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Page 1: Carlo Rovelli - Web Education
Page 2: Carlo Rovelli - Web Education
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CarloRovelli

REALITY IS NOT WHAT IT SEEMS

TheJourneytoQuantumGravity

TranslatedbySimonCarnellandEricaSegre

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Contents

Author’sNotePreface:WalkingalongtheShore

PARTONERoots

1Grains2TheClassics

PARTTWOTheBeginningoftheRevolution

3Albert4Quanta

PARTTHREEQuantumSpaceandRelationalTime

5SpacetimeisQuantum6QuantaofSpace7TimeDoesNotExist

PARTFOURBeyondSpaceandTime

8BeyondtheBigBang9EmpiricalConfirmations?10QuantumBlackHoles11TheEndofInfinity12Information13Mystery

NotesAnnotatedBibliographyFollowPenguin

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ABOUTTHEAUTHOR

CarloRovelliisatheoreticalphysicistwhohasmadesignificantcontributionstothephysicsofspaceandtime.HehasworkedinItalyandtheUS,andiscurrentlydirectingthequantumgravityresearchgroupoftheCentredePhysiqueThéoriqueinMarseille,France.HisSevenBriefLessonsonPhysicsisaphenomenalinternationalbestsellertranslatedintoforty-onelanguages.

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Author’sNote

Duringmyentireresearchlife,friendsandcuriouspeoplehaveaskedmetoexplainwhatwasgoingoninquantumgravityresearch.Howwasitpossibletostudynewwaysofthinkingaboutspaceandtime?OverandoveragainIhavebeenaskedtowriteapopularaccountofthisresearch.Whilebooksoncosmologyorstringtheoryabound,abookdescribingtheresearchonthequantumnatureofspaceandtime,andonloopquantumgravityinparticular,didnotyetexist.Ihavelonghesitated,becauseIwantedtoconcentrateonresearch.Someyearsago,aftercompletingmytechnicalbookonthesubject,Ifeltthatthecollectiveworkofmanyscientistshadmovedthetopictoastagematureenoughforapopularbook.Thelandscapeweareexploringisenchanting:whykeepithidden?ButIstilldelayedtheproject,becauseIcouldnot‘see’thebookinmyhead.Howtoexplaina

worldwithoutspaceandtime?Onenightin2012,duringalongsolitarydrivefromItalytoFrance,Irealizedthattheonlywaytoexplaininacomprehensiblemannertheongoingmodificationsofthenotionsofspaceandtimewastotellthestoryfromthebeginning:startingfromDemocritus,allthewaythroughtothequantaofspace.Afterall,thisishowIunderstandthestory.Ibegantodesigntheentirebookinmymindwhiledriving,andgotincreasinglyexcited,untilIheardapolicecar’ssirenstellingmetopullover:Iwasdrivingfarabovethespeedlimit.TheItalianpolicemenaskedmepolitelyifIwascrazytodriveatthatspeed.IexplainedthatIhadjustfoundtheideaI’dbeenseekingforsolong;thepolicemanletmegowithoutaticket,andwishedmegoodluckwiththebook.Thisisthebook.ThisbookwaswrittenandfirstpublishedinItalianatthebeginningof2014.Shortlyafterwards,I

wroteafewarticlesonfundamentalphysicsforanItaliannewspaper.AprestigiousItalianpublisher,Adelphi,askedmeforanextendedversionofthesearticles,toappearasasmallbooklet.ThisistheoriginoftheshortbookSevenBriefLessonsonPhysics,whichtomyimmensesurprisehasbecomeaninternationalbestsellerandhasopenedabeautifulchannelofcommunicationbetweenmeandsomanywonderfulreadersallovertheworld.TheSevenLessonswerethuswrittenafterthisbook,andtosomeextenttheyareasynthesisofsomeofthetopicsyoufindhere.IfyouhavereadSevenBriefLessonsonPhysicsandwanttoknowmore,tojourneydeeperintothestrangeworldthatbooksketched,hereyoucanfindmore.WhiletheaccountofestablishedphysicsIgivehereispresentedfromthepeculiarperspectivein

whichIunderstandit,itislargelyuncontroversial.However,thepartofthisbookthatdescribescurrentresearchinquantumgravityismyownpersonalunderstandingofthestateoftheart.Thisistheregionattheboundarybetweenwhatwehaveunderstoodandwhatwedonotyetunderstand,andisstillfarfromachievingconsensus.SomeofmyphysicistcolleagueswillagreewithwhatIwritehere;otherswon’t.Thisistrueforallpresentationsofongoingresearchatthefrontiersofknowledge,butIprefertostateitupfrontandclearly.Thisisnotabookaboutcertainties:itisabookabouttheadventureofmovingtowardstheunknown.Asawhole,thisisatravelbookdescribingoneofthemostspectacularjourneysthathumanityhas

taken:ajourneyoutofourlimitedandparochialviewsofreality,towardsanincreasinglyvastunderstandingofthestructureofthings.Amagicaljourneyoutofourcommon-senseviewofthings,farfromcomplete.

Marseille,4May2016

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Preface:WalkingalongtheShore

Weareobsessedwithourselves.Westudyourhistory,ourpsychology,ourphilosophy,ourgods.Muchofourknowledgerevolvesaroundmanhimself,asifwewerethemostimportantthingintheuniverse.IthinkIlikephysicsbecauseitopensawindowthroughwhichwecanseefurther.Itgivesmethesenseoffreshairenteringthehouse.Whatweseeouttherethroughthewindowisconstantlysurprisingus.Wehavelearnedagreat

dealabouttheuniverse.Inthecourseofthecenturieswehavecometorealizejusthowverymanywrongideaswehad.WethoughtthattheEarthwasflat,andthatitwasthestillcentreofourworld.Thattheuniversewassmall,andunchanging.Webelievedthatmanwasabreedapart,withoutkinshiptotheotheranimals.Wehavelearnedoftheexistenceofquarks,blackholes,particlesoflight,wavesofspace,andoftheextraordinarymolecularstructuresineverycellofourbodies.Thehumanraceislikeagrowingchildwhodiscoverswithamazementthattheworldconsistsnotjustofhisbedroomandplayground,butthatitisvast,andthatthereareathousandthingstodiscover,andinnumerableideasquitedifferentfromthosewithwhichhebegan.Theuniverseismultiformandboundless,andwecontinuetostumbleuponnewaspectsofit.Themorewelearnabouttheworld,themoreweareamazedbyitsvariety,beautyandsimplicity.Butthemorewediscover,themoreweunderstandthatwhatwedon’tyetknowisgreaterthan

whatweknow.Themorepowerfulourtelescopes,thestrangerandmoreunexpectedaretheheavenswesee.Thecloserwelookattheminutedetailofmatter,themorewediscoverofitsprofoundstructure.TodayweseealmosttotheBigBang,thegreatexplosionfromwhich,14billionyearsago,allthegalaxieswereborn–butwehavealreadybeguntoglimpsesomethingbeyondtheBigBang.Wehavelearnedthatspaceiscurved,butalreadyforeseethatthissamespaceiswovenfromvibratingquantumgrains.Ourknowledgeoftheelementarygrammaroftheworldcontinuestogrow.Ifwetrytoput

togetherwhatwehavelearnedaboutthephysicalworldinthecourseofthetwentiethcentury,thecluespointtowardssomethingprofoundlydifferentfromwhatweweretaughtatschool.Anelementarystructureoftheworldisemerging,generatedbyaswarmofquantumevents,wheretimeandspacedonotexist.Quantumfieldsdrawspace,time,matterandlight,exchanginginformationbetweenoneeventandanother.Realityisanetworkofgranularevents;thedynamicwhichconnectsthemisprobabilistic;betweenoneeventandanother,space,time,matterandenergymeltinacloudofprobability.Thisstrangenewworldisslowlyemergingtodayfromthestudyofthemainopenquestionposed

infundamentalphysics:quantumgravity.It’stheproblemofcoherentlysynthesizingwhatwehavelearnedabouttheworldwiththetwomajordiscoveriesoftwentieth-centuryphysics:generalrelativityandquantumtheory.Toquantumgravity,andthestrangeworldthatthisresearchisunfolding,thisbookisdedicated.Thebookisalivecoverageoftheongoingresearch:whatwearelearning,whatwealreadyknow,

andwhatwethinkwearebeginningtounderstand,abouttheelementarynatureofthings.Itstartsfromthedistantoriginofsomekeyideasthatweusetodaytoorderourunderstandingoftheworldanddescribesthetwogreatdiscoveriesofthetwentiethcentury–Einstein’sgeneralrelativityandquantummechanics–tryingtoputintofocusthecoreoftheirphysicalcontent.Ittellsofthepictureoftheworldwhichisemergingtodayfromresearchinquantumgravity,takingintoaccountthelatestindicationsgivenbynature,suchastheconfirmationofthecosmologicalstandardmodelobtainedfromthePlancksatelliteandthefailureatCERNtoobservethesuper-symmetricparticlesthatmanyexpected.Anditdiscussestheconsequencesoftheseideas:thegranularstructureofspace;thedisappearanceoftimeatsmallscale;thephysicsoftheBigBang;theoriginofblack-holeheat–uptotheroleofinformationinthefoundationofphysics.InafamousmythrelatedbyPlatointheseventhbookofTheRepublic,somemenarechainedat

thebottomofadarkcaveandseeonlyshadowscastuponawallbyafirebehindthem.Theythinkthatthisisreality.Oneofthemfreeshimself,leavesthecaveanddiscoversthelightoftheSun,andthewiderworld.Atfirstthelight,towhichhiseyesareunaccustomed,stunsandconfuseshim.Buteventuallyhecansee,andreturnsexcitedlytohiscompanionstotellthemwhathehasseen.Theyfindithardtobelieve.Weareallinthedepthsofacave,chainedbyourignorance,byourprejudices,andourweak

sensesrevealtousonlyshadows.Ifwetrytoseefurther,weareconfused:weareunaccustomed.Butwetry.Thisisscience.Scientificthinkingexploresandredrawstheworld,graduallyofferingusbetterandbetterimagesofit,teachingustothinkinevermoreeffectiveways.Scienceisa

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continualexplorationofwaysofthinking.Itsstrengthisitsvisionarycapacitytodemolishpreconceivedideas,torevealnewregionsofreality,andtoconstructnovelandmoreeffectiveimagesoftheworld.Thisadventurerestsupontheentiretyofpastknowledge,butatitsheartischange.Theworldisboundlessandiridescent;wewanttogoandseeit.Weareimmersedinitsmysteryandinitsbeauty,andoverthehorizonthereisunexploredterritory.Theincompletenessandtheuncertaintyofourknowledge,ourprecariousness,suspendedovertheabyssoftheimmensityofwhatwedon’tknow,doesnotrenderlifemeaningless:itmakesitinterestingandprecious.Ihavewrittenthisbooktogiveanaccountofwhatformeisthewonderofthisadventure.I’ve

writtenwithaparticularreaderinmind:someonewhoknowslittleornothingabouttoday’sphysicsbutiscurioustofindoutwhatweknow,butalsowhatwedon’tyetunderstand,abouttheelementaryweaveoftheworld–andwherewearesearching.AndIhavewrittenittotrytocommunicatethebreathtakingbeautyofthepanoramaofrealitywhichcanbeseenfromthisperspective.I’vealsowrittenitformycolleagues,fellowtravellersdispersedthroughouttheworld,aswellas

fortheyoungwomenandmenwithapassionforscience,eagertosetoutonthisjourneyforthefirsttime.I’vesoughttooutlinethegenerallandscapeofthestructureofthephysicalworld,asseenbythedoublelightsofrelativityandofquantumphysics,andtoshowhowtheycanbecombined.Thisisnotonlyabookofdivulgation;it’salsoonewhicharticulatesapointofview,inafieldofresearchwheretheabstractionofthetechnicallanguagemaysometimesobscurethewide-anglevision.Scienceismadeupofexperiments,hypotheses,equations,calculationsandlongdiscussions;buttheseareonlytools,liketheinstrumentsofmusicians.Intheend,whatmattersinmusicisthemusicitself,andwhatmattersinscienceistheunderstandingoftheworldwhichscienceprovides.TounderstandthesignificanceofthediscoverythattheEarthturnsaroundtheSun,itisnotnecessarytofollowCopernicus’scomplicatedcalculations;tounderstandtheimportanceofthediscoverythatalllivingbeingsonourplanethavethesameancestors,itisnotnecessarytofollowthecomplexargumentsofDarwin’sbooks.Scienceisaboutreadingtheworldfromagraduallywideningpointofview.Thisbookgivesanaccountofthecurrentstateofthesearchforournewimageoftheworld,asI

understandittoday.ItisthereplyIwouldgivetoacolleagueandfriendaskingme,‘So,whatdoyouthinkisthetruenatureofthings?’,aswewalkalongtheshore,onalongmidsummer’sevening.

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PartOne

ROOTS

ThisbookbeginsinMiletus,twenty-sixcenturiesago.Whybeginabookaboutquantumgravitywithevents,peopleandideassoancient?Ihopethereadereagertogetontoquantaofspacewillnotholdthisagainstme.Foritiseasiertounderstandideasbystartingwiththerootsfromwhichtheyhavegrown,andanimportantnumberoftheideaswhichturnedouttobeeffectiveforunderstandingtheworldoriginatedovertwothousandyearsago.Ifwebrieflyretracetheirbirth,theybecomeclearer,andthelaterstepsturnouttobesimplerandnatural.

Butthere’smore.Certainproblemsfirstposedinantiquitycontinuetobecrucialtoourunderstandingoftheworld.Someofthemostrecentideasaboutthestructureofspaceutilizeconceptsandissuesintroducedthen.Inspeakingofthesedistantideas,Iputontothetablequestionswhicharegoingtobecentraltoquantumgravity.Thismakesitalsopossible,whentreatingofquantumgravity,todistinguishbetweentheideaswhichgobacktotheveryoriginofscientificthought,evenifweareunfamiliarwiththem,andthosewhichareradicallynew.Theconnectionbetweenproblemsposedbythescientistsofantiquity,andsolutionsfoundbyEinsteinandquantumgravity,is,asweshallsee,surprisinglyclose.

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1.Grains

Accordingtotradition,intheyear450BCE,amanembarkeduponashiptravellingfromMiletustoAbdera.Itwastobeacrucialjourneyforthehistoryofknowledge.

ThemanwasprobablyfleeingpoliticalturmoilinMiletus,wherethearistocracywasviolentlyseizingbackpower.MiletushadbeenaprosperousandflourishingGreekcity,perhapstheprincipalcityoftheGreekworldbeforethegoldenageofAthensandSparta.Ithadbeenabusycommercialhub,dominatinganetworkofalmostahundredcoloniesandcommercialoutposts,stretchingfromtheBlackSeatoEgypt.CaravansfromMesopotamiaandshipsfromallovertheMediterraneanarrivedatMiletus,andideascirculated.

Figure1.1ThejourneymadebyLeucippusofMiletus,thefounderoftheatomistschool(circa450BCE).

Duringtheprecedingcentury,arevolutioninthinkingwhichwouldprovefundamentaltohumanityhadtakenplaceinMiletus.Agroupofthinkershadreformulatedthewayquestionswereaskedabouttheworld,andthewayanswersweresought.ThegreatestofthesethinkerswasAnaximander.

Fromtimeimmemorial,oratleastsincehumanityhadleftwrittentextswhichhavecomedowntous,menhadaskedthemselveshowtheworldhadcomeintobeing,whatitwascomposedof,howitwasordered,andwhynaturalphenomenaoccurred.Forthousandsofyearstheyhadgiventhemselvesanswerswhichallresembledoneanother:answerswhichreferredtoelaboratestoriesofspirits,deities,imaginaryandmythologicalcreatures,andothersimilarthings.FromcuneiformtabletstoancientChinesetexts;fromhieroglyphicwritinginthePyramidstothemythsoftheSioux;fromthemostancientIndiantextstotheBible;fromAfricanstoriestothoseofaboriginalAustralians,itwasallacolourfulbutbasicallyquitemonotonousflow–ofPlumedSerpentsandGreatCows,ofirascible,litigious,orkindlydeitieswhocreatetheworldbybreathingoverabysses,uttering‘Fiatlux’,oremergingoutofastoneegg.

Then,atMiletus,atthebeginningofthefifthcenturybeforeourera,Thales,hispupilAnaximander,Hecataeusandtheirschoolfindadifferentwayoflookingforanswers.Thisimmense

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revolutioninthoughtinauguratesanewmodeofknowledgeandunderstanding,andsignalsthefirstdawnofscientificthought.

TheMilesiansunderstandthatbyshrewdlyusingobservationandreason,ratherthansearchingforanswersinfantasy,ancientmythsorreligion–and,aboveall,byusingcriticalthoughtinadiscriminatingway–itispossibletorepeatedlycorrectourworldview,andtodiscovernewaspectsofrealitywhicharehiddentothecommonview.Itispossibletodiscoverthenew.

Perhapsthedecisivediscoveryisthatofadifferentstyleofthinking,wherethediscipleisnolongerobligedtorespectandsharetheideasofthemasterbutisfreetobuildonthoseideaswithoutbeingafraidtodiscardorcriticizethepartthatcanbeimproved.Thisisanovelmiddleway,placedbetweenfulladherencetoaschoolandgenericdeprecationofideas.Itisthekeytothesubsequentdevelopmentofphilosophicalandscientificthinking:fromthismomentonwards,knowledgebeginstogrowatavertiginouspace,nourishedbypastknowledgebutatthesametimebythepossibilityofcriticism,andthereforeofimprovingknowledgeandunderstanding.ThedazzlingincipitofHecataeus’sbookofhistorygoestotheheartofthiscriticalthinking,includingasitdoestheawarenessofourownfallibility:‘Iwrotethingswhichseemtruetome,becausetheaccountsoftheGreeksseemtobefullofcontradictoryandridiculousthings.’

Accordingtolegend,HeraclesdescendedtoHadesfromCapeTenaro.HecataeusvisitsCapeTenaro,anddeterminesthatthereisinfactnosubterraneanpassageorotheraccesstoHadesthere–andthereforejudgesthelegendtobefalse.Thismarksthedawnofanewera.

Thisnewapproachtoknowledgeworksquicklyandimpressively.Withinamatterofafewyears,AnaximanderunderstandsthattheEarthfloatsintheskyandtheskycontinuesbeneaththeEarth;thatrainwatercomesfromtheevaporationofwateronEarth;thatthevarietyofsubstancesintheworldmustbesusceptibletobeingunderstoodintermsofasingle,unitaryandsimpleconstituent,whichhecallsapeiron,theindistinct;thattheanimalsandplantsevolveandadapttochangesintheenvironment,andthatmanmusthaveevolvedfromotheranimals.Thus,gradually,wasfoundedthebasisofagrammarforunderstandingtheworldwhichissubstantiallystillourowntoday.

SituatedatapointofconjunctionbetweentheemergentGreekcivilizationandtheancientempiresofMesopotamiaandEgypt,nourishedbytheirknowledgebutimmersedinthelibertyandthepoliticalfluiditywhichistypicallyGreek;inasocialspacewithoutimperialpalaces,orpowerfulpriestlycastes,whereindividualcitizensdiscusstheirdestiniesinopenagoras,Miletusistheplacewhere,forthefirsttime,mendecidecollectivelytheirownlaws;wherethefirstparliamentinthehistoryoftheworldgathers–thePanionium,meeting-placeofthedelegatesoftheIonianLeague–andwhereforthefirsttimemendoubtthatonlythegodsarecapableofaccountingforthemysteriesoftheworld.Throughdiscussion,itispossibletoreachthebestdecisionsforthecommunity;throughdiscussion,itispossibletounderstandtheworld.ThisistheimmenselegacyofMiletus,cradleofphilosophy,ofthenaturalsciences,andofgeographicalandhistoricalstudies.Itisnoexaggerationtosaythattheentirescientificandphilosophicaltradition,Mediterraneanandthenmodern,hasacrucialrootinthespeculationsofthethinkersofMiletusinthesixthcenturyBCE.1

ThisluminousMiletusshortlyafterwardscametoacalamitousend.ThearrivalofthePersianEmpire,andafailedanti-imperialrevolt,ledtoaferociousdestructionofthecityin494BCEandtotheenslavementofalargenumberofitsinhabitants.InAthens,thepoetPhrynichuswritesthetragedyTheTakingofMiletus,whichsodeeplymovestheAtheniansthatitsre-stagingisprohibited,onaccountofitcausingtoomuchdistress.Buttwentyyearslater,theGreeksrepelthePersianmenace;Miletusisreborn,repopulated,andreturnedtobeingacentreofcommerceandideas,radiatingitsthoughtandspiritoncemore.

Thepersonwithwhomwebeganthischaptermusthavebeenmovedbythisspiritwhen,in450,accordingtotradition,heembarkedfromMiletusforAbdera.HisnamewasLeucippus.Littleisknownabouthislife.2HewroteabookentitledTheGreatCosmology.OnhisarrivalinAbdera,hefoundedascientificandphilosophicalschooltowhichhesoonaffiliatedayoungdisciple,Democritus,whoselongshadowwastobecastoverthethoughtofallsubsequenttimes.

Together,thesetwothinkershavebuiltthemajesticcathedralofancientatomism.Leucippuswastheteacher.Democritus,thegreatpupilwhowrotedozensofworksoneveryfieldofknowledge,andwasdeeplyveneratedinantiquity,whenpeoplewerefamiliarwiththeseworks.‘ThemostsubtleoftheAncients’,Senecacalledhim.3‘Whoistherewhomwecancomparewithhimforthegreatness,notmerelyofhisgenius,butalsoofhisspirit?’asksCicero.4

What,then,hadLeucippusandDemocritusdiscovered?TheMilesianshadunderstoodthattheworldcanbecomprehendedusingreason.Theyhadbecomeconvincedthatthevarietyofnaturalphenomenamustbeattributabletosomethingsimple,andhadtriedtounderstandwhatthissomethingmightbe.Theyhadconceivedofakindofelementarysubstanceofwhicheverythingwasmade.Anaximenes,amongtheMilesians,hadimaginedthissubstancecouldcompressandrarefy,thustransformingfromonetoanotheroftheelementsofwhichtheworldisconstituted.Itwasafirstgermofphysics,roughandelementary,butintherightdirection.Anideawasneeded,agreatidea,agrandvision,tograspthehiddenorderoftheworld.LeucippusandDemocrituscameupwiththisidea.

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Figure1.2DemocritusofAbdera.

TheideaofDemocritus’ssystemisextremelysimple:theentireuniverseismadeupofaboundlessspaceinwhichinnumerableatomsrun.Spaceiswithoutlimits;hasneitheranabovenorabelow;iswithoutacentre,oraboundary.Atomshavenoqualitiesatall,apartfromtheirshape.Theyhavenoweight,nocolour,notaste.‘Byconventionissweet,byconventionbitter,byconventionhot,byconventioncolour;butbyverityatomsandvoid.’5

Atomsareindivisible;theyaretheelementarygrainsofreality,whichcannotbefurthersubdivided,andeverythingismadeofthem.Theymovefreelyinspace,collidingonewithanother;theyhookontoandpushandpulleachother.Similaratomsattracteachotherandjoin.

Thisistheweaveoftheworld.Thisisreality.Everythingelseisnothingbutaby-product,randomandaccidental,ofthismovementandthiscombiningofatoms.Theinfinitevarietyofthesubstancesofwhichtheworldismadederivessolelyfromthiscombiningofatoms.

Whenatomsaggregate,theonlythingthatmatters,theonlythingthatexistsattheelementarylevel,istheirshape,theirarrangement,andtheorderinwhichtheycombine.Justasbycombiningthelettersofthealphabetindifferentwayswemayobtaincomediesortragedies,ridiculousstoriesorepicpoems,soelementaryatomscombinetoproducetheworldinitsendlessvariety.ThemetaphorisDemocritus’sown.6

Thereisnofinality,nopurpose,inthisendlessdanceofatoms.We,justliketherestofthenaturalworld,areoneofthemanyproductsofthisinfinitedance.Theproduct,thatis,ofanaccidentalcombination.Naturecontinuestoexperimentwithformsandstructures;andwe,liketheanimals,aretheproductsofaselectionwhichisrandomandaccidental,overthecourseofeonsoftime.Ourlifeisacombinationofatoms,ourthoughtsaremadeupofthinatoms,ourdreamsaretheproductsofatoms;ourhopesandouremotionsarewritteninalanguageformedbycombinationsofatoms;thelightwhichweseeiscomprisedofatomswhichbringusimages.Theseasaremadeofatoms,asareourcities,andthestars.It’sanimmensevision;boundless,incrediblysimpleandincrediblypowerful,oneonwhichtheknowledgeofacivilizationwouldlaterbebuilt.

OnthisfoundationDemocrituswrotedozensofbooksarticulatingavastsystem,dealingwithquestionsofphysics,philosophy,ethics,politicsandcosmology.Hewritesonthenatureoflanguage,onreligion,ontheoriginsofhumansocieties,andonmuchelsebesides.(TheopeningofhisLittleCosmologyisimpressive:‘InthisworkItreatofallthings.’)Allofthesebookshavebeenlost.Weknowofhisthoughtonlythroughthequotationsandreferencesmadebyotherancientauthors,andbytheirsummariesofhisideas.7Thethoughtwhichthusemergesisakindofintensehumanism,rationalistandmaterialist.8Democrituscombinesakeenattentiontonature,illuminatedbyanaturalisticclarityinwhicheveryresidualsystemofmythicideasisclearedaway,withagreatattentiontohumanityandadeepethicalconcernforlife–anticipatingbysometwothousandyearsthebestaspectsoftheeighteenth-centuryEnlightenment.TheethicalidealofDemocritusisthatofaserenityofmindreachedthroughmoderationandbalance,bytrustinginreasonandnotallowingoneselftobeoverwhelmedbypassions.

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PlatoandAristotlewerefamiliarwithDemocritus’sideas,andfoughtagainstthem.Theydidsoonbehalfofotherideas,someofwhichwerelater,forcenturies,tocreateobstaclestothegrowthofknowledge.BothinsistedonrejectingDemocritus’snaturalisticexplanations,infavouroftryingtounderstandtheworldinfinalisticterms–believing,thatis,thateverythingthathappenshasapurpose;awayofthinkingthatwouldrevealitselftobeverymisleadingforunderstandingthewaysofnature–orintermsofgoodandevil,confusinghumanissueswithmatterswhichdonotrelatetous.

AristotlespeaksextensivelyabouttheideasofDemocritus,andwithrespect.PlatonevercitesDemocritus,butscholarssuspecttodaythatthiswasoutofdeliberatechoiceandnotforlackofknowledgeofhisworks.CriticismofDemocritus’sideasisimplicitinseveralofPlato’stexts,asinhiscritiqueof‘physicists’,forexample.InapassageinhisPhaedo,PlatohasSocratesarticulateareproachtoall‘physicists’whichwillhavealastingresonance.Hecomplainsthatwhen‘physicists’hadexplainedthattheEarthwasround,herebelledbecausehewantedtoknowwhat‘good’itwasfortheEarthtoberound;howitsroundnesswouldbenefitit.Plato’sSocratesrecountshowhehadatfirstbeenenthusiasticaboutphysics,buthadcometobedisillusionedbyit:

IhadexpectedtobefirsttoldthattheEarthwasflatorround,butalsothat,afterwards,thereasonforthenecessityofthisshapewouldbeexplainedtome,startingfromtheprincipleofthebest,provingtomethatthebestthingfortheEarthistohavethisshape.AndifhehadsaidthattheEarthwasatthecentreoftheworld,thentoshowmehowbeingatthecentrewasofbenefittotheEarth.9

HowcompletelyofftrackthegreatPlatowashere!

Istherealimittodivisibility?Thegreatestphysicistofthesecondhalfofthetwentiethcentury,RichardFeynman,wroteatthebeginningofhiswonderfulintroductorylessonsonphysics:

If,insomecataclysm,allscientificknowledgeweretobedestroyed,andonlyonesentencepassedontothenextgenerationofcreatures,whatstatementwouldcontainthemostinformationinthefewestwords?Ibelieveitistheatomichypothesis,ortheatomicfact,orwhateveryouwishtocallit,thatallthingsaremadeofatoms–littleparticlesthatmovearoundinperpetualmotion,attractingeachotherwhentheyarealittledistanceapart,butrepellinguponbeingsqueezedintooneanother.Inthatonesentenceyouwillseeanenormousamountofinformationabouttheworld,ifjustalittleimaginationandthinkingareapplied.10

Withoutneedinganythingfrommodernphysics,Democritushadalreadyarrivedattheideathateverythingismadeupofindivisibleparticles.Howdidhedoit?

Hehadargumentsbaseduponobservation;forexample,heimagined,correctly,thatthewearingdownofawheel,orthedryingofclothesonaline,couldbeduetotheslowflightofparticlesofwoodorofwater.Buthealsohadargumentsofaphilosophicalkind.Let’sfocusonthese,becausetheirpotencyreachesallthewaytoquantumgravity.

Democritusobservedthatmattercouldnotbeacontinuouswhole,becausethereissomethingcontradictoryinthepropositionthatitshouldbeso.WeknowofDemocritus’sreasoningbecauseAristotlereportsit.11Imagine,saysDemocritus,thatmatterisinfinitelydivisible,thatistosay,itmaybebrokendownaninfinitenumberoftimes.Imaginethenthatyoubreakupapieceofmatteradinfinitum.Whatwouldbeleft?

Couldsmallparticlesofextendeddimensionremain?No,becauseifthiswerethecasethepieceofmatterwouldnotyetbebrokenuptoinfinity.Therefore,onlypointswithoutextensionwouldremain.Butnowletustrytoputtogetherthepieceofmatterstartingfromthesepoints:byputtingtogethertwopointswithoutextensionyoucannotobtainathingwithextension,norcanyouwiththree,orevenwithfour.Nomatterhowmanyyouputtogether,infact,youneverhaveextension,becausepointshavenoextension.Therefore,wecannotthinkthatmatterismadeofpointswithoutextension,becausenomatterhowmanyofthesewemanagetoputtogether,weneverobtainsomethingwithanextendeddimension.Theonlypossibility,Democritusconcludes,isthatanypieceofmatterismadeupofafinitenumberofdiscretepieceswhichareindivisible,eachonehavingfinitesize:theatoms.

Theoriginofthissubtlemodeofargumentationpre-datesDemocritus.ItcomesfromtheCilentoregioninthesouthofItaly,fromatownnowcalledVelia,whichinthefifthcenturyBCEwasaflourishingGreekcolonycalledElea.Parmenideslivedthere,thephilosopherwhohadtakentotheletter–perhapstoomuch–therationalismofMiletusandtheideaborntherethatreasoncanrevealtoushowthingscanbeotherthantheyappear.Parmenideshadexploredanavenuetotruthviapurereasonalone,apathwhichledhimtodeclarethatallappearancesareillusory,thusopeningadirectionofthinkingthatwouldprogressivelymovetowardsmetaphysicsanddistanceitselffromwhatwouldcometobeknownas‘naturalscience’.HispupilZeno,alsofromElea,hadbroughtsubtleargumentstobearinsupportofthisfundamentalistrationalism,whichrefutesthecredibilityofappearancesradically.Amongtheseargumentstherewasaseriesofparadoxesthatbecamecelebratedas‘Zeno’sparadoxes’;theyseektoshowhowallappearanceisillusory,arguingthatthecommonplacenotionofmotionisabsurd.12

ThemostfamousofZeno’sparadoxesispresentedintheformofabrieffable:thetortoisechallengesAchillestoarace,startingoutwithaten-metreadvantage.WillAchillesmanagetocatch

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upwiththetortoise?Zenoarguesthatrigorouslogicdictatesthathewillneverbeabletodoso.Beforecatchingup,ineffect,Achillesneedstocoverthetenmetres,andinordertodothishewilltakeacertainamountoftime.Duringthistime,thetortoisewillhaveadvancedafewcentimetres.Tocoverthesecentimetres,Achilleswillhavetotakealittlemoretimebut,meanwhile,thetortoisewillhaveadvancedfurther,andsoon,adinfinitum.Achillesthereforerequiresaninfinitenumberofsuchtimestoreachthetortoise,andaninfinitenumberoftimes,arguesZeno,isaninfiniteamountoftime.Consequently,accordingtostrictlogic,Achilleswilltakeaninfinitetimetoreachthetortoise;orrather,wewillneverseehimdoso.Since,however,wedoseetheswiftAchillesreachingandovertakingasmanytortoisesashelikes,itfollowsthatwhatweseeisirrational,andthereforeillusory.

