FoundationsofPhysicsmanuscriptNo. (willbeinsertedbytheeditor) Particles, Cutoffs and Inequivalent Representations FraserandWallaceonQuantumFieldTheory MatthiasEgg · Vincent Lam · Andrea Oldofredi 7 1 0 2 Received:date/Accepted:date n a Abstract We critically review the recentdebate betweenDoreen Fraser and David J Wallace on the interpretation of quantum field theory, with the aim of identifying 3 where the core of the disagreement lies. We show that, despite appearances, their 2 conflictdoesnotconcerntheexistenceofparticlesortheoccurrenceofunitarilyin- ] equivalentrepresentations.Instead,thedisputeultimatelyturnsontheverydefinition h ofwhata quantumfield theoryis. We furtherillustrate the fundamentaldifferences p between the two approaches by comparing them both to the Bohmian program in - t quantumfieldtheory. s i h Keywords Algebraicquantumfield theory· Particle physics· Renormalization· s. Unitarilyinequivalentrepresentations c i s y 1 Introduction h p [ Theinterpretationofrelativisticquantumfieldtheory(QFT)isadifficulttask,since itinvolvesspecificfield-theoreticandrelativisticissuesontopoftheusualquantum 1 puzzlesfoundinnon-relativisticquantummechanics(QM),suchasthemeasurement v problemandnon-locality.(MerelymovingfromQMtoQFTobviouslydoesnotsolve 0 0 5 MatthiasEgg 6 UniversityofBern,InstituteofPhilosophy,Laenggassstrasse49a,Postfach,3000Bern9,Switzerland E-mail:[email protected] 0 . VincentLam 1 UniversityofGeneva,DepartmentofPhilosophy,RuedeCandolle2,1211Gene`ve4,Switzerland 0 E-mail:[email protected] 7 & 1 TheUniversityofQueensland,SchoolofHistoricalandPhilosophicalInquiry,StLuciaQLD4072,Aus- : v tralia i E-mail:[email protected] X AndreaOldofredi r UniversityofLausanne,DepartmentofPhilosophy,1015Dorigny,Lausanne,Switzerland a E-mail:[email protected] 2 MatthiasEggetal. thelatterpuzzles,soanyrealistinterpretationofQFTwillhavetoprovideanaccount ofthem.)SimilarlytotheQMcase(tosomeextent),differentstancesontheseQFT issuesleadtodifferentvariantsofthetheory.DoreenFraserandDavidWallacehave recently debated over the best approachto QFT for foundationaland interpretative work,theformerarguinginfavorofthemathematicallyrigorousandphysicallyam- bitiousapproachofalgebraicquantumfieldtheory(AQFT),thelatterdefendingthe morepragmaticapproachof ‘conventional’quantumfield theory(CQFT), whichis describedinmostQFTtextbooksandusedbymostworkingphysicistsinthedomain. The debate between Fraser and Wallace has the great merit of highlightingand fo- cusing on some of the foundationaland interpretativeissues that appear within the QFTframework,butwhichhavebeenfarlessinvestigatedthantheusualonesalready presentinQM.TheupshotoftheirdebatefortheinterpretationofQFTis,however, farfromclearandevensomewhatconfusing. Inthispaper,wecriticallyreviewthisdebate,mainlyasexpressedin[1],[2]and [3][4],hopefullyclarifyingalongthewayseveralimportantpointsfortheinterpre- tationofQFT.Insection2,wewillarguethat,despiteappearances,thedisagreement betweenFraserandWallaceisnotreallyconcernedwiththeexistenceofparticles(or quanta).We will thenprogressivelyapproachwhatwe claim to be the coreof their disagreement,discussingfirsttheirdiverginginterpretationsofrenormalization(sec- tion3)andthereaftertheirdifferingviewsonthepermissibilityofcutoffs(section4). ThiscrucialissueisintimatelyconnectedtothedefinitionoftheveryprojectofQFT, whichwewilldiscussinsection5.Insection6,wewillspelloutwhatwetaketobe theupshotofthedebateforthefoundationofQFT. 