mul$ple reflec$ons, 23 may, 2016 · life at harvard in 1970’s • robert led an unconven$onal...
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Mul$pleReflec$ons,23May,2016
ThreeStagesofReflec$on
• Somereflec$onsonRobert• Somereflec$onsonreflec$onposi$vity• Somereflec$onsaAerRobert
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Reflec$onsonRobert
• IfirstmetRobert48yearsagoinMay1968– IwasguestprofessoratETH,courtesyJostandHepp– Robertwasworkingonhisthesis,completed1969– KonradOsterwalderwasmyassistant
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Robert Konrad
SeminarfürTheore$schePhysik
Hochstrasse60
Zürich
Reflec$onsonRobert• RobertcametoHarvardasmyfirstpost-doctoralfellowinthefallof1970(easytoarrange)• Wespentthespring1971togetherinPrinceton.• FirstprojectwasinfinitevolumelimitofYukawa2.• Konradjoinedusinfall1971atHarvardaspost-d.
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LifeatHarvardin1970• RobertlivedonHuronAvenuein“westCambridge.”• Hebecamegoodfriendswiththepar$clephysicists:ShellyGlashow,SidneyColeman,andJohnIliopolous($meofGIMmechanism).
• Manyunmarriedfaculty:Coleman,Glashow,AJ.Facultyandpostdoctoralfellowssocializedalot,occasionallyalsoCallan,Gross,Mar$n,Glauber,Schwinger.
• RegularSundaylunchatthePekingonMys$c.
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LifeatHarvardin1970’s• Robertledanunconven$onallife.• HisfriendChrista,whomhemetonaskiingtrip,wasEastGerman.SoRobertoAenwenttovisit.
• TheStasiaeemptedtorecruitRobertasaspy,interroga$nghimandthreateningherfamily.
• RobertwantedtogetChristatotheWestlegally.Igotintothemiddleofthatmess,theFBIgotinvolved,andsodidtheGermangovernment.
• Thistook5years,2kids,andalotofmoney.• In1992,aAerthewallcamedown,IgaveatalkinBerlin.Roberttookmetoseetheprison(thenamuseum)wherehehadbeeninterrogatedsome20yearbefore.
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Reflec$onsonReflec$onPosi$vityOntheDiscovery
• Summer1972,Roberthadgoneonvaca$ontovisitChrista.• IbegandiscussingwithKonradthepaperofEdwardNelsononMarkov
fieldreconstruc$on.• Iwasinterestedtoknowiftherewasabasicprinciplethatworked
independentoftheMarkovproperty,andmightapplytoothersitua$ons.
• ThenRobertcamebackfromhisvaca$on.• HehadleAhiscarinNorthampton,whereIhadanapartment.There
onecouldparkonthestreetwithoutlimitof$me.• IdroveRoberttopickuphiscar.Wehadaveryinteres$ng2-hour
trip!• IleAfromNothamptontoChicagoforaconference.• WhenIreturnedtoBoston4dayslater,O&ShaddiscoveredRP!• Wri$ngituptookalmosttwoyears.• Iwasdeeply$edupinmyworkonclusterexpansionsandonthree
dimensions.
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Reflec$onsonReflec$onPosi$vityWhatisRP?
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– BasicSchrödingerEqua$on:
– Hamiltonian(Energy).Wantstability:
– Ensuresanaly$cityinacomplexhalft-plane.
i}@ @t = H
H > 0
ModifyEqua$on:Con$nueanaly$callyintHeatDiffusionEqua$on:
Imaginary(unphysical)$me:Easiertostudy,BUT:Ques$on:Howcanonegetback?Answer:RP.18May2016 DIASSTPSPL 10
�}@ @t = H
t ! �it
PictorialInterpreta$onHorizontal($me)Reflec$on
t = 0
Posi$veTimeHalf-spaceRd+
= A⇥(A) =
⇥x
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Reflec$onsonReflec$onPosi$vityWhatisRP?
An$linearIsomorphism
Reflec$onPosi$vityHypothesis
ConsequenceStability(Osterwalder-Schrader)
0 6 h⇥(A)A(t)i| {z }classical
= hA, e�tH AiH| {z }quantum
6 hA, AiH| {z }quantum
0 6 h⇥(A)Ai| {z }classical
= hA, AiH| {z }quantum
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O-SAxioms≅WightmanAxioms
EnabledtheFirstNon-TrivialQFTModel
• Weakly-coupledpolynomialscalarfieldsintwodimensions.Glimm,AJ,Spencer(1974)– Wightmanaxiomsforrela$vis$cQFT– Uniquevacuum– Massgap– Uppergap(isolatedone-par$clestate)
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Mul$pleReflec$onBound
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|hAi| 6 h⇥(A)Ai1/2
= |h(⇥(A))(t)A(t)i|1/2
…..h| {z }n boxes
i2�n
Pictorialupperbound:
h| {z }n boxes
i2�n
AlsoReflectVer$cally
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…..
Pictorialupperbound:
…..
…..