Let’sbehonest:thisishardlyconvincing.Wheredoestheerrorlie?OnepossibleansweristhatZenoiswrongbecauseitisnottruethatbyaccumulatinganinfinitenumberofthingsoneendsupwithaninfinitething.Thinkoftakingapieceofstring,cuttingitinhalf,andthenagaininhalf,andagain,adinfinitum.Attheend,youwillobtainaninfinitenumberofsmallpiecesofstring;thesumofthese,however,willbefinite,becausetheycanonlyadduptothelengthoftheoriginalpieceofstring.Hence,aninfinitenumberofstringscanmakeafinitestring;aninfinitenumberofincreasinglyshorttimesmaymakeafinitetime,andthehero,evenifhewillhavetocoveraninfinitenumberofdistances,eversmaller,willtakeafinitetimetodoso,andwillendupcatchingthetortoise.

Itseemsthattheparadoxisresolved.Thesolution,thatis,isintheideaofthecontinuum–arbitrarilysmalltimesmayexist,aninfinitenumberofwhichmakeupafinitetime.Aristotleisthefirsttointuitthispossibility,whichwassubsequentlydevelopedbyancientandmodernmathematics.fn1

Butisthisreallythecorrectsolutionintherealworld?Doarbitrarilyshortstringsreallyexist?Canwereallycutapieceofstringanarbitrarynumberoftimes?Doinfinitelysmallamountsoftimeexist?Thisispreciselytheproblemthatquantumgravitywillhavetoface.

Accordingtotradition,ZenohadmetLeucippusandhadbecomehisteacher.LeucippuswasthereforefamiliarwithZeno’sriddles.Buthehaddevisedadifferentwayofresolvingthem.Maybe,Leucippussuggests,nothingarbitrarilysmallexists:thereisalowerlimittodivisibility.

Theuniverseisgranular,notcontinuous.Withinfinitelysmallpoints,itwouldbeimpossibleevertoconstructextension–asinDemocritus’sargument,reportedbyAristotleandmentionedabove.Therefore,theextensionofthestringmustbeformedbyafinitenumberoffiniteobjectswithfinitesize.Thestringcannotbecutasmanytimesaswewant;matterisnotcontinuous,itismadeofindividual‘atoms’ofafinitesize.

Whetherthisabstractargumentiscorrectornot,itsconclusion–asweknowtoday–containsagreatdealoftruth.Matterdoesindeedhaveanatomicstructure.IfIdivideadropofwaterintwo,Iobtaintwodropsofwater.Icandivideeachoneofthesetwodropsagain,andsoon.ButIcannotcontinuetoinfinity.AtacertainpointIhaveonlyonemolecule,andIhavefinished.Nodropsofwaterexistsmallerthanasinglemoleculeofwater.

Howdoweknowthistoday?Evidencehasaccumulatedovercenturies,muchofitfromchemistry.Chemicalsubstancesaremadeupofcombinationsofafewelementsandareformedbyproportions(ofweight)givenbywholenumbers.Chemistshaveconstructedawayofthinkingaboutsubstancesascomposedofmoleculesmadeupoffixedcombinationsofatoms.Water,forexample–H2O–iscomposedoftwopartshydrogenandonepartoxygen.

Buttheseareonlyclues.Stillatthebeginningofthelastcenturynumerousscientistsandphilosophersdidnotconsidertheatomichypothesistobecredible.AmongthemwastherenownedphysicistandphilosopherErnstMach,whoseideasonspacewouldcometohavegreatimportanceforEinstein.AttheendofalecturebyLudwigBoltzmannattheImperialAcademyofScienceinVienna,Machpubliclydeclared,‘Idonotbelievethatatomsexist!’Thiswasin1897.Many,likeMach,understoodchemicalnotationonlyasaconventionalmethodofsummarizinglawsofchemicalreactions–notasevidencethatthereactuallyweremoleculesofwatercomposedoftwoatomsofhydrogenandoneofoxygen.Youcan’tseeatoms,theywouldsay.Atomswillneverbeseen,theywouldsay.Andthen,theyasked,howbigwouldanatombe?Democrituscouldnevermeasurethesizeofhisatoms…

Butsomebodyelsecould.Thedefinitiveproofofthe‘atomichypothesis’hadtowaituntil1905.Itwasfoundbyarebellioustwenty-five-year-old,whohadstudiedphysicsbuthadnotbeenabletofindemploymentasascientistandwasmakingendsmeetbyworkingasanemployeeinthepatentofficeinBerne.Iwillspeakalotaboutthisyoungmanintherestofthisbook,andaboutthethreearticleshesenttothemostprestigiousphysicsjournalofthetime,theAnnalenderPhysik.Thefirstofthesearticlescontainedthedefinitiveproofthatatomsexist,andcalculatedtheirdimensions,solvingtheproblemposedbyLeucippusandDemocritustwenty-threecenturiesearlier.

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Figure1.3AlbertEinstein.

Thenameofthistwenty-five-year-old,obviously,isAlbertEinstein.Howdoeshedoit?Theideaissurprisinglysimple.Anyonecouldhavearrivedatit,fromthetime

ofDemocritusonwards,ifhehadhadEinstein’sacumenandasufficientmasteryofmathematicstomakewhatwasnotaneasycalculation.Theideagoeslikethis:ifweobserveattentivelyverysmallparticles,suchasaspeckofdustoragrainofpollen,suspendedinstillairorinaliquid,weseethemtrembleanddance.Pushedbythistrembling,theymove,randomlyzigzagging,andsotheydriftslowly,graduallymovingawayfromtheirstartingpoint.ThismotionofparticlesinafluidiscalledBrownianmotion,afterRobertBrown,abiologistwhodescribeditindetailinthenineteenthcentury.Thetypicaltrajectoryofaparticledancinginthismannerisillustratedinfigure1.4.Itisasifthesmallparticleisreceivingblowsrandomlyfromeachsideofit.Infact,itisn’t‘asif’itwerebeinghit,itreallyishit.Ittremblesbecauseitishitbyindividualmoleculesofair,whichcollidewiththeparticleattimesfromtherightandattimesfromtheleft.

Figure1.4TypicalBrownianmotion.

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Thesubtlepointisthefollowing.Thereareanenormousnumberofmoleculesofair.Onaverage,asmanyhitthegranulefromtheleftasthosethathititfromtheright.Iftheair’smoleculeswereinfinitelysmallandinfinitelynumerous,theeffectofthecollisionsfromrightandfromleftwouldbalanceandthuscanceloutateachinstant,andthegranulewouldnotmove.Butthefinitesizeofthemolecules–thefactthatthesearepresentinfiniteratherthaninfinitenumber–causestheretobefluctuations(thisisthekeyword):thatistosay,thecollisionsneverbalanceoutexactly;theyonlybalanceoutonaverage.Imagineforamomentthemoleculeswereveryfewinnumberandlargeinsize:thegranulewouldclearlyreceiveablowonlyoccasionally;nowoneontheright,thenoneontheleft…Betweenonecollisionandtheotheritwouldmovehereandtheretoasignificantdegree,likeafootballkickedbyboysrunningaroundaplayingfield.Thesmallerthemolecules,ontheotherhand,theshortertheintervalbetweencollisionsandthebetterthehitsfromdifferentdirectionsbalanceandcanceleachotherout.Andthelessthegranulemoves.

Itisindeedpossible,withalittlemathematics,toworkbackfromtheamountofmovementofthegranule,whichcanbeobserved,tothedimensionsofthemolecules.Einsteindoesthis,asImentionedabove,attheageoftwenty-five.Fromobservationsofgranulesdriftinginfluids,fromthemeasurementofhowmuchthese‘drift’–thatis,moveawayfromaposition–hecalculatesthedimensionsofDemocritus’satoms,theelementarygrainsofwhichmatterismade.Heprovides,after2,300years,theproofoftheaccuracyofDemocritus’sinsight:matterisgranular.

ThenatureofthingsSublimeLucretius’sworkwillnotdie,Untilthedaytheworlditselfpassesaway.

–Ovid13

IoftenthinkthatthelossoftheworksofDemocritusintheirentiretyfn2isthegreatestintellectualtragedytoensuefromthecollapseoftheoldclassicalcivilization.Takealookatthelistofhisworksinthefootnote;itisdifficultnottobedismayed,imaginingwhatwehavelostofthevastscientificreflectionsofantiquity.

WehavebeenleftwithallofAristotle,bywayofwhichWesternthoughtreconstructeditself,andnothingbyDemocritus.Perhaps,ifalloftheworksofDemocritushadsurvived,andnothingofAristotle’s,theintellectualhistoryofourcivilizationwouldhavebeenbetter…

ButcenturiesdominatedbymonotheismhavenotpermittedthesurvivalofDemocritus’snaturalism.TheclosureoftheancientschoolssuchasthoseofAthensandAlexandriaandthedestructionofallthetextsnotinaccordancewithChristianideaswerevastandsystematic,atthetimeofthebrutalanti-paganrepressionfollowingfromtheedictsofEmperorTheodosius,which,in390–1declaredthatChristianitywastobetheonlyandobligatoryreligionoftheempire.PlatoandAristotle,paganswhobelievedintheimmortalityofthesoulorintheexistenceofaPrimeMover,couldbetoleratedbyatriumphantChristianity.NotDemocritus.

Butatextsurvivedthedisaster,andhasreachedusinitsentirety.Throughit,weknowalittleaboutancientatomismand,aboveall,weknowthespiritofthatscience.ItisthesplendidpoemDererumnatura(TheNatureofThings,orOntheNatureoftheUniverse),bytheLatinpoetLucretius.

LucretiusadherestothephilosophyofEpicurus,apupilofapupilofDemocritus.EpicurusisinterestedmoreinethicalthanscientificquestionsanddoesnothaveDemocritus’sdepth.HesometimestranslatesDemocriteanatomismalittlesuperficially.ButhisvisionofthenaturalworldissubstantiallythatofthegreatphilosopherofAbdera.LucretiusdecantsinversethethoughtofEpicurusandtheatomismofDemocritus,andinthiswayapartofthisprofoundphilosophywassavedfromtheintellectualcatastropheoftheDarkAges.Lucretiussingsofatoms,thesea,thesky,ofnature.Heexpressesinluminousversephilosophicalquestions,scientificideas,refinedarguments.

…Iwillexplainbywhatforcesnaturesteersthecoursesofthesunandthejourneyingsofthemoon,sothatweshallnotsupposethattheyruntheiryearlyracesbetweentheheavenandearthoftheirownfreewill…orthattheyarerolledroundinfurtheranceofsomedivineplan…14

Thebeautyofthepoemliesinthesenseofwonderwhichpervadesthevastatomisticvision–thesenseoftheprofoundunityofthings,derivedfromtheknowledgethatweareallmadeofthesamesubstanceasarethestars,andthesea:

…weareallsprungfromheavenlyseed.Allalikehavethesamefather,fromwhomall-nourishingmotherearthreceivestheshoweringdropsofmoisture.Thusfertilized,shegivesbirthtosmilingcropsandlustytrees,tomankindandallthebreedsofbeasts.Sheitisthatyieldsthefoodonwhichtheyallfeedtheirbodies,leadtheirjoyouslivesandrenewtheirrace…15

Thereisasenseofluminouscalmandserenityaboutthepoem,whichcomesfromunderstandingthattherearenocapriciousgodsdemandingofusdifficultthings,andpunishingus.Thereisavibrantandairyjoyfulness,beginningwiththemarvellousopeningversesdedicatedtoVenus,aradiantsymbolofthecreativeforceofnature:

Beforeyouthewindsflee,andatyourcomingthecloudsforsakethesky.Foryoutheoceanlevelslaugh,theskyiscalmedandglowswithdiffusedradiance.16

Thereisadeepacceptanceofthelifeofwhichweareanintegralpart:

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Doyounotseethatnatureisclamouringfortwothingsonly,abodyfreefrompain,amindreleasedfromworryandfearfortheenjoymentofpleasurablesensations?17

Andthereisasereneacceptanceoftheinevitabilityofdeath,whichcancelseveryevilandaboutwhichthereisnothingtofear.ForLucretius,religionisignorance;reasonisthetorchthatbringslight.

Lucretius’stext,forgottenforcenturies,wasrediscoveredinJanuary1417bythehumanistPoggioBracciolini,inthelibraryofaGermanmonastery.Poggiohadbeenthesecretaryofmanypopes,andwasapassionatehunterofancientbooks,inthewakeofthecelebratedrediscoveriesmadebyFrancescoPetrarch.HisrediscoveryofatextbyQuintilianmodifiedthecourseofthestudyoflawthroughoutthefacultiesofEurope;hisdiscoveryofthetreatiseonarchitecturebyVitruviustransformedthewayinwhichfinebuildingsweredesignedandconstructed.ButhistriumphwasrediscoveringLucretius.TheactualcodexfoundbyPoggiohasbeenlost,butthecopymadebyhisfriendNiccolòNiccoli(nowknownastheCodexLaurenziano35.30)isstillpreservedinitsentiretyinFlorence’sBibliotecaLaurenziana.

ThegroundwasalreadysurelypreparedforsomethingnewwhenPoggiogaveLucretius’sbookbacktohumanity.Already,fromDante’sgeneration,ithadbeenpossibletohearmarkedlynewaccents:

Youreyeswentshootingthroughmyhearttowakemydormantthought.Looknow,I’mdesperateanddistraughtwithlovethattearsmylifeapart.18

ButtherediscoveryofDererumnaturahadaprofoundeffectupontheItalianandEuropeanRenaissance,19anditsechoresounds,directlyorindirectly,inthepagesofauthorsrangingfromGalileo20toKepler,21andfromBacontoMachiavelli.InShakespeare,acenturyafterPoggio,atomsmakeadelightfulappearance:

MERCUTIOO,thenIseeQueenMabhathbeenwithyou:Sheisthefairies’midwife,andshecomesInshapenobiggerthananagate-stoneOntheforefingerofanalderman,DrawnwithalittleteamofatomiesOvermen’snosesastheylieasleep…22

Montaigne’sEssaysincludeatleastahundredquotationsfromLucretius.ButthedirectinfluenceofLucretiusextendedtoNewton,Dalton,Spinoza,Darwin–andallthewayuptoEinstein.TheveryideaofEinstein’sthattheexistenceofatomsisrevealedbytheBrownianmotionofminuteparticlesimmersedinafluidmaybetracedbacktoLucretius.HereisapassageinwhichLucretiusprovidesa‘livingproof’ofthenotionofatoms:

Thisprocessisillustratedbyanimageofitthatiscontinuallytakingplacebeforeourveryeyes.Observewhathappenswhensunbeamsareadmittedintoabuildingandshedlightonitsshadowyplaces.Youwillseeamultitudeoftinyparticlesminglinginamultitudeofwaysintheemptyspacewithinthelightofthebeam,asthoughcontendingineverlastingconflict,rushingintobattlerankuponrankwithneveramoment’spauseinarapidsequenceofunionsanddisunions.Fromthisyoumaypicturewhatitisfortheatomstobeperpetuallytossedaboutintheillimitablevoid.Tosomeextentasmallthingmayaffordanillustrationandanimperfectimageofgreatthings.Besides,thereisafurtherreasonwhyyoushouldgiveyourmindtotheseparticlesthatareseendancinginasunbeam:theirdancingisanactualindicationofunderlyingmovementsofmatterthatarehiddenfromoursight.Thereyouwillseemanyparticlesundertheimpactofinvisibleblows,changingtheircourseanddrivenbackupontheirtracks,thiswayandthat,inalldirections.Youmustunderstandthattheyallderivethisrestlessnessfromtheatoms.Itoriginateswiththeatoms,whichmoveofthemselves.Thenthosesmallcompoundbodiesthatareleastremovedfromtheimpetusoftheatomsaresetinmotionbytheimpactoftheirinvisibleblowsandinturncannonagainstslightlylargerbodies.Sothemovementmountsupfromtheatomsandgraduallyemergestothelevelofoursenses,sothatthosebodiesareinmotionthatweseeinsunbeams,movedbyblowsthatremaininvisible.23

Einsteinresuscitatedthe‘livingproof’presentedbyLucretius,andprobablyfirstconceivedofbyDemocritus,andmadeitsolidbytranslatingitintomathematicalterms,thusmanagingtocalculatethesizeoftheatoms.

TheCatholicChurchattemptedtostopLucretius:intheFlorentineSynodofDecember1516itprohibitedthereadingofLucretiusinschools.In1551,theCouncilofTrentbannedhiswork.Butitwastoolate.AnentirevisionoftheworldwhichhadbeensweptawaybymedievalChristianfundamentalismwasre-emerginginaEuropewhichhadreopeneditseyes.Itwasnotjusttherationalism,atheismandmaterialismofLucretiusthatwerebeingproposedinEurope.Itwasnotmerelyaluminousandserenemeditationonthebeautyoftheworld.Itwasmuchmore:itwasanarticulateandcomplexstructureofthinkingaboutreality,anewmodeofthinking,radicallydifferentfromwhathadbeenforcenturiesthemind-setoftheMiddleAges.24

ThemedievalcosmossomarvellouslysungbyDantewasinterpretedonthebasisofahierarchicalorganizationoftheuniversewhichreflectedthehierarchicalorganizationofEuropeansociety:asphericalcosmicstructurewiththeEarthatitscentre;theirreducibleseparationbetweenEarthandheavens;finalisticandmetaphoricalexplanationsofnaturalphenomena.FearofGod,fearofdeath;littleattentiontonature;theideathatformsprecedingthingsdeterminethestructureoftheworld;theideathatthesourceofknowledgecouldonlybethepast,inrevelationandtradition…

ThereisnoneofthisintheworldofDemocritusassungbyLucretius.Thereisnofearofthegods;noendsorpurposesintheworld;nocosmichierarchy;nodistinctionbetweenEarthandheavens.

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Thereisadeeploveofnature,asereneimmersionwithinit;arecognitionthatweareprofoundlypartofit;thatmen,women,animals,plantsandcloudsareorganicthreadsofamarvellouswhole,withouthierarchies.ThereisafeelingofdeepuniversalisminthewakeofthesplendidwordsofDemocritus:‘Toawiseman,thewholeearthisopen,becausethetruecountryofavirtuoussoulistheentireuniverse.’25

Thereis,too,theambitionofbeingabletothinkabouttheworldinsimpleterms.Ofbeingabletoinvestigateandunderstandthesecretsofnature.Toknowmorethanourparents.AndthereareextraordinaryconceptualtoolsonwhichGalileo,KeplerandNewtonwillbuild:theideaoffreerectilinearmotioninspace;theideaofelementarybodiesandtheirinteractions,outofwhichtheworldisconstructed;theideaofspaceasacontaineroftheworld.

Andthereisthesimpleideaofthefinitedivisibilityofthings.Thegranularqualityoftheworld.Theideawhichstopstheinfinitebetweenourfingers.Thisideaisattherootoftheatomichypothesis,butitwillalsoreturnwithaugmentedforcewithquantummechanics,andtodayisrevealingitselftobepowerfulagain–asthekeystoneofquantumgravity.

ThefirstpersontomakethepartsofthemosaicwhichbegintoemergefromRenaissancenaturalismcohere–andtoreprisetheDemocriteanvision,immenselyreinforced,placingitatthecentreofmodernthought–willbeanEnglishman,thegreatestscientistofalltimeandthefirstprotagonistofthefollowingchapter.

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2.TheClassics

IsaacandthelittlemoonIfinthepreviouschapterIappearedtobesayingthatPlatoandAristotlehaveonlydoneharmtothedevelopmentofscience,Iwouldliketocorrectthisimpression.Aristotle’sstudiesofnature–ofbotanyandzoology,forexample–areextraordinaryscientificworks,groundeduponmeticulousobservationsofthenaturalworld.Theconceptualclarity,theattentiontothevarietyofnature,theimpressiveintelligenceandtheopennessofmindofthegreatphilosophermadehimanauthorityforcenturiestocome.ThefirstsystematicphysicsthatweknowofisAristotle’s,anditisnotbadphysicsatall.

Aristotlepresentsitinabookentitled,precisely,Physics.Thebookdidn’ttakeitstitlefromthenameofadiscipline:itwasthedisciplinewhichgotitsnamefromAristotle’sbook.ForAristotle,physicsworksasfollows.First,itisnecessarytodistinguishbetweentheheavensandEarth.Intheheavens,everythingismadeupofacrystallinesubstancewhichmovesinacircularmotionandturnseternallyaroundtheEarthingreatconcentriccircles,withthesphericalEarthatthecentreofeverything.OntheEarth,itisnecessarytodistinguishbetweenforcedmotionandnaturalmotion.Forcedmotioniscausedbyathrustandendswhenthethrustends.Naturalmotionisvertical–upwardsordownwards–anddependsbothonthesubstanceandthelocation.Eachsubstancehasa‘naturalplace’,thatistosay,aproperaltitudetowhichitalwaysreturns:earthatthebottom,wateralittlewayaboveit,airalittlehigherstill,andfireevenhigher.Whenyoupickupastoneandletitfall,thestonemovesdownwardsbecauseitwantstoreturntoitsnaturallevel.Airbubblesinwater,fireintheair;andchildren’sballoonsmoveupwards,seekingtheirnaturalplace.

Donotsmileatthistheory,ordismissit,becauseitisverysoundphysics.It’sagoodandcorrectdescriptionofthemotionofbodiesimmersedinafluidandsubjecttogravityandfriction,namely,therealthingswemeetinoureverydayexperience.It’snotwrongphysics,asisfrequentlysaid.fn3

It’sanapproximation.ButthephysicsofNewton,too,isanapproximationofgeneralrelativity.Andprobablyeverythingthatweknowtodayaswellisanapproximationofsomethingelsewhichwedon’tyetknow.ThephysicsofAristotleisstillrough,itisnotquantitative(wecannotcomputewithit),butitiscoherentandrationalandenablescorrectqualitativepredictionstobemade.Itisnotfornothingthatitremainedforcenturiesthebestavailablemodelforunderstandingmotion.1

PerhapsevenmoreimportantforthefuturedevelopmentofscienceisPlato.ItishewhounderstoodthevalueoftheintuitionofPythagorasandPythagorism:thatthekeyto

movingforwards,togoingbeyondMiletus,wasmathematics.PythagoraswasbornonSamos,asmallislandnotfarfromMiletus.Hisfirstbiographers,

IamblichusandPorphyry,reporthowyoungPythagoraswasadiscipleofelderlyAnaximander.EverythingoriginatesinMiletus.Pythagorastravelledwidely,probablyinEgyptandasfarasBabylon,beforefinallysettlinginthesouthofItaly,atCrotone,wherehefoundedareligio-politico-scientificsectwhichplayedanimportantroleinthepoliticsofthelittletownbutleftamomentouslegacytotheentireworld:thediscoveryofthetheoreticalutilityofmathematics:‘Number’–heissaidtohaveasserted–‘governsformsandideas.’2

PlatodivestedPythagorismofitscumbersomeanduselessmysticalbaggage.Heabsorbedanddistilleditsusefulmessage:mathematicsisthelanguagebestadaptedtounderstandanddescribetheworld.Thereachofthisinsightisimmense;itisoneofthereasonsforthesuccessofWesternscience.Accordingtotradition,Platocarvedonthedoorofhisschoolthephrase,‘Letnooneenterherewhoisignorantofgeometry.’

Drivenbythisconviction,itwasPlatowhoposedthemomentousquestion:thequestionoutofwhich,afteralongdetour,modernsciencewouldemerge.Ofhisdiscipleswhostudiedmathematicsheaskediftheycouldfindthemathematicallawsfollowedbythecelestialbodiesvisibleintheheavens.Venus,MarsandJupitercanbeeasilyobservedinthenightsky.Theyseemtomovealittleatrandom,backandforthamongtheotherstars.Isitpossibletofindamathematicswhichisabletodescribeandpredicttheirmovements?

TheexercisebeginswithEudoxusinPlato’sschool,andispursuedthroughoutthefollowingcenturiesbyastronomerssuchasAristarchusandHipparchus,bringingancientastronomyontoanextremelyhighscientificlevel.Weknowofthetriumphsofthissciencethankstoasinglebook,theonlyonetohavesurvived:theAlmagestofPtolemy.PtolemywasanastronomerwholivedinAlexandriainthefirstcenturyofourera,undertheRomanEmpire,whensciencewasalreadyin

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declineandabouttodisappearaltogether,overwhelmedbythecollapseoftheHellenisticworldandsuffocatedbytheChristianizationoftheempire.

Ptolemy’sbookisamajorworkofscience.Rigorous,precise,complex,itpresentsamathematicalsystemofastronomycapableofpredictingtheseeminglyrandommovementsoftheplanetsinthesky,withalmostcompleteprecision,giventhelimitationsofthehumaneye.ThebookistheproofthattheintuitionofPythagoraswascorrect.Mathematicsallowstheworldtobedescribedandthefuturetobepredicted:theapparentlywanderinganddisorderlymovementsoftheplanetscanbepreciselypredictedbyusingmathematicalformulaethatPtolemy,summarizingtheresultsofcenturiesofworkbyGreekastronomers,presentsinasystematicandmasterlyway.Eventoday,withalittlestudying,itispossibletoopenPtolemy’sbook,learnitstechniquesandcalculate,forexample,thepositionwhichMarswillhaveinafuturesky.Today:twothousandyears,thatis,afterthebookwaswritten.TherealizationthatworkingthismagicisreallypossibleisthebasisofmodernscienceandowesnotalittletoPythagorasandPlato.

Afterthecollapseofancientscience,noonethroughouttheMediterraneanwascapableofunderstandingPtolemy–oranyoftheothersmallnumberofmajorscientificworksthatsurvivedthecatastrophe,suchastheElementsofEuclid.InIndia,whereGreeklearninghadarrived,thankstorichcommercialandculturalexchanges,thesebookswerestudiedandunderstood.

FromIndiathisknowledgereturnedtotheWest,thankstolearnedPersianandArabscientistswhowereabletounderstandandpreserveit.Butastronomydidnottakeanyverysignificantstepforwardforoverathousandyears.

MoreorlessatthesametimeasPoggioBracciolinidiscoveredthemanuscriptofLucretius,theheadyatmosphereofItalianhumanismandtheenthusiasmforancienttextsalsointoxicatedayoungPolewhohadcometostudyinItaly,firstatBologna,thenatPadua.HesignedhimselfintheLatinmanner:NicolausCopernicus.TheyoungCopernicusstudiesPtolemy’sAlmagestandfallsinlovewithit.Hedecidestospendhislifedoingastronomy,followinginthefootstepsofthegreatPtolemy.

Thetimeisnowripeand,morethanathousandyearsafterPtolemy,CopernicusisabletomaketheleapforwardthatgenerationsofIndian,ArabandPersianastronomershadnotbeenabletomake:notsimplylearning,applyingandaddingsmallameliorationstothePtolemaicsystembutthoroughlyimprovingit–withthecouragetochangeitindepth.InsteadofdescribingheavenlybodiesturningaroundtheEarth,CopernicuswritesasortofrevisedandcorrectedversionofPtolemy’sAlmagest,inwhichtheSunisatthecentreandtheEarth,togetherwiththeotherplanets,runsaroundit.

Inthisway–Copernicushopes–thecalculationswouldworkevenbetter.Inreality,theydidnotworkmuchbetterthanthoseofPtolemy;infact,intheend,theyturnedouttoworklesswell.Buttheideawas,nevertheless,sound:inthesucceedinggeneration,JohannesKeplershowsthattheCopernicansystemcanreallybemadetofunctionbetterthanPtolemy’s.Painstakinglyanalysingnew,preciseobservations,KeplershowsthatafewnewmathematicallawscandescribewithexactitudethemovementsoftheplanetsaroundtheSun,withadegreeofaccuracyevengreaterthananyobtainedinantiquity.Wearenowin1600,andforthefirsttimehumanityfindsouthowtodosomethingbetterthanwhatwasdoneinAlexandriamorethanathousandyearsearlier.

WhileKeplerinthecoldnorthcalculatesmovementsinthesky,inItalyitiswithGalileoGalileithatthenewsciencebeginstotakeoff.Exuberant,Italian,polemical,argumentative,highlycultured,exceptionallyintelligentandoverflowingwithinventiveness,GalileogetssentfromHollandanewinvention–thetelescope–andmakesagesturewhichchangeshumanhistory.Hepointsittowardsthesky.

LikeRoyinBladeRunner,heseesthingswepeoplewouldn’tbelieve:ringsaroundSaturn,mountainsonthemoon,phasesofVenus,moonsorbitingJupiter…EachofthesephenomenarendersCopernicus’sideasmoreplausible.Scientificinstrumentsbegintoopenthemyopiceyesofhumankindontoaworldvasterandmorevariedthanithadasyetbeenabletoconceiveof.

ButGalileo’sgrandideaistomakethelogicaldeductionfromthecosmicrevolutionbroughtaboutbyCopernicus:convincedthattheEarthisaplanetlikeallothers,Galileoreasonsthat,ifmovementsintheheavensfollowprecisemathematicallaws,andiftheEarthisaplanetlikeallothers,andthuspartoftheheavens,thentheremustalsoexistprecisemathematicallawsgoverningthemovementsofobjectsonEarth.

ConfidentoftherationalityofnatureandofthePythagorean-Platonicvisionthatnatureisunderstandablethroughmathematics,GalileodecidestostudyhowobjectsmoveonEarthwhentheyaresetfree–thatis,whentheyfall.Convincedthatarelevantmathematicallawmustexist,hesetsouttosearchforit,bytrialanderror.Forthefirsttimeinthehistoryofmankind,anexperimentismade.ExperimentalsciencebeginswithGalileo.Hisexperimentissimple:heletsobjectsfall;thatis,heletsthemfollowwhatforAristotlewastheirnaturalmovementandseekstomeasurepreciselytheirfallingspeed.

Theresultismomentous:objectsdonotfallalwaysataconstantspeed,aseverybodythought.Theirspeed,instead,graduallyincreasesduringtheearlypartofthefall.Inthisphase,whatisconstantisnotthespeedofthefallbutrathertheacceleration,thatistosay,therateatwhichspeedincreases.And,magically,thisaccelerationturnsouttobethesameforallobjects.Galileocompletesafirstroughmeasurementofthisaccelerationandfindsitconstant.Itsvalueis

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approximately9.8metrespersecondpersecond,whichistosaythatforeverysecondthatanobjectfalls,itsspeedincreasesby9.8metrespersecond.Keepinmindthisnumber.

Thisisthefirstmathematicallawdiscoveredforearthlybodies:thelawoffallingbodies.fn4Upuntilthispoint,onlymathematicallawsforthemovementsoftheplanetshadbeendiscovered.Mathematicalperfectionisnolongerconfinedtotheheavens.

Butthegreatestresultisstilltocome,anditwillbeIsaacNewtonwhowillattainit.NewtonstudiesindepththeresultsofGalileoandKeplerandbycombiningthem,findsthehiddendiamond.Wecanfollowhisreasoningintermsofthe‘littlemoon’,ashehimselftellsitinTheMathematicalPrinciplesofNaturalPhilosophy,thebookthatformsthefoundationofmodernscience.

ImaginethattheEarth,writesNewton,hadmanymoons,likeJupiter.InadditiontotherealMoon,imagineothermoons,andinparticularalittlemoonwhichorbitstheEarthataminimaldistancefromit,justabovethemountainpeaks.Atwhatspeedwouldthislittlemoontravel?OneofthelawsdiscoveredbyKeplerrelatestheradiusoforbitwithperiod,namely,thetimetakentocompleteanorbit.fn5WeknowtheradiusoftheorbitoftherealMoon(Hipparchushadmeasureditinantiquity)anditsperiod(onemonth).Weknowtheradiusoftheorbitofthelittlemoon(theradiusoftheEarth,measuredbyEratosthenesinantiquity).Withasimpleproportionwecancomputetheperiodoftheorbitofthelittlemoon.Theresultisonehourandahalf.ThelittlemoonwouldcompleteitsorbitaroundtheEartheveryonehourandahalf.