2 Wheretheconflictdoesnotlie:Particleontology OneofthemostdiscussedinterpretativeissuesofQFTconcernstheextenttowhich thetheorycanbeunderstoodintermsofparticles(orintermsoftheweakernotion of quanta).A superficiallook at the debate between Fraser and Wallace could lead totheimpressionthatoneoftheirmainpointsofdisagreementpreciselyisthepar- ticleinterpretationofQFT.Afterall,oneofFraser’scentralclaimsisthataparticle ontologyis incompatible with the theoreticalprinciples of (any conceptuallyrigor- ous) QFT (see especially her [5] paper), while Wallace’s defense of CQFT seems tosupportjustsuchanontology,byarguingfortheconceptualrespectabilityofthe standardmodelofparticlephysics. Oncloserinspection,however,thisconflictturnsouttobeunsubstantial,because FraserandWallaceareactuallyspeakingaboutdifferentthings.WhatFraser[5]crit- icizes is the idea that QFT could be interpreted in terms of a fundamental particle notion, or, more precisely, in terms of quanta, that is, countable entities allowing physicalstatestobecharacterizedbywell-definedFockspaceoccupationnumbers. Indeed,sherightlyarguesthataFockspacerepresentationisingeneralandstrictly speaking only available for free, non-interactingQFT systems; central to her argu- mentisthefactthatfreeandinteractingQFTsystemsinvolveunitarilyinequivalent representations(thiscanalsobeseenasaconsequenceofHaag’stheorem). Particles,CutoffsandInequivalentRepresentations 3 DoesWallace [4] have in mind arguingfor a fundamentalontologyof particles whenhespeaksof“takingparticlephysicsseriously”?Clearlynot.Foronething,he doesnotdenythefoundationalimportanceoftheexistenceof(‘infra-red’)unitarily inequivalent representations ([4, 123-124];we will come back to the role unitarily inequivalentrepresentationsplayinthedebatebelow).Moreimportantly,hedoesnot criticizeFraser’santi-quantaargumentperse,butonlytheconclusionshedrawsfrom itasregardsthequestionwhethertherecouldbeapproximateagreementonmattersof ontologybetweenCQFT and(a possiblefutureinteractingmodelof)AQFT. Fraser has given a negative answer to that question, insisting that “a theory according to which quanta exist is not approximatelyequivalent to a theory according to which quantadonotexist”[1,560].Wallacereplies: Well, maybenot,butitisapproximatelyequivalentto atheoryaccordingto which quanta do approximatelyexist! And if AQFT (more precisely, if this supposedinteracting algebraic quantumfield theory)does not admit quanta inatleastsomeapproximatesense,thensomuchtheworseforit:theevidence fortheelectronisreasonablyconclusive.[4,123] ThisshowsthatWallaceisnotinthebusinessofdefendingafundamentalparticle interpretation of QFT. He is merely committed to the “approximate existence” of particles,whichmeansthatparticlesarenotbasicentitiesintheirownright,but“just certainpatternsofexcitationsinthe[quantum]fields”[3,73].1Therefore,evenifone takesCQFTtobethebesttheoryoftherelevantdomain(andacceptsthatweshould beontologicallycommittedtotheentitiesthataresaidtoexistaccordingtoourbest scientifictheories),theresultingfundamentalontologywillnotbeoneofparticles.2 Wethushaveanargumentagainstthefundamentalityofparticleswhichisinternalto CQFT. Furthermore, there is a second, more general argument, which is external to CQFT. It derives from the fact that CQFT, by its very nature, cannot even pretend tobeafundamentaltheory:aswewillseebelow,CQFTgenerallygoestogetherwith ashort-distancecutoffassumption(itmakesthetheoryfarbetterdefinedby‘freezing out’thehighenergyorshortlenghtscaledegreesoffreedomthatareresponsiblefor divergent behavior), which is naturally understood as indicating the breakdown of thetheoryatsomeshortlengthscales(renormalizationtheorythenensurestheepis- temic innocuousness of this assumption). This is usually expressed by saying that CQFT is an effective (as opposed to fundamental) theory. [3, 46-50] illustrates the status of CQFT by means of an analogy to classical mechanics (CM) and its rela- tionto non-relativisticQM. Whenoneapproachestheclassical