Fundamental Theories of Physics 190 Edward Anderson The Problem of Time Quantum Mechanics Versus General Relativity Fundamental Theories of Physics Volume 190 SeriesEditors HenkvanBeijeren,Utrecht,TheNetherlands PhilippeBlanchard,Bielefeld,Germany PaulBusch,York,UnitedKingdom BobCoecke,Oxford,UnitedKingdom DennisDieks,Utrecht,TheNetherlands BiancaDittrich,Waterloo,Canada DetlefDürr,München,Germany RuthDurrer,Geneva,Switzerland RomanFrigg,London,UnitedKingdom ChristopherFuchs,Boston,USA GiancarloGhirardi,Trieste,Italy DomenicoJ.W.Giulini,Bremen,Germany GreggJaeger,Boston,USA ClausKiefer,Köln,Germany NicolaasP.Landsman,Nijmegen,TheNetherlands ChristianMaes,Leuven,Belgium MioMurao,Tokyo,Japan HermannNicolai,Potsdam,Germany VesselinPetkov,Montreal,Canada LauraRuetsche,AnnArbor,USA MairiSakellariadou,London,UnitedKingdom AlwynvanderMerwe,Denver,USA RainerVerch,Leipzig,Germany ReinhardF.Werner,Hannover,Germany ChristianWüthrich,Geneva,Switzerland Lai-SangYoung,NewYorkCity,USA The international monograph series “Fundamental Theories of Physics” aims to stretchtheboundariesofmainstreamphysicsbyclarifyinganddevelopingthethe- oretical and conceptual framework of physics and by applying it to a wide range ofinterdisciplinaryscientificfields.Originalcontributionsinwell-establishedfields such as Quantum Physics, Relativity Theory, Cosmology, Quantum Field Theory, Statistical Mechanics and Nonlinear Dynamics are welcome. The series also pro- videsaforumfornon-conventionalapproachestothesefields.Publicationsshould presentnewandpromisingideas,withprospectsfortheirfurtherdevelopment,and carefullyshow howthey connectto conventionalviews of the topic. Althoughthe aim of this series is to go beyond established mainstream physics, a high profile andopen-mindedEditorialBoardwillevaluateallcontributionscarefullytoensure ahighscientificstandard. Moreinformationaboutthisseriesathttp://www.springer.com/series/6001 Edward Anderson The Problem of Time Quantum Mechanics Versus General Relativity EdwardAnderson DAMTP CentreforMathematicalSciences Cambridge,UK ISSN0168-1222 ISSN2365-6425(electronic) FundamentalTheoriesofPhysics ISBN978-3-319-58846-9 ISBN978-3-319-58848-3(eBook) DOI10.1007/978-3-319-58848-3 LibraryofCongressControlNumber:2017954323 ©SpringerInternationalPublishingAG2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface ThisbookconcernsthefoundationsofQuantumGravity,inparticularfromacon- ceptualpointofview.Itprovidesaself-containedintroductiontothistopic,resting on particular features of the accepted Paradigms of Physics: Newtonian Physics, SpecialRelativity(SR),QuantumMechanics(QM),QuantumFieldTheory(QFT) andGeneralRelativity(GR).InapproachingQuantumGravity,manyconceptualis- suesturnouttoberelatedtonotionsoftime.Thisoccursbecausenotionsoftimeare substantiallydifferentacrosstheseParadigms.Afirstexampleinwhichthisoccurs isQMversusGR.IshamandKuchaˇrformalizedthestudyofsuchdiscrepanciesbe- tweennotionsoftime.Theydidsobygivingaconceptualclassificationofthemany time-relatedreasonswhyawiderangeofattemptedapproachestoQuantumGravity fail to be satisfactory, in two seminal Reviews in the early 1990’s [483, 586]. The currentbook’stitular‘ProblemofTime’referstothisconceptualclassification.This isamulti-facetedcollectionofveryinterestingproblemswhichturnouttobeheav- ilyinterlinked.Quiteafewoftheseproblemswerefirstglimpsedinthepioneering