Discretesymmetrybreaking
QuantumMechanics:UniqueNode-lessGroundState(Tunnelling)
FieldTheory:Non-UniqueGroundState(PossibleLackofTunnelling,asinIsingmodel)Dobrushin-Minlos:announcement1973
- +
Reflec$onsonReflec$onPosi$vityPhaseTransi$ons
• Firstproofofphasetransi$onsinQFT.StronglycoupledGlimm,AJ,Spencer(1975)
• Idea:useRPandmul$plereflec$onboundtoes$matedevia$onofmeanfieldfrom“Peierlsargument”studiedbyDobrushinandbyGriffiths.Therebyonecanshowthismethodalsoappliesinthefieldcase.
• Dobrushin-Minlosretrac$on1976
• Wightmanaxioms,uniquevacuum,massgapGJS1976
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Reflec$onsonReflec$onPosi$vityPhaseTransi$ons
Con$nuousSymmetryBreaking
Fröhlich-Simon-Spencer(1976):Beau$ful“Infra-RedBound.”
d≥3,N≥1integrable
Reflec$onsonReflec$onPosi$vityPhaseTransi$ons
Fermions:Osterwalder-SchraderOsterwalder-Eckmann,BorthwickLawceGaugeTheory:Osterwalder-E.SeilerHigherSpin:OzkaynakComplexFields:AJ,Jäkel,Mar$nez Spectralcondi$ononatorus
Superspace:Osterwalder,(Lesniewski),AJ,inprogress
Reflec$onsonReflec$onPosi$vityOtherFieldTheories
SpinSystems:Fröhlich,Dyson,Israel,Simon,Lieb,Brydges,Sokal
Characteriza$onMajoranas:AJ-Pedrocchi,AJ-Janssens
Parafermions:AJ-PedrocchiAJ-Janssens
Reflec$onsonReflec$onPosi$vitySta$s$calMechanics
New:GeometricInterpreta$onofRP(workwithZhengweiLiu)
Ver$calReflec$on
= A
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= A⇤
x
x
Asan$-isomorphism(planaralgebra).Theoperatoradjoint.
StringFourierTransform
(FsT )(x1, . . . , xn) = T (x2, . . . , xn, x1)
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Par$alRota$on
23
Horizontal/Ver$calRela$on
hA⇤Ai > 0=
ParaisotopyRota$on+Horiz.-Reflec$on=Vert.-Reflec$on
=
TheoremConsiderascalar,fermion,orparafermionsta$s$calmechanicsinadoubledvonNeumannalgebra(e.g.pos./neg.$me).LetΘdenotetheTomitareflec$on.IfHisaHamiltonwithposi$veSFT,thenposi$vetemperaturestatessa$sfyRP.AJ-LiuPlanarparaalgebras(2016),arxiv1602.02662Usestwistedproduct:AJ-Pedrocchi,CMP337(2015),455–472AJ-Janssens,CMP(2016)hep://dx.doi.org/10.1007/s00220-015-2545-z. 23May2016 Mul$pleReflec$ons 24
IdeasconnecttoQuantumInforma$on(withZhengweiLiuandAlexWozniakowski)
• Teleporta$on
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hep://arxiv.org/abs/1605.00127hep://arxiv.org/abs/1605.00321
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State
SFT
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=MaximallyEntangledResourceState
Fs|~0 i = |Maxi
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ClassicTeleporta$onCircuitofCharlesBenneeet.al.
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AFurtherReflec$on
IstheRP-C*rela$onhelpfulforunderstandingRobert’sproofofRPinthecontextofReggecalculus?
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AndNowSomePhotos!
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RobertandJiminCargèse1979
9September1992,WhiteHartInn,Salisbury,CT 34
23May2016 AliceMackeyandRobertSeptember9,1992 35
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CapriMay1993(PhilippeBlanchard)
37RobertwithShouchengZhang,hisformerdiplomastudentinthe1980’s
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AulaJune2002
38
23May2016 March19,2005atHarvard 39
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Robert’sRe$rement16December2005
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19May2006Berlin,DürersMueer
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27February2009inBerlinbeforeconcertwithZehetmair&BerlinPhilharmonicconductedbyHeinzHolliger 46
23May2016 2006Berlin 47
23May2016
February28,2009BerlinPhilharmonicconductor’sroom
EdicsonRuizlessonfromHeinzHolligeron
“PreludioeFugafürSolokontrabass”
48
SchlossBrunegg,May14,2016
Skypecall,May232015
Kahlendamm3,Hannover50
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23May2016
February5,2016Hamburg
55
IngridandherdaughterHaraldandhiswife
SomeQues$ons1 WhyisOsterwalder-Schraderposi$vitynota
centralfeatureinEVERYquantumphysicstextbook?
2 WhywasRobertnevera“full”professor?3 WhydidFUBnotregardRobertasoneof
theirmostdis$nguishedfacultymembers?4 Robert’sLastTheorem“Reflec$onposi$vity
insimplicialgravity”J.Phys.A,April2016doi:10.1088/1751-8113/49/21/215202
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Hopefullysomeprogresswillbemade!
ManyThanks!!!
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ETH
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