arXiv:hep-th/9709212v129Sep1997
Whatisquantumfieldtheoryandwhyhavesomephysicistsabandonedit?
R.Jackiw MIT AbstractThepresent-daycrisisinquantumfieldtheoryisdescribed. ContributiontoPulse,thenewsletteroftheLaboratoryforNuclearScience MIT-CTP#2671hep-th/9709212September1997 Quantumfieldtheoryoffersphysicistsatremendouslywidevarietyofapplications:itisbothalanguagewithwhichmanyphysicalprocessescanbediscussed,andalsoitprovidesamodelforfundamentalphysics,theso-called“standardmodel,”whichthusfarhaspassedeveryexperimentaltest.Nootherframeworkexistsinwhichonecancalculatesomanyphenomenawithsucheaseanduracy. Arisingfromamathematicalountofthepropagationoffluids(both“ponderable”and“imponderable”),fieldtheoryemergedoverahundredyearsagointhedescriptionwithinclassicalphysicsofismandsoonthereafterofgravity.Schro¨dinger’swavemechanicsbecameabridgebetweenclassicalandquantumfieldtheory:thequantummechanicalwavefunctionisalsoalocalfield,whichwhen“second”quantizedgaverisetoatruequantumfieldtheory,albeitanonrelativisticone.Quantizationoficwavesproducedthefirstrelativisticquantumfieldtheory,whichwhensupplementedbythequantizedDiracfieldgaveusquantumelectrodynamics,whosefurthergeneralizationtomatricesoffieldsisthepresent-daystandardmodelofelementaryparticles.Thisdevelopmentcarrieswithitanextrapolationoverenormousscales:initialapplicationswereatmicroscopicdistancesoratenergiesofafewelectronvolts,whilecontemporarystudiesofelementaryparticlesinvolve1011electronvoltsorshortdistancesof10−16cm.The“quantization”procedure,whichextendedclassicalfieldtheory’srangeofvalidity,consistsofexpandingaclassicalfieldinnormalmodes,andtakingeachmodetobeaquantaloscillator. Fieldtheoreticideasalsoreachforthecosmosthroughthedevelopmentofthe“inflationaryscenario”—aspeculative,pletelyphysicalanalysisoftheearlyuniverse,whichappearstobeconsistentwithavailableobservations.Additionally,quantumfieldtheoriesprovideeffectivedescriptionsofmany-body,condensedmatterphysics.Heretheexcitationsarenotelementaryparticlesandfundamentalinteractionsarenotprobed,butthecollectivephenomenathataredescribedbymany-bodyfieldtheoryexhibitmanyinterestingeffects,whichinturnhavebeenrecognizedasimportantforelementaryparticletheory.Suchexchangesofideasbetweendifferentsubfieldsofphysicsdemonstratevividlythevitalityandflexibilityoffieldtheory. Butinspiteoftheseesses,todaythereislittleconfidencethatfieldtheorywilladvanceourunderstandingofNatureatitsfundamentalworkings,beyondwhathasbeenachieved.Whileinprincipleallobservedphenomenacanbeexplainedbypresent-dayfieldtheory(intermsofthequantalstandardmodelforparticlephysicsandtheclassicalNewton–Einsteinmodelforgravity),theseountsarestillimperfect.Theparticlephysicsmodelrequiresalistofadhocinputsthatgiverisetoconceptual,generalquestionssuchas:Whyisthedimensionalityofspace-timefour?
Whyaretheretwotypesofelementaryparticles(bosonsandfermions)?
Whatdeterminesthenumberofspeciesoftheseparticles?
Thestandardmodelalsoleavesuswithspecifictechnicalquestions:Whatfixesthematrixstructure,variousmassparameters,mixingangles,andcouplingstrengthsthatmustbespecifiedforconcreteprediction?