worksofWheelerandDeWittinthe1960’s[237,897,899]onthegeometrodynam- icalformulationofGR. The Problem of Time is, in greater generality, a consequence of the mismatch betweenBackgroundDependentandBackgroundIndependent[12,363]Paradigms ofPhysics.NewtonianPhysics,SR,QM,andQFTareallBackgroundDependent, whereas GR is Background Independent and many approaches to Quantum Grav- ity expect this to be Background Independent as well. So, whereas there has been quite widespread belief among theoretical physicists that the Problem of Time is a quantum matter, this is a misconception since clearly also Classical Physics can exhibitmismatchesbetweenBackgroundDependentandBackgroundIndependent Paradigms. Once this is taken into account, models exhibiting classical versions of the Problem of Time turn out to provide substantial conceptual insight into the harderquantumversionsoftheProblemofTime. Itisthusclearthatfurtherexplanationofwhatthisbook(and[483,586])takes theProblemofTimetoconsistofisbestdoneafterthefollowing. v vi Preface A) Presenting the standard Paradigms of Physics and explaining how notions of timedifferacrossthese. B) OutliningwhateachofQuantumGravityandBackgroundIndependenceare. N.B. that A)—in Chaps. 1 to 8’s account of notions of time and of space and of the diversity of physical laws across accepted Paradigms in Physics—serves as a preamble.Itisnottobemistakenforintroductionofthematerialwhichtherestof thebookgreatlyexpandsupon,whichis,rather,Chaps.9,10and12ontheProblem ofTimeandBackgroundIndependenceissueswhichunderliethis.Chapters1to8 enter,rather,intoassemblingcheckliststotestfoundationalandQuantumGravita- tionalcandidatetimesagainst,toseeifthesemerittobecalledtimefunctions,and towardbuildinguptowardplausibleQuantumGravitationallawsinChap.11.While these may be somewhat unexpected and indirect uses of Chaps. 1 to 8’s material, thisistheintendeduseoftheseChaptersinwritingthisbook.Bywayofexplana- tion, this book’s main topic happens to benefit from a preliminary presentation of thetypesoflawandnotionsoftimeandspacethateachoftheestablishedtheories has. This is prudent given that this book’s main topic is a systematic analysis of a widerangeofmorespeculativefoundationalandQuantumGravitationalprograms in which only subsets of standard theories’ laws and temporal and spatial notions arekept. FromtheAcceptedParadigmsofPhysicstoQuantum Gravity Thisbookthusbeginsbyconsideringtimeandclockconcepts,alongsidesupporting notionsofspace,length-measuringdevices,spacetimeandframes.Chapter1givesa largelytheory-freeconceptualoutlineofthese,intendedforaverywideanddiverse multidisciplinaryaudience. Each of the Newtonian Paradigm, SR, QM, QFT and GR are then covered in turn,inChaps.2to7.ThistreatmentincludesinoutlinehowtheseParadigmShifts affecttime,clock,space,length-measuring,spacetimeandframeconcepts. This theory by theory development has the further complication of not being a linearventure:theseParadigmsofPhysicsfanoutfromNewtonianMechanicsasin- dicatedinFig.1.a).Threedistincttheoreticaldevelopmentseachbringinoneofthe threeknownfundamentalconstantsofNature:1 Newton’sgravitationalconstantG, thereciprocalofthespeedoflightc,andPlanck’sconstant(cid:2),asfollows. 