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The Road to Reality.pdf

The Road to Reality

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2010-06-06 0人阅读 0 0 0 暂无简介 举报

简介:本文档为《The Road to Realitypdf》,可适用于人文社科领域

THEROADTOREALITYBYROGERPENROSETheEmperor’sNewMind:ConcerningComputers,Minds,andtheLawsofPhysicsShadowsoftheMind:ASearchfortheMissingScienceofConsciousnessRogerPenroseTHEROADTOREALITYACompleteGuidetotheLawsoftheUniverseJONATHANCAPELONDONPublishedbyJonathanCapeCopyright�RogerPenroseRogerPenrosehasassertedhisrightundertheCopyright,DesignsandPatentsActtobeidentifiedastheauthorofthisworkThisbookissoldsubjecttotheconditionthatitshallnot,bywayoftradeorotherwise,belent,resold,hiredout,orotherwisecirculatedwithoutthepublisher’spriorconsentinanyformofbindingorcoverotherthanthatinwhichitispublishedandwithoutasimilarconditionincludingthisconditionbeingimposedonthesubsequentpurchaserFirstpublishedinGreatBritaininbyJonathanCapeRandomHouse,VauxhallBridgeRoad,LondonSWVSARandomHouseAustralia(Pty)LimitedAlfredStreet,MilsonsPoint,Sydney,NewSouthWales,AustraliaRandomHouseNewZealandLimitedPolandRoad,Glenfield,Auckland,NewZealandRandomHouseSouthAfrica(Pty)LimitedEndulini,AJubileeRoad,Parktown,SouthAfricaTheRandomHouseGroupLimitedRegNowwwrandomhousecoukACIPcataloguerecordforthisbookisavailablefromtheBritishLibraryISBN–––PapersusedbyTheRandomHouseGroupLimitedarenatural,recyclableproductsmadefromwoodgrowninsustainableforeststhemanufacturingprocessesconformtotheenvironmentalregulationsofthecountryoforiginPrintedandboundinGreatBritainbyWilliamClowes,Beccles,SuffolkContentsPrefacexvAcknowledgementsxxiiiNotationxxviPrologueTherootsofscienceThequestfortheforcesthatshapetheworldMathematicaltruthIsPlato’smathematicalworld‘real’ThreeworldsandthreedeepmysteriesTheGood,theTrue,andtheBeautifulAnancienttheoremandamodernquestionThePythagoreantheoremEuclid’spostulatesSimilarareasproofofthePythagoreantheoremHyperbolicgeometry:conformalpictureOtherrepresentationsofhyperbolicgeometryHistoricalaspectsofhyperbolicgeometryRelationtophysicalspaceKindsofnumberinthephysicalworldAPythagoreancatastropheTherealnumbersystemRealnumbersinthephysicalworldDonaturalnumbersneedthephysicalworldDiscretenumbersinthephysicalworldMagicalcomplexnumbersThemagicnumber‘i’SolvingequationswithcomplexnumbersvConvergenceofpowerseriesCasparWessel’scomplexplaneHowtoconstructtheMandelbrotsetGeometryoflogarithms,powers,androotsGeometryofcomplexalgebraTheideaofthecomplexlogarithmMultiplevaluedness,naturallogarithmsComplexpowersSomerelationstomodernparticlephysicsRealnumbercalculusWhatmakesanhonestfunctionSlopesoffunctionsHigherderivativesCsmoothfunctionsThe‘Eulerian’notionofafunctionTherulesofdiVerentiationIntegrationComplexnumbercalculusComplexsmoothnessholomorphicfunctionsContourintegrationPowerseriesfromcomplexsmoothnessAnalyticcontinuationRiemannsurfacesandcomplexmappingsTheideaofaRiemannsurfaceConformalmappingsTheRiemannsphereThegenusofacompactRiemannsurfaceTheRiemannmappingtheoremFourierdecompositionandhyperfunctionsFourierseriesFunctionsonacircleFrequencysplittingontheRiemannsphereTheFouriertransformFrequencysplittingfromtheFouriertransformWhatkindoffunctionisappropriateHyperfunctionsContentsviSurfacesComplexdimensionsandrealdimensionsSmoothness,partialderivativesVectorWeldsandformsComponents,scalarproductsTheCauchy–RiemannequationsHypercomplexnumbersThealgebraofquaternionsThephysicalroleofquaternionsGeometryofquaternionsHowtocomposerotationsCliVordalgebrasGrassmannalgebrasManifoldsofndimensionsWhystudyhigherdimensionalmanifoldsManifoldsandcoordinatepatchesScalars,