Moreover,classicalgravitytheoryhasnotbeenintegratedintothequantumfielddescriptionofnongravitationalforces,againbecauseofconceptualand
2 technicalobstacles:quantumtheorymakesuseofafixedspace-time,soitisunclearhowtoquantizeclassicalgravity,whichallowsspace-timetofluctuate;evenifthisisignored,quantizingthemetrictensorofEinstein’stheoryproducesaquantumfieldtheorybesetbyinfinitiesthatcannotbecontrolled. Buttheseingsareactuallysymptomsofadeeperlackofunderstandingthathastodowithsymmetryandsymmetrybreaking.PhysicistsmostlyagreethatultimatelawsofNatureenjoyahighdegreeofsymmetry,thatis,theformulationoftheselawsisunchangedwhenvarioustransformationsareperformed.Presenceofsymmetryimpliesabsenceplicatedandirrelevantstructure,andourconvictionthatthisisfundamentallytruereflectsanancientaestheticprinciple:physicistsarehappyinthebeliefthatNatureinitsfundamentalworkingsisessentiallysimple.However,wemustalsorecognizethatactual,observedphysicalphenomenararelyexhibitoverwhelmingregularity.Therefore,attheverysametimethatweconstructaphysicaltheorywithintrinsicsymmetry,wemustfindawaytobreakthesymmetryinphysicalconsequencesofthemodel. Progressinphysicscanfrequentlybeseenastheresolutionofthistension.Inclassicalphysics,theprincipalmechanismforsymmetrybreakingisthroughboundaryandinitialconditionsondynamicalequationsofmotion.Forexample,Newton’srotationallysymmetricgravitationalequationsadmittherotationallynonsymmetricsolutionsthatdescribeactualorbitsinthesolarsystem,whenappropriate,rotationallynonsymmetric,initialconditionsareposited. Theconstructionofphysicallyessfulquantumfieldtheoriesmakesuseofsymmetryforyetanotherreason.Quantumfieldtheorymodelsarenotoriouslydifficulttosolveandalsoexplicitcalculationsarebesetbyinfinities.Thusfarwehavebeenabletoethesetwoobstaclesonlywhenthemodelspossessahighdegreeofsymmetry,whichallowsunravelingplicateddynamicsandtamingtheinfinitiesbyrenormalization.Ourpresent-daymodelforquarks,leptons,andtheirinteractionsexemplifiesthisbyenjoyingavarietyofchiral,scale/conformal,andgaugesymmetries.Buttoagreewithexperiment,mostofthesesymmetriesmustbeabsentinthesolutions.Atpresentwehaveavailabletwomechanismsforachievingthisnecessaryresult.Oneisspontaneoussymmetrybreaking,whichreliesonenergydifferencesbetweensymmetricandasymmetricsolutions:thedynamicsmaybesuchthattheasymmetricsolutionhaslowerenergythanthesymmetricone,thereforetheformerisrealizedinNaturewhilethelatterisunstable.Thesecondisanomalousorquantummechanicalsymmetrybreaking,whichusestheinfinitiesofquantumtheorytoeffectaviolationofthecorrespondenceprinciple:thesymmetriesthatappearinthemodelbeforequantizationdisappearafterquantization,becausetherenormalizationprocedure—neededtotametheinfinitiesandwelldefinethetheory—cannotbecarriedoutinafashionthatpreservesthesymmetries. Whilethesetwomethodsofsymmetrybreakingessfullyreducethesymmetriesofthestandardmodeltoaphenomenologicallyeptablelevel,thisreductionisachievedinanadhocmanner,andmuchofthepreviouslymentionedarbitrariness,
3 whichmustbefixedforphysicalprediction,arisespreciselybecauseoftheuncertaintiesinthesymmetry-breakingmechanisms.Spontaneoussymmetrybreakingisadoptedfrommany-body,condensedmatterphysics,whereitiswellunderstood:thedynamicalbasisfortheinstabilityofsymmetricconfigurationscanbederivedfromfirstprinciples.Intheparticlephysicsapplication,wehavenotfoundthedynamicalreasonfortheinstability.Rather,wehavepostulatedthatadditionalfieldsexist,whicharedestabilizingandplishthesymmetrybreaking.Butthisadhocextensionintroducesadditional,aprioriunknownparametersandyet-unseenparticles,theHiggsmesons.Anomaloussymmetrybreakingalsocarrieswithitarbitrariness:theQCD“renormalizationscale”aswellasyet-unseenparticles,theaxions,whichfixanarbitraryCP-violatingangle.Moreover,thefieldtheoreticinfinities,whichgiverisetoanomaloussymmetrybreaking,preventtheconstructionofaneptablequantumgravityfieldtheory,soitispeculiartorelyonthemsocriticallyfortheviabilityofthestandardmodel. Advancingourunderstandingoftheabovehasbeenatanimpasseforovertwodecades.Intheabsenceofnewexperimentstochanneltheoreticalspeculation,somephysicistshaveconcludedthatitwillnotbepossibletomakeprogressonthesequestionswithinfieldtheory,andhaveturnedtoanewstructure,stringtheory.Infieldtheorythequantizedexcitationsarepointparticleswithpointinteractionsandthisgivesrisetotheinfinities.Instringtheory,theexcitationsareextendedobjects—strings—withnonlocalinteractions;therearenoinfinities,andthisenormousdefectoffieldtheoryisabsent.Notonlydoesquantumgravityexistinthenewcontext,butitappearsthatsomepuzzleshavingtodowithblackholescanbeanswered.Moreover,stringtheoryaddressespreciselysomeofthequestionsthatremainedunansweredinfieldtheory:dimensionalityofspace-timecannotbearbitrarybecausestringtheorycannotbeformulatedinarbitrarydimensions;fermionsmustcoexistwithbosonsbecauseofsupersymmetry—anecessaryingredientofstringtheory;andsoon. Yetinspiteofthesepositivefeatures,uptonowstringtheoryprovidesonlyaframework,ratherthanadefinitestructure.Whilepresent-dayphysicsshouldbefoundinthelow-energylimitofstringtheory,aprecisederivationofthestandardmodelhasyettobegiven.Onethinksagainaboutsymmetryandsymmetrybreaking.Thesymmetriesofquantumfieldtheorysurpassthoseofclassicalphysicsandrequireelaboratesymmetrybreakingmechanisms.Thesymmetriesofstringtheoryagainvastlyoutpacethoseoffieldtheory,andmustbebrokenbyyet-to-be-developedprocedures,inordertoexplaintheworldaroundus. Onpreviousasionswhenitappearedthatquantumfieldtheorywasincapableofadvancingourunderstandingoffundamentalphysics,newapproachesandnewideasdispelledthepessimism.Todaywedonotknowwhetherourlackofprogressisduetoafailureofimagination,orwhetherindeedwehavetopresentfundamentalphysicallawsinanewframework,therebyreplacingthefieldtheoreticone,whichhasserveduswellforoverahundredyears.
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R.Jackiw MIT AbstractThepresent-daycrisisinquantumfieldtheoryisdescribed. ContributiontoPulse,thenewsletteroftheLaboratoryforNuclearScience MIT-CTP#2671hep-th/9709212September1997 Quantumfieldtheoryoffersphysicistsatremendouslywidevarietyofapplications:itisbothalanguagewithwhichmanyphysicalprocessescanbediscussed,andalsoitprovidesamodelforfundamentalphysics,theso-called“standardmodel,”whichthusfarhaspassedeveryexperimentaltest.Nootherframeworkexistsinwhichonecancalculatesomanyphenomenawithsucheaseanduracy. Arisingfromamathematicalountofthepropagationoffluids(both“ponderable”and“imponderable”),fieldtheoryemergedoverahundredyearsagointhedescriptionwithinclassicalphysicsofismandsoonthereafterofgravity.Schro¨dinger’swavemechanicsbecameabridgebetweenclassicalandquantumfieldtheory:thequantummechanicalwavefunctionisalsoalocalfield,whichwhen“second”quantizedgaverisetoatruequantumfieldtheory,albeitanonrelativisticone.Quantizationoficwavesproducedthefirstrelativisticquantumfieldtheory,whichwhensupplementedbythequantizedDiracfieldgaveusquantumelectrodynamics,whosefurthergeneralizationtomatricesoffieldsisthepresent-daystandardmodelofelementaryparticles.Thisdevelopmentcarrieswithitanextrapolationoverenormousscales:initialapplicationswereatmicroscopicdistancesoratenergiesofafewelectronvolts,whilecontemporarystudiesofelementaryparticlesinvolve1011electronvoltsorshortdistancesof10−16cm.The“quantization”procedure,whichextendedclassicalfieldtheory’srangeofvalidity,consistsofexpandingaclassicalfieldinnormalmodes,andtakingeachmodetobeaquantaloscillator. Fieldtheoreticideasalsoreachforthecosmosthroughthedevelopmentofthe“inflationaryscenario”—aspeculative,pletelyphysicalanalysisoftheearlyuniverse,whichappearstobeconsistentwithavailableobservations.Additionally,quantumfieldtheoriesprovideeffectivedescriptionsofmany-body,condensedmatterphysics.Heretheexcitationsarenotelementaryparticlesandfundamentalinteractionsarenotprobed,butthecollectivephenomenathataredescribedbymany-bodyfieldtheoryexhibitmanyinterestingeffects,whichinturnhavebeenrecognizedasimportantforelementaryparticletheory.Suchexchangesofideasbetweendifferentsubfieldsofphysicsdemonstratevividlythevitalityandflexibilityoffieldtheory. Butinspiteoftheseesses,todaythereislittleconfidencethatfieldtheorywilladvanceourunderstandingofNatureatitsfundamentalworkings,beyondwhathasbeenachieved.Whileinprincipleallobservedphenomenacanbeexplainedbypresent-dayfieldtheory(intermsofthequantalstandardmodelforparticlephysicsandtheclassicalNewton–Einsteinmodelforgravity),theseountsarestillimperfect.Theparticlephysicsmodelrequiresalistofadhocinputsthatgiverisetoconceptual,generalquestionssuchas:Whyisthedimensionalityofspace-timefour?