limitofQM, oneis consideringa certainregimewhereCM isapproximatelyisomorphictoQM, in the sensethatpredictionsofCMarereasonablygoodapproximations,orisomorphicto, those of the lower level theory, QM. Wallace now applies the same argumentative structure to CQFT, assuming that it breaks down at some small length scale where 1 WallaceandTimpson[6,707]alsoemphasizethenon-fundamentalcharacterofparticlesinQFT.In contrastto[3,4],theyevenseemtoadvocateAQFTasaguidetofundamentalontology(711-712).This reinforcestheclaimthatanAQFT-fueledrefusalofparticlesatthefundamentallevelisfullycompatible withacommitmenttoparticlesasnon-fundamentalentities. 2 Notethat,despiteitsname,QFTdoesnotstraightforwardlysupportafieldontologyeither[7]. 4 MatthiasEggetal. another,as-yet-unknownphysicaltheoryXissupposedtocomeintothescene.Then, thereshouldbeanisomorphismbetweenthetwo,inthesamewayasthereisacer- tainregimewhereCMisapproximatelyisomorphictoQM,orinotherwords,there should be a particular length-scale above which the predictionsof CQFT are good approximationsto those of the theory X. According to Wallace, then, the ontology of CQFT is the kindof patternor structurethatemergesfromthe deepertheoryX. Therefore,evenif(contrarytowhatwesawinthepreviousparagraph)thebasicon- tologyofCQFTincludedparticles,particleswouldstillnotbetrulyfundamental,due totheinherentlynon-fundamentalcharacterofCQFT.3 Tosumup,Wallaceclearlydoesnotadvocatethekindof(fundamental)particle ontology that Fraser primarily attacks. However, the following passage shows that shetakesissueevenwithhismodestcommitmentto(non-fundamental)particles: Wallaceconsidersthe‘particle’concepttobeanemergentconcept[...].For thistobeaviableresponse,thecogencyofthedistinctionbetweenfundamen- talandlessfundamentalentitiesmustbedefendedandacasemustbemade foradmittingadditional,non-fundamentalentitiesintoourontology.[5,858] Butthisobviouslyisageneralconcernaboutnon-fundamentalentities;itapplies to molecules,organisms,orplanetsnoless thanto particles(in Wallace’ssense). If that were the main point of conflict between Fraser and Wallace, it would be hard tounderstandwhytheirdebateshouldrevolvearoundquantumfieldtheoryasmuch asitdoes.Therefore,the existenceofparticlescannotbewhatthisdebateis about; ultimately,both Fraser and Wallace—relying on argumentsfromAQFT and CQFT respectively—agreeaboutparticlesbeingnon-fundamentalinQFT. 3 Renormalization:Field-theoreticbreakdownorunderdetermination? What,then,isthecoreofthedisagreementbetweenFraserandWallace?Onapurely methodologicallevel, the central question is easily identified by looking at the ab- stracts of [2] and [4]: is it AQFT or CQFT that should be subject to foundational analysis?NowtheprevioussectionhasshownthatFraser’sandWallace’sdisagree- mentonthis methodologicalquestionneednotimplya disagreementon mattersof ontology, because AQFT and CQFT simply address different kinds of ontological questions:whiletheformerisconcernedwithfundamentalontology,thelatteryields whatcouldbecalledaneffectiveontology. However,thereisstillasubstantialpointofontologicaldisagreement,whichhas todowiththeinterpretationofrenormalizationtechniques,andinparticular,theap- plicabilityoffieldtheoreticconceptsatlengthscalesbeyondthecutofflength.Con- siderthefollowingargumentbyWallace: 1. We have a verywell confirmedtheory (the Standard Model, understood asaCQFT),oneofwhosecentralclaimsisthatfielddegreesoffreedom arefrozenoutatsufficientlyshortlengthscales. 3 Thisdoesnotmean,however,thatnoontologicallessonscanbedrawnfromaneffectivetheorylike CQFT.Wewillreturntothispointinsection5. Particles,CutoffsandInequivalentRepresentations 5 2. Asgoodscientificrealists,weshouldtentativelyacceptthatclaimasap- proximatelytrue. 