An effective quantum mechanism for mass generation in diffeomorphism-invariant theories

fPhysicsLettersA,

WorldScientificPublishingCompanyANEFFECTIVEQUANTUMMECHANISMFORMASSGENERATION

INDIFFEOMORPHISM-INVARIANTTHEORIES∗

J.L.JARAMILLO

V.ALDAYA

InstitutoAstrof´ısicaAndaluc´ıa(CSIC),ApartadoPostal3004

Granada18080,Spain

InstitutoCarlosIdeF´ısicaTe´oricayComputacional,FacultaddeCiencias,

UniversidaddeGranada,CampusdeFuentenueva

Granada18002,Spain

Weproposeascenarioforparticle-massgeneration,assumingtheexistenceofaphysicalregimewhere,firstly,physicalparticlescanbeconsideredaspoint-likeobjectsmovinginabackgroundspace-timeand,secondly,theirmerepresencespoilstheinvarianceunderthelocaldiffeomorphismgroup,resultinginananomalousrealizationofthelatter.Underthesehypotheses,wedescribemassgenerationstartingfromthemasslessfreetheory.Themechanismisnotsensitivetothedetaileddescriptionoftheunderlyingtheoryathigherenergies,leaningonlyongeneralstructuralfeaturesofit,specificallydiffeomorphisminvariance.

Theproblemweaddressinthisworkistheoriginofparticlemasses.Eventhoughastrongemphasishasbeenplacedonthisissuethroughoutthedevelopmentofmodernphysics,thesubjectseemsfarfrombeingresolved.

Thecorrectionofthemassofaparticleasaneffectiveconsequenceofitsin-teractionwiththesurroundingenvironmentisaveryoldideathatcanbetrackedtonineteenth-centuryhydrodynamics.Infact,formanydifferentphysicalsystemsdescribingthemotionofanobjectinsideaclassicalcontinuumfluid,thesolutionofhydrodynamicalequationsadmitsaneffectivetreatmentinwhichtheobjectbe-havesasinfreemotionwithacorrectedorrenormalizedmasswhichdependsongeneralfeaturessuchasboundaryconditions.Theextensionoftheseideastoelec-trodynamicsledJ.J.Thomsontotheintroductionofthenotionofelectromagneticmassofachargeasaconsequenceoftheinteractionwithitsownelectromagneticfield,fundamentalelementinthelaterLorentz’stheoryoftheelectron1.WiththearrivaloftheQuantumTheory,theeffortsbyKramers(largelyinspiredinLorentz’sinsights)resultedintheconnectionbetweenthepreviousclassicalideasandthenew

∗Work

partiallysupportedbytheDGICYT.

1

problemsrelatedtothedivergencesappearinginthecalculationoftheelectronself-energy,leadingtoaradiativemassrenormalization.IntheearlydaysofQuantumElectrodynamicstherewashopeofdevelopingafundamentaltheorythatwouldeliminatethedivergencesandsuccessfullyderivetheactualvaluesofitscharacter-isticparameters.However,theeventualresolutionoftheproblembyimplementingtherenormalizationprogramfinallyledtoasituationinwhichQuantumFieldThe-ory(QFT)appearsasaneffectivetheory.Infact,physicsbeyondacertainenergyscaleisnotprobed,thoughtherenormalizationofcertainparametersofthemodel,amongthemthemassesoftheparticles,non-triviallyaffectslower-energyphysics.Inthisscenario,theideaofmassasself-energyhaswitheredawaytomootstatus.Nevertheless,conceptuallydifferentmechanismscanbedevisedforaddressingpar-tialyetfundamentalaspectsofthemass-originproblem,anexampleofwhichistheuseoflatticeQCDtechniquesforlighthadrons2.Inanycase,questionssuchasleptonmassesorcosmologicaldarkmatter,remainopen.