Now,anobjectwhichorbitsdoesnotgostraight:itcontinuallychangesdirection,andachangeofdirectionisanacceleration.ThelittlemoonacceleratestowardsthecentreoftheEarth.Thisaccelerationiseasytocompute.fn6Newtonmakesthesimplecalculation,andtheresultis…9.8metrespersecondpersecond!ThesameaccelerationasinGalileo’sexperimentsforfallingbodiesonEarth.

Coincidence?Itcan’tbe,reasonsNewton.Iftheeffectisthesame–adownwardsaccelerationof9.8metrespersecondpersecond–thecausemustbethesame.Andso:theforcewhichcausesthelittlemoontoturnarounditsorbitmustbethesameasthatwhichcausesobjectstofalltothegroundonEarth.

Wecalltheforcecausingobjectstofallgravity.NewtonunderstandsthatitisthissamegravitythatmakesthelittlemoonturnaroundtheEarth.Withoutthisgravityitwouldrunawayinastraightline.Butthen,also,therealMoonmustorbittheEarthbecauseofgravity!AndthemoonswhichorbitJupiterareattractedbyJupiter,andtheplanetswhichturnaroundtheSunareattractedbytheSun!Withoutthisattraction,everycelestialbodywouldmoveinastraightline.Sotheuniverse,then,isalargespacewherebodiesattractoneanotherbymeansofforces;andthereisauniversalforce:gravity–everybodyattractseveryotherbody.

Animmensevisiontakesshape.Suddenly,aftermillennia,thereisnolongerseparationbetweentheheavensandEarth;thereisno‘naturallevel’forthings,asAristotlepresumed;thereisnocentretotheworld;thingsletfreenolongerseektheirnaturalplacebutmoveinastraightline,forever.

AsimplecalculationwiththelittlemoonallowsNewtontodeducehowtheforceofgravitychangeswithdistanceandtodetermineitsstrength,*givenbywhatwecalltodayNewton’sconstant,indicatedbytheletter‘G’for‘gravity’.OnEarth,thisforcecausesthingstofall;intheheavensitholdsplanetsandsatellitesontheirorbits.Theforceisthesame.

It’sasubversionoftheconceptualstructureoftheAristotelianworld,thedominantvisionoftheworldthroughouttheMiddleAges.ThinkofDante’suniverse,forexample:justasforAristotle,Earthisaballatthecentreoftheuniverse,orbitedbythecelestialspheres.Notanymore.Theuniverseisanimmense,infinitespacestuddedwithstars,withoutlimitandwithoutcentre.Withinit,materialbodiesrunfreeandstraight,unlessaforce,generatedbyanotherbody,deviatesthem.ReferencetoancientatomismisclearinNewton,evenifstillformulatedinconventionalterms:

ItseemsprobabletomethatGod,inthebeginning,formedmatterinsolid,massy,hard,impenetrable,moveableparticles,ofsuchsizesandfigures,andwithsuchotherproperties,andinsuchproportionstospace…3

TheworldofNewtonianmechanicsissimple,andissummarizedinfigures2.1and2.2.ItistheworldofDemocritusreborn.Aworldmadeofavast,undifferentiatedspace,alwaysequaltoitself,whereparticlesrunforeverandactuponeachother–andnothingelse.TheworldsungbyLeopardi:

…sittinghereandgazing,IfindthatendlessSpacesbeyondthathedge,andmore-than-humanSilences,andthedeepestpeaceandquietArefashionedinmythought…4

ButnowthevisionisimmenselymorepowerfulthaninDemocritus,becauseitisnotjustamentalimagewithwhichtoordertheworld:itisnowcombinedwithmathematics,withthelegacyofPythagoras,andwiththemightytraditionoftheAlexandrianastronomers’mathematicalphysics.TheworldofNewtonistheworldofDemocritus,renderedmathematical.

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Figure2.1Whatistheworldmadeof?

Figure2.2TheworldofNewton:particleswhichmoveinspace,astimepasses,attractedbyforces.

Newtonacknowledgeswithouthesitationthedebtthatthenewscienceowestothescienceofantiquity.InthefirstlinesofhisTheSystemoftheWorld,forexample,heattributestoantiquity(correctly)theoriginsoftheideaatthebasisoftheCopernicanrevolution:‘Itwastheopinionoftheancientphilosophersthatinthehighestpartsoftheworldthestarsremainfixedandmotionless,andthattheEarthturnsaroundtheSun’;althoughheisabitconfusedastowhodidwhatinthepast,andquotes–sometimesappositely,sometimesoutofcontext–Philolaus,AristarchusofSamos,Anaximander,Plato,Anaxagoras,Democritusand(!)‘thelearnedNumaPompilius,kingoftheRomans’.

ThepowerofthenewNewtonianintellectualframe-workprovestobebeyondallexpectation.TheentiretechnologyofthenineteenthcenturyandofourownmodernworldrestslargelyuponNewton’sformulae.Threecenturieshavepassed,butit’sstillthankstotheoriesbaseduponNewton’sequationsthattodaywebuildbridges,trainsandskyscrapers,enginesandhydraulicsystems;thatweknowhowtoflyplanes,makeweatherforecasts,predicttheexistenceofaplanetbeforeseeingitandsendspaceshipstoMars…ThemodernworldwouldnothavebeenbornwithoutpassingbywayofNewton’slittlemoon.

Anewconceptionoftheworld,awayofthinkingwhichraisedtheenthusiasmoftheEnlightenmentofVoltaireandKant,aneffectivewayforpredictingthefuture:thishasbeenandcontinuestobetheimmenselegacyoftheNewtonianrevolution.

Itseemedthusthatthefinalkeytounderstandingrealityhadbeendiscovered:theworldconsistsonlyofagreat,infinitespacewhere,astimepasses,particlesmoveandattractoneanotherbymeansofforces.Wecanwritepreciseequationsthatdescribetheseforces.Theseequationsprovetobeimmenselyeffective.Stillinthenineteenthcentury,peoplesaidthatNewtonwasnotonlyoneofthemostintelligentandfar-seeingofmenbutalsothemostfortunate–becausethereisonlyonesystemoffundamentallaws,andhewasgiventhegoodfortuneofdiscoveringit.Everythingseemedclear.

Butisthatreallyall?

Michael:fieldsandlightNewtonknewthathisequationsdidnotdescribealltheforcesthatexistinnature.Thereareforcesotherthangravitythatactuponbodies.Thingsdon’tmovejustwhentheyfall.ThefirstproblemleftopenbyNewtonwastounderstandtheotherforcesthatdeterminewhathappensaroundus.Thishadtowaituntilthenineteenthcentury,anditledtotwosurprises.

Thefirstsurpriseisthatalmostallphenomenaweseearegovernedbyasingleforce,otherthangravity:theforcethattodaywecallelectromagnetism.Itisthisforcewhichholdstogetherthematterthatformssolidbodies;holdstogetheratomsinmolecules,andelectronsinatoms.Thisiswhatmakeschemistryandlivingmatterwork.Itisthisforcewhichoperatesintheneuronsofourbrainandgovernsourprocessingoftheinformationontheworldweperceive,andthewaywethink.Andit’salwaysthisforcethatcreatesthefrictionwhichstopsaslidingobject,whichsoftensthelandingofaparachutist,whichturnselectricmotorsandcombustionengines,fn7orthatallowsustoturnonlightsandlistentotheradio.

Thesecondandbiggestsurprise,crucialtothestoryI’mtelling,isthatunderstandingthisforcerequiresanimportantmodificationtotheworldofNewton:themodificationoutofwhichmodern

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physicswasborn,andthemostimportantnotiontokeepinfocus,tounderstandtherestofthisbook–thenotionoffield.

TheunderstandingofhowelectromagneticforceworkswasmadebyanotherBriton,orratherbytwo:science’soddestcouple–MichaelFaradayandJamesClerkMaxwell.

MichaelFaradayisanimpoverishedLondonerwithoutformaleducation,whoworksfirstinabookbindery,theninalaboratory,whereheexcels,gainshismaster’sconfidenceandgrowsintothemostbrilliantexperimenterofnineteenth-centuryphysicsanditsgreatestvisionary.Withoutknowingmathematics,hewritesoneofthebestbooksofphysicseverwritten,virtuallydevoidofequations.Heseesphysicswithhismind’seye,andwithhismind’seyecreatesworlds.JamesClerkMaxwellisarichScottisharistocratandoneofthegreatestmathematiciansofthecentury.Despitebeingseparatedbyagulfinintellectualstyleaswellassocialorigin,theysucceedinunderstandingeachother–and,together,combiningtwokindsofgenius,theyopenthewaytomodernphysics.

Figure2.3MichaelFaradayandJamesClerkMaxwell.

Whatwasknownaboutelectricityandmagnetismatthebeginningoftheeighteenthcenturyconsistedoflittleotherthanafewamusingsideshowtricks:glassrodswhichattractpiecesofpaper;magnetswhichrepelandattract.Thestudyofelectricityandmagnetismcontinuedslowlythroughouttheeighteenthcenturyandintothenineteenth,wherewefindFaradayworkinginaLondonlaboratoryfullofbobbins,needles,knivesandironcages,exploringhowelectricalandmagneticthingsattractandrepel.AgoodNewtonian,hetriestounderstandtheforcewhichactsbetweenchargedandmagneticthings.Butslowly,guidedbyhishands,inclosecontactwiththeseobjects,heisledtoanintuitionthatwillbecomethebasisofmodernphysics.He‘sees’somethingnew.

Hisintuitionisthis:wemustnotthinkofforcesactingdirectlybetweendistantobjects,asNewtonpresumed.Wemustinsteadthinkthatthereexistsanentitydiffusedthroughoutspace,whichismodifiedbyelectricandmagneticbodiesandwhich,inturn,actsupon(pushesandpulls)thebodies.Thisentity,whoseexistenceFaradayintuits,istodaycalledthefield.

Whatisit,then,afield?Faradayseesitasformedbybundlesofverythinlines(infinitelythin),whichfillspace;aninvisible,giganticcobwebfillingeverythingaroundus.Hecallstheselines‘linesofforce’because,insomeway,theselines‘carrytheforce’:theytransmittheelectricandthemagneticforcesfromonebodytoanother,asiftheywerecablespullingandpushing(figure2.4).

Anobjectwithanelectriccharge(arubbedglassrod,forinstance)distortstheelectricandmagneticfields(thelines)arounditself,andinturnthesefieldsproduceaforceoneachchargedobjectimmersedinthem.Thus,twodistantchargedobjectsdonotattractorrepeleachotherdirectlybutonlyviathemediuminterposedbetweenthem.

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Figure2.4Thefield’slinesfillspace.Throughthem,twoobjectswithelectricalchargeinteract.Theforceofthetwoobjectsis‘carried’bythefield’slinesofforce.

Ifyoutaketwomagnetsinyourhandsandplaywiththem,bringingthemtogetherandapartrepeatedly,feelingtheforcewithwhichtheyattractandrepel,itisnotdifficulttoexperiencethesameintuitionasFaraday,to‘feel’viatheseeffectsthefieldinterposedbetweenthemagnets.

ThisisaradicallydifferentideafromtheNewtoniannotionofaforceactingbetweendistantbodies.ButitwouldhaveappealedtoNewton.Newton,indeed,wasperplexedregardingthisveryattractionatadistancethathehimselfhadintroduced.HowdoestheEarthmanagetoattracttheMoon,whichissodistant?HowcantheSunattracttheEarthwithoutcomingintocontactwithit?Hehadwritteninaletter:

Itisinconceivablethatinanimatebrutemattershould,withouttheinterventionofsomethingelsewhichisnotmaterial,operateuponandaffectothermatter,andhaveaneffectuponit,withoutmutualcontact.5

And,furtherdownthepage,weevenfind:ThatGravityshouldbeinnate,inherentandessentialtoMatter,sothatoneBodymayactuponanotherataDistancethro’aVacuum,withouttheMediationofanythingelse,byandthroughwhichtheirActionandForcemaybeconveyedfromonetoanother,istomesogreatanAbsurdity,thatIbelievenoManwhohasinphilosophicalMattersacompetentFacultyofthinking,caneverfallintoit.GravitymustbecausedbyanAgentactingconstantlyaccordingtocertainLaws;butwhetherthisAgentbematerialorimmaterial,IhavelefttotheConsiderationofmyReaders.6

Newtonisdeeminghisveryownmasterworktobeabsurd–theverysameworkwhichwastobepraisedforcenturiestocomeastheultimateachievementofscience!Heunderstandsthatbehindtheactionatadistanceofhistheorytheremustbesomethingelse,buthehasnoideawhat,andleavesthequestion…‘totheConsiderationofmyReaders’!

Itischaracteristicofgeniustobeawareofthelimitationsofitsownfindings,eveninthecaseofsuchmomentousoutcomesasNewton’sdiscoveryofthelawsofmechanicsanduniversalgravity.Newton’stheoryworkedsowell,itturnedouttobesouseful,thatfortwocenturiesnoonebotheredanylongertoquestionit–untilFaraday,the‘reader’towhomNewtonhadbequeathedtheunansweredquestion,foundthekeytounderstandinghowbodiescanattractandrepeleachotheratadistanceinareasonablemanner.EinsteinwilllaterapplyFaraday’sbrilliantsolutiontoNewton’sowntheoryofgravity.

Introducingthenewentity–thefield–FaradaydepartsradicallyfromNewton’selegantandsimpleontology:theworldisnolongermadeuponlyofparticlesthatmoveinspacewhiletimepasses.Anewactor–thefield–appearsonthescene.Faradayisawareoftheimportanceofthestepheistaking.Therearebeautifulpassagesinhisbookwhereheaskswhethertheselinesofforcecouldbethingswitharealexistence.Afterdoubtsanddifferentconsiderations,heconcludesthathethinkstheyareindeedreal,butwith‘thehesitationthatisnecessarywhenfacedwiththedeepestquestionsofscience’.7Heisconsciousthatheissuggestingnothinglessthanamodificationofthestructureoftheworld,aftertwocenturiesofuninterruptedsuccessesforNewtonianphysics(figure2.5).

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Figure2.5TheworldofFaradayandMaxwell:particlesandfieldswhichmoveinspace,withthepassageoftime.

Maxwellquicklyrealizesthatgoldhasbeenstruckwiththisidea.HetranslatesFaraday’sinsight,whichFaradayexplainsonlyinwords,intoapageofequations.fn8ThesearenowknownasMaxwell’sequations.Theydescribethebehaviouroftheelectricandthemagneticfields:themathematicalversionofthe‘Faradaylines’.fn9

Today,Maxwell’sequationsareuseddailytodescribeallelectricandmagneticphenomena,todesignantennae,radios,electricenginesandcomputers.Andthisisnotall:thesesameequationsareneededtoexplainhowatomsfunction(theyareheldtogetherbyelectricalforces),andwhytheparticlesofthematerialthatformsastoneadheretogether,orhowtheSunworks.Theydescribeanamazingnumberandrangeofphenomena.Almosteverythingthatwewitnesstakingplace–withtheexceptionofgravity,butlittleelsebesides–iswelldescribedbyMaxwell’sequations.

Butthereismore.Thereisstillwhatisperhapsthemostbeautifulsuccessofscience:Maxwell’sequationstelluswhatlightis.

MaxwellrealizesthathisequationspredictthatFaraday’slinescantrembleandundulate,justlikethewavesofthesea.HecomputesthespeedatwhichtheundulationsofFaraday’slinesmoveandtheresultturnsouttobe…thesameasforlight!Why?Maxwellunderstands:becauselightisnothingotherthanthisrapidtremblingofFaraday’slines!NotonlyhaveFaradayandMaxwellfiguredouthowelectricityandmagnetismworkbut,withthesamestroke,asacollateraleffect,theyhavefiguredoutwhatlightis.

Weseetheworldaroundusincolour.Whatiscolour?Putsimply,itisthefrequency(thespeedofoscillation)oftheelectromagneticwavethatlightis.Ifthewavevibratesmorerapidly,thelightisbluer.Ifitvibratesalittlemoreslowly,thelightisredder.Colour,asweperceiveit,isthepsychophysicalreactionofthenervesignalgeneratedbythereceptorsofoureyes,whichdistinguishelectromagneticwavesofdifferentfrequencies.

IwonderhowMaxwellfeltwhenherealizedthathisequations–writtentodescribebobbins,smallcagesandlittleneedlesinFaraday’slab–turnedouttoexplainthenatureoflightandcolour…

LightisthusnothingmorethanarapidvibrationofthespiderwebofFaraday’slines,whichripplelikethesurfaceofalakeasthewindblows.Itisn’ttruethatwe‘donotsee’Faradaylines.WeonlyseevibratingFaradaylines.‘Tosee’istoperceivelight,andlightisthemovementofFaradaylines.Nothingleapsfromonelocationinspacetoanotherwithoutsomethingtransportingit.Ifweseeachildplayingonthebeach,itisonlybecausebetweenthemandourselvesthereisthislakeofvibratinglineswhichtransporttheirimagetous.Istheworldnotmarvellous?

Thediscoveryisextraordinary,butthereisevenmore.Theultimatecorollaryofthediscoveryhasaconcretevalueforusthatisunequalled.MaxwellrecognizesthattheequationsforeseethatFaraday’slinescanalsovibrateatmuchlowerfrequencies,thatistosay,moreslowlythanlight.Therefore,theremustbeotherwaveswhichnobodyhadyetseen,producedbythemovementofelectricalcharges,whichinturnmoveelectricalcharges.Itmustbepossibletoshakeanelectricchargehere,andtoproduceawavewhichwilldriveanelectriccurrentthere.Onlyafewyearslater,thesewaves,anticipatedtheoreticallybyMaxwell,willberevealedbytheGermanphysicistHeinrichHertz;andjustafewyearslaterstill,GuglielmoMarconibuildsthefirstradio.

Allmoderncommunicationstechnology–radio,television,telephones,computers,satellites,wi-fi,theinternet,etc.–isanapplicationofMaxwell’sprediction;theMaxwellequationsarethebasisforallcalculationsmadebytelecommunicationsengineers.Thecontemporaryworld,basedoncommunications,emergesfromtheintuitionsofapoorLondonbookbinder–askilfulexplorerofideaswithavividimagination–whosawsomelinesinhismind’seye;andtheworkofagoodmathematicianwhotranslatedthisvisionintoequations,understandingthatintheblinkofaneyethewavesoftheselinescancarrynewsfromonesideoftheplanettotheother.

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Figure2.6Whatistheworldmadeof?

Ourentirecurrenttechnologyisfoundedontheuseofaphysicalthing–electromagneticwaves–whichwasnotdiscoveredempirically:itwaspredictedbyMaxwell,simplybysearchingforthemathematicaldescriptionaccountingfortheintuitionFaradaygotfrombobbinsandneedles.Thisistheoutstandingpoweroftheoreticalphysics.

Theworldhaschanged:itisnolongermadeupofparticlesinspacebutofparticlesandfieldsinspace(figure2.6).Itseemsaminorchange,butafewdecadeslaterayoungJewishman,acitizenoftheworld,willdrawfromitconsequencesthatwillgofarbeyondMichaelFaraday’salreadyfervidimagination,andwhichwillshakeNewton’sworldevenfurthertoitscore.

*

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PartTwo

THE BEGINNING OF THE REVOLUTION

Twentieth-centuryphysicshasradicallymodifiedtheNewtonianimageoftheworld.Thenewstepsarethebasisofagreatdealoftoday’stechnology.Thedeepeningofourunderstandingoftheworldisbasedontwotheories:generalrelativityandquantummechanics.Bothdemandadaringre-evaluationofourconventionalideasabouttheworld:spaceandtimeinrelativity;matterandenergyinquantumtheory.

InthispartofthebookIdescribethetwotheoriesinsomedetail,tryingtoclarifytheircoremeaningandhighlighttheconceptualrevolutiontheybroughtabout.It’sherethatthemagicoftwentieth-centuryphysicsbegins.Studyingandtryingtounderstandthemindepthisabewitchingadventure.

Thesetwotheories–relativityandthequanta–providethebasisonwhichwearetodaybuildingaquantumtheoryofgravity.Theyarethegroundfromwhichwearetryingtomoveforwards.

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3.Albert

ThefatherofAlbertEinsteinbuiltpowerstationsinItaly.WhenAlbertwasayoungboy,theMaxwellequationswereonlyafewdecadesold–butItalywasenteringitsindustrialrevolution,andtheturbinesandtransformersthathisfatherconstructedwerealreadybasedonthem.Thepowerofthenewphysicswasobvious.Albertwasarebel.HisparentslefthiminGermanytoattendhighschool,buthefoundthe

Germanschoolsystemtoorigidandmilitaristic;hecouldnotstandtheauthorityoftheschoolandabandonedhisstudies.HejoinedhisparentsinItaly,inPavia,andspenthistimeloafing.Later,hewenttostudyinSwitzerland,initiallyfailingtogetintotheZurichPolytechnic,ashewished.Afterhisuniversityyearshecouldnotfindaresearchpositionand,inordertolivewiththegirlheloved,hefoundemploymentinthepatentofficeofBerne.Itwasn’tmuchofaprofessionforaphysicsgraduate,butitgaveAlberttimetothink,andtowork

independently.Andhedidthinkandwork.Afterall,thisiswhathehaddonesincehisearlyyouth:hewouldreadEuclid’sElementsandKant’sCritiqueofPureReasoninsteadofattendingtowhathewasbeingtaughtatschool.Youdon’tgettonewplacesbyfollowingestablishedtracks.Attheageoftwenty-five,EinsteinsendstotheAnnalenderPhysikthreearticles.Eachwasworthy

ofaNobelPrize,andmore.Eachoneofthethreeisapillarsupportingourunderstandingoftheworld.Ihavealreadyspokenofthefirstarticle,inwhichtheyoungAlbertcalculatesthedimensionsofatomsandproves,aftertwenty-threecenturies,thattheideasofDemocrituswerecorrect:matterisgranular.ThesecondarticleistheoneforwhichEinsteinismostfamous–thearticleinwhichheintroduces

thetheoryofrelativity–anditistothetheoryofrelativitythatthischapterisdevoted.Infact,therearetwotheoriesofrelativity.Theenvelopesentbythetwenty-five-year-oldEinstein

containedtheexpositionofthefirstofthese:thetheoryknowntodayas‘specialrelativity’.Thisisanimportantclarificationofthestructureofspaceandtime,whichIillustrateherebeforeturningtotheother,andmostimportantofEinstein’stheories:generalrelativity.Specialrelativityisasubtleandconceptuallydifficulttheory.Itismoredifficulttodigestthan

generalrelativity.Reader,don’tbecomedemoralizedifthenextfewpagessoundabitabstruse.Thetheoryshows,forthefirsttime,thatintheNewtonianvisionoftheworldthereisn’tjustsomethingmissing:rather,itmustberadicallymodified–inawaythatgoescompletelyagainstcommonsense.Itisthefirstrealleapintotherevisionofourmostintuitiveunderstandingoftheworld.

TheextendedpresentThetheoriesofNewtonandofMaxwellappeartocontradicteachotherinasubtleway.Maxwell’sequationsdetermineavelocity:thevelocityoflight.ButNewton’smechanicsisnotcompatiblewiththeexistenceofafundamentalvelocity,becausewhatentersNewton’sequationsisacceleration,notvelocity.InNewton’sphysics,velocitycanonlybevelocityofsomethingwithrespecttosomethingelse.GalileohadunderlinedthefactthattheEarthmoveswithrespecttotheSun,evenifwedonotperceivethismovement,becausewhatweusuallyterm‘velocity’isvelocity‘withrespecttoEarth’.Velocity,wesay,isarelativeconcept,thatis,thereisnomeaningtothevelocityofanobjectbyitself:theonlyvelocitywhichexistsisthevelocityofanobjectwithrespecttoanotherobject.Thisiswhatphysicsstudentslearnedinthenineteenthcentury,andwhattheylearntoday.Butifthisisso,thenthespeedoflightdeterminedbyMaxwell’sequationsisvelocitywithrespecttowhat?Onepossibilityisthatthereisakindofuniversalsubstratuminrelationtowhichlightmovesand

hasitsspeed.ButthepredictionsofMaxwell’stheoryseemtobeindependentofthissubstratum.TheexperimentalattemptstomeasurethespeedoftheEarthwithrespecttothishypotheticalsubstratumtriedattheendofthetwentiethcenturyallfailed.Einsteinhasclaimedthathewasnotputontherighttrackbyanyexperimentsbutonlyby

reflectingontheapparentcontradictionbetweenMaxwell’sequationsandNewton’smechanics.HeaskedhimselfwhethertherewasawayofrenderingNewton’sandGalileo’scorediscoveriesandMaxwell’stheoryconsistent.Indoingso,Einsteinarrivesatastupefyingdiscovery.Tounderstandit,thinkofallthepast,

presentandfutureevents(withrespecttothemomentinwhichyouarereading)andimaginethemdistributedasinfigure3.1.Well,Einstein’sdiscoveryisthatthisdiagramisincorrect.Inreality,thingsareactuallyasthey

aredepictedinfigure3.2.

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Betweenthepastandthefutureofanevent(forexample,betweenthepastandthefutureforyou,whereyouare,andintheprecisemomentinwhichyouarereading)thereexistsan‘intermediatezone’,an‘expandedpresent’;azonethatisneitherpastnorfuture.Thisisthediscoverymadewithspecialrelativity.

Figure3.1SpaceandtimebeforeEinstein.

Figure3.2Thestructureof‘spacetime’.Foreveryobserver,the‘extendedpresent’istheintermediatezonebetweenthepastandthefuture.

Thedurationofthisintermediatezone,fn10whichisneitherinyourpastnorinyourfuture,isverysmallanddependsonwhereaneventtakesplacerelativetoyou,asillustratedinfigure3.2:thegreaterthedistanceoftheeventfromyou,thelongerthedurationoftheextendedpresent.Atadistanceofafewmetresfromyournose,dearreader,thedurationofwhatforyouistheintermediatezone,neitherpastnorfuture,isnomorethanafewnanoseconds:nexttonothing(thenumberofnanosecondsinasecondisthesameasthenumberofsecondsinthirtyyears).Thisismuchlessthanwecouldpossiblynotice.Ontheothersideoftheocean,thedurationofthisintermediatezoneisathousandthofasecond,stillwellbelowthethresholdofourperceptionoftime–theminimumamountoftimeweperceivewithoursenses–whichissomewhereintheorderofatenthofasecond.ButontheMoonthedurationoftheexpandedpresentisafewseconds,andonMarsitisaquarterofanhour.Thismeanswecansaythat,onMars,thereareeventsthatinthisprecisemomenthavealreadyhappened,eventsthatareyettohappen,butalsoaquarterofanhourduringwhichthingsoccurthatareneitherinourpastnorinourfuture.Theyareelsewhere.Wehadneverbeforebeenawareofthis‘elsewhere’because,nexttous,this

‘elsewhere’istoobrief;wearenotquickenoughtonoticeit.Butitexists,anditisreal.ThisiswhyitisimpossibletoholdasmoothconversationbetweenhereandMars.SayIamon

Marsandyouarehere.Iaskyouaquestionandyoureplyassoonasyou’veheardwhatIsaid;yourreplyreachesmeaquarterofanhourafterIposedthequestion.Thisquarterofanhouristimethatisneitherpastnorfuturetothemomentinwhichyou’verepliedtome.ThekeyfactaboutnaturethatEinsteinunderstoodisthatthisquarterofanhourisinevitable:thereisnowayofreducingit.Itiswovenintothetextureoftheeventsofspaceandoftime:wecannotabbreviateit,anymorethanwecansendalettertothepast.

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Figure3.3Therelativityofsimultaneity.

It’sstrange,butthisishowtheworldhappenstobe.AsstrangeasthefactthatinSydneypeopleliveupsidedown:strange,buttrue.Onegetsaccustomedtothefact,whichthenbecomesnormalandreasonable.Itisthestructureofspaceandtimethatismadelikethis.ThisimpliesthatitmakesnosensetosayofaneventonMarsthatitistakingplace‘justnow’,

because‘justnow’doesnotexist(figure3.3).fn11Intechnicalterms,wesaythatEinsteinhasunderstoodthat‘absolutesimultaneity’doesnotexist:thereisnocollectionofeventsintheuniversewhichexist‘now’.Thecollectionofalltheeventsintheuniversecannotbedescribedasasuccessionof‘now’s,ofpresents,onefollowingtheother;ithasamorecomplexstructure,illustratedinfigure3.2.Thefiguredescribesthatwhichinphysicsiscalledspacetime:thesetofallpastandfutureevents,butalsothosethatare‘neither-past-nor-future’;thesedonotformasingleinstant:theyhavethemselvesaduration.IntheAndromedaGalaxy,thedurationofthisexpandedpresentis(withrespecttous)2million

years.Everythingthathappensduringthese2millionyearsisneitherpastnorfuturewithrespecttoourselves.IfafriendlyadvancedAndromedacivilizationdecidedtosendafleetofspacecrafttovisitus,itwouldmakenosensetoaskwhether‘now’thefleethasalreadyleft,ornotyet.Theonlymeaningfulquestioniswhenwereceivethefirstsignalfromthefleet:fromthatmomenton–notearlier–thedepartureofthefleetisinourpast.ThediscoveryofthestructureofspacetimemadebytheyoungEinsteinin1905hasconcrete

consequences.Thefactthatspaceandtimeareintimatelyconnected,asinfigure3.2,impliesasubtlerestructuringofNewton’smechanics,whichEinsteinrapidlycompletesin1905and1906.Afirstresultofthisrestructuringisthat,asspaceandtimefusetogetherinasingleconceptofspacetime,sotheelectricfieldandthemagneticfieldsfusetogetherinthesameway,mergingintoasingleentitywhichtodaywecalltheelectromagneticfield.ThecomplicatedequationswrittenbyMaxwellforthetwofieldsbecomesimplewhenwritteninthisnewlanguage.Thereisanotherimplicationofthetheory,freightedwithheavyconsequences.Theconceptsof

‘energy’and‘mass’becomecombinedinthesamewayastimeandspace,andelectricandmagneticfields,arefusedtogetherinthenewmechanics.Before1905twogeneralprinciplesappearedcertain:conservationofmass,andconservationofenergy.Thefirsthadbeenextensivelyverifiedbychemists:massneverchangesinachemicalreaction.Thesecond–conservationofenergy–followeddirectlyfromNewton’sequationsandwasconsideredoneofthemostincontrovertiblelaws.ButEinsteinrealizesthatenergyandmassaretwofacetsofthesameentity,justastheelectricandmagneticfieldsaretwofacetsofthesamefield,andasspaceandtimearetwofacetsoftheonething:spacetime.Thisimpliesthatmass,byitself,isnotconserved;andenergy–asitwasconceivedatthetime–isnotindependentlyconservedeither.Onemaybetransformedintotheother:onlyonesinglelawofconservationexists,nottwo.Whatisconservedisthesumofmassandenergy,noteachseparately.Processesmustexistthattransformenergyintomass,ormassintoenergy.ArapidcalculationteachesEinsteinhowmuchenergyisobtainedbytransformingonegramof

mass.TheresultisthecelebratedformulaE=mc².Sincethespeedoflightcisaverylargenumber,andc²anevengreaternumber,theenergyobtainedfromtransformingonegramofmassisenormous;itistheenergyofmillionsofbombsexplodingatthesametime–enoughenergytoilluminateacityandpowertheindustriesofacountryformonthsor,conversely,capableofdestroyinginasecondhundredsofthousandsofhumanbeings,inacitysuchasHiroshima.

