(R. Penrose)On The Nature Of Quantum Geometry

RogerPenroseOntheNatureofQuantumGeometryAsawayofhonoringProfessorWheeleronhissixtiethbirthday,IproposetotakethisopportunitytoelaborateuponcertainsomewhatspeculativeideaswhichIhavetriedtohintatonoccasion,concerningthepossiblenatureofaquantizedspace-time.ThereaderwillnotneedtobetoodiscerningtorecognizesomesubstantialdierencesbetweentheideasIamproposinghereandthosewhichProfessorWheelerhasonmanyoccasionssoeloquentlyandforcefullyputforward.Nonetheless,thereislittledoubtinmyownmindastotheverygreatinspirationalinuencethatProfessorWheeler'sownviewshavehadinthedevelopmentofseveralofthethoughtswhichIamexpressinghere.Tobeginwith,letmemakeclearthatIdonotnecessarilymeanby\quan-tizedspace-timesomethingwhichcouldbeobtainedbyapplyingstandard(orevennon-standard)techniquesofquantizationtoEinstein'sgeneralthe-oryofrelativity.WhatIwishtosayhasitsrootsinsomethingwhichisreallymoreprimitivethaneitherquantumtheoryorrelativityassuch.Thisisthequestionofthefundamentalroleplayedbythemathematicalconceptofcontinuuminvirtuallythewholeofacceptedpresent-dayphysicaltheory.Notonlydoesthecontinuumoccupyabasicpositioninourmathematicalmodelsofspaceandtime(withtheconcomitantimplicationofacontinuousnatureformanyrelatedphysicalconceptssuchasvelocity,energy,momen-tum,temperature,etc.),butsoalsodoespresent-dayquantumtheoryrestcruciallyonacontinuumconcept,namelyonthetwo-dimensionalcomplexcontinuumofprobabilityamplitudes,thisbeingthecontinuumwhichalsooccursinthesuperpositionlaw.LetmesayattheoutsetthatIamnothappywiththisstateofaairsinphysicaltheory.Themathematicalcontinuumhasalwaysseemedtometocontainmanyfeatureswhicharereallyveryforeigntophysics.Thispointhasbeenarguedforcefully,particularlybySchrodinger[1]andalsobyanumberofotherphysicistsandphilosophers[2].Ifoneistoacceptthephysicalrealityofthecontinuum,thenonemustacceptthatthereareas0ThispaperoriginallyappearedinMagicWithoutMagic,editedbyJ.Klauder,Free-man,SanFrancisco,1972,pp.333{354.1manypointsinavolumeofdiameter1013cmor1033cmor101000cmasthereareintheentireuniverse.Indeed,onemustaccepttheexistenceofmorepointsthantherearerationalnumbersbetweenanytwopointsinspacenomatterhowclosetogethertheymaybe.(Andwehaveseenthatquantumtheorycannotreallyeliminatethisproblem,sinceitbringsinitsowncomplexcontinuum.)Itseemsclearthatsuch\pointshaveactuallyverylittletodowithphysicalreality.Nevertheless,theirpostulatedexistenceisvirtuallyessentialtocontemporaryphysicaltheory.Theconceptsofopenandclosedsets,forexample,dependvitallyonthiscontinuumofpointsandareessentiallymeaninglessinstrictlyphysicalterms,butsuchconceptscanbeusedtogreateectwhenmathematicaltheoryisappliedtophysics(Forexample,inprovingglobal\singularitytheoremsaboutspace-times,etc.[3,4,5]).Ithinkitmustbethecasethattheall-pervadinguseofthecontinuuminphysicsstemsfromitsmathematicalutilityratherthanfromanyessentialphysicalrealitythatitmaypossess.However,itisnotevenquiteclearthatsuchuseofthecontinuumisnot,tosomeextentahistoricalaccident.Foral-thoughtheessentialmathematicalideascanbetracedbacktoEudoxus(4thcenturyB.C.),itwasnotuntilsometimeafterNewtonandLeibnizinventedthecalculusthatitwasfelttobenecessarytoformalizeandmakecom-pletelyrigorousthemathematicalcontinuumconcept.Butthereareother\nonstandardcontinua[6,7]dierentfromtheonethathasnowbecomeconventional,whichcouldequallywellhavebeenadoptedinordertomakethecalculusrigorous.Inthesenonstandardcontinua,\innitesimaland\inniteelementsareintroducedandaretreatedasbeingjustas\realasthe\realnumbersofconventionalanalysis.Infact,ithasoccasionallybeenarguedthatnonstandardanalysismightreallyhavebeenamorenaturalde-velopmentoftheideasofNewton,Leibniz,andEulerthantheactualanalysiswhichCauchy,Weierstrass,andothersnallyformalized.(Inaddition,non-standardanalysisallowsonetodenesuchconceptsasDiracdelta-functionssothattheybecomeeectively\ordinary'functions[8].)Ifthehistoryofmathematicshaddevelopeddierently,thenwemight,bynow,haveformedaverydierentviewfromtheonenowprevalentofthenatureofspaceandtime,andofmanyotherphysicalconcepts.Idonotwanttoimplyherethatnon-standardanalysisoughttobeem-ployedinphysicaltheory.Iwishmerelytopointoutthelackofrmfounda-tionforassigninganyphysicalrealitytotheconventionalcontinuumconcept.Myownviewisthatultimatelyphysicallawsshouldndtheirmostnatural2expressionintermsofessentiallycombinatorialprinciples,thatistosay,intermsofniteprocessessuchascountingorotherbasicallysimplemanipula-tiveprocedures.Thus,inaccordancewithsuchaview,shouldemergesomeformofdiscreteorcombinatorialspace-time.Idonotmeanthatnecessarilyweshouldarriveataspace-timecontainingadiscretesetofpoints,suchasthelatticespace-timeofSchild[9],orsomediscretecausalspace[10]or,forexample,somestructurebasedonAhmavaara'slargeniteeld[11].Iwouldexpect,rather,thattheconceptofaspace-timecomposedofpointsshouldceasetobeanappropriateone|exceptinsomekindoflimitingsense.Butifpointsarenottobethebasicelementsofthediscretespace-time,thenhowarewetodecidewhatthesebasicelementsshouldinfactbe?Itismyviewthatanessentialinsightintothenatureoftheappropri-atecombin