Fromthishistoricaldetour(see3forfurtherdetails),weextractourtwomainguidelines.Firstly,wetakeuptheoldideaofemphasizingtheinteractionwiththesurroundingfieldsasfundamentalinthegenerationofmassand,secondly,weadoptaneffectiveaproachinwhichphysicsbeyondacertainscaleisnotdiscussed.Thepresenceofunansweredquestionssuggeststheintroductionofphysicsoftenignoredinthemassgenerationproblem.Anappealing(andobvious)candidateforthemissingphysicalingredientisGravitywithitsassociateddiffeomorphisminvariance,generallynotconsideredinhigh-energyparticlephysics.Therefore,theonlyexplicitconditionweshallimposeontheunderlyingfundamentaltheoryisanessentialroleforthenotionofdiffeomorphisminvariance.

Whenadoptingtheabove-mentionedeffectiveattitude,itseemsreasonabletoadmittheexistenceofascaleofenergiesinwhichstandardQFTisagoodapprox-imation,anditsnotionofaparticleasalocalexcitationofthevacuumresultingfromtheactionofalocalfieldoperatorapplies.Wearealsoimplicitlyacceptinganotionofspace-timeasadifferentiablemanifoldmakingupthebackgroundinwhichparticlesmove.Weshallphenomenologicallyseparatetheintrinsicdynamicsofthiseffectivebackground,governedbyclassicalGeneralRelativity,fromtheeffectthattheunderlyingdiffeomorphisminvariancecouldexertinthequantumproccessofparticlecreation.

Wearethereforestudyingaregimeinwhichphysicalparticlescanbeconsideredaspoint-likeobjectsandtheclassicaldynamicsofspace-timeisdecoupled(adiabaticcondition).Theadjectivephysicalappearinghereisessential,asopossedtotheidealtestparticles,causinganeffectivebreakdownofthespace-timenotionatthepointitselfonwhichtheparticlelies.Wearesuggestingthatphysicalparticlesliterallypiercespace-time,producingahole.Thishasprofoundconsequencesinthequan-tummodeldescribingparticlecreation.Itcanbeshown4thatthepresenceofaholeinatwo-dimensionalmanifoldinducesanomalous(central)termsinthequantumcommutatorsbetween(someof)thegeneratorsofdiffeomorphisminvariance,thusspoilingthisclassicalsymmetry(eventhoughthiscanbeproperlyhealedinspe-2

cifictheories).Weproposethatthisphenomenongeneralizestorealisticspace-timedimensions,inducingananomalousrealizationofclassicaldiffeomorphismsymme-tryintheeffectivequantumprocessofparticlecreation,somethingthatcouldbesupportedbytheanalysisoftheleadingtermsofappropiateOperatorProductEx-pansions.Thisdoesnotcontradictanexactimplementationofthissymmetryathigherenergies,whenusingamorefundamentalmodelforthecouplingbetweenthegravitationalandmatterdegreesoffreedom.Itsimplymeansthatthepricewemustpayforavoidingsuchadetaileddescription,andadmittinganeffectivetreat-mentinwhichclassicalspace-timeisdecoupled,istheacceptanceofabreakdownofclassicaldiffeomorphismgaugeinvariance.

Thepreviousheuristicmotivationscanbesynthesizedinthefollowinghypoth-esis:thereexistsaneffectiveregimeinwhichphysicalparticlesarepoint-likeandtheircreationprocessentailsabreakdownofclassicaldiffeomorphisminvariance,thelatterbeingrealizedinananomalousway.

Thepresenceofananomalyinalocalgaugetheoryobstructsthereductionofdegreesoffreedomforwhichthegaugesymmetryisdevised,entailinganenlarge-mentofthephysicalphasespaceperformedbythespurious(inprinciple)modesa.Thisissueposesseriousconcernsfortheconsistencyofthetheory,atleastwhenap-plyingstandardtechniques,somethingespeciallycriticalwhenaddressingthegaugetheoryasfundamental(attemptstoconstructconsistentanomaloustheoriesdoex-ist5andtheabove-mentionedexplicitappearanceofextradegreesoffreedomcanbemadeapparent).However,thepresenceofananomalycanalsobeinterpretedasasignatureforunderstandingthetheoryasalow-energyeffectivemodel,indicatingtheexistenceofnewphysicsathigherenergies6.Thisispreciselythesituationwearedealingwithhere.Theinfluenceofhigher-energydegreesoffreedomisencodedinsomeeffectivedegreesoffreedomarisingintheanomalouslow-energytheory.Astandardwayinwhichananomalymanifestsitself,inaccordancewiththeconsiderationsabove,isthroughtheappearanceofextratermsinthequantumcommutatorswithrespecttotheonesdefiningtheclassicalsymmetry.Therefore,weproposethatthediffeomorphismsymmetryisrealizedintheeffectivetheoryasanextension(notnecessarilycentral)oftheclassicallocaldiffeomorphismalgebra.Forconcretenesswefocusonthetensorialextensions,whichareclassifiedin7anddiscussedin8.Weshallworkinmomentumspaceanddenotethediffeomorphism

