Download What is the correct framework for Quantum Field Theories?

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