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ThetheoreticalspeculationsoftheyoungEinsteinhadtransportedhumanityintoanewera:theeraofnuclearpower,aneraofnewpossibilities,andnewdangers.Today,thankstotheintelligenceofarebelliousyoungmanwhowouldnotabiderules,wehavetheinstrumentstobringlighttothehomesofthe10billionhumanbeingswhowillsooninhabittheplanet,totravelinspacetowardsotherstars,ortodestroyeachotheranddevastatetheplanet.Itdependsonourchoices;onwhichleaderswecallupontodecideforus.TodaythestructureofspacetimeproposedbyEinsteiniswellunderstoodandrepeatedlytestedin

laboratories;itisconsideredconclusivelyestablished.TimeandspacearedifferentfromthewaytheyhadbeenconceivedsinceNewton.Spacedoesnotexistindependentlyfromtime.Intheexpandedspaceoffigure3.2thereisnoparticularslicehavingabetterclaimthanotherstobecalled‘spacenow’.Ourintuitiveideaofthepresent–theensembleofalleventshappening‘now’intheuniverse–isaneffectofourblindness:ourinabilitytorecognizesmalltemporalintervals.Itisanillegitimateextrapolationfromourparochialexperience.ThepresentisliketheflatnessoftheEarth:anillusion.WeimaginedaflatEarthbecauseofthe

limitationsofoursenses,becausewecannotseemuchbeyondourownnoses.Hadwelivedonanasteroidofafewkilometresindiameter,liketheLittlePrince,wewouldhaveeasilyrealizedwewereonasphere.Hadourbrainandoursensesbeenmoreprecise,hadweeasilyperceivedtimeinnanoseconds,wewouldneverhavemadeuptheideaofa‘present’extendingeverywhere.Wewouldhaveeasilyrecognizedtheexistenceoftheintermediatezonebetweenpastandfuture.Wewouldhaverealizedthatsaying‘hereandnow’makessense,butthatsaying‘now’todesignateevents‘happeningnow’throughouttheuniversemakesnosense.Itislikeaskingwhetherourgalaxyis‘aboveorbelow’thegalaxyofAndromeda:aquestionthatmakesnosense,because‘above’or‘below’hasmeaningonthesurfaceoftheEarth,notintheuniverse.Thereisn’tan‘up’ora‘down’intheuniverse.Similarly,thereisn’teitheralwaysa‘before’andan‘after’betweentwoeventsintheuniverse.Theresultingknittedstructurethatspaceandtimeformtogether,depictedinfigureslike3.2and3.3,iswhatphysicistscall‘spacetime’(figure3.4).WhentheAnnalenderPhysikpublishedthearticlebyEinsteininwhichallthiswassuddenly

clarified,theimpactupontheworldofphysicswasmomentous.TheapparentcontradictionbetweentheequationsofMaxwellandNewtonianphysicswerewellknown,andnooneknewhowtoresolvethem.Einstein’ssolution,astonishingandextremelyelegant,tookeveryonebysurprise.ThestorygoesthatinthedimlylitoldhallsofCracowUniversity,anaustereprofessorofphysicscameoutofhisstudy,wavingaroundEinstein’sarticle,screaming,‘ThenewArchimedesisborn!’ButdespitetheoutcryprovokedbythestepforwardsmadebyEinsteinin1905,wearenotyetat

hismasterpiece.Einstein’striumphisthesecondtheoryofrelativity,thetheoryofgeneralrelativity,publishedtenyearslater,whenhewasthirty-five.

Figure3.4Whatistheworldmadeof?

Thetheoryof‘generalrelativity’isthemostbeautifultheoryproducedbyphysics,andthefirstofthepillarsofquantumgravity.Itisattheheartofthenarrativeofthisbook.Here,therealmagicoftwentieth-centuryphysicsbegins.

ThemostbeautifuloftheoriesAfterpublishingthetheoryofspecialrelativity,Einsteinbecomesarenownedphysicistandreceivesoffersofworkfromnumerousuniversities.Butsomethingtroubleshim:specialrelativitydoesnotsquarewithwhatwasknownaboutgravity.Herealizesthiswhilewritingareviewonhistheory,andwonderswhetherthevenerabletheoryofthe‘universalgravity’ofthefatherofphysics,Newton,shouldnotbereconsideredaswell,tomakeitcompatiblewithhisrelativity.

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Theoriginoftheproblemiseasytounderstand.Newtonhadtriedtoexplainwhythingsfallandplanetsrevolve.Hehadimagineda‘force’thatdrawsallbodiestowardsoneanother:the‘forceofgravity’.Howthisforcemanagedtodrawdistantthingstogetherwithoutanythingbetweenthemwasnotunderstood.Newtonhimself,aswehaveseen,hadsuspectedthatintheideaofaforceactingbetweendistantbodiesthatdonottouchtherewassomethingmissing;andthatinorderfortheEarthtoattracttheMoonsomethingthatcouldtransmitthisforcehadtobetherebetweenthetwo.Twohundredyearslater,Faradayhadfoundthesolution–notfortheforceofgravity,butfortheelectricandmagneticforces:thefield.Electricandmagneticfields‘carryaround’theelectricandmagneticforce.It’sclear,atthisstage,toanyreasonableperson,thattheforceofgravitymusthaveitsFaraday

linesaswell.It’sclearalso,byanalogy,thattheforceofattractionbetweentheSunandtheEarth,orbetweentheEarthandfallingobjects,mustbeattributedtoafield–inthiscase,agravitationalfield.ThesolutiondiscoveredbyFaradayandMaxwelltothequestionastowhatcarriestheforcemustreasonablybeappliednotonlytoelectricitybutalsotogravity.Theremustbeagravitationalfield,andsomeequationsanalogoustoMaxwell’s,capableofdescribinghowFaraday’sgravitationallinesmove.Inthefirstyearsofthetwentiethcenturythisiscleartoanysufficientlyreasonableperson;thatistosay,onlytoAlbertEinstein.Einstein,fascinatedsinceadolescencebytheelectromagneticfieldthatpushedtherotorsinhis

father’spowerstations,beginstolookintothisgravitationalfieldandsearchforwhatkindofmathscoulddescribeit.Heimmerseshimselfintheproblem.Itwouldtaketenyearstoresolve.Tenyearsofmanicstudies,attempts,mistakes,confusion,brilliantideas,wrongideas,alongseriesofarticlespublishedwithincorrectequations,furthermistakesandstress.Finally,in1915,hecommitstoprintanarticlecontainingthecompletesolution,whichhenamestheGeneralTheoryofRelativity:hismasterpiece.ItisLevLandau,themostoutstandingtheoreticalphysicistoftheSovietUnion,whocalledit‘themostbeautifuloftheories’.Thereasonforthebeautyofthetheoryisnothardtosee.Insteadofsimplyinventingthe

mathematicalformofthegravitationalfieldandseekingtodevisetheequationsforit,EinsteinfishesouttheotherunresolvedquestioninthefurthestdepthsofNewton’stheoryandcombinesthetwoquestions.NewtonhadreturnedtoDemocritus’sidea,accordingtowhichbodiesmoveinspace.Thisspace

hadtobealarge,emptycontainer,arigidboxfortheuniverse;animmensescaffoldinginwhichobjectsruninstraightlines,untilaforcecausesthemtocurve.Butwhatisthis‘space’whichcontainstheworldmadeof?Whatisspace?Tous,theideaofspaceseemsnatural,butitisourfamiliaritywithNewtonianphysicsthatmakes

itso.Ifyouthinkaboutit,emptyspaceisnotpartofourexperience.FromAristotletoDescartes,thatistosay,fortwomillennia,theDemocriteanideaofspaceasapeculiarentity,distinctfromthings,hadneverbeenseenasreasonable.ForAristotle,asforDescartes,thingshaveextension:extensionisapropertyofthings;extensiondoesnotexistwithoutsomethingbeingextended.Icantakeawaythewaterfromaglass,butairwillfillit.Haveyoueverseenareallyemptyglass?Ifbetweentwothingsthereisnothing,Aristotlereasoned,thenthereisnothing.Howcantherebe

atthesametimesomething(space)andnothing?Whatisthisemptyspacewithinwhichparticlesmove?Isitsomething,orisitnothing?Ifitisnothing,itdoesn’texist,andwecandowithoutit.Ifitissomething,canitbetruethatitsonlypropertyistobethere,doingnothing?Sinceantiquity,theideaofemptyspace,halfwaybetweenathingandanon-thing,hadtroubled

thinkers.Democritushimself,whohadplacedemptyspaceatthebasisofhisworldwhereatomscourse,certainlywasn’tcrystalclearontheissue:hewrotethatemptyspaceissomething‘betweenbeingandnon-being’:‘Democrituspostulatedthefullandtheempty,callingone“Being”,andtheother“Non-Being”,’saysSimplicius.1Atomsarebeing.Spaceisnon-being–a‘non-being’that,nevertheless,exists.Itisdifficulttobemoreobscurethanthis.Newton,whoresuscitatedtheDemocriteanideaofspace,hadtriedtopatchthingsupbyarguing

thatspacewasGod’ssensorium.NoonehaseverunderstoodwhatNewtonmeantby‘God’ssensorium’,perhapsnotevenNewtonhimself.Certainly,Einstein,whogavelittlecredittotheideaofaGod(withorwithoutasensorium),exceptasaplayfulrhetoricaldevice,foundNewton’sexplanationofthenatureofspaceutterlyunconvincing.Newtonstruggledconsiderablytoovercomethescientists’andphilosophers’resistancetohis

revivingtheDemocriteanconceptofspace;atfirstnobodytookhimseriously.Onlytheextraordinaryefficacyofhisequations,whichturnedouttopredictalwaysthecorrectoutcome,endedupsilencingcriticism.ButdoubtsconcerningtheplausibilityoftheNewtonianconceptofspacepersisted,andEinstein,whoreadphilosophers,waswellawareofthem.ErnstMach,whoseinfluenceEinsteinreadilyacknowledged,wasthephilosopherwhohighlightedtheconceptualdifficultiesoftheNewtonianideaofspace–thesameMachwhodidnotbelieveintheexistenceofatoms.(Agoodexample,incidentally,ofhowthesamepersoncanbeshort-sightedinonerespectandfar-seeinginanother.)Thus,Einsteinaddressesnotonebuttwoproblems.First,howcanwedescribethegravitational

field?Second,whatisNewton’sspace?

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Andit’sherethatEinstein’sextraordinarystrokeofgeniusoccurs,oneofthegreatestflightsinthehistoryofhumanthinking:whatifthegravitationalfieldturnedoutactuallytobeNewton’smysteriousspace?WhatifNewton’sspacewasnothingmorethanthegravitationalfield?Thisextremelysimple,beautiful,brilliantideaisthetheoryofgeneralrelativity.

Figure3.5Whatistheworldmadeof?

Theworldisnotmadeupofspace+particles+electromagneticfield+gravitationalfield.Theworldismadeupofparticles+fields,andnothingelse;thereisnoneedtoaddspaceasanextraingredient.Newton’sspaceisthegravitationalfield.Orviceversa,whichamountstosayingthesamething:thegravitationalfieldisspace(figure3.5).But,unlikeNewton’sspace,whichisflatandfixed,thegravitationalfield,byvirtueofbeinga

field,issomethingwhichmovesandundulates,subjecttoequations–likeMaxwell’sfield,likeFaraday’slines.Itisamomentoussimplificationoftheworld.Spaceisnolongerdifferentfrommatter.Itisoneof

the‘material’componentsoftheworld,akintotheelectromagneticfield.Itisarealentitywhichundulates,fluctuates,bendsandcontorts.Wearenotcontainedwithinaninvisible,rigidscaffolding:weareimmersedinagigantic,flexible

mollusc(themetaphorisEinstein’s).TheSunbendsspacearounditself,andtheEarthdoesnotcirclearounditdrawnbyamysteriousdistantforcebutrunsstraightinaspacethatinclines.It’slikeabeadwhichrollsinafunnel:therearenomysteriousforcesgeneratedbythecentreofthefunnel,itisthecurvednatureofthefunnelwallwhichguidestherotationofthebead.PlanetscirclearoundtheSun,andthingsfall,becausespacearoundthemiscurved(figure3.6).

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Figure3.6TheEarthturnsaroundtheSunbecausespacetimearoundtheSuniscurved,ratherlikeabeadwhichrollsonthecurvedwallofafunnel.

Alittlemoreprecisely,whatcurvesisnotspacebutspacetime–thatspacetimewhich,tenyearspreviously,Einsteinhimselfhadshowntobeastructuredwholeratherthanasuccessionofinstants.Thisistheidea.Einstein’sonlyproblemwastofindtheequationstomakeitconcrete.Howto

describethisbendingofspacetime?AndhereEinsteinislucky:theproblemhadalreadybeensolvedbythemathematicians.Thegreatestmathematicianofthenineteenthcentury,CarlFriedrichGauss,the‘princeof

mathematicians’,hadwrittenmathstodescribecurvedsurfaces,suchasthesurfacesofhills,orsuchastheoneportrayedinfigure3.7.

Figure3.7Acurved(bidimensional)surface.

Thenhehadaskedatalentedstudentofhistogeneralizethismathstocurvedspacesinthreeormoredimensions.Thestudent,BernhardRiemann,producedaponderousdoctoralthesisofthekindthatseemscompletelyuseless.Riemann’sresultwasthatthepropertiesofacurvedspace(orspacetime)inanydimensionare

describedbyaparticularmathematicalobject,whichwenowcallRiemanncurvatureandindicatewiththeletter‘R’.Ifyouthinkofalandscapeofplains,hillsandmountains,thecurvatureRofthesurfaceiszerointheplains,whichareflat–‘withoutcurvature’–anddifferentfromzerowheretherearevalleysandhills;itisatitsmaximumwheretherearepointedpeaksofmountains,thatistosay,wherethegroundisleastflat,ormostcurved.UsingRiemann’stheory,itispossibletodescribetheshapeofcurvedspacesinthreeorfourdimensions.Withagreatdealofeffort,seekinghelpfromfriendsbetterversedinmathematicsthanhimself,

EinsteinlearnsRiemann’smaths–andwritesanequationwhereRisproportionaltotheenergyofmatter.Inwords:spacetimecurvesmorewherethereismatter.Thatisit.TheequationistheanalogueoftheMaxwellequations,butforgravityratherthanelectricity.Theequationfitsintohalfaline,andthereisnothingmore.Avision–thatspacecurves–becomesanequation.Butwithinthisequationthereisateeminguniverse.Andherethemagicalrichnessofthetheory

opensupintoaphantasmagoricalsuccessionofpredictionsthatresemblethedeliriousravingsofa

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madmanbutwhichhaveallturnedouttobetrue.Evenuptothebeginningofthe1980s,almostnobodytookthemajorityofthesefantasticalpredictionsentirelyseriously.Andyet,oneafteranother,theyhaveallbeenverifiedbyexperience.Let’sconsiderafewofthem.Tobeginwith,EinsteinrecalculatestheeffectofamassliketheSunonthecurvatureofthespace

thatsurroundsit,andtheeffectofthiscurvatureonthemovementsoftheplanets.HefindsthemovementsoftheplanetsaspredictedbyKepler’sandNewton’sequations,butnotexactly:inthevicinityoftheSun,theeffectofthecurvatureofspaceisstrongerthantheeffectofNewton’sforce.EinsteincomputesthemovementofMercury,theplanetclosesttotheSunandhencetheoneforwhichthediscrepancybetweenthepredictionsofhisandNewton’stheoriesisgreatest.Hefindsadifference:thepointoftheorbitofMercuryclosesttotheSunmoveseveryyear0.43secondsofarcmorethanthatpredictedbyNewton’stheory.Itisasmalldifference,but,withinthescopeofwhatastronomerswereabletomeasure,andcomparingthepredictionswiththeobservationsofastronomers,theverdictisunequivocal:MercuryfollowsthetrajectorypredictedbyEinstein,nottheonepredictedbyNewton.Mercury,thefleet-footedmessengerofthegods,thegodofthewingedsandals,followsEinstein,notNewton.Einstein’sequation,then,describeshowspacecurvesveryclosetoastar.Duetothiscurvature,

lightdeviates.EinsteinpredictsthattheSuncauseslighttocurvearoundit.In1919themeasurementisachieved;adeviationoflightismeasuredwhichturnsouttobeexactlyinaccordancewiththeprediction.Butitisnotonlyspacethatcurves:timedoes,too.EinsteinpredictsthattimeonEarthpasses

morequicklyathigheraltitude,andmoreslowlyatloweraltitude.Thisismeasured,andalsoprovestobethecase.Todaywehaveextremelypreciseclocks,inmanylaboratories,anditispossibletomeasurethisstrangeeffectevenforadifferenceinaltitudeofjustafewcentimetres.Placeawatchonthefloorandanotheronatable:theoneonthefloorregisterslesspassingoftimethantheoneonthetable.Why?Becausetimeisnotuniversalandfixed,itissomethingwhichexpandsandshrinks,accordingtothevicinityofmasses:theEarth,likeallmasses,distortsspacetime,slowingtimedowninitsvicinity.Onlyslightly–buttwotwinswhohavelivedrespectivelyatsea-levelandinthemountainswillfindthat,whentheymeetupagain,onewillhaveagedmorethantheother(figure3.8).Thiseffectoffersaninterestingexplanationastowhythingsfall.Ifyoulookatamapoftheworld

andtheroutetakenbyanaeroplaneflyingfromRometoNewYork,itdoesnotseemtobestraight:theaeroplanemakesanarctowardsthenorth.Why?Because,theEarthbeingcurved,crossingnorthwardsisshorterthankeepingtothesameparallel.Thedistancesbetweenmeridiansareshorterthemorenortherlyyouare;therefore,itisbettertoheadnorthwards,toshortentheroute(figure3.9).Well,believeitornot,aballthrownupwardsfallsdownwardsforthesamereason:it‘gainstime’

movinghigherup,becausetimepassesatadifferentspeedupthere.Inbothcases,aeroplaneandballfollowastraighttrajectoryinaspace(orspacetime)thatiscurved(figure3.10).fn12

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Figure3.8Twotwinsspendtheirtimeoneatsea-levelandtheotherinthemountains.Whentheymeetupagain,thetwinwholivedinthemountainsisolder.Thisisthegravitationaldilationoftime.

Butthepredictionsofthetheorygowellbeyondtheseminuteeffects.Starsburnaslongastheyhaveavailablehydrogen–theirfuel–thendieout.Theremainingmaterialisnolongersupportedbythepressureoftheheatandcollapsesunderitsownweight.Whenthishappenstoalargeenoughstar,theweightissostrongthatmatterissquasheddowntoanenormousdegreeandspacecurvessointenselyastoplungedownintoanactualhole.Ablackhole.

Figure3.9Thefurthernorthyougo,thesmallerthedistancebetweentwomeridians.

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Figure3.10Thehigherupsomethingis,themorequicklytimepassesforit.

WhenIwasauniversitystudent,blackholeswereregardedasascarcelycredibleimplicationofanesoterictheory.Todaytheyareobservedintheirhundredsandstudiedindetailbyastronomers.Oneoftheseblackholes,withamassamilliontimesgreaterthantheSun,islocatedatthecentreofourgalaxy–wecanobservestarsorbitingaroundit.Some,passingtooclose,aredestroyedbyitsviolentgravity.Furtherstill,thetheorypredictsthatspacerippleslikethesurfaceofthesea,andthatthese

ripplesarewavessimilartotheelectromagneticoneswhichmaketelevisionpossible.Theeffectsofthese‘gravitationalwaves’canbeobservedintheskyonbinarystars:theyradiatesuchwaves,losingenergyandslowlyfallingtowardseachother.fn13GravitationalwavesproducedbytwoblackholesfallingintooneanotherweredirectlyobservedbyanantennaonEarthinlate2015,andtheannouncement,giveninearly2016,hasonceagainlefttheworldspeechless.Oncemore,theseeminglymadpredictionsofEinstein’stheoryturnouttobepreciselytrue.Andfurtherstill,thetheorypredictsthattheuniverseisexpandingandemergedfromacosmic

explosion14billionyearsago–asubjectIwilldiscussinmoredetailshortly.Thisrichandcomplexrangeofphenomena–bendingofraysoflight,modificationofNewton’s

force,slowingdownofclocks,blackholes,gravitationalwaves,expansionoftheuniverse,theBigBang–followfromunderstandingthatspaceisnotadull,fixedcontainerbutpossessesitsowndynamic,itsown‘physics’,justlikethematterandtheotherfieldsitcontains.Democritushimselfwouldhavesmiledwithpleasure,hadhebeenabletoseethathisideaofspacewouldturnouttohavesuchanimpressivefuture.Itistruethathetermeditnon-being,butwhathemeantbybeing(δέν)wasmatter;andhewrotethathisnon-being,thevoid,nevertheless‘hasacertainphysics(ϕύσιν)andasubstantialityofitsown’.fn14Howrighthewas.WithoutthenotionoffieldsintroducedbyFaraday,withoutthespectacularpowerofmathematics,

withoutthegeometryofGaussandRiemann,this‘certainphysics’wouldhaveremainedincomprehensible.Empoweredbynewconceptualtoolsandbymathematics,EinsteinwritestheequationswhichdescribeDemocritus’svoidandfindsforits‘certainphysics’acolourfulandamazingworldwhereuniversesexplode,spacecollapsesintobottomlessholes,timeslowsdowninthevicinityofaplanet,andtheboundlessexpansesofinterstellarspacerippleandswaylikethesurfaceofthesea…Allofthissoundslikeataletoldbyanidiot,fullofsoundandfury,signifyingnothing.Andyet,

instead,itisaglancetowardsreality.Orbetter,aglimpseofreality,alittlelessveiledthanourblurredandbanaleverydayviewofit.Arealitywhichseemstobemadeofthesamestuffourdreamsaremadeof,butwhichisneverthelessmorerealthanourcloudeddailydreaming.Andallthisistheresultonlyofanelementaryintuition–thatspacetimeandthegravitationalfield

areoneandthesamething–andasimpleequationwhichIcan’tresistcopyingouthere,evenifmostofmyreaderswillcertainlynotbeabletodecipherit.Idoso,anyway,inthehopethattheymightbeabletocatchaglimpseofitsbeautifulsimplicity:

In1915theequationwassimplerstill,becausetheterm+Λgab,whichEinsteinaddedtwoyearslater(andwhichIdiscussbelow)didnotyetexist.fn15RabdependsonRiemann’scurvature,andtogetherwith½Rgabrepresentsthecurvatureofspacetime;Tabstandsfortheenergyofmatter;GisthesameconstantthatNewtonfound:theconstantthatdeterminesthestrengthoftheforceofgravity.

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That’sit.Avisionandanequation.

Mathematicsorphysics?Iwouldliketopause,beforecontinuingwithphysics,tomakeafewobservationsaboutmathematics.Einsteinwasnogreatmathematician.Hestruggledwithmaths.Hesaysthishimself.In1943herepliedinthefollowingwaytoanine-year-oldchildwiththenameofBarbarawhowrotetohimaboutherdifficultieswiththesubject:‘Don’tworryaboutexperiencingdifficultieswithmaths,Icanassureyouthatmyownproblemsareevenmoreserious!’2Itseemslikeajoke,butEinsteinwasnotkidding.Withmathematics,heneededhelp:hehaditexplainedtohimbypatientfellowstudentsandfriends,suchasMarcelGrossman.Itwashisintuitionasaphysicistthatwasprodigious.Duringthelastyearinwhichhewascompletingtheconstructionofhistheory,Einsteinfound

himselfcompetingwithDavidHilbert,oneofthegreatestmathematiciansofalltime.Einsteinhadgivenalecture,attendedbyHilbert,inGöttingen.HilbertimmediatelyunderstoodthatEinsteinwasintheprocessofmakingamajordiscovery,graspedtheideaandtriedtoovertakeEinsteinandbethefirsttowritethecorrectequationsofthenewtheoryEinsteinwasslowlybuilding.Thesprinttothefinishlinebetweenthetwogiantswasanail-bitingaffair,eventuallydecidedbyamatterofjustafewdays.Einstein,inBerlin,endedupgivingapubliclecturealmosteveryweek,eachtimepresentingadifferentequation,anxiousthatHilbertwouldnotgettothesolutionbeforehim.Theequationwasincorrecteverytime.Until,thatis,byahair’sbreadth–justmarginallyaheadofHilbert–Einsteinfoundtherightone.Hehadwontherace.Hilbert,agentleman,neverquestionedEinstein’svictory,eventhoughhewasworkingonvery

similarequationsatthetime.Infact,heleftagentleandbeautifulphrasewhichcapturesperfectlyEinstein’sdifficultrelationshipwithmathematics,and,perhaps,thedifficultrelationshipwhichexistsgenerallybetweenthewholeofphysicsandmathematics.Themathsthatwasnecessarytoformulatethetheorywasgeometryinfourdimensions,andHilbertwrites:

AnyyoungsteronthestreetsofGöttingenfn16understandsgeometryinfourdimensionsbetterthanEinstein.Andyet,itwasEinsteinwhocompletedthetask.

Why?BecauseEinsteinhadauniquecapacitytoimaginehowtheworldmightbeconstructed,to‘see’itinhismind.Theequations,forhim,cameafterwards;theywerethelanguagewithwhichtomakeconcretehisvisionsofreality.ForEinstein,thetheoryofgeneralrelativityisnotacollectionofequations:itisamentalimageoftheworldarduouslytranslatedintoequations.Theideabehindthetheoryisthatspacetimecurves.Ifspacetimehadonlytwodimensions,and

welivedonasortofplane,itwouldbeeasytoimaginewhatitmeanstosaythat‘physicalspacecurves’.Itwouldmeanthatthephysicalspaceinwhichweliveisnotlikeaflattablebutresemblesinsteadasurfacewithmountainsandvalleys.Buttheworldweinhabitdoesnothaveonlytwodimensions,ithasthree.Four,infact,whentimeisincluded.Toimagineacurvedspaceinfourdimensionsismorecomplicated,becauseinourhabitualperceptionwedonothaveanintuitionofa‘largerspace’withinwhichspacetimecancurve.ButEinstein’simaginationhadnodifficultyinintuitingthecosmicmolluscinwhichweareimmersed,whichcanbesquashed,stretchedandtwisted–andthatconstitutesthespacetimearoundus.ItisthankstothisvisionaryclaritythatEinsteinmanagedtobethefirsttowritethetheory.Intheend,adegreeoftensionbetweenHilbertandEinsteindiddevelop.Afewdaysbefore

Einsteinmadehissuccessfulequationpublic,Hilberthadsentanarticletoaperiodicalwhichshowsjusthowclosehehadcometothesamesolution–andeventodayhistoriansofsciencearefacedwithdoubtswhentryingtoevaluatetherespectivecontributionsofthesetwogiants.Atsomepointtheirrelationscooled,andEinsteinfearedthatHilbert,moreseniorandpowerfulthanhim,wouldseektoattributetohimselftoomuchofthemeritfortheconstructionofthetheory.ButHilbertneverclaimedtobethefirsttodiscovergeneralrelativity–andinaworldsuchasthatofscience,whereoften,toofrequently,disputesoverprecedencebecomepoisonous–thetwogaveatrulywonderfulexampleofwisdom,clearingthefieldofallnegativetension.EinsteinwritesamarvellouslettertoHilbert,summarizingtheprofoundsenseoftheshared

coursetheyhadtaken:Therewasamomentinwhichsomethinglikeanirritationcamebetweenus,theoriginofwhichInolongerwanttoanalyse.Ihavefoughtagainstthebitternesswhichitprovokedinme,andhavesucceededcompletelyindoingso.Iagainthinkofyouwithuncloudedfriendship,andIaskyoutodothesameforme.Itisreallyapityifcompanionssuchasweare,whohavemanagedtoforgeapathasidefromthepettinessofthisworld,couldfindanythingotherthanjoyineachother’scompany.3

ThecosmosTwoyearsafterthepublicationofhisequation,Einsteindecidestouseittodescribethespaceoftheentireuniverse,consideredatthelargestscale.Andherehehasanotherofhisamazingideas.Forthousandsofyears,menhadaskedthemselveswhethertheuniversewasinfinite,orhada

limit.Bothhypothesesentailthornyproblems.Aninfiniteuniversedoesnotseemtostandtoreason:ifitisinfinite,forexample,theremustexistsomewhereareaderjustlikeyouwhoisreadingthe

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verysamebook(infinityistrulyvast,andtherearenotsufficientcombinationsofatomstofillitwiththingsalwaysdifferentfromeachother).Infact,theremustbenotonlyonebutaninfiniteseriesofreadersidenticaltoyourself…Butifthereisalimittotheuniverse,whatisthatboundary?Whatsenseisthereinaborderwithnothingontheotherside?AlreadyinthefourthcenturyCE,inTaranto,thePythagoreanphilosopherArchytashadwritten:

IfIfoundmyselfinthefurthestsky,thatofthefixedstars,wouldIbeabletostretchmyhand,orarod,outbeyondit–ornot?ThatIshouldnotbeabletoisabsurd;butifIamableto,thenanoutsideexists,beitofmatter,orspace.Inthiswayonecouldproceedeverfurther,towardstheend,fromtimetotimeaskingthesamequestion,astowhethertherewillalwaysbesomethingintowhichtoextendtherod.4

Thesetwoabsurdalternatives–theabsurdityofaninfinitespace,andtheabsurdityofauniversewithafixedborder–didn’tseemtoleaveanyreasonablechoicebetweenthem.ButEinsteinfindsathirdway:theuniversecanbefiniteandatthesametimehavenoboundary.

How?JustasthesurfaceoftheEarthisnotinfinitebutdoesnothaveaboundaryeither,whereit‘ends’.Thiscanhappen,naturallyenough,ifsomethingiscurved:thesurfaceoftheEarthiscurved.Andinthetheoryofgeneralrelativity,ofcourse,three-dimensionalspacecanalsobecurved.Consequently,ouruniversecanbefinitebutborderless.OnthesurfaceoftheEarth,ifIweretokeepwalkinginastraightline,Iwouldnotadvancead

infinitum:IwouldeventuallygetbacktothepointIstartedfrom.Ouruniversecouldbemadeinthesameway:ifIleaveinaspacecraftandjourneyalwaysinthesamedirection,IflyaroundtheuniverseandeventuallyendupbackonEarth.Athree-dimensionalspaceofthiskind,finitebutwithoutboundary,iscalleda3-sphere.

Figure3.11Aspherecanberepresentedastwodiscswhichinrealityaresmoothlyjoinedallalongtheiredges.

Tounderstandthegeometryofa‘3-sphere’,letusreturntotheordinarysphere;thesurfaceofaball,ortheEarth.TorepresentthesurfaceoftheEarthonaplane,wecandrawtwodiscs,asiscustomarywhendrawingthecontinents(figure3.11).Noticethataninhabitantofthesouthernhemisphereisinacertainsense‘surrounded’bythe

northernhemisphere,sinceinwhicheverdirectionshegoestoexitherhemisphere,shewillalwaysarriveintheotherone.Butthecontraryisobviouslytrueaswell:eachhemisphere‘surrounds’,andissurroundedby,theother.A3-spheremayberepresentedinasimilarfashion,butwitheverythinggivenanadditionaldimension:twoballsstucktogetherallalongtheiredges(figure3.12).Whenweleaveoneballweenterintotheother,justaswhenweleaveoneofthediscsinthe

representationoftheglobeweenterintotheotherdisc.Eachballsurroundsandissurroundedbytheother.Einstein’sideaisthatspacecouldbea3-sphere:somethingwithafinitevolume(thesumofthevolumeofthetwoballs),butwithoutborders.fn17The3-sphereisthesolutionwhichEinsteinproposesinhisworkof1917totheproblemoftheborderoftheuniverse.Thisarticleinitiatesmoderncosmology,thestudyoftheentirevisibleuniverse,studiedatthegrandestscale.Fromitwillarisethediscoveryoftheexpansionoftheuniverse;thetheoryoftheBigBang;theproblemofthebirthoftheuniverse,andmuchelsebesides.IspeakaboutallofthisinChapter8.

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Figure3.12A3-spherecanberepresentedastwoballsjoinedtogether.