ˆµ(m),thefieldscorrespondingtotheparticlesgenericallybyΦˆa(m)generatorsbyL

ˆi(m)(µisaspace-timeindex,aandiinternalandthetensorialextensionsbyA

indicesandmavectorlabellingmomentumspace).Inthisnotation,thequantumbracketsaregivenby:

󰀉󰀁

ˆν(m+n)+ciˆˆµ(m+n)−nµLˆˆLµ(m),Lν(n)=mνLµν(m,n)Ai(m+n)

󰀁󰀉

ˆa(m+n)ˆµ(m),Φˆa(n)=−nµΦL

aAfamiliarexampleofthisphenomenoninstringtheoryistheLiouvillemodeinthenon-critical

string.

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󰀉󰀁ˆˆΦa(m),Φb(n)=

αˆab(m,n),(1)

whereciµν(m,n)isthecocyclelinkedtotheanomalousextensiongivingeffectivedynamicalcontenttothediffeomorphismsandαˆab(m,n)providesthestandardcommutatorsofthefreefields.

Togiveaspecificmeaningtotheentireforegoingdiscussions,weneedtocon-structexplicitlyaphysicalmodeldescribingdynamicsconsistentwiththealgebra(1).Aparticularlywellsuitedformalismforsuchataskistheso-calledGroupApproachtoQuantization(GAQ,see9).Inshort,themainachievementofthisap-proachistheconstructionofphysicaldynamicsoutofagivenLiealgebraicstructuretakenastheonlyphysicalinput.Thetechnique,insomepoints,resemblesKirillov’sconstructionofdynamicsonthecoadjointorbitsofagroup10andsharessomeim-portantgeneralfeatureswithGeometricQuantization11.ThefinaloutcomeisanexplicitunitaryandirreduciblerepresentationoftheoperatorsinthestartingLiealgebra.

Whenapplyingthesetechniquestoalgebrasofthetype(1),weobtainmaximum-weightrepresentations(possessingauniquevacuumintheHilbertspace),wherethe

ˆµ(m)actandgenuinelyraiseandlowercorrespondingdiffeomorphismoperatorsL

thephysicalstates,accordingtotheirgaineddynamicalcontent.Amostimpor-tantpreciseandgeneral(perturbativeb)resultistheconstructionofan(effective)Hamiltonianoperatorforthesystemwiththegeneralform:

󰀃󰀆󰀂†ˆ,Φˆ+ˆeff=HˆfreeΦˆµ)†(m)Lˆν(m)+Hθµν(m)(L

ˆ†,Φˆ,(Lˆµ)†,LˆµˆmixingΦ+H

󰀃

m

ˆfreeistheHamiltoniancorrespondingtothefreemasslessfieldtheory,thewhereH

secondtermisapuredynamical-diffeomorphismquadraticcontributiontotheen-ergy(θµν(m)isac-numberfunctiononmwhichcloselydependsontheinverseof

ˆthecocycleciµν(m,n))andHmixingcorrespondstohigher-powertermsinvolving

apotentialmixingamongalltheoperators.Appearingperhapsasanoddphe-ˆmixing,nomenon,thelowest-ordertermproducinginteractionisnotfoundinsideH

butalreadyinthequadraticdiffeomorphismone,thereasonbeingthenon-canonicalformofthecommutatorsin(1),inparticularthesecondone.Thiswillbeappar-entinaspecificexamplebelow.Regardingtheexpression(2),ourclaimisthatthetermscorrectingthefreeHamiltonian,couldaccountforthemasstermsintheeffectivetheory.