3. SinceAQFTdeniesthatclaim,itsbasicstructureiswrong. 4. Sonowonderthatwecan’tconstructempiricallyadequatetheorieswithin thatstructure![4,120] Inherresponse,Fraser [2,133]criticizesthemovefrom1to 2.Thismove,she claims,isbasedontheso-called‘nomiracles’argumentforscientificrealism,accord- ing to which the empirical success of our well-confirmed scientific theories would be inexplicable unless these theories were approximately true. Against this, Fraser advances the argument from underdetermination,which states that, if mutually in- compatibletheoriesmakethesameempiricalpredictions,empiricalsuccessdoesnot supporttheclaimthatanyparticularoneofthemisapproximatelytrue. Beforediscussingthedifferentrolesthatrenormalizationplaysinthesetwoop- posingarguments,letusnotethatbothofthemhaveaninternalweakness,whichis acknowledgedbytheirrespectiveproponents.Ontheonehand,Wallaceadmitsthat his move from 1 to 2 is a little too quick, because “CQFT, in itself, doesn’t actu- ally require the existence of a cutoff; it just tells us that assuming a cutoff suffices tomakethetheorywell-defined”[4,120].Indeed,oneofthekeyinsightsofmodern renormalizationtheory (repeatedly stressed by both Wallace and Fraser) is that the empiricalcontentofCQFT islargelyinsensitivetowhatpreciselyhappensatsmall lengthscales.Therefore,“itisconsistentwiththeCQFTframeworkthatthetheory’s degreesoffreedomafterallremaindefinedonarbitrarilyshortlengthscales”(ibid.). On the other hand,Fraser hasto acknowledgethatthere is no actualunderdetermi- nation between AQFT and CQFT, because so far, only the latter makes empirical predictionsaboutinteractingsystemsinfourspacetimedimensions[2,132].Weare thusatpresentconfrontedwithmerelyapotentialunderdetermination,premisedon theassumptionthatonedayarealisticAQFTmodelwillbefound. Thattheinterpretationandtheimplicationsoftherenormalizationgroupmethods constituteacentralpointofdisagreementintheFraser-Wallacedebateishighlighted bytheopposedroletheyplayintheabovenomiraclesargumentbyWallace andin Fraser’smainobjectiontoit.AccordingtoWallace,renormalizationtheoryremoves allthemotivationfordevelopingrealisticAQFTmodelsinthedomainofCQFT(i.e. well above the short lengthscale cutoff), since the renormalization group methods showthatthe(hypothetical)AQFTmodelswillhavethesameempiricalpredictions as CQFT (in the relevantdomain).So, evenif CQFT is strictly speakingconsistent withafullfield-theoreticdescriptionatalllenghtscales,thispossibilitydoesnotre- allymatter,thankstorenormalization.However,accordingtoFraser,thisconstitutes aclearcaseoftheoryunderdeterminationbyempiricalevidence,sothatsheregards renormalization theory as supporting her objection to Wallace’s no miracles argu- ment. Atthisstage,twoimportantremarksareinorder.First,letusrepeatthat,asWal- lace [4, 121] clearly points out, there is no actual underdetermination, since there is no AQFT model reproducing the empirical predictions of CQFT (e.g. the Stan- dard Model) for the time being. Second, as Fraser [2, 131] argues, there might be someothermotivationsbesidesnovelempiricalpredictionsforlookingforanalter- 6 MatthiasEggetal. nativeto CQFT (evenin the domainwell abovethe shortlengthscale cutoff),such ascoherencewithcertaintheoretical(e.g.relativistic)principlesorontologicalclar- ity.4(Forinstance,thislatteristheprincipalmotivationbehindBohmianmechanics, which makesthe same predictionsas ‘conventional’or textbookQM; indeed,Wal- lace[4,ftn.14],whoprivilegestheEverettinterpretation,seemstorecognizethathis pointaboutrenormalizationmightbeweakenedinthecasesoftheothermainrealist approachesto the measurementproblemsuch asBohmianmechanicsordynamical collapseapproaches,e.g.GRW.Wewillreturntothispointinsection5below.) AllthistendstoshowthatbothWallace’snomiraclesargumentandFraser’sar- gumentfromunderdetermination—aswellastherespectiverolesofrenormalization theoryin thesearguments—areinconclusive.Thismightalso showthattheirdiver- gent interpretation of the renormalization group methods—however central to the Fraser-Wallacedebate—doesnotconstitutetherealbottomlineofthediscussion.In- deed,asexplicitlyrecognizedbyboth[2,127]and[4,121],whatisreallyatstakeis thestatusofcutoffsinQFT,towhichwenowturn. 