ThereisonemoreobservationwhichIwouldliketomakeaboutEinstein’s3-sphere.Howeverincredibleitmightseem,thesameideahadalreadybeenconceivedbyanothergenius,fromanentirelydifferentculturaluniverse:DanteAlighieri,Italy’sgreatestpoet.IntheParadiso,thethirdpartofhismajorpoem,theCommedia,Danteoffersagrandiosevisionofthemedievalworld,calquedontheworldofAristotle,withthesphericalEarthatitscentre,surroundedbythecelestialspheres(figure3.13).

Figure3.13TraditionalrepresentationofDante’suniverse.

Accompaniedbyhisshininglovedone,Beatrice,Danteascendsthesespheresinthecourseofafantastic,visionaryjourneyuptotheoutermostsphere.Whenhereachesit,hecontemplatestheuniversebelowhimwithitsrotatingheavensandtheEarth,veryfardown,atitscentre.Butthenhelooksevenhigher–andwhatdoeshesee?Heseesapointoflightsurroundedbyimmensespheresofangels,thatistosay,byanotherimmenseball,which,inhiswords‘surroundsandisatthesametimesurroundedby’thesphereofouruniverse!HereareDante’sversesfromCantoXXVIIoftheParadiso.Questaaltrepartedell’Universod’uncerchioluicomprendesicomequestolialtri:‘This

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otherpartoftheuniversesurroundsthefirstinacirclelikethefirstsurroundstheothers.’Andinthenextcanto,stillonthelast‘circle’,parendoinchiusodaquelch’elliinchiude:‘appearingtobetobeenclosedbythosethatitencloses’.Thepointoflightandthesphereofangelsaresurroundingtheuniverse,andatthesametimetheyaresurroundedbytheuniverse!Itisanexactdescriptionofa3-sphere!TheusualrepresentationsofDante’suniversecommoninItalianschoolbooks(suchasfigure3.13)

placetheangelicspheresseparatefromthecelestialones.ButDantewritesthatthetwoballs‘surroundandaresurroundedby’eachother.Dantehasacleargeometricalintuitionofa3-sphere.fn18ThefirsttonoticethattheParadisodescribestheuniverseasa3-spherewasanAmerican

mathematician,MarkPeterson,in1979.Ingeneral,scholarsofDantearenotveryfamiliarwith3-spheres.Today,everyphysicistandmathematiciancouldeasilyrecognizethe3-sphereinDante’sdescriptionoftheuniverse.HowisitpossiblethatDantehadanideathatsoundssomodern?Ithinkitwaspossible,inthe

firstplace,duetotheprofoundintelligenceofItaly’sfinestpoet.ThisintelligenceisoneofthereasonswhytheCommediaissofascinating.ButitisalsoduetothefactthatDantewaswritingwellbeforeNewtonconvincedeveryonethattheinfinitespaceofthecosmoswastheflatoneofEuclideangeometry.DantewasfreeoftherestraintsuponourintuitionwehaveasaresultofourNewtonianschooling.Dante’sscientificculturewasbasedprincipallyontheteachingsofhismentorandtutor,Brunetto

Latini,whohasleftusasmall,enchantingtreatise,Litresor,whichisasortofencyclopaediaofmedievalknowledge,writteninadelightfulcombinationofoldFrenchandItalian.InLitresor,BrunettoexplainsindetailthefactthattheEarthisround.Buthedoesso,curiouslytotheeyesofamodernreader,intermsof‘intrinsic’ratherthan‘extrinsic’geometry.Thatistosay,hedoesnotwrite,‘theEarthislikeanorange’,astheEarthwouldlookifseenfromtheoutside,butwritesinstead,‘Twoknightswhocouldgallopsufficientlyfarinoppositedirectionswouldmeetupontheotherside.’And:‘Ifhewerenotimpededbytheseas,amanwhosetouttowalkforeverwouldreturntothepointontheEarthfromwhichhedeparted.’Inotherwords,headoptsaninternal,notanexternal,pointofview:theperspectiveofsomeonewhowalkstheEarth,notofsomeonewholooksatitfromafar.Atfirstglanceitmightseemlikeapointless,complicatedwayofexplainingthattheEarthisaball.Whydoesn’tBrunettosimplysaythattheEarthislikeanorange?But,onreflection:if,say,anantwalksonanorange,itwillatsomepointfinditselfupsidedown,andmustkeepitselfattachedbymeansofthetinysuctionpadsonitslegs,toavoidfallingoff.AndyetatravellerwhowalkstheEarthneverfindshimselfupsidedown,andneedsnosuctionpadsonhislegs.Brunetto’sdescriptionisnotsoquaintafterall.Now,thinkaboutit.Forsomeonewhohaslearnedfromhisteacherthattheformofthesurfaceof

ourplanetissuchthatbywalkingalwaysinastraightlinewereturntothepointwestartedfrom,itisperhapsnotsodifficulttotakethenextobviousstep,andimaginethattheformoftheentireuniverseissuchthat,flyingalwaysinastraightline,wereturntothesamepointofdeparture:a3-sphereisaspaceinwhich‘twowingedknightsthatcouldflyinoppositedirectionswouldmeetupontheotherside’.Intechnicalterms,thedescriptionofthegeometryoftheEarthofferedbyBrunettoLatiniinLitresorisgivenintermsofintrinsicgeometry(seenfromtheinside)ratherthanextrinsic(seenfromtheoutside),andthisisexactlythedescriptionthatissuitabletogeneralizethenotionof‘sphere’fromtwodimensionstothree.Thebestwayofdescribinga3-sphereisnottotryto‘seeitfromtheoutside’,butrathertodescribewhathappenswhenmovingwithinit.ThemethoddevelopedbyGausstodescribecurvedsurfaces,andgeneralizedbyRiemannto

describethecurvatureofspacesinthreeormoredimensions,basically,amountstoBrunettoLatini’sway.Thatistosay,theideaistodescribeacurvedspacenotas‘seenfromtheoutside’,statinghowitcurvesinanexternalspace,butinsteadintermsofwhatmaybeexperiencedbysomebodywithinthatspace,whoismovingandalwaysremainingwithinit.Forinstance,thesurfaceofanordinarysphere,asBrunettoobserves,isasurfacewhereallthe‘straight’linesgetbacktothestartingpointaftertraversingthesamedistance(thelengthoftheequator).A3-sphereisathree-dimensionalspacewiththesameproperty.Einstein’sspacetimeisnotcurvedinthesensethatitcurves‘inanexternalspace’.Itiscurvedin

thesensethatitsintrinsicgeometry,thatistosay,thewebofdistancesbetweenitspoints,whichcanbeobservedbystayingwithinit,isnotthegeometryofaflatspace.ItisaspacewherePythagoras’stheoremisnotvalid,justasPythagoras’stheoremisnotvalidonthesurfaceoftheEarth.fn19Thereisawayofunderstandingthecurvatureofspacefromwithinit,andwithoutlookingatit

fromoutside,whichisimportantforwhatfollows.ImagineyouareattheNorthPoleandwalksouthwardsuntilyoureachtheequator,carryingwithyouanarrowpointingahead.Onceyoureachtheequator,turntotheleftwithoutchangingthedirectionofthearrow.Thearrowstillpointssouth,whichisnowtoyourright.Advancealittletowardstheeastalongtheequatorandthenturnagaintowardsthenorth–againwithoutchangingthedirectionofthearrow,whichwillnowbepointingbehindyou.WhenyoureachtheNorthPoleagain,youhaveexecutedaclosedcircuit–a‘loop’,asit

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istermed–andthearrowdoesnotpointinthesamedirectionaswhenyoustartedout(figure3.14).Theanglethroughwhichthearrowhasturnedinthecourseoftheloopmeasuresthecurvature.Iwillreturnlatertothismethodofmeasuringcurvaturebymakingaloopinspace.Thesewillbe

theloopsthatgivethenametothetheoryofloopquantumgravity.

Figure3.14Anarrowcarriedparalleltoitselfalongacircuit(aloop)inacurvedspacearrivesbackrotatedatthepointofdeparture.

DanteleavesFlorencein1301,whilethemosaicsinthecupolaoftheBaptisteryarebeingcompleted.Themosaic,representingHell(theworkofCoppodiMarcovaldo,theteacherofCimabue),probablyterrifyingintheeyesofamedievalperson,hasoftenbeenindicatedasasourceofinspirationtoDante(figure3.15).Shortlybeforestartingtowritethisbook,IvisitedtheBaptisteryinthecompanyofEmanuela

Minnai,thefriendwhoconvincedmetowriteit.EnteringtheBaptisteryandlookingup,youseeashiningpointoflight(thelightsourcefromthelanternatthesummitofthecupola)surroundedbynineordersofangels,withthenameofeachorderwritten:Angels,Archangels,Principalities,Powers,Virtues,Domains,Thrones,CherubimandSeraphim.ThiscorrespondsexactlytothestructureofthesecondsphereofParadise.ImaginethatyouareanantontheflooroftheBaptisteryandareabletowalkinanydirection;regardlessofwhichdirectionyoufollowtoclimbthewall,youwouldreachtheceilingatthesamepointoflightsurroundedbyangels:thepointoflightanditsangelsboth‘surround’and‘aresurroundedby’therestofthedecoratedinterioroftheBaptistery(figure3.16).

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Figure3.15ThemosaicdepictingHell,byCoppodiMarcovaldo,intheBaptisteryofFlorence.

LikeeverycitizenofFlorenceattheendofthethirteenthcentury,Dantemusthavebeenprofoundlyawe-struckbytheBaptistery,thegrandiosearchitecturalenterprisehiscitywascompleting.IbelievethathemayhavebeeninspiredbytheBaptistery,notonlybyCoppodiMarcovaldo’sInferno,butalsobyitsoverallarchitecture,forhisvisionofthecosmos.TheParadisoreproducesitsstructureremarkablyprecisely,includingtheninecirclesofangelsandthepointoflight,justtranslatingitfromtwotothreedimensions.AfterdescribingthesphericaluniverseofAristotle,Brunettohadalreadywrittenthatbeyonditliestheplaceofdivinity–andmedievaliconographyhadalreadyimaginedParadiseasGodsurroundedbyspheresofangels.Intheend,DantedoesnomorethanmountthepiecesthatalreadyexistedintoacoherentarchitecturalwholewhichfollowsthesuggestivearchitectureoftheBaptisteryandresolvestheancientproblemofthebordersoftheuniverse.Insodoing,DanteanticipatesbysixcenturiesEinstein’s3-sphere.

Figure3.16TheinterioroftheBaptistery.

Idon’tknowiftheyoungEinsteinhadencounteredtheParadisoduringhisintellectualwanderingsinItaly,andwhetherornotthevividimaginationoftheItalianpoetmayhavehadadirectinfluenceonhisintuitionthattheuniversemightbebothfiniteandwithoutboundary.Whetherornotsuchinfluenceoccurred,Ibelievethatthisexampledemonstrateshowgreatscienceandgreatpoetryarebothvisionary,andmayevenarriveatthesameintuitions.Ourcultureisfoolishtokeepscienceandpoetryseparated:theyaretwotoolstoopenoureyestothecomplexityandbeautyoftheworld.Dante’s3-sphereisonlyanintuitionwithinadream.Einstein’s3-spherehasmathematicalform

andfollowsfromthetheory’sequations.Theeffectofeachisdifferent.Dantemovesusdeeply,touchingthesourcesofouremotions.Einsteinopensaroadtowardstheunsolvedmysteriesofouruniverse.Butbothcountamongthemostbeautifulandsignificantflightsthatthemindcanachieve.Butlet’sreturnto1917,whenEinsteintriestoinserttheideaofthe3-sphereintohisequations.

Hereheencountersaproblem.Heisconvincedthattheuniverseisfixedandimmutable,buthisequationstellhimthatthisisnotpossible.Itisn’tdifficulttounderstandwhy.Everythingattracts,thereforetheonlywayforafiniteuniversenottocollapseonitselfisforittobeexpanding:justas

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theonlywaytopreventafootballfromfallingtothegroundistokickitupwards.Iteithergoesup,orfallsdown–itcan’tstaystill,suspendedintheair.ButEinsteindoesnotbelievewhathisownequationsaretellinghim.Heevenmakesasilly

physicsmistake(hedoesnotrealizethatthesolutionheconsidersisunstable)justtoavoidacceptingwhathistheorypredicts:theuniverseiseithercontractingorexpanding.Hemodifieshisequations,tryingtoavoidtheimplicationthatitisexpanding.ItisforthisreasonthatheaddsthetermΛgabintheequationwrittenabove.Butitisafurthermistake:theaddedtermiscorrect,butitdoesnotchangethefactthattheequationpredictsthattheuniversemustbeexpanding.Forallhisbravery,Einsteinthegeniuslacksthecouragetobelievehisownequations.AfewyearslaterEinsteinisforcedtogiveup:itishistheorythatisright,nothisreservations

aboutit.Astronomersrealizethatallgalaxiesareindeedmovingawayfromus.Theuniverseisexpanding,exactlyastheequationspredicted.Fourteenbillionyearsago,theuniversewasconcentratedalmosttoasingle,furiouslyhotpoint.Fromthereitexpandedinacolossal‘cosmic’explosion–andheretheterm‘cosmic’isnotusedinanyrhetoricalsense:itis,literally,acosmicexplosion.Thisisthe‘BigBang’.Todayweknowtheexpansionisreal.ThedefinitiveproofofthescenarioforeseenbyEinstein’s

equationsarrivesin1964,whentwoAmericanradio-astronomers,ArnoPenziasandRobertWilson,discoverbyaccidentaradiationdiffusedthroughouttheuniversewhichturnsouttobepreciselywhatremainsoftheoriginalimmenseheatoftheearlyuniverse.Onceagain,thetheoryturnsouttohavebeencorrect,uptoitsmostamazingpredictions.

Figure3.17Einstein’sworld:particlesandfieldswhichmoveonotherfields.

EversincewediscoveredthattheEarthisroundandturnslikeamadspinning-top,wehaveunderstoodthatrealityisnotwhatitseems:everytimeweglimpseanewaspectofit,itisadeeplyemotionalexperience.Anotherveilhasfallen.ButtheleapmadebyEinsteinisunparalleled:spacetimeisafield;theworldismadeonlyoffieldsandparticles;spaceandtimearenotsomethingelse,somethingdifferentfromtherestofnature:theyarejustafieldamongtheothers(figure3.17).In1953,aprimaryschoolchildwritestoAlbertEinstein,‘Ourclassisstudyingtheuniverse.Iam

veryinterestedinspace.Iwouldliketothankyouforallthatyouhavedonesothatwemightunderstandit.’5Ifeelthesameway.

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4.Quanta

Thetwopillarsoftwentieth-centuryphysics–generalrelativityandquantummechanics–couldnotbemoredifferentfromeachother.Generalrelativityisacompactjewel:conceivedbyasinglemind,basedoncombiningprevioustheories,itisasimpleandcoherentvisionofgravity,spaceandtime.Quantummechanics,orquantumtheory,ontheotherhand,emergesfromexperimentsinthecourseofalonggestationoveraquarterofacentury,towhichmanyhavecontributed;achievesunequalledexperimentalsuccessandleadstoapplicationswhichhavetransformedoureverydaylives(thecomputeronwhichIwrite,forinstance);but,morethanacenturyafteritsbirth,itremainsshroudedinobscurityandincomprehensibility.

Thischapterillustratesthestrangephysicsofthistheory,relateshowthetheorycameintobeingandthethreeaspectsofrealityithasunveiled:granularity,indeterminismandrelationality.

AlbertagainIt’ssaidthatquantummechanicswasbornpreciselyin1900,virtuallyusheringinacenturyofintensethought.In1900theGermanphysicistMaxPlancktriestocomputetheamountofelectromagneticwavesinequilibriuminahotbox.Toobtainaformulareproducingtheexperimentalresults,heendsupusingatrickwhichdoesnotappeartomakemuchsense:heassumesthattheenergyoftheelectricfieldisdistributedin‘quanta’,thatistosay,insmallpackets,littlebricksofenergy.Thesizeofthepackets,heassumes,dependsonthefrequency(thatis,thecolour)oftheelectromagneticwaves.Forwavesoffrequencyν,everyquantum,oreverypacket,hasenergy

E=hνThisformulaisthefirstofquantummechanics;hisanovelconstantwhichtodaywecallthe

Planckconstant.Itfixeshowmuchenergythereisineachpacketofenergy,forradiationoffrequency(colour)ν.Theconstanthdeterminesthescaleofallquantumphenomena.

Theideathatenergycouldbemadeupoffinitepacketsisatoddswitheverythingthatwasknownatthetime:energywasconsideredsomethingthatcouldvaryinacontinuousmanner,andtherewasnoreasontotreatitasifitweremadeupofgrains.Forexample,theenergyofapendulummeasurestheamplitudeoftheswing.Thereseemstobenoreasonforapendulumtooscillateonlywithcertaindeterminedamplitudesandnotothers.ForMaxPlanck,takingenergyinfinite-sizepacketswasonlyastrangetrickwhichhappenedtoworkforthecalculation–thatis,toreproducelaboratorymeasurements–butforutterlyunclearreasons.

FiveyearslateritisAlbertEinstein–himagain–whocomestounderstandthatPlanck’spacketsofenergyareinfactreal.ThisisthesubjectofthethirdofthethreearticlessenttotheAnnalenderPhysikin1905.Andthisisthetruedateofbirthofquantumtheory.

Inthearticle,Einsteinarguesthatlighttrulyismadeupofsmallgrains,particlesoflight.Heconsidersaphenomenonthathadbeenrecentlyobserved:thephotoelectriceffect.Therearesubstancesthatgenerateaweakelectriccurrentwhenstruckbylight.Thatistosay,theyemitelectronswhenlightshinesonthem.Todayweusethem,forexample,inthephotoelectriccellswhichopendoorswhenweapproachthembydetectingiflightarrives,ornot,inasensor.Thatthishappensisnotstrange,becauselightcarriesenergy(itwarmsus,forexample),anditsenergymakestheelectrons‘jumpout’oftheiratoms;itgivesthemapush.

Butsomethingisstrange:itseemsreasonabletoexpectthatiftheenergyoflightisscarce–namely,ifthelightisdim–thephenomenonwouldnottakeplace;andthatitwouldtakeplacewhentheenergyissufficient–namely,whenthelightisbright.Butitisn’tlikethis:whatisobservedisthatthephenomenonhappensonlyifthefrequencyoflightishighanddoesnothappenifthefrequencyislow.Thatistosay,ithappensordoesn’thappendependingonthecolouroflight(thefrequency)ratherthanitsintensity(energy).Thereisnowayofmakingsenseofthiswithstandardphysics.

EinsteinusesPlanck’sideaofthepacketsofenergy,withasizethatdependsuponfrequency,andrealizesthatifthesepacketsarereal,thephenomenoncanbeexplained.Itisn’tdifficulttounderstandwhy.Imaginethatthelightarrivesintheformofgrainsofenergy.Anelectronwillbesweptoutofitsatomiftheindividualgrainhittingithasagreatdealofenergy.Whatmattersistheenergyofeachgrain,notthenumberofgrains.If,asinPlanck’shypothesis,theenergyofeachgrainisdeterminedbyfrequency,thephenomenonwilloccuronlyiffrequencyissufficientlyhigh,

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thatistosay,iftheindividualgrainsofenergyaresufficientlylarge,independentlyfromthetotalamountofenergythat’saround.

Itislikewhenithails:whatdetermineswhetheryourcarwillbedentedisnotthetotalquantityofhailthatfallsbutthesizeoftheindividualhailstones.Therecouldbeanenormousamountofhail,butitwilldonodamageifallthestonesaresmall.Inthesameway,eveniflightisintense–whichamountstosayingthatthereareagreatdealoflightpackets–theelectronswouldnotbeextractedfromtheiratomsiftheindividualgrainsoflightaretoosmall,thatis,ifthefrequencyoflightistoolow.Thisexplainswhyitisthecolourandnottheintensitywhichdetermineswhetherthephotoelectriceffectoccursornot.ForthissimplereasoningEinsteinwasawardedtheNobelPrize.Itiseasytounderstandthingsoncesomeonehasthoughtthemthrough.Thedifficultyliesinthinkingthemthroughinthefirstplace.

Todaywecallthesepacketsofenergy‘photons’,fromtheGreekwordforlight:ϕώς.Photonsarethegrainsoflight,its‘quanta’.InthearticleEinsteinwrites:

Itseemstomethattheobservationsassociatedwithblackbodyradiation,fluorescence,theproductionofcathoderaysbyultravioletlight,andotherrelatedphenomenaconnectedwiththeemissionortransformationoflightaremorereadilyunderstoodifoneassumesthattheenergyoflightisdiscontinuouslydistributedinspace.Inaccordancewiththeassumptiontobeconsideredhere,theenergyofalightrayspreadingoutfromapointsourceisnotcontinuouslydistributedoveranincreasingspacebutconsistsofafinitenumberof‘energyquanta’whicharelocalizedatpointsinspace,whichmovewithoutdividing,andwhichcanonlybeproducedandabsorbedascompleteunits.1

Thesesimpleandclearlinesaretherealbirthcertificateofquantumtheory.Notethewonderfulinitial‘Itseemstome…’,whichrecallsthehesitationsofFaraday,orthoseofNewton;ortheuncertaintyofDarwininthefirstpagesofOntheOriginofSpecies.Truegeniusisawareofthemomentousnessofthestepsitistaking,andisalwayshesitant…

ThereisaclearrelationbetweenEinstein’sworkonBrownianmotion(discussedinChapter1)andhisworkonthequantaoflight,bothcompletedin1905.Inthefirst,Einsteinhadmanagedtofindademonstrationoftheatomichypothesis,thatistosay,ofthegranularstructureofmatter.Inthesecondheextendsthissamehypothesistolight:lightmusthaveagranularstructureaswell.

Atfirst,Einstein’sideathatlightcouldbemadeupofphotonsisregardedbyhiscolleaguesasnomorethanyouthfulwaywardness.Everyonecommendshimforhistheoryofrelativity,buteverybodyjudgesthenotionofphotonstobeoutlandish.Scientistshadonlyrecentlybeenpersuadedthatlightwasawaveintheelectromagneticfield:howcoulditbemadeupofgrains?InaletteraddressedtotheGermanMinistry,recommendingthatEinsteinshouldhaveaprofessorshipinauguratedforhiminBerlin,themostdistinguishedphysicistsofthedaywritethattheyoungmanissobrilliantthathe‘maybeexcused’certainexcesses,suchastheideaofphotons.Notmanyyearslater,theverysamecolleaguesawardhimtheNobelPrize,preciselyforhavingunderstoodthatphotonsexist.Lightfallsonasurfacelikeagentlehailshower.

Tocomprehendhowlightmaybesimultaneouslyanelectromagneticwaveandaswarmofphotonswillrequiretheentireconstructionofquantummechanics.Butthefirstbuildingblockofthistheoryhasbeenestablished:thereexistsafundamentalgranularityinallthings,includinglight.

Niels,WernerandPaulIfPlanckisthebiologicalfatherofthetheory,Einsteinistheparentwhogavebirthtoandnurturedit.Butasisoftenthecasewithchildren,thetheorythenwentitsownway,barelyrecognizedbyEinsteinashisown.

Duringthefirsttwodecadesofthetwentiethcentury,itistheDaneNielsBohrwhoisresponsibleforguidingitsdevelopment.Bohrstudiesthestructureofatoms,whichwasbeginningtobeexploredattheturnofthecentury.Experimentshadshownthatanatomislikeasmallsolarsystem:themassisconcentratedinaheavycentralnucleus,aroundwhichlightelectronsrevolve,moreorlessliketheplanetsaroundtheSun.Thispicture,however,didnotaccountforasimplefact:matteriscoloured.

Saltiswhite,pepperisblack,chilliisred.Why?Studyingthelightemittedbyatoms,itisapparentthatsubstanceshavespecificcolours.Sincecolouristhefrequencyoflight,lightisemittedbysubstancesatcertainfixedfrequencies.Thesetofthefrequenciesthatcharacterizesagivensubstanceisknownasthe‘spectrum’ofthissubstance.Aspectrumisacollectionoffinelinesofdifferenthues,inwhichthelightemittedbyagivensubstanceisdecomposed(forinstance,byaprism).Thespectraofafewelementsareshowninfigure4.2.

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Figure4.1NielsBohr.

Spectraofnumeroussubstanceshadbeenstudiedandcataloguedinmanylaboratoriesattheturnofthecentury,andnobodyknewhowtoexplainwhyeachsubstancehadthisorthatspectrum.Whatdeterminesthecolourofthoselines?

Figure4.2Thespectraofsomeelements:sodium,mercury,lithiumandhydrogen.

ColouristhespeedatwhichFaraday’slinesvibrate,andthisisdeterminedbythevibrationsoftheelectricchargeswhichemitlight.Thesechargesaretheelectronsthatmoveinsidetheatoms.Therefore,studyingspectra,wecanunderstandhowelectronsmovearoundnuclei.Theotherwayaround,wecouldpredictthespectrumofeachatombycomputingthefrequenciesoftheelectronscirclingtheirnucleus.Easytosay,butinpracticenobodywasabletodoso.Infact,thewholethingseemedimpossible,becauseinNewton’smechanicsanelectroncanrevolvearounditsnucleusatanyspeed,andhenceemitlightatanyfrequency.Butthenwhydoesthelightemittedbyanatomnotcontainallcolours,ratherthanjustafewparticularones?Whyareatomicspectranotacontinuumofcolours,insteadofjustafewseparatelines?Why,intechnicalparlance,arethey‘discrete’insteadofcontinuous?Fordecades,physicistsseemedincapableoffindingananswer.

Bohrfindsatentativesolution,bywayofastrangehypothesis.Herealizesthateverythingcouldbeexplainediftheenergyofelectronsinatomscouldonlyassumecertain‘quantized’values–certainspecificvalues,justaswashypothesizedbyPlanckandbyEinsteinfortheenergyofthequantaoflight.Onceagain,thekeyisagranularity,butnotnowfortheenergyoflightbutratherfortheenergyoftheelectronsintheatom.Itbeginstobecomeclearthatgranularityissomethingwidespreadinnature.

Bohrmakesthehypothesisthatelectronscanexistonlyatcertain‘special’distancesfromthenucleus,thatis,onlyoncertainparticularorbits,thescaleofwhichisdeterminedbyPlanck’sconstanth.Andthatelectronscan‘leap’betweenoneorbitwiththepermittedenergytoanother.Thesearethefamous‘quantumleaps’.Thefrequencyatwhichtheelectronmovesontheseorbitsdeterminesthefrequencyoftheemittedlightand,sinceonlycertainorbitsareallowed,itfollowsthatonlycertainfrequenciesareemitted.

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ThesehypothesesdefineBohr’s‘atomicmodel’,whosecentenarywascommemoratedin2013.Withtheseassumptions(outlandish,butsimple)Bohrmanagestocomputethespectraofallatoms,andeventopredictaccuratelyspectranotyetobserved.Theexperimentalsuccessofthissimplemodelisastonishing.

Clearly,theremustbesometruthintheseassumptions,eveniftheyruncontrarytoallcontemporarynotionsofmatteranddynamics.Butwhyaretherealwaysonlyjustcertainorbits?Andwhatdoesitmeantosaythatelectrons‘leap’?

InBohr’sinstituteinCopenhagen,themostbrilliantyoungmindsofthecenturygathertotrytogiveordertothisjumbleofincomprehensiblebehavioursintheatomicworld,andtoconstructacoherenttheory.Theresearchisarduousandprotracted,untilayoungGermanfindsthekeytounlockthedoorofthemysteryofthequantumworld.

WernerHeisenbergistwenty-fiveyearsoldwhenhewritestheequationsofquantummechanics,thesameageasEinsteinwaswhenhewrotehisthreemajorarticles.Hedoessoonthebasisofdizzyingideas.

Figure4.3WernerHeisenberg.

TheintuitioncomestohimonenightintheparkbehindtheCopenhagenInstituteofPhysics.TheyoungWernerwalksaboutpensivelyinthepark.Itisreallydarkthere;wearein1925.Thereisonlyanoccasionalstreetlamp,castingdimislandsoflighthereandthere.Thepoolsoflightareseparatedbylargeexpansesofdarkness.Suddenly,Heisenbergseesafigurepassby.Actually,hedoesnotseehimpass:heseeshimappearbeneathalamp,thendisappearintothedarkbeforereappearingbeneathanotherlamp,andthenvanishingbackintothedarkagain.Andsoon,frompooloflighttopooloflight,untilheeventuallydisappearsaltogetherintothenight.Heisenbergthinksthat,‘evidently’,themandoesnotactuallyvanishandreappear:inhismind,hecaneasilyreconstructtheman’strajectorybetweenonestreetlampandanother.Afterall,amanisasubstantialobject,bigandheavy–andbig,heavyobjectsdonotsimplyappearandvanish…

Ah!Theseobjects,whicharesubstantial,largeandheavy,don’tvanishandreappear…butwhatdoweknowaboutelectrons?Alightflashesoninhismind.Whyshouldsmallobjectssuchaselectronsdothesame?Whatif,effectively,electronscouldvanishandreappear?Whatifthesewerethemysteriousquantumleapswhichappearedtounderliethestructureoftheatomicspectra?Whatif,betweenoneinteractionwithsomething,andanotherwithsomethingelse,theelectroncouldliterallybenowhere.

Whatiftheelectroncouldbesomethingthatmanifestsitselfonlywhenitinteracts,whenitcollideswithsomethingelse;andthatbetweenoneinteractionandanotherithadnopreciseposition?Whatifalwayshavingaprecisepositionissomethingwhichisacquiredonlyifoneissubstantialenough–largeandheavylikethemanthatpassedbyalittlewhileago,likeaghostinthedark,andthendisappearedintothenight…?

Onlysomeoneinhistwentiescantakesuchdeliriouspropositionsseriously.Youhavetobeatwenty-somethingtobelievethattheycanbeturnedintoatheoryoftheworld.Andperhapsyouhavetobethisyoungtounderstandbetterthananyoneelse,forthefirsttime,thedeepstructureofnature.JustasEinsteinwasinhistwentieswhenherealizedthattimedoesnotpassinthesamewayforeveryone,so,too,wasHeisenbergonthatCopenhagennight.Perhaps,itisnolongeragoodideatotrustyourintuitionsaftertheageofthirty…

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Heisenbergreturnshomegrippedbyfeverishemotion,andplungesintocalculations.Heemerges,sometimelater,withadisconcertingtheory:afundamentaldescriptionofthemovementofparticles,inwhichtheyaredescribednotbytheirpositionateverymomentbutonlybytheirpositionatparticularinstants:theinstantsinwhichtheyinteractwithsomethingelse.

Thisisthesecondcornerstoneofquantummechanics,itshardestkey:therelationalaspectofthings.Electronsdon’talwaysexist.Theyexistwhentheyinteract.Theymaterializeinaplacewhentheycollidewithsomethingelse.Thequantumleapsfromoneorbittoanotherconstitutetheirwayofbeingreal:anelectronisacombinationofleapsfromoneinteractiontoanother.Whennothingdisturbsit,anelectrondoesnotexistinanyplace.Insteadofwritingthepositionandvelocityoftheelectron,Heisenbergwritestablesofnumbers(matrices).Hemultipliesanddividestablesofnumbersrepresentingpossibleinteractionsoftheelectron.And,asiffromthemagicalabacusofamagus,theresultscorrespondexactlywithwhatwasobserved.Thesearethefirstfundamentalequationsofquantummechanics.Fromhereon,theseequationswilldonothingbutwork,work,work.Upuntilnow,incredibleasitmayseem,theyhaveneverfailed.

Intheend,itisanothertwenty-five-year-oldwhopicksuptheworkinitiatedbyHeisenberg,takesthenewtheoryinhishandsandconstructsitsentireformalandmathematicalscaffolding:theEnglishmanPaulAdrienMauriceDirac,consideredbymanytobethegreatestphysicistofthetwentiethcenturyafterEinstein.