Finally,weareinthepositionofunambiguoslystatingourconjecture:acrucialpartofmassgenerationcanbephenomenologicallydescribedasa(radiative)cor-rectionresultingfromtheinteractionbetweenthemasslessfieldsandsomeeffectivedegreesoffreedomappearingfromthemereexistenceofparticles.

bAcrucialstepofGAQconsistsinexponentiatingthestartingLiealgebra.Whenthelatter

󰀆

,(2)

isinfinite-dimensionalthisisaenormoustaskandonlyanorder-by-orderprocedureisgenerallyfeasible,leadingtoperturbativethoughrenormalizedresults.

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Ofcourse,arealpredictionofthiscontributiontothemasswouldrequireaknowledgeoftheunderlyingfundamentaltheory,sinceitplaystheroleoffixingthevaluesoftheextensionsinthealgebra(1)andthereforeofthekeyθµν(m).Beyondthat,themechanismisnotsensitivetothehigher-energydetaileddescriptionwhichcouldfindsupportonstringstheory,loopquantumgravity,anon-commutativeversionofgeometry,amoresophisticatedQFToranothereffectiveyetmorefunda-mentalmodel,suchasworm-holes(see12)inEuclideanquantumgravity.Aseriousattempttoprovidearealisticexampleinthiscontextdeservesacarefulanalysisonthepotentialanomalousbreakingofdiffeomorphisminvarianceincurrentcan-didatesforfundamentaltheories.Forthetimebeing,wepresentanover-simplifiedillustrativeexample,consistingofafreerealscalarfieldinone(compact)spatialdi-mension.Withtheansatzthatonlythespatialdiffeomorphismsbecomedynamical,therelevantalgebrais:

󰀉

Lˆ󰀉m,Lˆn

󰀁=(m−n)Lˆm+n+cm3δm+n,0Lˆm,aˆn

[ˆa󰀁=−naˆm+nm,aˆn]=

mδm+n,0.

(3)

TheHilbertspaceisconstructedfromauniquevacuumstate|0thecreationoperatorsαˆ†naˆandLˆnannihilationoperatorsare≡ngivenby−n

≡Lˆ>,byapplying

1†−n,withn>0,andwheretheαˆnn

αˆ†n,

αˆ≡1aˆnandLˆn(n≥0).Theoperatorsnnowsatisfythestandardcanonicalcommutators:[ˆαn,αˆ†m]=δn,m.The

perturbativecalculationoftheHamiltonianofthesystem(whichcanbederivedfromNoetherinvariantsin13,togetherwithapropersettinginthevaluesofthecentralextensionsthere)yields:

Hˆ=󰀂󰀈nαˆ†1n

αˆn+n>0

M2+n2αˆ†nα

ˆn(weareexplicitlyomittingzero-energyterms).Takingintoaccountthattheexpressionin(4)isonlyperturbative,lookforaregimeinwhichwecancoherentlycompareitwithH

ˆwemust

fieldc.Thiscanbeachievedbyexpandingthefielddispersionrelationforlargen.Thuswehave

Hˆfield=󰀂n󰀇αˆ†αˆ=󰀂󰀈nαˆ†M2nαˆn+n>0n2nnn>0

raisethequestionabouttheconvenienceofusing|k>=αˆ†kphysical1-particlestatesinthetheorywithdynamicaldiffeomorphisms,|0>toimplementwherethetheexcitationoftheseeffectivemodeslinearcombinationcontainingLˆwouldsuggestthepossibilityofamoregeneral

†k

spiritofperturbationtheoryover|0>afreestates.masslessAtworst,scalaronefield,couldthusconsiderusing(4)theinnon-theperturbated|k>=αˆ†kThisistheapproachwe|0>shalltouseevaluatehere.

first-ordercorrectionstotheenergylevels.Thecorrectiontotheexcitationenergyofthemasslessparticle,calculatedinthequantumeffectivetheoryderivedfrom(3),accountsfortheenergyoftheinteractionwiththeeffectivediffeomorphismdegreesoffreedom.Eventhoughthisenergycouldshowacomplicatedbehaviourinthemomentumoftheparticle,weseparatetheintermediate(low)andveryhigh-momentadependenceandattempttoextracttheenergyrelatedtothemassoutoftheformoftheinteractionenergyatthelowestappearingmomenta.shalldenotebyH

ˆTherefore,whenevalutingthesecondtermin(4)(whichwe

L)weexpecttofindanexpressionthatcanbeidentifiedwiththeonecomingfromthesecondtermin(5)plusanenergycorrespondingtoveryhigh-momentadependence(Eh−m):

ˆL|k>2k

+Eh−m.