4 Thestatusofcutoffs Animportantsourceofinspiration(andjustification)forrenormalizationmethodsin CQFT is the successful application of such methods in condensed matter physics, wherethereisgoodreasontoassumethatthefieldtheoretictreatmentfailsatlength- scales small enoughfor the atomic structureof matter to become relevant.Wallace claims that “nothing prevents us telling exactly the same story in particle physics, provided only that something freezes out the short-distance degrees of freedom on some lengthscalefar belowwhatcurrentexperimentalphysicscan probe”[4, 118]. Inthislineofthought,assumingtheshort-distancebreakdownofQFT,i.e.assuming ashort-distancecutoff,isacrucialstepintherenormalizationprocessandconstitutes thereforea key elementof the conceptualgroundingand of the explanatoryframe- workofCQFT: This,inessence,ishowmodernparticlephysicsdealswiththerenormaliza- tion problem[footnotedeleted]: it is taken to presage an ultimate failure of quantumfield theoryat some shortlengthscale,and oncethe bare existence of that failure is appreciated, the whole of renormalization theory becomes unproblematic,andindeedpredictivelypowerfulinitsownright.[4,119] However,Fraserdetectsanimportantdisanalogybetweencondensedmatterphysics andCQFT:“inthecondensedmattercase,independentevidencefortheexistenceof atoms plays a pivotal role. [...] There is no analogue of this evidence in the QFT case;wedonotpossessevidenceofthissortthatQFTbreaksdownatshortdistance scales”[4,118]. Actually,WallacedoescitesomeevidenceforaneventualbreakdownofQFTat shortdistancescales: 4 [8]alsoprivilegesAQFTonontologicalgrounds. Particles,CutoffsandInequivalentRepresentations 7 OncewegetdowntoPlanckianlengthscales,thefictionthatspacetimeisnon- dynamicalandthatgravitycanbeignoredwillbecomeunsustainable.What- everoursub-Planckianphysicslookslike(stringtheory?twistortheory?loop quantumgravity?non-commutativegeometry?causalsettheory?something as-yet-undreamed-of?)there are pretty powerful reasons not to expect it to looklikequantumfieldtheoryonaclassicalbackgroundspacetime.[4,120- 121] Presumably,Fraserdoesnotacceptthisasevidenceinthesamesenseastheevi- dencefortheexistenceofatoms,becauseitisnotexperimentalevidence.Nowweare notopposedtotheideathatexperimentalevidenceshouldbegivenmoreweightthan theoreticalevidence,5butonemustbecarefulnottoexaggeratetheprivilegegranted toexperimentalevidence,becausesuchanattitude(exemplifiedby[11])wouldun- dermine the motivation of giving any relevance to the results of AQFT in the first place.Instead,theoreticalconsiderationsofthetypecitedbyWallaceshouldbetaken seriously,especiallyincontextswherenodirectexperimentalevidenceisavailable, asisthecaseforthequestionthatconcernsushere(thebreakdownofQFTatsmall lengthscales). Once thisis acknowledged,the theoreticalevidencein favor of a breakdownof QFT atthe Planck scale can be evaluatedagainstthe (equallytheoretical)evidence in favor of the basic commitments of AQFT, which include the claim that a field theoretic treatment is adequate for all lengthscales. Fraser’s general argument for acceptingtheontologicalcommitmentsofAQFT(anddisregardingthoseofCQFT) “isthatQFTisaunificationproject;thegoaloftheprojectistoformulateatheorythat incorporatesbothspecialrelativisticandquantumprinciples”[2,131].6Shegoeson