Whyaretheretwotypesofelementaryparticles(bosonsandfermions)?
Whatdeterminesthenumberofspeciesoftheseparticles?
Thestandardmodelalsoleavesuswithspecifictechnicalquestions:Whatfixesthematrixstructure,variousmassparameters,mixingangles,andcouplingstrengthsthatmustbespecifiedforconcreteprediction?
Moreover,classicalgravitytheoryhasnotbeenintegratedintothequantumfielddescriptionofnongravitationalforces,againbecauseofconceptualand
2 technicalobstacles:quantumtheorymakesuseofafixedspace-time,soitisunclearhowtoquantizeclassicalgravity,whichallowsspace-timetofluctuate;evenifthisisignored,quantizingthemetrictensorofEinstein’stheoryproducesaquantumfieldtheorybesetbyinfinitiesthatcannotbecontrolled. Buttheseingsareactuallysymptomsofadeeperlackofunderstandingthathastodowithsymmetryandsymmetrybreaking.PhysicistsmostlyagreethatultimatelawsofNatureenjoyahighdegreeofsymmetry,thatis,theformulationoftheselawsisunchangedwhenvarioustransformationsareperformed.Presenceofsymmetryimpliesabsenceplicatedandirrelevantstructure,andourconvictionthatthisisfundamentallytruereflectsanancientaestheticprinciple:physicistsarehappyinthebeliefthatNatureinitsfundamentalworkingsisessentiallysimple.However,wemustalsorecognizethatactual,observedphysicalphenomenararelyexhibitoverwhelmingregularity.Therefore,attheverysametimethatweconstructaphysicaltheorywithintrinsicsymmetry,wemustfindawaytobreakthesymmetryinphysicalconsequencesofthemodel. Progressinphysicscanfrequentlybeseenastheresolutionofthistension.Inclassicalphysics,theprincipalmechanismforsymmetrybreakingisthroughboundaryandinitialconditionsondynamicalequationsofmotion.Forexample,Newton’srotationallysymmetricgravitationalequationsadmittherotationallynonsymmetricsolutionsthatdescribeactualorbitsinthesolarsystem,whenappropriate,rotationallynonsymmetric,initialconditionsareposited. Theconstructionofphysicallyessfulquantumfieldtheoriesmakesuseofsymmetryforyetanotherreason.Quantumfieldtheorymodelsarenotoriouslydifficulttosolveandalsoexplicitcalculationsarebesetbyinfinities.Thusfarwehavebeenabletoethesetwoobstaclesonlywhenthemodelspossessahighdegreeofsymmetry,whichallowsunravelingplicateddynamicsandtamingtheinfinitiesbyrenormalization.Ourpresent-daymodelforquarks,leptons,andtheirinteractionsexemplifiesthisbyenjoyingavarietyofchiral,scale/conformal,andgaugesymmetries.Buttoagreewithexperiment,mostofthesesymmetriesmustbeabsentinthesolutions.Atpresentwehaveavailabletwomechanismsforachievingthisnecessaryresult.Oneisspontaneoussymmetrybreaking,whichreliesonenergydifferencesbetweensymmetricandasymmetricsolutions:thedynamicsmaybesuchthattheasymmetricsolutionhaslowerenergythanthesymmetricone,thereforetheformerisrealizedinNaturewhilethelatterisunstable.Thesecondisanomalousorquantummechanicalsymmetrybreaking,whichusestheinfinitiesofquantumtheorytoeffectaviolationofthecorrespondenceprinciple:thesymmetriesthatappearinthemodelbeforequantizationdisappearafterquantization,becausetherenormalizationprocedure—neededtotametheinfinitiesandwelldefinethetheory—cannotbecarriedoutinafashionthatpreservesthesymmetries. Whilethesetwomethodsofsymmetrybreakingessfullyreducethesymmetriesofthestandardmodeltoaphenomenologicallyeptablelevel,thisreductionisachievedinanadhocmanner,andmuchofthepreviouslymentionedarbitrariness,