Despitehisscientificstature,Diracismuchlesswell-knownthanEinstein.Thisisdue,inpart,totherarefiedabstractionofhisscience,andpartlyduetohisdisconcertingcharacter.Silentincompany,extremelyreserved,incapableofexpressingemotions,frequentlyunabletorecognizethefacesofacquaintances–incapableevenofconductinganordinaryconversation,orof,apparently,understandingsimplequestions–heseemedvirtuallyautistic,andperhapsfellwithinthespectrumofthiscondition.

Figure4.4PaulDirac.

Duringoneofhislectures,acolleaguesaidtohim,‘Idon’tunderstandthatformula.’Afterashort,silentpause,Diraccontinuedonregardless.Themoderatorinterruptedhim,askingifhewouldliketoreplytothequestion.Dirac,sincerelyastonished,replied,‘Question?Whatquestion?Mycolleaguehasmadeanassertion.’Andso,inaverypedanticsense,hehad.Itwasn’tarrogance:themanwhocoulddiscoversecretsofnaturewhichhadeludedeveryoneelsecouldnotunderstandtheimplicitmeaningoflanguage,couldnotgraspitsnon-literalusage,andtookeveryphraseatfacevalue.2Andyet,inhishands,quantummechanicsistransformedfromajumbleofintuitions,half-bakedcalculations,mistymetaphysicaldiscussionsandequationsthatworkwell,butinexplicably,intoaperfectarchitecture:airy,simpleandextremelybeautiful.Beautiful,butstratosphericallyabstract.

ThevenerableBohrsaidofhim,‘Ofallphysicists,Dirachasthepurestsoul.’Anddon’thiseyes,infigure4.4showso?Hisphysicshasthepristineclarityofasong.Forhim,theworldisnotmadeofthings,it’sconstitutedofanabstractmathematicalstructurewhichshowsushowthingsappearandhowtheybehavewhenmanifestingthemselves.It’samagicalencounterbetweenlogicandintuition.Deeplyimpressed,Einsteinremarked,‘Diracposesproblemsforme.Tomaintainanequilibriumalongthisvertiginouscourse,betweengeniusandmadness,isadauntingenterprise.’

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Dirac’squantummechanicsisthemathematicaltheoryusedtodaybyanyengineer,chemistormolecularbiologist.Init,everyobjectisdefinedbyanabstractspacefn20andhasnopropertyinitself,apartfromthosethatareunchanging,suchasmass.Itspositionandvelocity,itsangularmomentumanditselectricalpotential,andsoon,acquirerealityonlywhenitcollides–‘interacts’–withanotherobject.Itisnotjustitspositionwhichisundefined,asHeisenberghadrecognized:novariableoftheobjectisdefinedbetweenoneinteractionandthenext.Therelationalaspectofthetheorybecomesuniversal.

Whenitsuddenlyappears,inthecourseofaninteractionwithanotherobject,aphysicalvariable(velocity,energy,momentum,angularmomentum)doesnotassumejustanyvalue.Diracprovidesthegeneralrecipetocomputethesetofvaluesthataphysicalvariablecantake.fn21Thesevaluesareanalogoustothespectraofthelightemittedbyatoms.Todaywecallthesetoftheparticularvalueswhichavariablemayassumethe‘spectrum’ofthatvariable,byanalogywiththespectraintowhichthelightofelementsdecomposes–thefirstmanifestationofthisphenomenon.Forexample,theradiusoftheorbitalsofanelectronaroundanucleuscanacquireonlyspecificvalues,thosethatBohrhadhypothesized,whichformthe‘spectrumoftheradius’.

Thetheoryalsogivesinformationonwhichvalueofthespectrumwillmanifestitselfinthenextinteraction,butonlyintheformofprobabilities.Wedonotknowwithcertaintywheretheelectronwillappear,butwecancomputetheprobabilitythatitwillappearhereorthere.ThisisaradicalchangefromNewton’stheory,whereitispossible,inprinciple,topredictthefuturewithcertainty.Quantummechanicsbringsprobabilitytotheheartoftheevolutionofthings.Thisindeterminacyisthethirdcornerstoneofquantummechanics:thediscoverythatchanceoperatesattheatomiclevel.WhileNewton’sphysicsallowsforthepredictionofthefuturewithexactitude,ifwehavesufficientinformationabouttheinitialdataandifwecanmakethecalculations,quantummechanicsallowsustocalculateonlytheprobabilityofanevent.Thisabsenceofdeterminismatasmallscaleisintrinsictonature.Anelectronisnotobligedbynaturetomovetowardstherightortheleft;itdoessobychance.Theapparentdeterminismofthemacroscopicworldisdueonlytothefactthatthemicroscopicrandomnesscancelsoutonaverage,leavingonlyfluctuationstoominuteforustoperceiveineverydaylife.

Dirac’squantummechanicsthusallowsustodotwothings.Thefirstistocalculatewhichvaluesaphysicalvariablemayassume.Thisiscalled‘calculationofthespectrumofavariable’;itcapturesthegranularnatureofthings.Whenanobject(atom,electromagneticfield,molecule,pendulum,stone,star,andsoon)interactswithsomethingelse,thevaluescomputedarethosewhichitsvariablescanassumeintheinteraction(relationism).ThesecondthingthatDirac’squantummechanicsallowsustodoistocomputetheprobabilitythatthisorthatvalueofavariableappearsatthenextinteraction.Thisiscalled‘calculationofanamplitudeoftransition’.Probabilityexpressesthethirdfeatureofthetheory:indeterminacy–thefactthatitdoesnotgiveuniquepredictions,onlyprobabilisticones.

ThisisDirac’squantummechanics:arecipeforcalculatingthespectraofthevariablesandarecipeforcalculatingtheprobabilitythatoneoranothervalueinthespectrumwillappearduringaninteraction.That’sit.Whathappensbetweenoneinteractionandthenextisnotmentionedinthetheory.Itdoesnotexist.

Theprobabilityoffindinganelectronoranyotherparticleatonepointoranothercanbeimaginedasadiffusecloud,denserwheretheprobabilityofseeingtheparticleisstronger.Sometimesitisusefultovisualizethiscloudasifitwerearealthing.Forinstance,thecloudthatrepresentsanelectronarounditsnucleusindicateswhereitismorelikelythattheelectronappearsifwelookatit.Perhapsyouencounteredthematschool:thesearetheatomic‘orbitals’.fn22

Theefficacyofthetheorysoonprovesextraordinary.Iftodaywebuildcomputers,haveadvancedmolecularchemistryandbiology,lasersandsemiconductors,itisthankstoquantummechanics.ForacertainnumberofdecadesitwasasifitwereChristmaseverydayforphysicists:foreverynewproblem,therewasananswerwhichfollowedfromtheequationsofquantummechanics,anditwasalwaysthecorrectanswer.Oneexampleofthiswillsuffice.

Themattersurroundingusismadeupofathousanddifferentsubstances.Duringthenineteenthandtwentiethcenturieschemistsunderstoodthatallthesedifferentsubstancesarejustcombinationsofarelativelysmallnumber(lessthanahundred)ofsimpleelements:hydrogen,helium,oxygen,andsoon–touranium.Mendeleevputtheseelementsinorder(accordingtoweight)inthefamousperiodictablewhichispinnedtothewallsofsomanyclassroomsandwhichsummarizesthepropertiesoftheelementsofwhichtheworldismade–notonlyonEarthbutallovertheuniverseinallgalaxies.Whythesespecificelements?Whatexplainstheperiodicstructureofthetable?Whydoeseachelementhavecertainpropertiesandnotothers?Why,forinstance,dosomeelementscombineeasily,whereasothersdonot?WhatisthesecretofthecuriousstructureofMendeleev’stable?

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Figure4.5Lightisawaveonafield,butithasalsoagranularstructure.

Well,taketheequationofquantummechanicsthatdeterminestheformoftheorbitalsofanelectron.Thisequationhasacertainnumberofsolutions,andthesesolutionscorrespondexactlytohydrogen,helium,oxygen…andtheotherelements!Mendeleev’speriodictableisstructuredexactlylikethesetofthesesolutions.Thepropertiesoftheelements,witheverythingelse,followsfromthesolutionofthisequation.Quantummechanicsdeciphersperfectlythesecretofthestructureoftheperiodictableofelements.

PythagorasandPlato’sancientdreamisrealized:todescribealloftheworld’ssubstanceswithasingleformula.Theinfinitecomplexityofchemistry,capturedbythesolutionsofasingleequation!Andthisisjustoneoftheapplicationsofquantummechanics.

FieldsandparticlesarethesamethingShortlyaftercompletingthegeneralformulationofquantummechanics,Diracrealizesthatthetheorycanbedirectlyappliedtofieldssuchaselectromagneticones,andcanbemadeconsistentwithspecialrelativity.(Makingitconsistentwithgeneralrelativitywillprovemuchharder,andisthemainsubjectofthisbook).Indoingthis,Diracdiscoversanulterior,profoundsimplificationofourdescriptionofnature:theconvergencebetweenthenotionofparticlesusedbyNewtonandthenotionoffieldsintroducedbyFaraday.

Thecloudofprobabilitywhichaccompanieselectronsbetweenoneinteractionandanotherdoesresembleafield.FaradayandMaxwell’sfields,inturn,aremadeupofgrains:photons.Notonlyaretheparticlesinacertainsensediffusedinspacelikefieldsbutthefieldsinteractlikeparticles.Thenotionsoffieldsandparticles,separatedbyFaradayandMaxwell,endupmerginginquantummechanics.

Thewaythishappensinthetheoryiselegant:theequationofDiracdeterminesthevaluesavariablecantake.AppliedtotheenergyofFaraday’slines,theytellusthatthisenergycanonlytakeoncertainvalues,andnotothers.Sincetheenergyoftheelectromagneticfieldcantakeononlycertainvalues,thefieldbehaveslikeasetofpacketsofenergy.ThesearepreciselythequantaofenergyintroducedbyPlanckandEinsteinthirtyyearsearlier.Thecirclecloses,andthestoryiscomplete.Theequationsofthetheory,writtenbyDirac,accountforthegranularnatureoflight,whichPlanckandEinsteinhadintuited.

TheelectromagneticwavesarevibrationsofFaraday’slines,butalso,atasmallscale,swarmsofphotons.Whentheyinteractwithsomethingelse,asinthephotoelectriceffect,theymanifestthemselvesasparticles:tooureye,lightrainsinseparatedroplets,insinglephotons.Photonsarethequantaoftheelectromagneticfield.

Buttheelectronsandalltheotherparticlesofwhichtheworldismadeareequallyquantaofafield–a‘quantumfield’similartoFaradayandMaxwell’s,subjecttogranularityandtoquantumprobability.Diracwritestheequationsforthefieldoftheelectronsandoftheotherelementaryparticles.fn23ThesharpdistinctionbetweenfieldsandparticlesintroducedbyFaradayvanishes.

Thegeneralformofquantumtheorycompatiblewithspecialrelativityisthuscalledquantumfieldtheory,anditformsthebasisoftoday’sparticlephysics.Particlesarequantaofafield,justasphotonsarequantaoflight.Allfieldsdisplayagranularstructureintheirinteractions.

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Duringthecourseofthetwentiethcenturythelistoffundamentalfieldswasrepeatedlyupdated,andtodaywehaveatheorycalledthestandardmodelofelementaryparticleswhichdescribesalmostallwesee,withtheexceptionofgravity,fn24inthecontextofquantumfieldtheory.Thedevelopmentofthismodeloccupiedphysicistsforagoodpartofthelastcentury,andrepresentsinitselfawonderfuladventureofdiscovery.Idon’tpresentthissideofthestoryhere:itisquantumgravitythatIwouldliketogetonto.Thestandardmodeliscompletedbythe1970s.Thereareapproximatelyfifteenfields,whosequantaaretheelementaryparticles(electrons,quarks,muons,neutrinos,Higgs,andlittleelse),plusafewfieldssimilartotheelectromagneticone,whichdescribeelectromagneticforceandtheotherforcesoperatingatanuclearscale,whosequantaaresimilartothephotons.

Thestandardmodelwasnottakenveryseriouslyatfirst,duetoitssomewhatcobbled-togetheraspect,sodifferentfromtheairysimplicityofgeneralrelativityandMaxwell’sorDirac’sequations.Againstexpectations,however,allofitspredictionshavebeenconfirmed.Formorethanthirtyyears,everysingleexperimentofparticlephysicshasdonenothingbutrepeatedlyreconfirmthestandardmodel.ArecentconfirmationwasthediscoveryoftheHiggsparticle,whichcausedasensationin2013.Introducedtorenderthetheorycoherent,theHiggsfieldseemedabitartificial–untiltheHiggsparticle,thequantumofthisfield,wasactuallyobservedandfoundtohavepreciselythepropertiespredictedbythestandardmodel.fn25(Thefactthatithasbeencalled‘theGodparticle’issostupidastobeunworthyofcomment.)Inshort,despiteitsunjustlymodestname,thestandardmodelhasbeenatriumph.

Figure4.6Whatistheworldmadeof?

Quantummechanics,withitsfields/particles,offerstodayaspectacularlyeffectivedescriptionofnature.Theworldisnotmadeupoffieldsandparticlesbutofasingletypeofentity:thequantumfield.Therearenolongerparticleswhichmoveinspacewiththepassageoftime,butquantumfieldswhoseelementaryeventshappeninspacetime.Theworldisstrange,butsimple(figure4.6).

Quanta1:InformationisfiniteThetimehascometoattemptsomeconclusionsaboutwhatitis,precisely,thatquantummechanicstellsusabouttheworld.Itisn’taneasytask,becausequantummechanicsisnotconceptuallyclearanditstruemeaningremainscontroversial;butit’sanecessaryexercise,togainclarityandgoforward.Ithinkthatquantummechanicshasrevealedthreeaspectsofthenatureofthings:granularity,indeterminacyandtherelationalstructureoftheworld.Let’slookateachofthesemoreclosely.

Thefirstistheexistenceofafundamentalgranularityinnature.Thegranularityofmatterandlightisattheheartofquantumtheory.Itisn’tthesamegranularityintuitedbyDemocritus,however.ForDemocritus,atomswerelikelittlepebbles,whereasinquantummechanicsparticlesvanishandreappear.Buttherootoftheideaofthesubstantivegranularityoftheworldisstilltobefoundinancientatomism,andquantummechanics–strengthenedbycenturiesofexperiments,bypowerfulmathematics,andbyitsextraordinarycapacityformakingcorrectpredictions–isagenuinerecognitionoftheprofoundinsightsonthenatureofthingsreachedbythegreatphilosopherofAbdera.

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Saywemakemeasurementsonaphysicalsystemandfindthatthesystemisinaparticularstate.Forinstance,wemeasuretheamplitudeoftheoscillationsofapendulumandfindthatithasacertainvalue–say,somewherebetweenfivecentimetresandsixcentimetres(nomeasurementisexactinphysics).Beforequantummechanicswewouldhavesaidthat,sincethereareaninfinitenumberofpossiblevaluesbetweenfiveandsixcentimetres(forinstance5.1or5.101or5.101001…),thenthereareinfinitepossiblestatesofmotioninwhichthependulumcouldfinditself:theamountofourignoranceaboutthependulumstateisstillinfinite.

Instead,quantummechanicstellsusthatbetweenfiveandsixcentimetresthereisafinitenumberofpossiblevaluesoftheamplitude,henceourmissinginformationaboutthependulumisfinite.

Thisgoesforeverythingingeneral.fn26Therefore,thefirstmeaningofquantummechanicsistheexistenceofalimittotheinformationthatcanexistwithinasystem:alimittothenumberofdistinguishablestatesinwhichasystemcanbe.Thislimitationuponinfinity–thisgranularityofnatureglimpsedbyDemocritus–isthefirstcentralaspectofthetheory.Planck’sconstanthmeasurestheelementaryscaleofthisgranularity.

Quanta2:IndeterminacyTheworldisasequenceofgranularquantumevents.Thesearediscrete,granularandindividual;theyareindividualinteractionsofonephysicalsystemwithanother.Anelectron,aquantumofafieldoraphotondoesnotfollowatrajectoryinspacebutappearsinagivenplaceandatagiventimewhencollidingwithsomethingelse.Whenandwherewillitappear?Thereisnowayofknowingwithcertainty.Quantummechanicsintroducesanelementaryindeterminacytotheheartoftheworld.Thefutureisgenuinelyunpredictable.Thisisthesecondfundamentallessonlearnedwithquantummechanics.

Duetothisindeterminacy,intheworlddescribedbyquantummechanics,thingsareconstantlysubjecttorandomchange.Allthevariables‘fluctuate’continually,asif,atthesmallestscale,everythingisconstantlyvibrating.Wedonotseetheseomnipresentfluctuationsonlybecauseoftheirsmallscale;theycannotbeobservedatalargescale,aswhenweobservemacroscopicbodies.Ifwelookatastone,itstaysstill.Butifwecouldseeitsatoms,wewouldobservethemconstantlyspreadhereandthere,andinceaselessvibration.Quantummechanicsrevealstousthat,themorewelookatthedetailoftheworld,thelessconstantitis.Theworldisnotmadeupoftinypebbles.Itisaworldofvibrations,acontinuousfluctuation,amicroscopicswarmingoffleetingmicro-events.

Theatomismofantiquityhadanticipatedalsothisaspectofmodernphysics:theappearanceoflawsofprobabilityatadeeplevel.Democritusassumed(justlikeNewton)thatthemovementofatomswasrigorouslydeterminedbytheircollisions.Buthissuccessor,Epicurus,correctsthedeterminismofthemasterandintroducesintoatomismthenotionofindeterminacy–inthesamewayinwhichHeisenbergintroducesindeterminacyintoNewton’sdeterminism.ForEpicurus,atomscanonoccasiondeviatebychancefromtheircourse.Lucretiussaysthisinbeautifulwords:thisdeviationoccurs‘incertotempore…incertisqueloci’:3atanuncertainplace,atanuncertaintime.Thesamerandomness,thesameappearanceofprobabilityatanelementarylevel,isthesecondkeydiscoveryabouttheworldthatquantummechanicsexpresses.

So,howdowecomputetheprobabilitythatanelectroninacertaininitialpositionAwillreappear,afteragiventime,inoneoranotherfinalpositionB?

Inthe1950s,RichardFeynman,whoI’vealreadymentioned,foundasuggestivemethodofmakingthiscalculation:considerallpossibletrajectoriesfromAtoB,thatistosay,allpossibletrajectoriestheelectroncanfollow(straight,curved,zigzagging…).Eachtrajectorydeterminesanumber.Theprobabilityisobtainedfromthesumofallthesenumbers.Thedetailsofthiscalculationarenotimportant:whatmattersisthefactthatalltrajectoriesfromAtoBcontribute:itisasiftheelectron,inordertogofromAtoB,passed‘throughallpossibletrajectories’,or,inotherwords,unfurledintoacloudinorderthentoconvergemysteriouslyonpointB,whereitcollidesagainwithsomethingelse(figure4.7).

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Figure4.7InordertomovefromAtoBanelectronbehavesasifpassingthroughallpossibletrajectories.

ThistechniqueforcomputingtheprobabilityofaquantumeventiscalledFeynman’ssumoverpaths,fn27andweshallseethatitplaysaroleinquantumgravity.

Quanta3:RealityisrelationalThethirddiscoveryabouttheworldarticulatedbyquantummechanicsisthemostprofoundanddifficult–andonewhichwasnotanticipatedbytheatomismofantiquity.

Thetheorydoesnotdescribethingsastheyare:itdescribeshowthingsoccurandhowtheyinteractwitheachother.Itdoesn’tdescribewherethereisaparticlebuthowtheparticleshowsitselftoothers.Theworldofexistentthingsisreducedtoarealmofpossibleinteractions.Realityisreducedtointeraction.Realityisreducedtorelation.4

Inacertainsense,thisisjustanextensionofrelativity,albeitaradicalone.Aristotlewasfirsttoemphasizethatweonlyperceiverelativespeed.Onaship,forexample,wetalkofourspeedrelativetotheship;onland,relativetotheEarth.GalileounderstoodthatthisisthereasonwhytheEarthcanmovewithrespecttotheSunwithoutusfeelingthemovement.Speedisnotapropertyofanobjectonitsown:itisthepropertyofthemotionofanobjectwithrespecttoanotherobject.Einsteinextendedthenotionofrelativitytotime:wecansaythattwoeventsaresimultaneousonlyrelativelytoagivenmotion(seehere).Quantummechanicsextendsthisrelativityinaradicalway:allvariableaspectsofanobjectexistonlyinrelationtootherobjects.Itisonlyininteractionsthatnaturedrawstheworld.

Intheworlddescribedbyquantummechanicsthereisnorealityexceptintherelationsbetweenphysicalsystems.Itisn’tthingsthatenterintorelationsbut,rather,relationsthatgroundthenotionof‘thing’.Theworldofquantummechanicsisnotaworldofobjects:itisaworldofevents.Thingsarebuiltbythehappeningofelementaryevents:asthephilosopherNelsonGoodmanwroteinthe1950s,inabeautifulphrase,‘Anobjectisamonotonousprocess.’Astoneisavibrationofquantathatmaintainsitsstructureforawhile,justasamarinewavemaintainsitsidentityforawhilebeforemeltingagainintothesea.

Whatisawave,whichmovesonwaterwithoutcarryingwithitanydropofwater?Awaveisnotanobject,inthesensethatitisnotmadeofmatterthattravelswithit.Theatomsofourbody,aswell,flowinandawayfromus.We,likewavesandlikeallobjects,areafluxofevents;weareprocesses,forabrieftimemonotonous…

Quantummechanicsdoesnotdescribeobjects:itdescribesprocessesandeventswhicharejunctionpointsbetweenprocesses.

Tosummarize,quantummechanicsisthediscoveryofthreefeaturesoftheworld:

Granularity(figure4.8).Theinformationinthestateofasystemisfinite,andlimitedbyPlank’sconstant.Indeterminacy.Thefutureisnotdeterminedunequivocallybythepast.Eventhemorerigidregularitiesweseeare,ultimately,statistical.Relationality.Theeventsofnaturearealwaysinteractions.Alleventsofasystemoccurinrelationtoanothersystem.

Quantummechanicsteachesusnottothinkabouttheworldintermsof‘things’whichareinthisorthatstatebutintermsof‘processes’instead.Aprocessisthepassagefromoneinteractionto

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another.Thepropertiesof‘things’manifestthemselvesinagranularmanneronlyinthemomentofinteraction,thatistosay,attheedgesoftheprocesses,andaresuchonlyinrelationtootherthings.Theycannotbepredictedinanunequivocalwaybutonlyinaprobabilisticone.

Figure4.8The‘lightbox’inEinstein’smentalexperiment,asdrawnbyBohr.

ThisisthevertiginousdivetakenbyBohr,HeisenbergandDirac–intothedepthofthenatureofthings.

Butdowereallyunderstand?Certainly,quantummechanicsisatriumphofefficacy.Andyet…areyousure,dearreader,thatyouhavefullyunderstoodwhatquantummechanicsrevealstous?Anelectronisnowherewhenitisnotinteracting…mmm…thingsonlyexistbyjumpingfromoneinteractiontoanother…well…Doesitallseemalittleabsurd?

ItseemedabsurdtoEinstein.Ontheonehand,EinsteinproposedWernerHeisenbergandPaulDiracfortheNobelPrize,

recognizingthattheyhadunderstoodsomethingfundamentalabouttheworld.Ontheother,hetookeveryopportunitytogrumblethat,however,noneofthismademuchsense

TheyounglionsoftheCopenhagengroupweredismayed:howcouldthiscomefromEinsteinhimself?Theirspiritualfather,themanwhohadthecouragetothinktheunthinkable,nowpulledbackandfearedthisnewleapintotheunknown–theveryleapwhichhehadhimselftriggered.HowcoulditbethatthesameEinstein,whohadtaughtusthattimeisnotuniversalandthatspacebends,wasnowsayingthattheworldcouldnotbethisstrange?

NielsBohrpatientlyexplainedthenewideastoEinstein.Einsteinobjected.Bohr,intheend,alwaysmanagedtofindanswerstotheobjections.Thedialoguecontinuedforyears,bywayoflectures,letters,articles…Einsteindevisedmentalexperimentstoshowthatthenewideaswerecontradictory:‘Imagineaboxfilledwithlight,fromwhichisletescapeforabriefinstantasinglephoton…’:thusoneofthemostfamousexamplesofthesebegins(figure4.8).fn28

Duringthecourseoftheexchange,bothgreatmenhadtogiveway,toaltertheirideas.Einsteinwasobligedtorecognizethattherewasactuallynocontradictionwithinthenewideas.ButBohrhadtorecognizethatthingswerenotassimpleandasclearashethought.Einsteindidnotwanttorelentonwhatforhimwasthekeypoint:thenotionthatthereisanobjectivereality,independentofwhateverinteractedwithwhat.Herefusedtoaccepttherelationalaspectofthetheory,thefactthatthingsmanifestthemselvesonlythroughinteractions.Bohrdidnotwanttoconcedeonthevalidityoftheprofoundlynewwayinwhichtherealwasconceptualizedbythetheory.Ultimately,Einsteinacceptsthatthetheoryrepresentsagiganticleapforwardinourunderstandingoftheworld,andthatitiscoherent.Butheremainsconvincedthatthingscouldnotbeasstrangeasthistheoryproposed–andthat,‘behind’it,theremustbeafurther,morereasonableexplanation.

Acenturyhaspassed,andweareatthesamepoint.RichardFeynman,whomorethananyonehasknownhowtojugglewiththetheory,haswritten,‘IthinkIcanstatethatnobodyreallyunderstandsquantummechanics.’

Theequationsofthetheoryandtheirconsequencesareuseddailyinawidevarietyoffields:byphysicists,engineers,chemistsandbiologists.Buttheyremainmysterious:theydonotdescribephysicalsystemsbutonlyhowphysicalsystemsinteractwithandaffectoneanother.Whatdoesthismean?

Physicistsandphilosopherscontinuetoaskthemselveswhattherealmeaningofthetheorymightbeand,inrecentyears,articlesandconferencesontheissuehaveproliferated.Whatisquantumtheory,acenturyafteritsbirth?Anextraordinarydivedeepintothenatureofreality?Ablunder

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thatworks,bychance?Partofanincompletepuzzle?Oracluetosomethingprofoundregardingthestructureoftheworld,whichwehaveyettofullydecipher?

TheinterpretationofquantummechanicswhichIhavepresentedhereistheonewhichseemsleastunreasonabletome.Itiscalledthe‘relationalinterpretation’,andithasbeendiscussedbyseriousphilosopherssuchasBasvanFraassen,MichelBitbolandMauroDorato.5Butthereisnoconsensusonhowtothinkaboutquantummechanics:thereareotherwaysofthinkingaboutit,discussedbyotherphysicistsandotherphilosophers.Weareonthebrinkofthatwhichwedon’tknow,andopinionsdiverge.

Quantummechanicsisonlyaphysicstheory:perhapstomorrowitwillbecorrectedbyanunderstandingoftheworldwhichisdifferentandevenmoreprofound.Somescientiststodaytrytoironitoutabit,torenderitmoreinkeepingwithourintuition.Inmyopinion,itsdramaticempiricalsuccessshouldcompelustotakeitseriously,andtoaskourselvesnotwhatthereistochangeinthetheory–butratherwhatislimitedaboutourintuitionthatmakesitseemsostrangetous.

Ithinkthattheobscurityofthetheoryisnotthefaultofquantummechanicsbut,rather,isduetothelimitedcapacityofourimagination.Whenwetryto‘see’thequantumworld,weareratherlikemolesusedtolivingundergroundtowhomsomeoneistryingtodescribetheHimalayas.OrlikethemenimprisonedatthebackofPlato’scave.

WhenEinsteindied,hisgreatestrival,Bohr,foundforhimwordsofmovingadmiration.When,afewyearslater,Bohrinturndied,someonetookaphotographoftheblackboardinhisstudy.There’sadrawingonit.Itrepresentsthe‘boxoflight’ofEinstein’sthoughtexperiment.Totheverylast,thedesiretodebate,tounderstandmore.Totheverylast,doubt.

Thispermanentdoubt,thedeepsourceofscience.

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PartThree

QUANTUM SPACE AND RELATIONAL TIME

Ifyouhavefollowedmethisfar,younowhavealltheelementswithwhichtounderstandthecurrentimageoftheworldsuggestedbyfundamentalphysics–itspower,itsweaknesses,itslimits.

Thereisacurvedspacetimeborn14billionyearsago–nobodyknowshow–andstillexpanding.Thisspaceisarealobject,aphysicalfieldwithitsdynamicsdescribedbyEinstein’sequations.Spacebendsandcurvesundertheweightofmatterandplungesintoblackholeswhenmatteristooconcentrated.

Matterisdistributedin100billiongalaxies,eachcontaining100billionstars,andismadeupofquantumfieldswhichmanifestthemselvesintheformofparticles,suchaselectronsandphotons,oraswaves,suchastheelectromagneticonesthatbringustelevisionimagesandthelightoftheSunandthestars.

Thesequantumfieldsmakeupatoms,lightandthefullcontentsoftheuniverse.Theyarestrangeobjects:theirquantaareparticlesthatappearwhentheyinteractwithsomethingelse;leftalone,theyunfurlintoa‘cloudofprobability’.Theworldisaswarmingofelementaryevents,immersedintheseaofavastdynamicalspacewhichswayslikethewaterofanocean.

Withthisimageoftheworld,andthefewequationsthatmakeitconcrete,wecandescribealmosteverythingthatwesee.

Almost.Somethingismissing.Anditisthissomethingthatweareseeking.Therestofthebooktalksaboutthismissingpart.

Turningthepage,youpassfromwhat,forgoodorill,wecrediblyknowabouttheworld,towhatwedon’tyetknowbutaretryingtoglimpse.

Turningthepageislikeleavingthesecurityofoursmallspacecraftofnear-certaintiesandsteppingintotheunknown.

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5.SpacetimeisQuantum

Thereisaparadoxattheheartofourunderstandingofthephysicalworld.Generalrelativityandquantummechanics,thetwojewelsthatthetwentiethcenturyhasleftus,havebeenprolificingifts–forcomprehendingtheworldandfortoday’stechnology.Fromthefirstofthese,cosmologyhasdeveloped,aswellasastrophysics,thestudyofgravitationalwavesandofblackholes.Thesecondhasprovidedthefoundationforatomicphysics,nuclearphysics,thephysicsofelementaryparticlesandofcondensedmatter,andofmuchelsebesides.Andyetbetweenthetwotheoriesthereissomethingthatgrates.Theycannotbothbetrue,at

leastnotintheirpresentforms,becausetheyappeartocontradicteachother.Thegravitationalfieldisdescribedwithouttakingquantummechanicsintoaccount,withoutaccountingforthefactthatfieldsarequantumfields–andquantummechanicsisformulatedwithouttakingintoaccountthefactthatspacetimecurvesandisdescribedbyEinstein’sequations.Auniversitystudentattendinglecturesongeneralrelativityinthemorning,andotherson

quantummechanicsintheafternoon,mightbeforgivenforconcludingthathisprofessorsarefools,orthattheyhaven’ttalkedtoeachotherforatleastacentury.Inthemorning,theworldisacurvedspacetimewhereeverythingiscontinuous;intheafternoon,theworldisaflatonewherediscretequantaofenergyleapandinteract.Theparadoxresidesinthefactthatboththeoriesworkremarkablywell.Witheveryexperimentandeverytest,naturecontinuestosay‘youareright’togeneralrelativity,

andcontinuestosay‘youareright’toquantummechanicsaswell,despitetheseeminglyoppositeassumptionsonwhichthetwotheoriesarefounded.Itisclearthatsomethingstilleludesus.Inmostsituationswecanneglectquantummechanicsorgeneralrelativity(orboth).TheMoonis

toolargetobesensitivetominutequantumgranularity,sowecanforgetthequantawhendescribingitsmovements.Ontheotherhand,anatomistoolighttocurvespacetoasignificantdegree,andwhenwedescribeitwecanforgetthecurvatureofspace.Buttherearesituationswherebothcurvatureofspaceandquantumgranularitymatter,andforthesewedonotyethaveanestablishedphysicaltheorythatworks.Anexampleistheinteriorofblackholes.AnotheriswhathappenedtotheuniverseduringtheBig

Bang.Inmoregeneralterms,wedonotknowhowtimeandspacebehaveatverysmallscale.Inalltheseinstances,today’stheoriesbecomeconfusedandnolongertellusanythingreasonable:quantummechanicscannotdealwiththecurvatureofspacetime,andgeneralrelativitycannotaccountforquanta.Thisistheproblemofquantumgravity.Theproblemgoesevendeeper.Einsteinunderstoodthatspaceandtimearemanifestationsofa

physicalfield:thegravitationalfield.Bohr,HeisenbergandDiracunderstoodthatphysicalfieldshaveaquantumcharacter:granular,probabilistic,manifestingthroughinteractions.Itfollowsthatspaceandtimemustalsobequantumentitiespossessingthesestrangeproperties.What,then,isquantumspace?Whatisquantumtime?Thisistheproblemwecallquantum

gravity.Abandoftheoreticalphysicistsscatteredacrossfivecontinentsislaboriouslyseekingtosolvetheproblem.Theirobjectiveistofindatheory,thatistosay,asetofequations–but,aboveall,acoherentvisionoftheworld–withwhichtoresolvethecurrentschizophreniabetweenquantaandgravity.Itisn’tthefirsttimethatphysicshasfounditselffacedwithtwohighlysuccessfulbutapparently

contradictorytheories.Theefforttosynthesizehasinthepastbeenrewardedwithgreatstridesforwardinourunderstandingoftheworld.NewtondiscovereduniversalgravitypreciselybycombiningGalileo’sphysicsofhowthingsmoveonEarthwithKepler’sphysicsoftheheavens.MaxwellandFaradayfoundtheequationsofelectromagnetismbybringingtogetherwhatwasknownaboutelectricityandwhatwasknownaboutmagnetism.EinsteinfoundspecialrelativityinordertoresolvetheapparentconflictbetweenNewton’smechanicsandMaxwell’selectromagnetism–andthengeneralrelativityinordertoresolvetheresultingconflictbetweenNewton’smechanicsandhisownspecialrelativity.Theoreticalphysicistsarethusonlytoohappywhentheydiscoveraconflictofthistype:itisan

extraordinaryopportunity.Thequestiontoaskis:canweconstructaconceptualstructurecompatiblewithwhatwehavelearnedabouttheworldwithboththeories?Tocomprehendwhatquantumspaceandquantumtimeare,weneedoncemoretoreviseindepth

thewayweconceivethings.Weneedtorethinkthegrammarofourunderstandingoftheworld.JustashappenedwithAnaximander,whounderstoodthatEarthfliesinspace,andthat‘up’and‘down’donotexistinthecosmos;orwithCopernicus,whounderstoodthatwearemovingacrossthe

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heavensatgreatspeed;orwithEinstein,whounderstoodthatspacetimesquasheslikeamollusc,andtimepassesdifferentlyindifferentplaces…onceagain,inseekingacoherentvisionoftheworldinkeepingwithwhatwehavelearnedaboutit,ourideasaboutthenatureofrealityhavetochange.

Figure5.1MatveiBronštejn.

Thefirsttorealizethatourconceptualbasismustchangeinordertounderstandquantumgravitywasaromanticandlegendarycharacter:MatveiBronštejn,ayoungRussianwholivedduringtheStalineraanddiedtragically.

MatveiMatveiwasayoungerfriendofLevLandau–thescientistwhowouldgoontobecomethebesttheoreticalphysicistoftheSovietUnion.Colleagueswhoknewthembothwouldclaimthat,ofthetwo,Matveiwasthemorebrilliant.AtthepointwhenHeisenbergandDiracwereconstructingthebasesofquantummechanics,Landau,wrongly,thoughtthatfieldsbecameilldefinedduetoquanta:quantumfluctuationwouldpreventusfrommeasuringthevalueofacomponentofafieldatapoint(anarbitrarysmallregion)inspace.ThemasterlyBohrimmediatelysawthatLandauwaswrong,studiedtheissueindepthandwrotealonganddetailedarticletoshowthatfields,suchastheelectricone,remainwelldefinedevenwhenquantummechanicsisbroughttobear.1Landaudroppedtheissue.ButhisyoungfriendMatveiwasintrigued,realizingthatLandau’sintuition,thoughimprecise,

containedsomethingofimportance.HerepeatedthesamereasoningbywhichBohrhaddemonstratedthatthequantumelectricfieldwaswelldefinedatapointofspace,applyingitinsteadtothegravitationalfield,forwhichEinsteinhadjustafewyearspreviouslywrittentheequations.Andhere–surprise!–Landauwasright.Thegravitationalfieldatapointisnotwelldefined,whentakingquantaintoaccount.Thereisanintuitivewayofunderstandingwhathappens.Supposewewanttoobserveavery,

very,verysmallregionofspace.Todothis,weneedtoplacesomethinginthisarea,tomarkthepointthatwewishtoconsider.Sayweplaceaparticlethere.Heisenberghadunderstoodthatyoucan’tlocateaparticleatapointinspaceforlong.Itsoonescapes.Thesmallertheregioninwhichwetrytolocateaparticle,thegreaterthevelocityatwhichitescapes.(ThisisHeisenberg’suncertaintyprinciple.)Iftheparticleescapesatgreatspeed,ithasagreatdealofenergy.NowletustakeEinstein’stheoryintoaccount.Energymakesspacecurve.Alotofenergymeansthatspacewillcurveagreatdeal.Alotofenergyinasmallregionresultsincurvingspacesomuchthatitcollapsesintoablackhole,likeacollapsingstar.Butifaparticleplummetsintoablackhole,Icannolongerseeit.Icannolongeruseitasareferencepointforaregionofspace.Ican’tmanagetomeasurearbitrarilysmallregionsofspace,becauseifItrytodothistheseregionsdisappearinsideablackhole.

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Thisargumentcanbemademoreprecisewithalittlemathematics.Theresultisgeneral:quantummechanicsandgeneralrelativity,takentogether,implythatthereisalimittothedivisibilityofspace.Belowacertainscale,nothingmoreisaccessible.Moreprecisely,nothingexiststhere.Howsmallisthisminimalregionofspace?Thecalculationiseasy:weneedonlytocalculatethe

minimumsizeofaparticlebeforeitfallsintoitsownblackhole,andtheresultisstraightforward.Theminimumlengthisaround:

Underthesignofthesquareroottherearethethreeconstantsofnaturewehavealreadyencountered:Newton’sconstantG,discussedinChapter2,whichsetsthestrengthofgravity;thespeedoflightc,introducedinChapter3whendiscussingrelativity,whichopensuptheextendedpresent;andPlanck’sconstanth,foundinChapter4,whichdeterminesthescaleofthequantumgranularity.fn29Thepresenceofthesethreeconstantsconfirmsthefactthatwearelookingatsomethingwhichhastodowithgravity(G),relativity(c)andquantummechanics(h).ThelengthLP,determinedinthisfashion,iscalledthePlancklength.Itshouldbecalledthe

Bronštejnlength,butsuchisthewayoftheworld.Innumericalterms,itisequivalenttoapproximatelyonemillionthofabillionthofabillionthofabillionthofacentimetre(10-33centimetres).So,thatistosay…small.Itisatthisextremelyminutescalethatquantumgravitymanifestsitself.Togiveanideaofthe

smallnessofthescalewearediscussing:ifweenlargedawalnutshelluntilithadbecomeasbigasthewholeobservableuniverse,wewouldstillnotseethePlancklength.Evenafterhavingbeenenormouslymagnifiedthus,itwouldstillbeamilliontimessmallerthantheactualwalnutshellwasbeforemagnification.Atthisscale,spaceandtimechangetheirnature.Theybecomesomethingdifferent;theybecome‘quantumspaceandtime’,andunderstandingwhatthismeansistheproblem.MatveiBronštejnunderstandsallofthisinthe1930sandwritestwoshortandilluminating

articlesinwhichhepointsoutthatquantummechanicsandgeneralrelativity,takentogether,areincompatiblewithourcustomaryideaofspaceasaninfinitelydivisiblecontinuum.2Thereis,however,aproblem.MatveiandLevaresincerecommunists.Theybelieveinrevolution

astheliberationofmankind,theconstructionofagenuinelybettersociety,withoutinjustice,withouttheimmenseinequalitieswhichwestillseegrowingsystematicallythroughouttheworld.TheyareenthusiasticfollowersofLenin.WhenStalinassumespower,theyarebothperplexed,thencritical,thenhostile.Theywritearticleswhicharemildlybutopenlycritical…Thiswasnotthecommunismtheywanted…Buttheseareharshtimes.Landaugetsthroughthem,noteasily,buthesurvives.Matvei,theyear

afterhavingbeenthefirsttounderstandthatourideasonspaceandtimehadtochangeinaradicalway,isarrestedbyStalin’spoliceandcondemnedtodeath.Hisexecutiontakesplaceonthesamedayashistrial,18February,1938.3Heisthirtyyearsold.

JohnAfterMatveiBronštejn’sprematuredeath,manyofthecentury’seminentphysiciststriedtosolvethepuzzleofquantumgravity.Diracdedicatedthefinalyearsofhislifetotheproblem,openingavenuesandintroducingideasandtechniquesonwhichagoodpartofcurrentworkonquantumgravityisbased.Itisthankstothesetechniquesthatweknowhowtodescribeaworldwithouttime,asIwillexplainfurtheron.Feynmantried,attemptingtoadaptthetechniqueshehaddevelopedforelectronsandphotonstothecontextofgeneralrelativity,butwithoutsuccess:electronsandphotonsarequantainspace;quantumgravityissomethingelse:itisn’tenoughtodescribe‘gravitons’movinginspace,itisspaceitselfthathastobequantized.AfewNobelPrizeswereawardedtophysicistswhohappenedtoresolveotherproblems,almost

bymistake,duringthecourseoftheirattemptstodisentanglethepuzzleofquantumgravity.TwoDutchphysicists,Gerard’tHooftandMartinusVeltman,receivedtheNobelPrizein1999forhavingshowntheconsistencyofthetheorieswhichtodayareusedtodescribenuclearforces–apartofthestandardmodel–buttheirresearchprogrammewasactuallyaimingtodemonstratetheconsistencyofatheoryofquantumgravity.Theywereworkingonthetheoriesoftheseotherforcesonlyasapreliminaryexercise.The‘preliminaryexercise’earnedthemaNobelPrize,buttheydidnotsucceedinshowingtheconsistencyoftheirversionofquantumgravity.

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Thelistcouldgoonandwouldreadlikearollofhonourofthecentury’soutstandingtheoreticalphysicists.Aswellaslikeacatalogueoffailures.Verygradually,though,overthecourseofdecades,ideaswereclarifiedanddeadendsexploredandusefullyclosedoff;techniquesandgeneralideaswerestrengthened,andresultsbegantobuild,onedevelopingfromanother.Tomentionherethenumerousscientistswhohavecontributedtothisgradual,slow-moving,collectiveconstructionwouldrequireatediouslistofnames,eachoneofwhomhasaddedagrainorastonetotheprocess.Iwouldliketomentionjustone,whoforyearsheldtogetherthethreadsofthiscollective

research:theremarkable,eternallyyouthfulEnglishman–halfphilosopherandhalfphysicist–ChrisIsham.ItwaswhenreadingoneofhisarticlesreviewingthequestionofquantumgravitythatIfirstbecameenamouredwiththeproblem.Thearticleexplainedjustwhyitwassodifficult,howourconceptionofspaceandtimeneededtobemodified,andgavealucidoverviewofalltherouteswhichwerebeingfollowedatthetime,withtheresultsachieved,anddifficultiesentailed.Iwasinmythirdyearatuniversity,andthepossibilityofrethinkingspaceandtimefromsquareonefascinatedme.Thisfascinationhasneverdiminished.For,asPetrarchsings,‘Thewounddoesnothealduetotheweakeningofthebow.’

Figure5.2JohnWheeler.

ThescientistwhohasmostcontributedtoquantumgravityisJohnWheeler,alegendaryfigurewhohastraversedthephysicsofthepastcentury.ApupilofandcollaboratorwithNielsBohrinCopenhagen;acollaboratorwithEinsteinwhenEinsteinmovedtotheUnitedStates;ateacherwhocancountamonghisstudentsfiguressuchasRichardFeynman…Wheelerwasattheheartofthephysicsofthetwentiethcentury.Hewasgiftedwithafervidimagination.Itwashewhoinventedandmadepopulartheterm‘blackhole’.Hisnameisassociatedwiththeearlyextendedinvestigations–frequentlymoreintuitivethanmathematical–intohowtothinkaboutquantumspacetime.HavingabsorbedBronštejn’slessonthatquantumpropertiesofthegravitationalfieldimplyamodificationofthenotionofspaceatasmallscale,Wheelerlookedfornovelideastohelpconceiveofthisquantumspace.Heimagineditasacloudofsuperimposedgeometries,justaswecanthinkofaquantumelectronasacloudofpositions.Imaginethatyouarelookingattheseafromagreatheight:youperceiveavastexpanseofit,a

flat,ceruleantable.Nowyoudescendandlookatitmoreclosely.Youbegintomakeoutthegreatwavesswollenbythewind.Youdescendfurther,andyouseethatthewavesbreakupandthatthesurfaceoftheseaisaturbulentfrothing.Thisiswhatspaceislike,asimaginedbyWheeler.fn30Onourscale,immenselylargerthanthePlancklength,spaceissmooth.IfwemovedowntothePlanckscale,itshattersandfoams.Wheelersoughtawaytodescribethisfoamingofspace,thiswaveofprobabilityofdifferent

geometries.In1966,ayoungcolleagueofhiswholivedinCarolina,BryceDeWitt,providedthekey.4Wheelertravelledfrequently,andmetcollaboratorswhereverhecould.HeasksBrycetomeetatRaleighDurhamairport,inNorthCarolina,wherehehadafewhours’waitbetweenconnectingflights.Brycearrivesandshowshimanequationfor‘awavefunctionofspace’,obtainedbyusinga

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simplemathematicaltrick.fn31Wheelerisenthused.Fromthisconversationatypeof‘equationoforbitals’forgeneralrelativityisborn;anequationwhichshoulddeterminetheprobabilityofoneoranothercurvedspaces.Foralongtime,DeWittcalleditWheeler’sequationfn32–whileWheelercalledittheDeWittequation.EveryoneelsecallsittheWheeler-DeWittequation.Theideaisverygood,andbecomesabasisfortheattemptstoconstructthefulltheoryof

quantumgravity.Buttheequationitselfisriddledwithproblems–seriousones.Inthefirstplace,fromamathematicalpointofview,theequationisreallyquitebadlydefined.Ifwetrytouseittodocalculations,wesoonobtainresultsthatareinfinite,whichmakesnosense.Itmustbeimproved.Butitisalsodifficulttounderstandhowtointerpretthisequation,toknowwhatitmeans.Among

itsdisconcertingaspectsisthefactthatitnolongercontainsthetimevariable.Howcanitbeusedtocomputetheevolutionofsomethingwhichhappensintimeifitdoesnotincludeatimevariable?Dynamicalequations,inphysics,alwayscontainthevariablet,time.Whatdoesaphysicaltheorywithoutatemporalvariablesignify?Foryears,researchwillrevolvearoundsuchquestions,tryingtorevisetheequationindifferentmanners,inordertoimproveitsdefinitionandunderstandwhatitmightmean.

ThefirststepsoftheloopsThefogbeginstodissipatetowardstheendofthe1980s.Surprisingly,somesolutionsoftheWheeler–DeWittequationappear.DuringtheseyearsIfoundmyselffirstattheUniversityofSyracuse,inNewYorkState,visitingtheIndianphysicistAbhayAshtekar,andtheninConnecticut,atYaleUniversity,visitingtheAmericanphysicistLeeSmolin.Irememberaperiodofintensediscussionsandburningintellectualfervour.AshtekarhadrewrittentheWheeler–DeWittequationinasimplerform;andSmolin,togetherwithTedJacobsonoftheUniversityofMarylandinWashington,hadbeenthefirsttofindsomeofthesenewstrangesolutionsoftheequation.Thesolutionshadacuriouspeculiarity:theydependedonclosedlinesinspace.Aclosedlineisa

‘loop’.SmolinandJacobsoncouldwriteasolutiontotheWheeler–DeWittequationforeveryloop:foreverylineclosedonitself.Whatdidthismean?Thefirstworksofwhatwilllaterbecomeknownasloopquantumgravityemergefromthesediscussions,asthemeaningofthesesolutionsoftheWheeler–DeWittequationgraduallyclarify.Uponthesesolutions,littlebylittle,acoherenttheorybeginstobeerected,inheritingthename‘looptheory’fromthefirstsolutionsstudied.Todaytherearehundredsofscientistsworkingonthistheory,spreadthroughouttheworldfrom

ChinatoArgentina,fromIndonesiatotheUnitedStates.Whatisslowlybeingerectedisthetheorynowknownaslooptheory,orloopquantumgravity:thetheorytowhichthefollowingchaptersaredevoted.Itisnottheonlydirectionexploredinthesearchforaquantumtheoryofgravity,butitistheoneIconsiderthemostpromising.fn33

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6.QuantaofSpace

ThelastchapterclosedwiththesolutionsoftheWheeler–DeWittequationdiscoveredbyJacobsonandSmolin.Thesesolutionsdependonlinesthatcloseonthemselves,orloop.Whatdoesitallmean?

RememberFaraday’slines–thelineswhichcarrytheelectricforceandwhich,inFaraday’svision,fillspace?Thelinesfromwhichtheconceptof‘field’originates?Well,theclosedlinesthatappearinthesolutionsoftheWheeler–DeWittequationareFaradaylinesofthegravitationalfield.

ButtwonewingredientsarenowaddedtoFaraday’sideas.Thefirstisthatwearedealingwithquantumtheory.Inquantumtheory,everythingisdiscrete.

Thisimpliesthattheinfinitelyfine,continuousspiderwebofFaraday’slinesnowbecomessimilartoarealspiderweb:ithasafinitenumberofdistinctthreads.EverysinglelinedeterminingasolutionoftheWheeler–DeWittequationdescribesoneofthethreadsofthisweb.

Thesecondnewaspect,thecrucialone,isthatwearespeakingofgravityand,therefore,asEinsteinunderstood,wearenotspeakingoffieldsimmersedinspacebutoftheverystructureofspaceitself.Faraday’slinesofthequantumgravitationalfieldarethethreadsofwhichspaceiswoven.

Atfirst,theresearchwasfocusedontheselinesandhowtheycould‘weave’ourthree-dimensionalphysicalspace.Figure6.1representsanearlyattempttogiveanintuitiveideaofthediscretestructureofspacewhichwouldresultfromthis.

Soonafter,thankstotheintuitionsandmathematicalabilityofyoungscientistssuchastheArgentineJorgePullinandthePoleJurekLewandowski,itbecameclearthatthekeytounderstandingthephysicsofthesesolutionsliesinthepointswheretheselinesintersect.Thesepointsarecallednodes,andthelinesbetweennodesarecalledlinks.Asetofintersectinglinesformswhatiscalledagraph,thatistosay,acombinationofnodesconnectedbylinks,asinfigure6.3.

Acalculation,infact,demonstratesthat,withoutnodes,physicalspacehasnovolume.Inotherwords,itisinthenodesofthegraph,notinthelines,thatthevolumeofspace‘resides’.Thelines‘linktogether’individualvolumessittingatthenodes.

Gettingtoafullclarificationoftheresultingpictureofquantumspacetimetookyears.Itwasnecessarytotransformtheill-definedmathematicsoftheWheeler–DeWittequationintoastructuresufficientlywelldefinedtobeabletocomputewith.Withthis,itbecamepossibletoachievepreciseresults.Thekeytechnicalresultwhichclarifiesthephysicalmeaningofthegraphsisthecalculationofthespectraofvolumeandofarea.

Figure6.1ThequantumversionofFaraday’slinesofforce,whichweavespacelikeathree-dimensionalmeshofinterlinkedrings(loops).

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Figure6.2Thespectrumofthevolume:thevolumesofaregulartetrahedronthatarepossibleinnaturearelimitedinnumber.Thesmallest,atthebottom,isthesmallestvolumeinexistence.

SpectraofvolumeandareaTakeanyregionofspace,forexample,theroominwhichyouarereadingthis,ifyouareinaroom.Howbigisthisroom?Thesizeofthespaceoftheroomismeasuredbyitsvolume.Volumeisageometricalquantitywhichdependsonthegeometryofspace,butthegeometryofspace–asEinsteinunderstood,andasIrecountedinChapter3–isthegravitationalfield.Volumeisthereforeapropertyofthegravitationalfield,expressinghowmuchgravitationalfieldthereisbetweenthewallsoftheroom.Butthegravitationalfieldisaphysicalquantityand,likeallphysicalquantities,issubjecttothelawsofquantummechanics.Inparticular,likeallphysicalquantities,volumemaynotassumearbitraryvaluesbutonlycertainparticularones,asIdescribedinChapter4.Thelistofallpossiblevaluesiscalled,ifyouremember,thespectrum.Hencethereshouldexista‘spectrumofthevolume’(figure6.2)

Diracprovideduswiththeformulawithwhichtocomputethespectrumofeveryvariable.Thecalculationtooktime,firsttoformulateitandthentocompleteit,andmadeussuffer.Itwascompletedinthemid-1990s,andtheanswer,asexpected(Feynmanusedtosaythatweshouldneverdoacalculationwithoutfirstknowingtheresult),isthatthespectrumofthevolumeisdiscrete.Thatis,thevolumecanonlybemadeupof‘discretepackets’.Thesearesomewhatsimilartotheenergyoftheelectromagneticfield,whichisalsoformedofdiscretepackets:photons.

Thenodesofthegraphrepresentthediscretepacketsofvolumeand,asinthecaseofphotons,canonlyhavecertainsizes,whichcanbecomputedusingDirac’sgeneralquantumequation.fn34

Everynodeninthegraphhasitsownvolumevn:oneofthenumbersinthespectrumofthevolume.Thenodesaretheelementaryquantaofwhichphysicalspaceismade.Everynodeofthegraphisa‘quantumparticleofspace’.Thestructurethatemergesistheoneillustratedinfigure6.3.

Figure6.3Ontheleft,agraphformedbynodesconnectedbylinks.Ontheright,thegrainsofspacewhichthegraphrepresents.Thelinksindicatetheadjacentparticles,separatedbysurfaces.

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AlinkisanindividualquantumofaFaradayline.Nowwecanunderstandwhatitrepresents:ifyouimaginetwonodesastwosmall‘regionsofspace’,thesetworegionswillbeseparatedbyasmallsurface.Thesizeofthissurfaceisitsarea.Thesecondquantity,afterthevolume,whichcharacterizesthequantumwebsofspace,istheareaassociatedwitheachline.fn35

Thearea,justasinthecaseofthevolume,isaphysicalvariable,andhasaspectrumwhichmaybecalculatedusingDirac’sequation.*Areaisnotcontinuous,itisgranular.Thereisnosuchthingasanarbitrarilysmallarea.

Spaceappearscontinuoustousonlybecausewecannotperceivetheextremelysmallscaleoftheseindividualquantaofspace.JustaswhenwelookcloselyattheclothofaT-shirt,weseethatitiswovenfromsmallthreads.

Whenwesaythatthevolumeofaroomis,forexample,100cubicmetres,weareineffectcountingthegrainsofspace–thequantaofthegravitationalfield–whichitcontains.Inaroom,thisnumberhasmorethanahundreddigits.Whenwesaythattheareaofthispageis200squarecentimetres,weareactuallycountingthenumberoflinksintheweb,orloops,whichtraversethepage.Acrossthepageofthisbook,thereisanumberofquantawithmoreorlessseventydigits.

Theideathatmeasuringlength,areaandvolumeisaquestionofcountingindividualelementshadbeenproposedinthenineteenthcenturybyRiemannhimself.Themathematicianwhohaddevelopedthetheoryofcontinuouscurvedmathematicalspaceswasalreadyawarethatadiscretephysicalspaceis,ultimately,morereasonablethanacontinuousone.

Tosummarize,thetheoryofloopquantumgravity,orlooptheory,combinesgeneralrelativitywithquantummechanicsinaratherconservativeway,becauseitdoesnotemployanyotherhypothesisapartfromthoseofthetwotheoriesthemselves,suitablyrewrittentorenderthemcompatible.Buttheconsequencesareradical.

Generalrelativitytaughtusthatspaceissomethingdynamic,liketheelectromagneticfield:animmense,mobilemolluscinwhichweareimmersed,whichstretchesandbends.Quantummechanicsteachesusthateveryfieldofthissortismadeofquanta,thatistosay,ithasafine,granularstructure.Itfollowsthatphysicalspace,beingafield,ismadeofquantaaswell.Thesamegranularstructurecharacterizingtheotherquantumfieldsalsocharacterizesthequantumgravitationalfield,andthereforespace.Weexpectspacetobegranular.Weexpectquantaofgravity,justastherearequantaoflight,quantaoftheelectromagneticfield,andasparticlesarequantaofquantumfields.Butspaceisthegravitationalfield,andthequantaofthegravitationalfieldarequantaofspace:thegranularconstituentsofspace.

Thecentralpredictionoflooptheoryisthereforethatspaceisnotacontinuum,itisnotdivisibleadinfinitum,itisformedof‘atomsofspace’.Abillionbilliontimessmallerthanthesmallestofatomicnuclei.

Looptheorydescribesthisatomicandgranularquantumstructureofspaceinaprecisemathematicalform.ItisobtainedbyapplyingthegeneralequationsofquantummechanicswrittenbyDiractoEinstein’sgravitationalfield.

Inparticular,looptheoryspecifiesthatvolume(forexample,thevolumeofagivencube)cannotbearbitrarilysmall.Aminimumvolumeexists.Nospacesmallerthanthisminimumvolumeexists.Thereisaminimumquantumofvolume:anelementaryatomofspace.

AtomsofspaceRememberAchilleschasingafterthetortoise?ZenoobservedthatthereissomethingdifficulttoacceptintheideathatAchilleshastocoveraninfinitenumberofdistancesbeforereachingtheslow-movingcreature.Mathematicshadfoundapossibleanswertothisdifficulty,showinghowaninfinitenumberofprogressivelysmallerintervalscouldneverthelessamounttoafinitetotalinterval.

Butisthiswhattrulyhappensinnature?ArethereintervalsbetweenAchillesandthetortoisethatcanbearbitrarilyshort?Doesitreallymakesensetotalkofabillionthofabillionthofabillionthofamillimetre,andthentothinkofdividingitagainfurtherinnumerabletimes?

Thecalculationofthequantumspectraofgeometricquantitiesindicatesthattheanswerisnegative:arbitrarilysmallchunksofspacedonotexist.Thereisalowerlimittothedivisibilityofspace.Itisataverysmallscaleindeed,butitisthere.ThisiswhatMatveiBronštejnhadintuitedinthe1930s.ThecalculationofthespectraofvolumeandareaconfirmsBronštejn’sideaandframesitinamathematicallyprecisemanner.

Achillesdoesnotneedaninfinitenumberofstepstoreachthetortoisebecause,inaspacemadeofgrainsoffinitesize,infinitelysmallstepsdonotexist.Theherowillcomeeverclosertothecreatureuntil,intheend,hereachesitinasinglequantumleap.

But,onreflection,wasthisnotpreciselythesolutionproposedbyLeucippusandbyDemocritus?Theyspokeofthegranularstructureofmatter,andweareratherunsureastowhat,precisely,theysaidaboutspace.Unfortunately,wedonothavetheirtextsandmustmakedowiththesparsefragmentsinthecitationsofothers.ItisliketryingtoreconstructShakespeare’splaysfromalistofShakespearequotes.fn36Democritus’sargumentontheincongruityofthecontinuumasacollectionofpoints,reportedbyAristotle,maybeappliedtospace.IimaginethatifwecouldaskDemocritus

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ifitmakessensetosplitaspaceintervaladinfinitum,hisreplycouldonlybetorepeatthatdivisibilitymusthavealimit.ForthephilosopherofAbdera,matterismadeofatomsthatcannotbedivided.Havingonceunderstoodthatspaceisverymuchlikematter–space,ashehadsaidhimself,hasitsownnature,‘acertainphysics’–Isuspecthewouldnothavehesitatedtodeducethatspace,too,canonlybemadeofelementarychunksthatcannotbedivided.WeareperhapsjustfollowinginthefootstepsofDemocritus.

Icertainlydon’tmeantoimplythatthephysicsoftwomillenniawasuseless,thatexperimentsandmathematicsarepointlessandthatDemocrituscouldbeasconvincingasmodernscience.Obviouslynot.Withoutexperimentsandmathematics,wewouldneverhaveunderstoodwhatwehaveunderstood.Yetwedevelopourconceptualschemaforunderstandingtheworldbyexploringnewideasbutalsobybuildingonthepowerfulintuitionsofgiantfiguresfromthepast.Democritusisoneofthem,andwediscoverthenewsittingonhistitanicshoulders.

Butletusreturntoquantumgravity.

SpinnetworksThegraphswhichdescribethequantumstatesofspacearecharacterizedbyavolumevforeverynodeandahalf-integerjforeveryline.Agraphwiththisadditionalinformationiscalledaspinnetwork(figure6.4).(Half-integersinphysicsarecalled‘spin’becausetheyappearinthequantummechanicsofspinningobjects.)Aspinnetworkrepresentsaquantumstateofthegravitationalfield:aquantumstateofspace;agranularspaceinwhichareaandvolumearediscrete.Fine-meshgridsareusedelsewhereinphysicstoapproximatecontinuousspace.Here,thereisnospacecontinuumtoapproximate:spaceisgenuinelygranular.

Thecrucialdifferencebetweenphotons(thequantaoftheelectromagneticfield)andthenodesofthegraph(thequantaofgravity)isthatphotonsexistinspace,whereasthequantaofgravityconstitutespacethemselves.Photonsarecharacterizedby‘wheretheyare’.fn37Quantaofspacehavenoplacetobein,becausetheyarethemselvesthatplace.Theyhaveonlyonepieceofinformationwhichcharacterizesthemspatially:informationaboutwhichotherquantaofspacetheyareadjacentto,whichoneisnexttowhichother.Thisinformationisexpressedbythelinksinthegraph.Twonodesconnectedbyalinkaretwonodesinproximity.Theyaretwograinsofspaceincontactwitheachother:this‘touching’constructsthestructureofspace.

Figure6.4Aspinnetwork

Thequantaofgravity,thatis,arenotinspace,theyarethemselvesspace.Thespinnetworkswhichdescribethequantumstructureofthegravitationalfieldarenotimmersedinspace;theydonotinhabitaspace.Thelocationofsinglequantaofspaceisnotdefinedwithregardtosomethingelsebutonlybythelinksandtherelationtheseexpress.

IfIstepfromgraintograinalongthelinksuntilIcompleteacircuitandreturntothegrainfromwhichIstarted,Iwillhavemadea‘loop’.Thesearetheoriginalloopsofthelooptheory.InChapter4Ishowedthatthecurvatureofspacemaybemeasuredbylookingatwhetheranarrowtransportedacrossaclosedcircuitreturnspointinginthesamedirection,orturned.The

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mathematicsofthetheorydeterminesthiscurvatureforeveryclosedcircuitonaspinnetwork,andthismakesitpossibletoevaluatethecurvatureofspacetime,andhencetheforceofthegravitationalfield,fromthestructureofaspinnetwork.fn38

Now,quantummechanicsismorethangranularity.Thereisalsothefactthatevolutionisprobabilistic–thewayinwhichthespinnetworksevolveisrandom.I’llspeakaboutthisinthenextchapter,devotedtotime.

Andthereisthefactthatwhatmattersisnothowthingsare,butratherhowtheyinteract.Spinnetworksarenotentities;theydescribetheeffectofspaceuponthings.Justasanelectronisinnoplacebutdiffusedinacloudofprobabilityinallplaces,spaceisnotactuallyformedbyasinglespecificspinnetworkbutratherbyacloudofprobabilitiesoverthewholerangeofallpossiblespinnetworks.