toclaimthatAQFTsatisfiestheminimalcriteriaforsuccessinthisunificationproject (obviously,sinceitisoneofitsexplicitaims),butCQFTdoesnot(CQFTwithcutoff isstrictlyspeakingnotrelativistic,becauseitisnotPoincare´covariant;seesection5 below). Wearethusfacedwithtwoconflictingbodiesoftheoreticalevidenceconcerning thebreakdownofQFTatshortlengthscales,andbothofthemarebasedonconsider- ationsofunification:unificationofquantumtheoryandspecialrelativityinFraser’s case,unificationofquantumtheoryandgeneralrelativityinWallace’scase.Thecru- cialdifferenceisthatFrasertakestheunificationofquantumtheoryandspecialrela- tivitytobepartoftheverydefinitionofwhatQFTis,andthereforetobeprivileged overotherconsiderations,suchasgravitationalconsiderations,thatare“external”to theQFTproject[1,552].Bycontrast,Wallacementionstheunificationofquantum theoryandgeneralrelativitypreciselytoshowthatthesearchforatrulyfundamental theoryhastomovebeyondtheQFTframework. 5 See[9],[10]forarecentversionofscientificrealismbasedonthisidea.Section9.3ofthelatterwork alsocontainsadeeperinvestigationoftheinterplaybetweenexperimentalandtheoreticalconsiderations intheinterpretationofQFT. 6 Frasermightdisagreewithourcharacterizationofthiskindofevidenceastheoretical,assheclaims that “this unification project is alsoempirical, broadly construed, insofar as there is indirect empirical supportforspecialrelativityanditstheoreticalprinciplesandfornon-relativisticquantumtheoryandits theoreticalprinciples”[2,131].However,giventhatthereisindirectempiricalsupportforgeneralrelativity aswell,Wallace’sabove-mentionedargumentsalsocountas“empirical”inthissense. 8 MatthiasEggetal. ThisactuallytouchestheheartofthedisagreementbetweenFraserandWallace. Since Fraser primarilyconsidersQFT as the projectof combiningthe principlesof quantumtheorywiththoseofspecialrelativity,shecannotacceptcutoffssincethey destroyPoincare´covarianceandtherebycontradicttheveryideaofthisproject.(She might be willing to accept them in the last resort, after it has been shown that the QFT project as she understands it fails). On the other hand, according to Wallace, such in principle understandingof QFT is illegitimate in the face of its failure (for thetimebeing)toaccountfortheQFTphenomenology(i.e.theStandardModel).By contrast,thislatterisnaturallyaccountedforwithintheframeworkofCQFTunder- stoodasaneffectivetheorybreakingdownatshortlenthscales,andpossiblyapprox- imating some deeper, more fundamental theory (e.g. one that would take quantum gravitationaleffects into account).In this context,assuming a short-distance cutoff implementsthiseffectiveunderstanding,whichcanwellbe motivatedby‘external’ considerations.Renormalizationtheory ensuresthat the details of this implementa- tionareirrelevantforthepredictiveandexplanatorypowerofCQFT ([3]alludesto somebroadstructuralistunderstandingoftheoriesinordertojustifytheexplanatory powerofanapproximatetheorylikeCQFT). Therefore,we see that Fraser and Wallace ultimately disagree about what QFT is,andthisdisagreementmanifestsitselfintheiroppositeunderstandingoftheshort- distancecutoff.Beforecontinuingourdiscussionoftheirdisagreementaboutthevery natureofQFT,weendthissectionbyhighlightinganimportantpointofagreement asregardsthecutoffs.Somewhatparadoxicallyatfirstsight,FraserandWallacere- joinonthestatusofthelong-distance(or‘infra-red’)cutoff.Indeed,ifFraserholds both short- and long-distancecutoffsas unsatisfactory,Wallace seems to think that the(external,mainlycosmological)motivationsforthelatterarefarweakerthanfor theformer(observationaldata,i.e.experimentalevidence,actuallyseemstospeakin