(6)

Theevaluationofthefirstmembergivesthefiniteresult:

ˆL|k>k󰀂

−1k(k−n)

c

n=1

12

2k

∼

2

wedecoupledfromtheverybeginning)thatisnotseenbytherestofphysicalinteractions,andthusitiscompletelydark.Itistemptingtosuggestspeculativelythisasanavenuetowardsthedark-matterproblem.Whenstudyingthetwo-particlestates,wefind

ˆL|kl>k−1󰀂

k−n

󰀈

c

n=1

cn=11

l−1󰀂l−n

c(1+δn,k)(k−l)2

󰀊

,(9)

wherethefirsttwotermsinr.h.s.correspondtothemassesoftheparticlesand

itshigh-momentacorrectionswhilethethirdtermcanbeinterpretatedasanextraenergyneededtomaintaintheparticlesseparated(notethatitispositive,sothatwemustdosomeworktohaveseparatedparticles).

Finally,weshouldpointoutthattheuseoftheVirasoroalgebradoesnotturnthetwo-dimensionalexampleintoatoospecialcase,sinceadirectgeneralizationtohigherdimensionsisinfactprovidedbythenon-centralVirasoro-likeextension8ob-iiˆˆtainedbymakingciµν(m,n)=cmµnν(m−n)andAi(m)=Si(m)in(1),although

involvingmuchmorecumbersomeexpressions.Theonlyaimofthepresentedex-ampleisthatofprovidingatasteofandsparkingintuitionforthepotentialitiesofexpressions(2)and(4),therealpointwewishtoemphasize.Rigorousanalysesre-quireasubtleweavingtogetherofthepossibleextensionsin(1),thefielddispersionrelationweseektofitandthepossibilityofmorecomplexsettingsinwhichthoseelementsmayact.

Inconclusion,wehaveposedasimpleframeworkformassgenerationofparticlesbyproposingamechanismcapableofendowingmasslessfreefieldswithanon-zeromass.Evenifthisisatinyeffect,itwouldprovideanon-zerogermsuitableofbeingamplifiedwithothermechanismssuchasthemultiplicativerenormalizationappearinginQEDfortheelectron.Thepresentdescriptionhasaneffectivenatureandisquiteinsensitivetotheunderlyingfundamentaltheory,providedthatdif-feomorphisminvarianceplaysafundamentalrole.ThislastpointisreinforcedbyMach’sconceptualintuitionslinkinginertiaandGravity.Aheavyuseoftheimageofparticlecreationasaninherentlyquantumprocessisdisplayed.ThiscanbeexplicitlyseenintheexamplediscussedbyunderliningthedependenceofparticlemassesonthecentralchargeoftheVirasoroalgebrain(3),insuchawaythattheyvanishintheclassicallimitc→∞(see14),thusrevealingthemselvesasaquantumphenomenon.Letusfinallystatethat,thoughthepresenceof(non-gravitational)interactionsiscrucialinascribingapoint-likenaturetotheparticlesinQFT,onceweacceptsuchanaturewecangetridofthoseinteractionsanddealessentiallywithfreetheoriesasastartingpointfortheproposedmechanism.Therefore,evenifourconjectureofrelatingamassoriginforparticlestotheinteractionwiththesomeeffectivedegreesoffreedomdoesnotfullywork,theproposedradiativecorrectionscouldimplyimportantandobservableconsequencesontheenergyspectrumoffree

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fields,inparticularentailingmodificationsinthefieldpropagators.Acknowledgments

WewanttothankC.Barcel´oforcrucialdiscussionsandhiscontinuoussupport.WealsowanttothankA.P.Balachandranforstimulatingcomments.References

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