantumfieldtheories.

26/29

Intextbooksonquantumfieldtheories, it’softensaid:

Lagrangian: S =

!(somethingsimple)d4x

↓

Hamiltonians: Ht, Hx, Hy, Hz, stillcomplicated.

↓Equations ofmotion: complicated.

RichardFeynmanevengotaNobelPrizeforcomingupwitha

Lagrangianframeworkforquantumfieldtheories.

26/29

Intextbooksonquantumfieldtheories, it’softensaid:

Lagrangian: S =

!(somethingsimple)d4x

↓Hamiltonians: Ht, Hx, Hy, Hz, stillcomplicated.

↓Equations ofmotion: complicated.

RichardFeynmanevengotaNobelPrizeforcomingupwitha

Lagrangianframeworkforquantumfieldtheories.

26/29

Butinthelastseveralyears, welearnedthat

(Lagrangian: notavailable)

Hamiltonians: Ht, Hx, Hy, Hz, complicated.

↓Equations ofmotion: 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.

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