favorofanopen,infiniteuniverse,thatis,againstanygenuinelong-distancecutoff). Asaconsequence,Wallaceseriouslyconsidersthepossibilityoftheexistenceofin- finitelymanyQFTdegreesoffreedomandthereforeofinequivalentrepresentations. (SincetheStone-vonNeumanntheoremdoesnotapplytoinfinitedimensionalcases, the quantizationofa theorywithinfinitely manydegreesoffreedomleadsto many unitarilyinequivalentrepresentationsoftherelevantalgebraencodingtheappropri- atecommutationrelations.)Itisinterestingtonotethattheexistenceofinequivalent representationspreciselyconstitutesoneoftheoriginalmotivationsforAQFT,which focusesonthealgebraicratherthanrepresentationalstructuresofthetheory.Inshort, the foundational importance of unitarily inequivalent representations is part of the commongroundsharedbybothFraserandWallace.Wewillcomebacktothispoint insection6. 5 WhatexactlyisQFT? Aswehaveseenintheprevioussection,FraserintendsQFTasthetheorythatbest unifies the principles of quantum mechanics and special relativity (SR). From this naturallyfollowsthedefinitionofwhatQFTshouldbe:inordertobeanacceptable QFT, a theorymustincorporatetheprinciplesofbothquantummechanicsandspe- Particles,CutoffsandInequivalentRepresentations 9 cialrelativity,inshortQFT=QM+SR.Thisdefinitionfindsitsrootsinthehistory ofquantumfieldtheory,7sincealreadyfromtheTwentiesandtheThirtiesphysicists weretryingtoincorporatewithinthedomainofquantummechanicsthetreatmentof theelectromagneticfield,giventhefactthatstandardQMdoesnotprovideanytreat- mentofrelativisticparticles.Thus,aquantumtheoryabletodescribetheseparticles was needed, and QFT is the first theory which yields this results, providing a suc- cessfultreatmentofphotons.InadditiontothesehistoricalconsiderationsthatFraser considerstojustifyherdefinition,itshouldbenotedthatitiscongenialtoherargu- ment,sinceamongthevariantsofQFTsheconsidersonlyAQFTsatisfiesit,beinga projectdirectlyaimedtofindaconsistent(explicitly)relativisticQFT.Certainlythis factplaysapivotalroleinarguingwhyoneshouldprefertheaxiomaticvariantover therivals. Showinghowthe interactionpictureis inconsistentasa consequenceof Haag’s theorem,Fraser[1,544-553]discussesvariousmodalitiestosolvetheinconsistency: ontheonehandwehaveaformalor‘principled’response(AQFT),exemplifiedby theGlimmandJaffemodel,andontheotherhand,theapproaches,towhichCQFT belongs,relyingonrenormalizationmethodsandinparticularontheintroductionof cutoffsto solve the issue. The introductionof cutoffsimplies that the theory under considerationis not covariantunder Poincare´ transformationsand in brief that it is notrelativistic.8AccordingtoFraser’sdefinitionofQFT,ifatheoryisnotrelativistic, itcannotbeagoodcandidateforaQFT. Therefore,FraserdoesnotonlyclaimthatweshouldpreferAQFTforitsclarity insolvingtheinconsistencyoftheinteractionpicture,butshealsogivesanotherargu- ment,namelythatAQFTisgenuinelyrelativisticandthereby,unlikeitsrivals,fulfills the minimal criterion for being a QFT in the first place. This shows how Fraser’s definitionofaQFTplaysacrucialroleinherargument. Aradicalconsequenceofthisapproachisthatagreatvarietyofdifferentformu- lationsandapproachestoQFTaresimplyruledoutinprinciple.ApartfromCQFT, theBohmianapproachtoQFTconstitutesanothergoodexampleofsuchanalterna- tive ruled out from the start (for a recent and accessible overview of this approach to QFT, see [14]). This latter approach actually highlightsimportantaspects of the discussion, because the Bohmian versions of QFT share crucial features with both CQFTandAQFT. Letusdiscussthesesimilaritiesinturn. Ontheonehand,theBohmianQFTsrelycompletelyonthemathematicalstruc- turesofCQFT,meaningthatalltheperturbationandscatteringtheoryonwhichthe standard QFT is based is simply retained without modifications in this theoretical framework.Thisfactimpliesthat,asinthenon-relativisticcase,theBohmiantheo- riesaddtothestandardquantumtheoryasetof(so-called‘primitive’)variablesand providetheirdynamicallawsofmotion,leavinguntouchedthemathematicalmachin- eryofbothstandardQMandQFT.TheconsequencefortheseBohmianquantumfield theoriesis thatthey arenotgenuinelyrelativistic andthereforefalloutsideFraser’s 7 Thereaderinterestedinhistoricalaspectsofthetheoryshouldreferto[12]and[13] 8 See[3,50-52]fordifferentwaystoaddresstheproblemofPoincare´ non-covariance fromtheper- spectiveofCQFT. 