1Valuesof—anduncertaintiesin—thesefundamentalconstantsareasfollows[661].Thespeedof lightinvacuocisdefinedtobeexactly299,792,458ms−1duetothemetreitselfbeingdefinedin termsofc(seeChap.1.13).Planck’sconstant(cid:2)=1.054571800(13)×10−34kgm2s−1.Newton’s gravitationalconstantG=6.67408(31)×10−11 m3kg−1s−1.TheerroranalysisforthePlanck unitsisverystraightforward:sinceGisbyfartheleastaccuratelyknownofthefundamentalunits, theerrorinthisswampstheothers. Preface vii Fig.1 a)Planckiancubeoffundamentalphysicaltheories.Here,NMstandsforNewtonianMe- chanicsand(Q)NGstandsfor(Quantum)NewtonianGravity.b)indicatesthe‘Newton–Einstein’ (alias‘classical’,inthesenseof‘non-quantum’)plane,andthe‘ParticlePhysics’(i.e.‘non-grav- itational’)plane.c)Gordiancube:hereitisperceivedthatdifferentroutesalongtheedgestothe ‘finalQuantumGravityvertex’maynotcommute(inthealgebraicsense,uponviewingitsedges asmaps).d)CuttingtheGordiancube?[Or‘thinkingoutsideofthebox’thatisa)?]Here,(Q)RPM standsfor(Quantum)RelationalParticleMechanics;seethenextSubsectionforanoutlineofwhat (Quantum)Gestaltmeans.e)IndicatesthecorrespondingcoveragebyChaptersinPartI 1) G is significant when gravitational force is non-negligible in comparison to whichever forces dominate the physics. G was originally formulated in New- tonianGravity,whichlieswithinNewton’sParadigmofPhysics,whereas2)and 3)eachadditionallyrepresentintroducinganewParadigm. 2) cisnon-negligiblyfiniteinSR[736];thisisrelevanttoobjectswhosevelocities varenon-negligiblecomparedtoc. 3) (cid:2) is significant in Quantum Mechanics (QM) [599], due to certain quantities cominginminimum-sizedpackets.Forinstance,angularmomentumcomesin(cid:2) (or(cid:2)/2)sizedpackets.Thisisrelevantinsituationsinvolvingquantitiescompa- rableinsizetothecorrespondingminimumpackets. Pairwiseincorporationsoftheseconstantsarefurthermoreasfollows(Fig.1.a). 4) RelativisticQFT[712]involvescand(cid:2)together,correspondingtotheCompton wavelength l =(cid:2)/mc (1) C viii Preface length-scalefora‘particle’ofmassm. 5) GR [874]—in the sense of a Relativistic Theory of Gravitation—considers c and G together, corresponding to strongly gravitating fast-moving objects e.g. confinedtoaroundthescalegivenbytheSchwarzschildradius, r =GM/2c2. (2) Schw Each of the ‘Particle Physics’ and ‘Newton–Einstein’ planes indicated in Fig.1.b)areself-consistenttwo-stepParadigmShifts. 6) Considering(cid:2)andGtogethergives‘QuantumNewtonianGravity’;thisishow- evermuchlessrelevant(ExVI.0).Acharacteristiclengthscaleherewouldbe l :=(cid:2)2/2GMm2. (3) g 7) Finally, ‘Quantum Gravity’ [75, 154, 194, 237–239, 385, 471, 474, 475, 483, 485, 552, 586, 746, 845] is often held to be ‘the’ triple combination at the last vertexofFig.1.a)’s‘cube’oftheories.Thethreefundamentalconstantscombine heretoformthePlanckunits: (cid:2) l = (cid:2)G/c3(cid:2)1.616228(38)×10−35m, (4) Pl (cid:2) t = (cid:2)G/c5(cid:2)5.39116(13)×10−44s, (5) Pl (cid:3) m = (cid:2)c/G(cid:2)2.176470(51)×10−8kg. (6) Pl Thefirsttwooftheseareverysmallcomparedto‘ordinaryphysicalquantities’. [Compare l with the atomic (cid:2)10−10 m and nuclear (cid:2)10−15 m lengthscales, Pl and with the maximum precision of displacement detection in existing gravi- tational wave detectors corresponding to displacements of (cid:2)10−18 m. Com- pare also the ratio of t to the timescales of observational Physics with the Pl maximally accurate clock precision of currently around 1 part in 1016, as per Chap. 1.] On the other hand, m is very large upon considering its interpre- Pl tation as a single ‘fundamental particle’ mass: compare e.g. the proton mass 1.672621898(21)×10−27 kg.BySR’sE=mc2,m correspondstoanenergy Pl scaleE =1.220910(29)×1019 GeV.N.B.thisismuchlargerthanthe102 to Pl 104 GeVrangeofthemostpowerfulparticleacceleratortodate:CERN’sLarge Hadron Collider (LHC). Moreover, as detailed in Chap. 11, the Planck regime is expected to feature in some parts of Black Hole Physics and Early-Universe Cosmology. In particular, this book covers the Quantum Cosmology arena and simplermodelarenasthatexhibitfeaturesofthis. DifferingRolesofTimeand SpaceThroughout theParadigms ofPhysics Space and especially time are moreover not consistently conceived of throughout ‘the Planckian cube of theories’. Due to this, consideration of which units can be Preface ix builtoutoffundamentalconstantsmaynotsufficeasaconceptualframeworkwithin whichtoreconciletheParadigmShiftsofPhysics.Indeed,thisbookexpoundsthat GRinvolvesqualitativelydistinctconceptsoftime,space,spacetimeandframefrom thoseusedinParticlePhysics. On these grounds, this book contends that conceiving of ‘Quantum Gravity’ solely in terms of ‘the Planckian cube’ is a misleading simplification. This con- ceptualdisparitypointsinsteadtodifferentpathsaroundthis‘cube’notcommuting (Fig.1.c).Bythisdisparityandthesubsequentnotoriousdifficultywithitsresolu- tion, it might be more apt to name the cube not ‘Planckian’ but ‘Gordian’: after a notoriousknotoftheancientworldthatwaspresentedtoAlexandertheGreat.Heis supposedtohavedealtwiththisknotby‘thinkingoutofthebox’.Accountsdiffer, however,astowhetherthisinvolvedcuttingitorremovingitfromthewoodenpole itwasmountedupon.Indeed,suggestionsforapproaching‘QuantumGravity’differ amongstthemselvesaswell. A further interplay is that the Theoretical Physics literature often pays little at- tentiontothepropertiesentailedincallinganentityatimeoraclock.Thisisunfor- tunate,becauseanumberofsuchpurportedtimequantitiesdonotstandupagainst a suitable list of temporal properties. We emphasize this point in this book firstly byattributingmathematicalpropertiesto‘timefunctions’,andoperationalcharacter to ‘clocks’, in contradistinction to physical, philosophical and conceptual discus- sionofaspectsoftime.2 Secondly,werefertocandidatetimes,timefunctionsand clocksuntilenoughpropertiesofthesehavebeenestablished.Thereismoreovernot auniquelistofpropertiestocheckagainst,sincedifferentphysicaltheoriesinvolve differentlistsofproperties,asthe‘Gordiancube’inFig.1.cforeshadows. Background Independence inMechanics,GRand Quantum Gravity Following Einstein, a second perspective on the nature of—and motivation for— GR is as a freeing from absolute or background structures. From this perspective, GR is more than just a Relativistic Theory of Gravitation. Such perspectives have subsequentlybeendubbedBackgroundIndependence[12,363,483,485,552,752]; contrastwithhowBackgroundDependentabsolutestructurespervadeallsixofthe othernon-finalverticesofthecube.ThisbookconsidersGRasembodyingbothof theseperspectivesatonce,phrasingthisintheshorthandthatGRisa‘gestalt’ofa RelativisticTheoryofGravitationandofBackgroundIndependence. 2Thisbookprioritizesconceptualmattersconcerningtimeovergivingdetailedspecificsofaccu- ratetimekeeping[82,783].Itdoescontainsomecommentsonsiderealandephemerisastronom- icaltimes,atomicclocks,andSRandGRimplicationsfortimekeeping.Someofthetheoretical conceptsoutlinedinthisbook,moreover,mayeventuallybecomerelevanttotimekeeping:space clocks, extending Earth or Solar System based reference systems to galactic and cosmological scales,andclocksinphysicallyextremeregimes.