The Implications of a Cosmological Information Bou

1THEIMPLICATIONSOFACOSMOLOGICALINFORMATIONBOUNDFORCOMPLEXITY,QUANTUMINFORMATIONANDTHENATUREOFPHYSICALLAWP.C.W.DaviesBEYOND:CenterforFundamentalConceptsinScience,ArizonaStateUniversityP.O.Box876505,Tempe,AZ85287-6505Whereofonecannotspeak,thereofonemustremainsilent.LudwigWittgenstein1Isouruniverseapolynomialoranexponentialplace?ScottAaronson2AbstractThefiniteageoftheuniverseandtheexistenceofcosmologicalhorizonsprovidesastrongargumentthattheobservableuniverserepresentsafinitecausalregionwithfinitematerialandinformationalresources.Asimilarconclusionfollowsfromtheholographicprinciple.InthispaperIaddressthequestionofwhetherthecosmologicalinformationboundhasimplicationsforfundamentalphysics.OrthodoxphysicsisbasedonPlatonism:thelawsaretreatedasinfinitelyprecise,perfect,immutablemathematicalrelationshipsthattranscendthephysicaluniverseandremaintotallyunchangedbyphysicalprocesses,howeverextreme.Ifinsteadthelawsofphysicsareregardedasakintocomputersoftware,withthephysicaluniverseasthecorrespondinghardware,thenthefinitecomputationalcapacityoftheuniverseimposesafundamentallimitontheprecisionofthelawsandthespecifiabilityofphysicalstates.Thatlimitdependsontheageoftheuniverse.Iexaminehowtheimprecisionofthelawsimpactsontheevolutionofhighlyentangledstatesandontheproblemofdarkenergy.21.Whatarethelawsofphysics?GregoryChaitinisundoubtedlyoneofthemostprofoundthinkersofourtime.Ihavedrawnonhisworkatseveralstagesinmyowncareerdevelopment,andespeciallyinformulatingtheideasthatfollow.ItisanhonortocontributetothisvolumetocelebrateChaitin’simportantinsightsintomathematics,computingandphysicalreality.IshouldliketostartwithaquotationfromChaitin’srecentbookMeta-Math:“WhyshouldIbelieveinarealnumberifIcan’tcalculateit,ifIcan’tprovewhatitsbitsare,andifIcan’tevenrefertoit?…Thereallinefrom0to1looksmoreandmorelikeaSwisscheese.”3Inotherwords,thereallineisausefulfiction,anunattainableidealization.ThequestionIwishtoaddresshereishowthissweepingconclusionimpactsonmyownfieldoftheoreticalphysicsandcosmology.Therealline,itsextensiontothecomplexplane,togetherwiththerelatedpropertiesofdifferentiability,playanabsolutelycentralroleintheoreticalphysics,onaccountofthefactthatalltheknownfundamentallawsofphysicsareexpressedintermsofdifferentiablefunctionsdefinedoverthesetofrealorcomplexnumbers.SoIwanttostartbyaskingaverybasic,butsurprisinglylittleaddressedquestion:Whatarethelawsofphysicsandwheredotheycomefrom?Thesubsidiaryquestion,Whydotheyhavetheformthattheydo?Ihavediscussedindetailelsewhere.4Firstletmearticulatetheorthodoxposition,adoptedbymosttheoreticalphysicists,whichisthatthelawsofphysicsareimmutable:absolute,eternal,perfectmathematicalrelationships,infinitelypreciseinform.Thelawswereimprintedontheuniverseatthemomentofcreation,i.e.atthebigbang,andhavesinceremainedfixedinbothspaceandtime.Thepropertiesofthephysicaluniversedependinanobviouswayonthelawsofphysics,butthebasiclawsthemselvesdependnotoneiotaonwhathappensinthephysicaluniverse.Thereisthusafundamentalasymmetry:thestatesoftheworldareaffectedbythelaws,butthelawsarecompletelyunaffectedbythestates–adualismthatgoesbacktothefoundationofphysicswithGalileoandNewton.Theultimatesourceofthelawsisleftvague,butitistacitlyassumedtotranscendtheuniverseitself,i.e.toliebeyondthephysicalworld,andthereforebeyondthescopeofscientificinquiry.Thepropertaskofthephysicist,itisoftensaid,istodiscovertheformsofthelawsusingreasonandexperiment,adoptthempragmatically,andgetonwiththejobofdeterminingtheirconsequences.Inquiryintotheiroriginisdiscouragedasaquasi-religiousquest.TheorthodoxviewofthenatureofphysicallawsconformswelltothemathematicaldoctrineofPlatonism.Platoregardedmathematicalformsandrelationshipsasenjoyingarealexistenceinanotherworldlyrealm,wheremathematicianscomeupontheminavoyageofintellectualdiscovery.APlatonistregardsmathematicsaspossessinganexistenceindependentofthephysicaluniverse,ratherthanbeingaproductofthehumanbrain.AnessentialqualityofthePlatonicheavenisthatthemathematicalformsitcontainsareperfect.Forexample,circlesareexactlyround,incontrasttocirclesinthephysicaluniverse,whicharealwaysflawedapproximationstotheidealizedPlatonicforms.MosttheoreticalphysicistsarebytemperamentPlatonists.Theyenvisagethelawsofphysicstooasperfectidealizedmathematicalrelationshipsandoperationsthatreallyexist,locatedinanabstractrealmtranscendingthephysicaluniverse.IshallcallthisviewpointphysicalPlatonismtodistinguishitfrommathematicalPlatonism.NewtonwasaphysicalPlatonist,andcasthislawsofmechanicsandgravitationintermsofwhatwewouldnowcallrealnumbersanddifferentiable3functions.TakingNewton’slawsseriouslyimpliesacceptinginfiniteandinfinitesimalquantities,andarbitraryprecision.Theidealized,PlatonicnotionofthelawsofphysicsreacheditszenithwiththefamousclaimofLaplace,concerninganomniscientdemon.Laplacepointedoutthatthestatesofacloseddeterministicsystem,suchasafinitecollectionofparticlessubjecttothelawsofNewtonianmechanics,arecompletelyfixedoncetheinitialconditionsarespecified5“Wemayregardthepresentstateoftheuniverseastheeffectofitspastandthecause