What is the correct framework for Quantum Field Theories? · Quantum Field Theory??? As a...
Transcript of What is the correct framework for Quantum Field Theories? · Quantum Field Theory??? As a...
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Whatisthecorrectframework
forQuantumFieldTheories?
YujiTachikawa
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• Myresearcharea
theoreticalhighenergyphysics/stringtheory/mathematics
• Myobsessionthesedays
whatisthe correctframework for quantumfieldtheories?
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Disclaimer:
I don’tyethaveananswer!
ButI’lltrytoexplainwhyit’sanimportantand/orinterestingquestion.
I don’thaveanycutefiguresintheslideseither.
AndI don’thaveanyexcuseforit. I’msorry.
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Disclaimer:
I don’tyethaveananswer!
ButI’lltrytoexplainwhyit’sanimportantand/orinterestingquestion.
I don’thaveanycutefiguresintheslideseither.
AndI don’thaveanyexcuseforit. I’msorry.
4/29
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Whatisa quantum field theory?
• Itdescribes quantum propertiesof fields,
• whereafieldis anythingthatisextendedalongthespacetime, e.g.
electromagneticfield, electronfield, …
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A prototypical quantumfieldtheory (QFT) is
the QuantumElectrodynamics
• whichdescribesthequantizedelectromagneticfield
interactingwithelectronsetc., and
• wasestablishedaround1950.
Sincethentherehasbeenasteadyprogress.
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Bynowweknowthat ourworld, atthemostmicroscopiclevelthatis
experimentallyaccessible, isdescribedbya quantumfieldtheory called
the StandardModel, establishedtheoreticallyinthe1970s.
Itsfinalpiece, theHiggsboson, wasconfirmedexperimentallyin2012,
andbothhighenergytheoristsandexperimentalistsarelookingfor
physicsbeyondtheStandardModel.
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Quantumfieldtheories alsoappearubiquitouslyin
condensedmatterphysics.
Forexample, physicsatthesecond-orderphasetransitionisoften
describedby conformalfieldtheories,
whichformacertainsubclassofquantumfieldtheories.
Thisubiquityisnotsurprising, since quantumfields arejust
the quantumversionofanythingthatareextendedalongthespace.
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Duetoitsubiquity, manypeoplehaveworkedonquantumfieldtheories
overitshistoryofmorethanhalfacentury.
Theoreticalpredictionsagreewellwithexperimentalresults. One
extremeexampleisthe anomalousmagneticmomentofelectron:
ae = 1159 652 181.7... × 10−12
Inthiscasetheagreementisto 12decimalplaces.
AgreementsinthecaseoftheStandardModelvs. theLargeHadron
Colliderexperimentsarequiteimpressivetoo.
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Evenwiththisimpressiveagreement, I say wedon’tyetknowwhat’sthe
correctframeworktostudyquantumfieldtheories.
Thisisincontrasttothesituationfor quantummechanicsorgeneral
relativity, forwhich I say weknowthecorrectframeworks.
WhydoI sayso?
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Inthecaseofquantummechanicsorgeneralrelativity, I canexplainat
leastoneframeworktomathematiciansinafewsentences.
QuantummechanicsIt’sastudyofunitaryoperatorsactingonaHilbertspace.
Generalrelativity
It’sastudyofadifferentialequationsatisfiedby
theRiemanntensorofaLorentzianmetriconamanifold.
Notbad.
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Butthen, whatisaquantumfieldtheory?
QuantumFieldTheory???
Asaphysicist, I thinkI know,
butI don’thaveawaytoexplainittomathematicians.
I believeit’snotjustaproblemaboutme.
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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
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Thatsaid, themethodtocomputethisvalue
ae = 1159 652 181.7... × 10−12
usingquantumelectrodynamicsis explainedin
quantumfieldtheorytextbooksforphysicists.
Doesn’titsufficetomathematicallyformulatewhatarewrittenthere?
I don’tthinkso.
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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.
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And themodernviewitselfisinflux inthelastfewyears.
I’dliketoillustratethispointintherestofmytalk,
usingmyownrecentresearchasanexample.