10 MatthiasEggetal. definition of QFT. Furthermore, the fact that these theories rely on the very same regularization techniques as CQFT makes them in some sense explicitly effective, ratherthanfundamentaltheories[15].Thisraisestheissueoftheontologicalimpact of(explicitly)effectivetheories. AccordingtotheBohmianapproach,oneshouldbeabletoprovideaclear(even ifnon-fundamental)ontologicalpictureforeffectivetheoriestoo.Moreparticularly, suchapictureshouldbeabletoprovideanexplicitaccountoftheempiricaldatathat constitute the basis for empirical confirmation, by introducing primitive variables referringtopositionsin3-dimensionalphysicalspace,thetemporaldevelopmentof which is then describedby the theory.(Thisis the spiritof John S. Bell’s questfor ‘local beables’, which is echoed by the so-called ‘primitive ontology’ approach in thecurrentdebateonthefoundationsofquantumtheory;see[16]foranup-to-date review.) On the other hand, Bohmian QFTs share with AQFT a striving for the kind of conceptualandontologicalclaritywhichFraserandothersfindlackinginCQFT(see section3above).ByaselectionofeitheraparticleorafieldontologytheBohmian QFTsaimtoconstructatheorywithoutambiguousnotionsappearingintheaxioms andabletosolvethemeasurementproblem,whichcarriesoverintactinthepassage fromnon-relativisticQMtoQFT(forarecentdiscussionofthemeasurementprob- leminQFTsee[17]),followingthestrategyappliedinthenon-relativisticcase.These theoriesare by constructionempiricallyequivalentto anyregularized QFT, so they are able to provide a qualitative description of the phenomenologyof the standard modelofparticlephysicsintermsofthemotionoflocalbeables,andthus,theyare abletoexplaininaclearmannerthephenomenaofparticlecreationandannihilation. Exactly as in the case of AQFT, these Bohmian quantum field theories try to pro- poseaclearsolutiontotheconceptualissuesaffectingCQFT.Theydosobystarting fromthedefinitionofthelocalbeablesofthetheoryandtheirrespectivedynamical equations. ClearlytheBohmianandthealgebraicapproachestoQFTaimtosolvedifferent problems of CQFT:9 on the one hand, one has the usual problems of QM which are simply inherited by QFT such as the measurement problem and the lack of a clear ontology. These problems imply that QFT is not a theory able to provide a clear description of the physical processes taking place at the subatomic level. For instance,betweenthefreeasymptoticstatesatminusandplusinfinity,CQFTdoesnot yieldadescriptionofthemotionandinteractionsofthequantumobjectsgoingonin scatteringprocesses.BohmianapproachestoQFTaimataddressingtheseproblems. On the other hand, AQFT starts from a rigorous unification of QM and SR trying to constructa theorywhichis genuinelyrelativistic andable to removethe cutoffs, givinglessimportancetotheontologicalshortcomingsofCQFT. However,oneshouldacknowledgethatboththesetworesearchprogramsarewor- thyofconsideration,sincetheymightprovidevaluablesolutionstothewidespectrum ofthewell-knownproblemsaffectingCQFT.Moreimportantly,sincethesetwoalter- nativeapproachestoQFTrelyondifferentassumptionsandsolvedifferentproblems 9 ThesetwoapproacheshaveclearlyadifferentscopeandinsomesenseBohmianQFTislessambitious thanAQFT,butfarmoreempiricallysuccessful,sinceitisbuilttobeempiricallyequivalenttoCQFTas explainedabove.