vectors,andcovectorsGrassmannproductsIntegralsofformsExteriorderivativeVolumeelementsummationconventionTensorsabstractindexanddiagrammaticnotationComplexmanifoldsSymmetrygroupsGroupsoftransformationsSubgroupsandsimplegroupsLineartransformationsandmatricesDeterminantsandtracesEigenvaluesandeigenvectorsRepresentationtheoryandLiealgebrasTensorrepresentationspacesreducibilityOrthogonalgroupsUnitarygroupsSymplecticgroupsCalculusonmanifoldsDiVerentiationonamanifoldParalleltransportCovariantderivativeCurvatureandtorsionContentsviiGeodesics,parallelograms,andcurvatureLiederivativeWhatametriccandoforyouSymplecticmanifoldsFibrebundlesandgaugeconnectionsSomephysicalmotivationsforWbrebundlesThemathematicalideaofabundleCrosssectionsofbundlesTheCliVordbundleComplexvectorbundles,(co)tangentbundlesProjectivespacesNontrivialityinabundleconnectionBundlecurvatureTheladderofinWnityFiniteWeldsAWniteorinWnitegeometryforphysicsDiVerentsizesofinWnityCantor’sdiagonalslashPuzzlesinthefoundationsofmathematicsTuringmachinesandGo¨del’stheoremSizesofinWnityinphysicsSpacetimeThespacetimeofAristotelianphysicsSpacetimeforGalileanrelativityNewtoniandynamicsinspacetimetermsTheprincipleofequivalenceCartan’s‘Newtonianspacetime’TheWxedWnitespeedoflightLightconesTheabandonmentofabsolutetimeThespacetimeforEinstein’sgeneralrelativityMinkowskiangeometryEuclideanandMinkowskianspaceThesymmetrygroupsofMinkowskispaceLorentzianorthogonalitythe‘clockparadox’HyperbolicgeometryinMinkowskispaceThecelestialsphereasaRiemannsphereNewtonianenergyand(angular)momentumRelativisticenergyand(angular)momentumContentsviiiTheclassicalWeldsofMaxwellandEinsteinEvolutionawayfromNewtoniandynamicsMaxwell’selectromagnetictheoryConservationandXuxlawsinMaxwelltheoryTheMaxwellWeldasgaugecurvatureTheenergy–momentumtensorEinstein’sWeldequationFurtherissues:cosmologicalconstantWeyltensorGravitationalWeldenergyLagrangiansandHamiltoniansThemagicalLagrangianformalismThemoresymmetricalHamiltonianpictureSmalloscillationsHamiltoniandynamicsassymplecticgeometryLagrangiantreatmentofWeldsHowLagrangiansdrivemoderntheoryThequantumparticleNoncommutingvariablesQuantumHamiltoniansSchro¨dinger’sequationQuantumtheory’sexperimentalbackgroundUnderstandingwave–particledualityWhatisquantum‘reality’The‘holistic’natureofawavefunctionThemysterious‘quantumjumps’ProbabilitydistributioninawavefunctionPositionstatesMomentumspacedescriptionQuantumalgebra,geometry,andspinThequantumproceduresUandRThelinearityofUanditsproblemsforRUnitarystructure,Hilbertspace,DiracnotationUnitaryevolution:Schro¨dingerandHeisenbergQuantum‘observables’yesnomeasurementsprojectorsmeasurementshelicitySpinandspinorsTheRiemannsphereoftwostatesystemsHigherspin:MajoranapictureSphericalharmonicsContentsixRelativisticquantumangularmomentumThegeneralisolatedquantumobjectTheentangledquantumworldQuantummechanicsofmanyparticlesystemsHugenessofmanyparticlestatespaceQuantumentanglementBellinequalitiesBohmtypeEPRexperimentsHardy’sEPRexample:almostprobabilityfreeTwomysteriesofquantumentanglementBosonsandfermionsThequantumstatesofbosonsandfermionsQuantumteleportationQuanglementDirac’selectronandantiparticlesTensionbetweenquantumtheoryandrelativityWhydoantiparticlesimplyquantumWeldsEnergypositivityinquantummechanicsDiYcultieswiththerelativisticenergyformulaThenoninvarianceof=tCliVord–DiracsquarerootofwaveoperatorTheDiracequationDirac’sroutetothepositronThestandardmodelofparticlephysicsTheoriginsofmodernparticlephysicsThezigzagpictureoftheelectronElectroweakinteractionsreXectionasymmetryChargeconjugation,parity,andtimereversalTheelectroweaksymmetrygroupStronglyinteractingparticles‘Colouredquarks’BeyondthestandardmodelQuantumWeldtheoryFundamentalstatusofQFTinmoderntheoryCreationandannihilationoperatorsInWnitedimensionalalgebrasAntiparticlesinQFTAlternativevacuaInteractions:LagrangiansandpathintegralsDivergentpathintegrals:Feynman’sresponseConstructingFeynmangraphstheSmatrixRenormalizationContentsxFeynmangraphsfromLagrangiansFeynmangraphsandthechoiceofvacuumTheBigBanganditsthermodynamiclegacyTimesymmetryindynamicalevolutionSubmicroscopicingredientsEntropyTherobustnessoftheentropyconceptDerivationofthesecondlawornotIsthewholeuniversean‘isolatedsystem’TheroleoftheBigBangBlackholesEventhorizonsandspacetimesingularitiesBlackholeentropyCosmologyConformaldiagramsOurextraordinarilyspecialBigBangSpeculativetheoriesoftheearlyuniverseEarlyuniversespontaneoussymmetrybreakingCosmictopologicaldefectsProblemsforearlyuniversesymmetrybreakingInXationarycosmologyArethemotivationsforinXationvalidTheanthropicprincipleTheBigBang’sspecialnature:ananthropickeyTheWeylcurvaturehypothesisTheHartle–Hawking‘noboundary’proposalCosmologicalparameters:observationalstatusThemeasurementparadoxTheconventionalontologiesofquantumtheoryUnconventionalontologiesforquantumtheoryThedensitymatrixDensitymatricesforspin:theBlochsphereThedensitymatrixinEPRsituationsFAPPphilosophyofenvironmentaldecoherenceSchro¨dinger’scatwith‘Copenhagen’ontologyCanotherconventionalontologiesresolvethe‘cat’WhichunconventionalontologiesmayhelpGravity’sroleinquantumstatereductionIstoday’squantumtheoryheretostayCluesfromcosmologicaltimeasymmetryContentsxiTimeasymmetryinquantumstatereductionHawking’sblackholetemperatureBlackholetemperaturefromcomplexperiodicityKillingvectors,energyXowandtimetravel!EnergyoutXowfromnegativeenergyorbitsHawkingexplosionsAmoreradicalperspectiveSchro¨dinger’slumpFundamentalconXictwithEinstein’sprinciplesPreferredSchro¨dinger–NewtonstatesFELIXandrelatedproposalsOriginofXuctuationsintheearlyuniverseSupersymmetry,supradimensionality,andstringsUnexplainedparametersSupersymmetryThealgebraandgeometryofsupersymmetryHigherdimensionalspacetimeTheoriginalhadronicstringtheoryTowardsastringtheoryoftheworldStringmotivationforextraspacetimedimensionsStringtheoryasquantumgravityStringdynamicsWhydon’tweseetheextraspacedimensionsShouldweacceptthequantumstabilityargumentClassicalinstabilityofextradimensionsIsstringQFTWniteThemagicalCalabi–YauspacesMtheoryStringsandblackholeentropyThe‘holographicprinciple’TheDbraneperspectiveThephysicalstatusofstringtheoryEinstein’snarrowerpathloopvariablesCanonicalquantumgravityThechiralinputtoAshtekar’svariablesTheformofAshtekar’svariablesLoopvariablesThemathematicsofknotsandlinksSpinnetworksStatusofloopquantumgravityMoreradicalperspectivestwistortheoryTheorieswheregeometryhasdiscreteelementsTwistorsaslightraysContentsxiiConformalgroupcompactiWedMinkowskispaceTwistorsashigherdimensionalspinorsBasictwistorgeometryandcoordinatesGeometryoftwistorsasspinningmasslessparticlesTwistorquantumtheoryTwistordescriptionofmasslessWeldsTwistorsheafcohomologyTwistorsandpositivenegativefrequencysplittingThenonlineargravitonTwistorsandgeneralrelativityTowardsatwistortheoryofparticlephysicsThefutureoftwistortheoryWhereliestheroadtorealityGreattheoriesofthcenturyphysicsandbeyondMathematicallydrivenfundamentalphysicsTheroleoffashioninphysicaltheoryCanawrongtheorybeexperimentallyrefutedWhencemayweexpectournextphysicalrevolutionWhatisrealityTherolesofmentalityinphysicaltheoryOurlongmathematicalroadtorealityBeautyandmiraclesDeepquestionsanswered,deeperquestionsposedEpilogueBibliographyIndexContentsxiiiIdedicatethisbooktothememoryofDENNISSCIAMAwhoshowedmetheexcitementofphysicsPrefaceThepurposeofthisbookistoconveytothereadersomefeelingforwhatissurelyoneofthemostimportantandexcitingvoyagesofdiscoverythathumanityhasembarkeduponThisisthesearchfortheunderlyingprinciplesthatgovernthebehaviourofouruniverseItisavoyagethathaslastedformorethantwoandahalfmillennia,soitshouldnotsurpriseusthatsubstantialprogresshasatlastbeenmadeButthisjourneyhasprovedtobeaprofoundlydiYcultone,andrealunderstandinghas,forthemostpart,comebutslowlyThisinherentdiYcultyhasledusinmanyfalsedirectionshenceweshouldlearncautionYetthethcenturyhasdeliveredusextraordinarynewinsightssomesoimpressivethatmanyscientistsoftodayhavevoicedtheopinionthatwemaybeclosetoabasicunderstandingofalltheunderlyingprinciplesofphysicsInmydescriptionsofthecurrentfundamentaltheories,thethcenturyhavingnowdrawntoitsclose,IshalltrytotakeamoresoberviewNotallmyopinionsmaybewelcomedbythese‘optimists’,butIexpectfurtherchangesofdirectiongreatereventhanthoseofthelastcenturyThereaderwillWndthatinthisbookIhavenotshiedawayfrompresentingmathematicalformulae,despitedirewarningsoftheseverereductioninreadershipthatthiswillentailIhavethoughtseriouslyaboutthisquestion,andhavecometotheconclusionthatwhatIhavetosaycannotreasonablybeconveyedwithoutacertainamountofmathematicalnotationandtheexplorationofgenuinemathematicalconceptsTheunderstandingthatwehaveoftheprinciplesthatactuallyunderliethebehaviourofourphysicalworldindeeddependsuponsomeappreciationofitsmathematicsSomepeoplemighttakethisasacausefordespair,astheywillhaveformedthebeliefthattheyhavenocapacityformathematics,nomatterathowelementaryalevelHowcoulditbepossible,theymightwellargue,forthemtocomprehendtheresearchgoingonatthecuttingedgeofphysicaltheoryiftheycannotevenmasterthemanipulationoffractionsWell,IcertainlyseethediYcultyxvYetIamanoptimistinmattersofconveyingunderstandingPerhapsIamanincurableoptimistIwonderwhetherthosereaderswhocannotmanipulatefractionsorthosewhoclaimthattheycannotmanipulatefractionsarenotdeludingthemselvesatleastalittle,andthatagoodproportionofthemactuallyhaveapotentialinthisdirectionthattheyarenotawareofNodoubttherearesomewho,whenconfrontedwithalineofmathematicalsymbols,howeversimplypresented,canseeonlythesternfaceofaparentorteacherwhotriedtoforceintothemanoncomprehendingparrotlikeapparentcompetenceaduty,andadutyaloneandnohintofthemagicorbeautyofthesubjectmightbeallowedtocomethroughPerhapsforsomeitistoolatebut,asIsay,IamanoptimistandIbelievethattherearemanyoutthere,evenamongthosewhocouldnevermasterthemanipulationoffractions,whohavethecapacitytocatchsomeglimpseofawonderfulworldthatIbelievemustbe,toasigniWcantdegree,genuinelyaccessibletothemOneofmymother’sclosestfriends,whenshewasayounggirl,wasamongthosewhocouldnotgraspfractionsThisladyoncetoldmesoherselfaftershehadretiredfromasuccessfulcareerasaballetdancerIwasstillyoung,notyetfullylaunchedinmyactivitiesasamathematician,butwasrecognizedassomeonewhoenjoyedworkinginthatsubject‘It’sallthatcancelling’,shesaidtome,‘Icouldjustnevergetthehangofcancelling’Shewasanelegantandhighlyintelligentwoman,andthereisnodoubtinmymindthatthementalqualitiesthatarerequiredincomprehendingthesophisticatedchoreographythatiscentraltoballetareinnowayinferiortothosewhichmustbebroughttobearonamathematicalproblemSo,grosslyoverestimatingmyexpositionalabilities,Iattempted,asothershaddonebefore,toexplaintoherthesimplicityandlogicalnatureoftheprocedureof‘cancelling’IbelievethatmyeVortswereasunsuccessfulaswerethoseofothers(Incidentally,herfatherhadbeenaprominentscientist,andaF

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