3 whichmustbefixedforphysicalprediction,arisespreciselybecauseoftheuncertaintiesinthesymmetry-breakingmechanisms.Spontaneoussymmetrybreakingisadoptedfrommany-body,condensedmatterphysics,whereitiswellunderstood:thedynamicalbasisfortheinstabilityofsymmetricconfigurationscanbederivedfromfirstprinciples.Intheparticlephysicsapplication,wehavenotfoundthedynamicalreasonfortheinstability.Rather,wehavepostulatedthatadditionalfieldsexist,whicharedestabilizingandplishthesymmetrybreaking.Butthisadhocextensionintroducesadditional,aprioriunknownparametersandyet-unseenparticles,theHiggsmesons.Anomaloussymmetrybreakingalsocarrieswithitarbitrariness:theQCD“renormalizationscale”aswellasyet-unseenparticles,theaxions,whichfixanarbitraryCP-violatingangle.Moreover,thefieldtheoreticinfinities,whichgiverisetoanomaloussymmetrybreaking,preventtheconstructionofaneptablequantumgravityfieldtheory,soitispeculiartorelyonthemsocriticallyfortheviabilityofthestandardmodel. Advancingourunderstandingoftheabovehasbeenatanimpasseforovertwodecades.Intheabsenceofnewexperimentstochanneltheoreticalspeculation,somephysicistshaveconcludedthatitwillnotbepossibletomakeprogressonthesequestionswithinfieldtheory,andhaveturnedtoanewstructure,stringtheory.Infieldtheorythequantizedexcitationsarepointparticleswithpointinteractionsandthisgivesrisetotheinfinities.Instringtheory,theexcitationsareextendedobjects—strings—withnonlocalinteractions;therearenoinfinities,andthisenormousdefectoffieldtheoryisabsent.Notonlydoesquantumgravityexistinthenewcontext,butitappearsthatsomepuzzleshavingtodowithblackholescanbeanswered.Moreover,stringtheoryaddressespreciselysomeofthequestionsthatremainedunansweredinfieldtheory:dimensionalityofspace-timecannotbearbitrarybecausestringtheorycannotbeformulatedinarbitrarydimensions;fermionsmustcoexistwithbosonsbecauseofsupersymmetry—anecessaryingredientofstringtheory;andsoon. Yetinspiteofthesepositivefeatures,uptonowstringtheoryprovidesonlyaframework,ratherthanadefinitestructure.Whilepresent-dayphysicsshouldbefoundinthelow-energylimitofstringtheory,aprecisederivationofthestandardmodelhasyettobegiven.Onethinksagainaboutsymmetryandsymmetrybreaking.Thesymmetriesofquantumfieldtheorysurpassthoseofclassicalphysicsandrequireelaboratesymmetrybreakingmechanisms.Thesymmetriesofstringtheoryagainvastlyoutpacethoseoffieldtheory,andmustbebrokenbyyet-to-be-developedprocedures,inordertoexplaintheworldaroundus. Onpreviousasionswhenitappearedthatquantumfieldtheorywasincapableofadvancingourunderstandingoffundamentalphysics,newapproachesandnewideasdispelledthepessimism.Todaywedonotknowwhetherourlackofprogressisduetoafailureofimagination,orwhetherindeedwehavetopresentfundamentalphysicallawsinanewframework,therebyreplacingthefieldtheoreticone,whichhasserveduswellforoverahundredyears.
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