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A bigtopicinourareais supersymmetric quantumfieldtheories.
Supersymmetryallowsustorelate
the bosons (photons, etc.) andthe fermions (electrons, etc.):
a boson a fermionsupersymmetry
Ifitisphysicallyrealized, itcansolvevariousunsettlingquestions
inthecurrentStandardModel.
Thereare(theoretical)condensed-matterrealizationstoo.
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ButI’mmoreinterestedinitstheoreticalstructure.
Thentherecanbequantumfieldtheoriesthatcanhave
multiplesupersymmetries:
a boson another boson
a fermion
another fermion
supersymmetry1
supersymmetry1supersymmetry2
supersymmetry2
Iftherearetwo, itiscalledan N=2 supersymmetry,
forhistoricalreasons.
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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 ?
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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.
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I myselfexplainedthusinalecturenote
publishedtwoyearsagofromSpringer:
Lecture Notes in Physics 890
Yuji Tachikawa
N=2 Supersymmetric Dynamics for Pedestrians
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Andstill, thispaperappearedon December20, 2015 !
where genuinely N=3 theorieswerefound, usingstringtheory.
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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?
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Inthecaseofanobjectattachedtoamechanicalspring:
Lagrangian: S =
!dt(
m
2(dx
dt)2 −
k
2x2)
↕
Hamiltonian: Ht =1
2mp2 +
k
2x2
↓
Equationofmotion: md2
dt2x = −kx
Wearethentaughtthat HamiltnianandLagrangianframeworksare
equivalent.
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Inthecaseofanobjectattachedtoamechanicalspring:
Lagrangian: S =
!dt(
m
2(dx
dt)2 −
k
2x2)
↕
Hamiltonian: Ht =1
2mp2 +
k
2x2
↓
Equationofmotion: md2
dt2x = −kx
Wearethentaughtthat HamiltnianandLagrangianframeworksare
equivalent.
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Inthecaseofanobjectattachedtoamechanicalspring:
Lagrangian: S =
!dt(
m
2(dx
dt)2 −
k
2x2)
↕
Hamiltonian: Ht =1
2mp2 +
k
2x2
↓
Equationofmotion: md2
dt2x = −kx
Wearethentaughtthat HamiltnianandLagrangianframeworksare
equivalent.
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Intextbooksonquantumfieldtheories, it’softensaid:
Lagrangian: S =
!(somethingsimple)d4x
↓Hamiltonians: Ht, Hx, Hy, Hz, stillcomplicated.
↓
Equations ofmotion: complicated.
RichardFeynmanevengotaNobelPrizeforcomingupwitha
Lagrangianframeworkforquantumfieldtheories.
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Intextbooksonquantumfieldtheories, it’softensaid:
Lagrangian: S =
!(somethingsimple)d4x
↓
Hamiltonians: Ht, Hx, Hy, Hz, stillcomplicated.
↓Equations ofmotion: complicated.
RichardFeynmanevengotaNobelPrizeforcomingupwitha
Lagrangianframeworkforquantumfieldtheories.
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Intextbooksonquantumfieldtheories, it’softensaid:
Lagrangian: S =
!(somethingsimple)d4x
↓Hamiltonians: Ht, Hx, Hy, Hz, stillcomplicated.
↓Equations ofmotion: complicated.
RichardFeynmanevengotaNobelPrizeforcomingupwitha
Lagrangianframeworkforquantumfieldtheories.
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Butinthelastseveralyears, welearnedthat
(Lagrangian: notavailable)
Hamiltonians: Ht, Hx, Hy, Hz, complicated.
↓Equations ofmotion: complicated.
formanyquantumfieldtheories.
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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.
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Letmesummarize:
Basically, allthetextbooksonquantumfieldtheoriesoutthere usean
oldframeworkthatis simplytoonarrow, inthat itassumestheexistence
ofaLagrangian.
Thisisaseriousissue, becausewhenyoutrytocomeupe.g. witha
theorybeyondtheStandardModel, peoplehabituallystartbywritinga
Lagrangian…but thatmightbeputtingtoostronganassumption.
Weneedtodosomething.
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