Atanextremelysmallscale,spaceisafluctuatingswarmofquantaofgravitywhichactuponeachother,andtogetheractuponthings,manifestingthemselvesintheseinteractionsasspinnetworks,grainsinterrelatedwitheachother(figure6.5).

Figure6.5Ataminutescale,spaceisnotcontinuous:itiswovenfrominterconnectedfiniteelements.

Physicalspaceisthefabricresultingfromtheceaselessswarmingofthiswebofrelations.Thelinesthemselvesarenowhere;theyarenotinaplacebutrathercreateplacesthroughtheirinteractions.Spaceiscreatedbytheinteractionofindividualquantaofgravity.

Thisisthefirststeptowardsunderstandingquantumgravity.Thesecondconcernstime.And,totime,thenextchapterisdevoted.

*Theresultofthecalculationissimple.IshowitheresoyoucanseehowDirac’sspectrawork.ThepossiblevaluesoftheareaAaregiveninthefollowingformula,wherejisa‘half-integer’,thatistosay,anumberwhichishalfofaninteger,suchas0,½,1,3⁄2,2,5⁄2,3…

Aistheareathatasurfaceseparatingtwograinsofspacecanhave.8isthenumbereight,nothingspecialaboutit.πistheGreekpiwhichwestudiedatschool:theconstantwhichgivestherelationbetweenthecircumferenceandthediameterofanycircle,andwhichappearseverywhereinphysics,Idon’tknowwhy.LpisthePlancklength,theextremelysmallscaleatwhichthephenomenaofquantumgravitytakeplace.L2pisthesquareofLp,whichisthe(extremelysmall)areaofatinysquarewithsidesequaltothePlancklength.Therefore8πL2pissimplya‘small’area:theareaofaminusculesquarewithasidewhichisaboutamillionthofabillionthofabillionthofabillionthofacentimetre(10-66cm²).Theinterestingaspectoftheformulaisthesquarerootandwhatiswithinit.Thekeypointisthatjisahalf-integer,thatistosay,itmayhaveonlyvalueswhicharemultiplesof½.Foreachoneofthese,theroothasacertainvalue,listedapproximatelyintable6.1.

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Table6.1Spin(half-integers)andcorrespondingvalueoftheareainunitsofminimalarea.

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Multiplyingthenumbersintheright-handcolumnbythearea8πL2p,weobtainthepossiblevaluesoftheareaofthesurface.Thesespecialvaluesareliketheoneswhichappearinthestudyoftheorbitsofelectronsinatoms,wherequantummechanicsallowsonlycertainorbits.Thepointisthatnootherareasapartfromthevaluesderivedbythisequationexist.Nosurfacecanhaveanareaonetenthof8πL2p.

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7.TimeDoesNotExist

Itmustnotbeclaimedthatanyonecansensetimebyitselfapartfromthemovementofthings.–Lucretius,Dererumnatura1

Thealertreaderwillhaverealizedthatintheprecedingchapterlittleattentionwasgiventotime.AndyetEinsteinshowed,overacenturyago,thatwecannotseparatetimeandspace,thatwemustthinkofthemtogetherasasinglewhole:spacetime.Themomenthascometorectifythisandbringtimebackintothepicture.

Researchonquantumgravityhasrevolvedforyearsaroundspatialequations,beforehavingthecouragetoconfronttime.Inthelastfifteenyears,awayofthinkingabouttimehasbeguntoemerge.I’lltrytoexplainit.

Spaceasanamorphouscontainerofthingsdisappearsfromphysicswithquantumgravity.Things(thequanta)donotinhabitspace,theydwelloneovertheother,andspaceisthefabricoftheirneighbouringrelations.Asweabandontheideaofspaceasaninertcontainer,similarly,wemustabandontheideaoftimeasaninertflowalongwhichrealityunfurls.Justastheideaofthespacecontinuumcontainingthingsdisappears,so,too,doestheideaofaflowingcontinuum‘time’duringthecourseofwhichphenomenahappen.

Inacertainsense,spacenolongerexistsinfundamentaltheory;thequantaofthegravitationalfieldarenotinspace.Inthesamesense,timenolongerexistsinthefundamentaltheory:thequantaofgravitydonotevolveintime.Timejustcountstheirinteractions.AsevidencedwiththeWheeler–DeWittequation,thefundamentalequationsnolongercontainthetimevariable.Timeemerges,likespace,fromthequantumgravitationalfield.

Thiswasalreadypartiallytrueforclassicalgeneralrelativity,wheretimealreadyappearsasanaspectofthegravitationalfield.Butaslongasweneglectquantumtheory,wecanstillthinkofspacetimeinaratherconventionalmanner,likethetapestryinwhichthestoryoftherestofrealityunfolds,evenifitisadynamical,movingtapestry.Themomentwetakequantummechanicsintoaccount,werecognizethattime,too,musthavethoseaspectsofprobabilisticindeterminacy,granularityandrelationalitywhicharecommontoallofreality.Itbecomesa‘time’markedlydifferentfromallthatwehavehithertomeantbytheword.

Thissecondconceptualconsequenceofthetheoryofquantumgravityismoreextremeeventhanthevanishingofspace.

Let’sattempttounderstandit.

TimeisnotwhatwethinkitisThatthenatureoftimeisdifferentfromthecommonideawhichwehaveofitwasalreadyclearoveracenturyago.Specialandgeneralrelativitymadethisexplicit.Today,theinadequacyofourcommon-senseviewoftimecanbeeasilyverifiedinalaboratory.

Let’sreconsider,forexample,thefirstconsequenceofgeneralrelativity,asillustratedinChapter3.Taketwowatches,ensurethattheymarkexactlythesametime,placeoneonthefloorandtheotheronapieceoffurniture.Waitforabouthalfanhourandthenbringthembacknexttoeachother.Willtheystilltellthesametime?

AsdescribedinChapter3,theanswerisno.Thewatcheswhichweusuallywearonourwrists,orhaveonourmobilephones,arenotsufficientlyprecisetoallowustoverifythisfact,butinphysicslaboratoriesallovertheworldtherearetimepiecespreciseenoughtodemonstratethediscrepancywhichoccurs:thewatchleftonthefloorisslowwhencomparedtotheonewhichhasbeenraisedaboveit.

Why?Becausetimedoesnotpassinthesamewayeverywhereintheworld.Insomeplaces,itflowsmorequickly;inothers,moreslowly.ThecloseryougettotheEarth,wheregravityfn39ismoreintense,theslowertimepasses.RememberthetwinsinChapter3,whoendedupwithdifferentagesasaresultofhavinglivedoneatsealevelandoneinthemountains?Theeffectisveryslight:thetimegainedduringalifespentbythesea,withrespecttoonepassedinthemountains,consistsoffractionsofasecond–butthesmallnessoftheamountdoesnotalterthefactthatthereisarealdifference.Timedoesnotworkaswecustomarilyimagineitdoes.

Wemustnotthinkoftimeasiftherewereagreatcosmicclockthatmarksthelifeoftheuniverse.Wehaveknownformorethanacenturythatwemustthinkoftimeinsteadasalocalizedphenomenon:everyobjectintheuniversehasitsowntimerunning,atapacedeterminedbythelocalgravitationalfield.

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Buteventhisnotionofalocalizedtimenolongerworkswhenwetakethequantumnatureofthegravitationalfieldintoaccount.QuantumeventsarenolongerorderedbythepassageoftimeatthePlanckscale.Time,inasense,ceasestoexist.

Whatdoesitmeantosaythattimedoesnotexist?First,theabsenceofthevariabletimefromthefundamentalequationsdoesnotimplythat

everythingisimmobileandthatchangedoesnothappen.Onthecontrary,itmeansthatchangeisubiquitous.Only:elementaryprocessescannotbeorderedalongacommonsuccessionofinstants.Attheextremelysmallscaleofthequantaofspace,thedanceofnaturedoesnotdeveloptotherhythmkeptbythebatonofasingleorchestralconductor:everyprocessdancesindependentlywithitsneighbours,followingitsownrhythm.Thepassingoftimeisintrinsictotheworld,itisbornoftheworlditself,outoftherelationsbetweenquantumeventswhicharetheworldandwhichthemselvesgeneratetheirowntime.

Infact,thenonexistenceoftimedoesnotmeananythingparticularlycomplicated.Let’strytounderstand.

ThecandlechandelierandthepulseTimeappearsinmostequationsofclassicphysics.Itisthevariableindicatedbythelettert.Theequationstellushowthingschangeintime.Ifweknowwhathashappenedinthepast,theyallowustopredictthefuture.Moreprecisely,wemeasuresomevariables–forexample,thepositionAofanobject,theangleBofaswingingpendulum,thetemperatureCofanobject–andtheequationsofphysicstellushowthesevariablesA,BandCwillchangewithtime.TheypredictthefunctionsA(t),B(t),C(t),andsoon,whichdescribethechangingofthesevariablesintimet.

GalileowasthefirsttounderstandthatthemovementofobjectsonEarthcouldbedescribedbyequationsforthefunctionsoftimeA(t),B(t),C(t)–andthefirsttowriteexplicitequationsforthesefunctions.ThefirstlawofterrestrialphysicsfoundbyGalileo,forexample,describeshowanobjectfalls,thatistosay,howitsaltitudexvarieswiththepassageoftimet.fn40

Todiscoverandverifythislaw,Galileoneededtwokindsofmeasurements.Hehadtomeasuretheheightxoftheobjectandthetimet.Therefore,heneeded,inparticular,aninstrumenttomeasuretime.Heneededaclock.

WhenGalileolivedtherewerenoaccurateclocks.Galileohimself,asayoungman,discoveredakeytomakingprecisetimepieces.Hediscoveredthattheoscillationsofapendulumallhavethesameduration(irrespectiveoftheamplitude).Thus,itispossibletomeasuretimebysimplycountingtheoscillationsofapendulum.Itseemssuchanobviousidea,butittookGalileotofindit;ithadnotoccurredtoanyonebeforehim.Soitgoes,withscience.

Butthingsarenotreallythisstraightforward.Accordingtolegend,GalileoalightedontheideainPisa’smarvellouscathedralwhilewatching

theslowoscillationsofagiganticcandlechandelier,whichisstillthere.(Thelegendisfalse,sincethechandelierwasactuallyfirsthungthereyearsafterGalileo’sdeath,butitmakesforagoodstory.Perhapstherewasanotheronehangingthereatthetime.)Thescientistwasobservingtheoscillationsduringareligiousserviceinwhichhewasevidentlynotparticularlyabsorbed,andhewasmeasuringthedurationofeachoscillationofthechandelierbycountingthebeatsofhisownpulse.Withmountingexcitement,hediscoveredthatthenumberofbeatswasthesameforeachoscillation:itdidnotchangewhenthechandelierslowedandoscillatedwithdiminishedamplitude.Theoscillationsallhadthesameduration.

It’safinestorybut,onreflection,itleavesusperplexed–andthisperplexitygoestotheheartoftheproblemoftime.HowcouldGalileoknowthathisownindividualpulse-beatsalllastedforthesameamountoftime?fn41

NotmanyyearsafterGalileo,doctorsbegantomeasuretheirpatients’pulsesbyusingawatch–whichisnothing,afterall,butapendulum.Soweusethebeatstoassureourselvesthatthependulumisregular,andthenthependulumtoascertaintheregularityofthepulse-beats.Isthisnotsomewhatcircular?Whatdoesitmean?

Itmeansthatwe,inreality,nevermeasuretimeitself;wealwaysmeasurethephysicalvariablesA,B,C…(oscillations,beats,andmanyotherthings)andcompareonevariablewithanother,thatistosay,wemeasurethefunctionsA(B),B(C),C(A),andsoon.Wecancounthowmanybeatsforeachoscillation;howmanyoscillationsforeverytickofmystopwatch;howmanyticksofmystopwatchbetweenintervalsoftheclockonthebell-tower…

Thepointisthatitisusefultoimaginethatavariabletexists–the‘truetime’–whichunderpinsallthosemovements,evenifwecannotmeasureitdirectly.Wewritetheequationsforthephysicalvariableswithregardtothisunobservablet,equationswhichtellushowthingschangeint;thatis,forinstance,howmuchtimeittakesforeachoscillation,andhowlongeachheart-beatlasts.Fromthis,wecanderivehowthevariableschangeinrelationtoeachother–howmanyheartbeatsthereareinoneoscillation–andcomparethispredictionwithwhatweobserveintheworld.Ifthepredictionsarecorrect,wetrustthatthiscomplicatedschemaisasoundoneand,inparticular,thatitisusefultoemploythevariableoftimet,evenifwecannotmeasureitdirectly.

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Inotherwords,theexistenceofthevariabletimeisausefulassumption,nottheresultofanobservation.

ItwasNewtonwhounderstoodallofthis:heunderstoodthatthiswasagoodwaytoproceed,andclarifiedanddevelopedthisschema.Newtonassertsexplicitlyinhisbookthatwecan’tevermeasurethetruetimetbut,ifweassumethatitexists,wecansetupanefficientframeworktodescribenature.

Havingclarifiedthis,wecanreturntoquantumgravityandthemeaningofthestatementthat‘timedoesnotexist’.ItsimplymeansthattheNewtonianschemanolongerworkswhenwearedealingwithsmallthings.Itwasagoodone,butonlyforlargethings.

Ifwewanttounderstandtheworldwidely,ifwewanttounderstandhowitfunctionsinthelessfamiliarsituationswherequantumgravitymatters,weneedtoabandonthisschema.Theideaofatimetwhichflowsbyitself,andinrelationtowhichallthingsevolve,isnolongerausefulone.Theworldisnotdescribedbyequationsofevolutionintimet.WhatwemustdoissimplytoenumeratethevariablesA,B,C…whichweactuallyobserve,andwriteequationsexpressingrelationsbetweenthesevariables,andnothingelse:thatis,equationsfortherelationsA(B),B(C),C(A)…whichweobserve,andnotforthefunctionsA(t),B(t),C(t)…whichwedonotobserve.

Intheexampleofthepulseandthecandlechandelier,wewillnothavethepulseandthecandelabrumevolvingintime,butonlyequationswhichtellushowthetwovariablesevolvewithrespecttoeachother.Thatistosay,equationswhichtellusdirectlyhowmanypulse-beatsthereareinanoscillation,withoutmentioningt.

‘Physicswithouttime’isphysicsinwhichwespeakonlyofthepulseandthechandelier,withoutmentioningtime.

It’sasimplechange–butfromaconceptualpointofview,it’sahugeleap.Wemustlearntothinkoftheworldnotassomethingwhichchangesintimebutinsomeotherway.Thingschangeonlyinrelationtooneanother.Atafundamentallevel,thereisnotime.Oursenseofthecommonpassageoftimeisonlyanapproximationwhichisvalidforourmacroscopicscale.Itderivesfromthefactthatweperceivetheworldinacoarse-grainedfashion.

Theworlddescribedbythetheoryisthusfarfromtheonewearefamiliarwith.Thereisnolongerspacewhichcontainstheworld,andnolongertimeduringthecourseofwhicheventsoccur.Thereareelementaryprocessesinwhichthequantaofspaceandmattercontinuouslyinteractwitheachother.JustasacalmandclearAlpinelakeismadeupofarapiddanceofamyriadofminusculewatermolecules,theillusionofbeingsurroundedbycontinuousspaceandtimeistheproductofalong-sightedvisionofadenseswarmingofelementaryprocesses.

SpacetimesushiHowdothesegeneralideasapplytoquantumgravity?Howcanwedescribechangewithouttheideasofspaceasacontainer,ortimealongwhichtheworldglides?

Consideraprocess:forexample,thecollisionoftwobilliardballsonatable’sgreenbaize.Imaginearedballplayedinthedirectionofayellowone;itgetsclose,collides,andthetwoballsmoveawayindifferentdirections.Thisprocess,likeallprocesses,takesplaceinafinitezoneofspace–let’ssayonatableapproximatelytwometreswide–andlastsforafiniteintervaloftime–let’ssaythreeseconds.Todealwiththisprocessinthecontextofquantumgravity,itisnecessarytoincludespaceandtimeintheprocessitself(figure7.1).

Wemustnot,inotherwords,describeonlythetwoballs,butalsoallthatisaroundthem:thetableandanyothermaterialobjects–andthespaceinwhichtheyareimmersedduringthetimethatelapsesbetweenthestartoftheshotandtheendoftheprocess.Spaceandtimearethegravitationalfield,Einstein’s‘mollusc’:wearealsoincludingthegravitationalfield,thatistosay,apieceofthemollusc,intheprocess.EverythingisimmersedinEinstein’sgreatmollusc:here,imaginethatyouareslicingasmall,finiteportionofit,likeapieceofsushi,whichencompassesthecollisionandwhatsurroundsit.

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Figure7.1Aregionofspaceinwhichablackballhitsastationarywhiteball,propelsitandrebounds.Theboxistheregionofspacetime.Withinitaredrawnthetrajectoriesoftheballs.

Whatweobtainfromthisisaspacetimebox(asinfigure7.1):afiniteportionofspacetimeafewcubicmetresindimensionbyafewsecondsoftime.Thisprocessdoesnotoccur‘in’time.Theboxisnotinspacetime,itincludesspacetime.Itisn’taprocessintime,inthesamewayinwhichgrainsofspacearenotinspace.Thepassageoftimeisonlythemeasureoftheprocessitself,justasquantaofgravityarenotinspace,astheythemselvesconstitutespace.

Thekeytounderstandinghowquantumgravityworksliesinconsideringnotsolelythephysicalprocessgivenbythetwoballsbutrathertheentireprocessdefinedbythewholeboxwithallthatitentails,includingthegravitationalfield.

NowletusreturntoHeisenberg’soriginalinsight:quantummechanicsdoesnottelluswhathappensduringthecourseofaprocess,buttheprobabilitywhichtiestogetherthedifferentinitialandfinalstatesoftheprocess.Inourcase,theinitialandfinalstatesaregivenbyallthathappensattheborderofthespacetimebox.

Whattheequationsofloopquantumgravitygiveusistheprobabilityassociatedwithagivenpossibleboundaryofthebox–theprobabilitythattheballswillcomeoutoftheboxinoneparticularconfigurationoranother,iftheyhaveentereditinanother.

Howisthisprobabilitycomputed?RecallFeynman’ssumoverpaths,whichIdescribedwhenspeakingaboutquantummechanics.Probabilities,inquantumgravity,canbecalculatedinthesameway.Byconsideringallthepossible‘trajectories’thathavethesameboundary.Sinceweareincludingthedynamicsofspacetime,thismeansconsideringallpossiblespacetimeswhichhavethesameboundaryasthebox.

Quantummechanicsassumesthatbetweentheinitialboundary,wherethetwoballsenter,andthefinalboundarywheretheyexit,thereisnodefinitespacetimenordefinitetrajectoryoftheballs.Thereisaquantum‘cloud’inwhichallthepossiblespacetimesandallpossibletrajectoriesexisttogether.Theprobabilityofseeingtheballsgoingoutinonewayoranothercanbecomputedbysummingoverallpossiblespacetimes.

SpinfoamIfquantumspacehasthestructureofaspinnetwork,whatstructurewillspacetimehave?Whatwilloneofthespacetimespreviouslyalludedtointhecalculationbelike?

Itmustbea‘history’ofaspinnetwork.Imaginethatyoutakethegraphofthespinnetworkandmoveit:everynodeinthewebdrawsaline,liketheballsinfigure7.1,andeverylineofthegraph,moving,drawsasurface(forexample,amovingsegmentdrawsarectangle).Butthereismore:anodecanopenupintotwoormorenodes,justasaparticlecansplitintotwoormoreparticles.Conversely,twoormorenodescancombineintoasingleone.Inthisway,agraphwhichevolvesdrawsanimageliketheoneinfigure7.2.

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Figure7.2Anevolvingspin-network:threenodescombineintoasinglenode,andthenseparateagain.Ontheright,thespinfoamrepresentingthisprocess.

Theimageportrayedontherightoffigure7.2isa‘spinfoam’.‘Foam’becauseitismadeofsurfaceswhichmeetonlines,whichinturnmeetonvertices,resemblingafoamofsoapbubbles(figure7.3).‘Spinfoam’becausethefacesofthefoamcarryspins,asdothelinksofthegraphswhoseevolutiontheydescribe.

Figure7.3Thefoamofsoapbubbles.

Tocomputetheprobabilityofaprocess,onemustsumupoverallthepossiblespinfoamswithintheboxwhichhavethesameboundaryasthatprocess.Theboundaryofaspinfoamisaspinnetworkandthematteronit.

Theequationsofloopquantumgravityexpresstheprobabilityofaprocessintermsofsumsoverspinfoamswithgivenboundaries.Inthiswayitispossibletocompute,inprinciple,theprobabilityofanyphysicalevent.fn42

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Figure7.4Avertexofspinfoam.CourtesyofGregEgan.

Atfirstsight,thiswayformakingcalculationsinquantumgravity,basedonspinfoams,seemsverydifferentfromtheusualwaysinwhichthingsarecomputedintheoreticalphysics.Thereisnogivenspace,nogiventime,andspinfoamsseemobjectsquiteremotefrom,say,theparticlesofthestandardmodel.Butinfacttherearestrongsimilaritiesbetweenthespinfoamtechniqueandthecalculationtechniquesusedinthestandardmodel.Infact,evenmorethanthis,thespinfoamtechniqueisactuallyabeautifulmergingofthetwomaincalculationtechniquesusedinthecontextofthestandardmodel:Feynmandiagramsandthelatticeapproximation.

Feynmandiagramsareused,forinstance,tocomputeprocessesdominatedbyelectromagneticorweakforces.AFeynmandiagramrepresentsasequenceofelementaryinteractionsamongparticles.Anexampleisinfigure7.5,whichrepresentstwoparticles,ortwoquantaofthefield,interacting.Theparticleontheleftsplitsintotwoparticles,oneofwhichsplitsinturnintotwoparticles,whichthenreunite,convergingwiththeparticleontheright.Thegraphportraysahistoryofthefield’squanta.

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Figure7.5AFeynmandiagram.

Thelatticeapproximationisusedwhentheforcesarestrongandtheparticlepictureisnolongereffectivefordescribingphysics,forinstanceincomputingthestrongforcesbetweenquarksinsidethenucleusofanatom.Thelatticetechniqueentailsapproximatingacontinuousphysicalspacebymeansofalattice,oragrid,asinfigure7.6.Thisgridisnotassumedtobeafaithfuldescriptionofspace,butonlyanapproximation,aswhenengineerscalculatetheresistanceofabridgebyapproximatingtheconcretewithafinitenumberofelements.Thesetwomethodsofmakingcalculations–Feynmandiagramsandthelattice–arethetwomostefficienttechniquesofquantumfieldtheory.

Figure7.6Agridapproximatingphysicalspacetime.

Inquantumgravity,somethingbeautifuloccurs:thetwomethodsofmakingcalculationsbecomeoneandthesame.Thespacetimefoamrepresentedinfigure7.2,usedtocomputeaphysicalprocessinquantumgravity,maybeinterpretedeitherasaFeynmandiagramorasalatticecalculation.fn43Therefore,thetwocalculationtechniquesusedforthestandardmodelturnouttobeparticularcasesofacommontechnique:summingoverthespinfoamsofquantumgravity.

Earlier,IsetoutEinstein’sequations.Again,Ican’tresistincludingherethecompletecollectionoftheequationsoflooptheory,evenifthereaderwillobviouslynotbeabletodecipherthem–not

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beforeundertakingthestudyofagooddealofmathematics.Someoneonceclaimedthatatheoryisn’tcredibleifitsequationscannotbesummarizedonaT-shirt.HereisthatT-shirtforloopquantumgravity(figure7.7).

Figure7.7Theequationsofloopquantumgravity,summarizedonaT-shirt.

Theseequationsfn44arethemathematicalversionofthepictureoftheworldIhavegiveninthelasttwochapters.Wearenotatallsureiftheyarethecorrectequations–but,inmyopinion,theyarethebestaccountofquantumgravitywehaveatpresent.

Spaceisaspinnetworkwhosenodesrepresentitselementarygrains,andwhoselinksdescribetheirproximityrelations.Spacetimeisgeneratedbyprocessesinwhichthesespinnetworkstransformintooneanother,andtheseprocessesaredescribedbysumsoverspinfoams.Aspinfoamrepresentsahistoryofaspinnetwork,henceagranularspacetimewherethenodesofthegraphcombineandseparate.

Thismicroscopicswarmingofquanta,whichgeneratesspaceandtime,underliesthecalmappearanceofthemacroscopicrealitysurroundingus.Everycubiccentimetreofspace,andeverysecondthatpasses,istheresultofthisdancingfoamofextremelysmallquanta.

Whatistheworldmadeof?Thebackdropofspacehasdisappeared,timehasdisappeared,classicparticleshavedisappeared,alongwiththeclassicfields.Sowhatistheworldmadeof?

Theanswernowissimple:theparticlesarequantaofquantumfields;lightisformedbyquantaofafield;spaceisnothingmorethanafield,whichisalsomadeofquanta;andtimeemergesfromtheprocessesofthissamefield.Inotherwords,theworldismadeentirelyfromquantumfields(figure7.8).

Thesefieldsdonotliveinspacetime;theylive,sotospeak,oneontopoftheother:fieldsonfields.Thespaceandtimethatweperceiveinlargescaleareourblurredandapproximateimageofoneofthesequantumfields:thegravitationalfield.

Fieldsthatliveonthemselves,withouttheneedofaspacetimetoserveasasubstratum,asasupport,andwhicharecapablebythemselvesofgeneratingspacetime,arecalled‘covariantquantumfields’.Thesubstanceofwhichtheworldismadehasbeenradicallysimplifiedinrecentyears.Theworld,particles,light,energy,spaceandtime–allofthisisnothingbutthemanifestationofasingletypeofentity:covariantquantumfields.

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Figure7.8Whatistheworldmadeof?Ofonlyoneingredient:covariantquantumfields.

Covariantquantumfieldshavebecometodaythebestdescriptionthatwehaveoftheἄπειρον,theapeiron,theprimalsubstanceofwhicheverythingisformedhypothesizedbythemanthatcouldperhapsbecalledthefirstscientistandthefirstphilosopher,Anaximander.fn45

TheseparationbetweenthecurvedandcontinuousspaceofEinstein’sgeneralrelativityandthediscretequantaofquantummechanicswhichdwellinaflatanduniformspacehasdissolved.Theapparentcontradictionisnolongerthere.Betweenthespacetimecontinuumandquantaofspace,thereisthesamerelationshipasbetweenelectromagneticwavesandphotons.Thewavesgiveanapproximatelarge-scalevisionofphotons.Photonsarethewayinwhichwavesinteract.Continuousspaceandtimeareanapproximatelarge-scalevisionofthedynamicofquantaofgravity.Thequantaofgravityarethewayinwhichspaceandtimeinteract.Thesamemathematicscoherentlydescribesthequantumgravitationalfieldasotherquantumfields.

Theconceptualpricepaidistherelinquishingoftheideaofspace,andoftime,asgeneralstructureswithinwhichtoframetheworld.Spaceandtimeareapproximationswhichemergeatalargescale.Kantwasperhapsrightwhenheaffirmedthatthesubjectofknowledgeanditsobjectareinseparable,buthewasdefinitelymistakenwhenheconsideredNewtonianspaceandtimeasaprioriformsofknowledge,partsofanindispensablegrammarforunderstandingtheworld.Thisgrammarhasevolved,andisstillintheprocessofevolving,withthegrowthofourknowledge.

Generalrelativityandquantummechanicsare,intheend,notasincompatibleastheyseemed.Oncloserinspection,theyshakehandsandengageinabeautifuldialogue.ThespatialrelationsthatweaveEinstein’scurvedspacearetheveryinteractionsweavingtherelationsbetweenthesystemsofquantummechanics.Thetwobecomecompatibleandconjoined,twosidesofthesamecoin,assoonasitisrecognizedthatspaceandtimeareaspectsofaquantumfield,andquantumfieldscanexistevenwithoutbeinggroundedinanexternalspace.

Thisrarefiedpictureofthefundamentalstructureofthephysicalworldisthevisionofrealityofferedtodaybyquantumgravity.

Themainrewardofthiskindofphysicsisthat,asweshallseeinthenextchapter,infinitydisappears.Theinfinitelysmallnolongerexists.Theinfinitieswhichplagueconventionalquantumfieldtheory,predicatedonthenotionofacontinuousspace,nowvanish,becausetheyweregeneratedpreciselybytheassumption,physicallyincorrect,ofthecontinuityofspace.ThesingularitieswhichrenderEinstein’sequationsabsurdwhenthegravitationalfieldbecomestoostrongalsodisappear:theyareonlytheresultofneglectingthequantizationofthefield.Littleby

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little,thepiecesofthepuzzlefindtheirplace.Inthefinalsectionsofthisbook,Idescribesomeofthephysicalconsequencesofthistheory.

Itmayappearstrangeanddifficulttothinkofdiscreteelementaryentitiesnotinspaceandtime,butweavingspaceandtimewiththeirrelations.ButhowstrangeitmusthaveseemedtolistentoAnaximander,whenheclaimedthatbeneathourfeettherewasonlythesameskythatwecanseeaboveourheads?OrtoAristarchus,whenhetriedtomeasurethedistancefromtheEarthtotheMoonandtheSun,discoveringthattheyareextremelydistant,andarethereforenotthesizeoflittleballs,butgigantic–andtheSunisimmensecomparedtotheEarth.OrtoHubble,whenherealizedthatthesmall,diaphanouscloudsbetweenstarsarevastseasofimmenselydistantstars…

Forcenturies,theworldhascontinuedtochangeandexpandaroundus.Weseefurther,understanditbetterandareastonishedbyitsvariety,bythelimitationsoftheimageswehadofit.Thedescriptionwemanagetoproducetoaccountforitbecomesincreasinglyrarefied,yetsimple.

Weareakintosmall,blindmolesundergroundwhoknowlittleornothingabouttheworld.Butwecontinuetolearn…ButallthestoryofthenighttoldoverAndalltheirmindstransfiguredsotogetherMorewitnesseththanfancy’simagesAndgrowstosomethingofgreatconstancy;But,howsoever,strangeandadmirable.2

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PartFour

BEYOND SPACE AND TIME

Ihaveillustratedthebasisofquantumgravity,andtheimageoftheworldwhichemergesfromit.InthefinalchaptersIdescribesomeconsequencesofthetheory:whatthetheorytellsusaboutphenomenasuchastheBigBangandblackholes.Ialsodiscussthecurrentstateofpossibleexperimentstotestthetheory,andwhatitseemstomenatureistellingus–inparticularwiththefailureoftheexpectedobservationofsupersymmetricparticles.

Iconcludewithafewreflectionsonwhatisstillmissingfromourunderstandingoftheworld:especiallythermodynamics,theroleofinformationinatheorywithouttimeandspacesuchasquantumgravity,andthere-emergenceoftime.

Allofthistakesustotheedgeofwhatweknow,tothevantagepointfromwhichwelookuponwhatwedefinitelydon’tknow,theimmensemysterythatsurroundsus.

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8.BeyondtheBigBang

ThemasterIn1927ayoungBelgianscientist,aJesuit-educatedCatholicpriest,studiesEinstein’sequationsandrealizes–justasEinsteinhad–thattheypredicttheuniversemustexpandorcontract.Butinsteadoffoolishlyrejectingtheresultandstubbornlytryingtoavoidit,asEinsteindid,theBelgianpriestbelievesitandlooksforastronomicaldatatotestit.Atthetime,galaxieswerenotcalledgalaxies.Theywerecallednebulaebecause,seenthrougha

telescope,theylookedlikesmall,opalescentcloudsamongthestars.Itwasnotyetknownthattheyaredistant,immenseislandsofstarslikeourveryowngalaxy.ButtheyoungBelgianpriestunderstandsthatthescarceavailabledataonthegalaxieswereindeedcompatiblewiththepossibilitythattheuniverseisexpanding:nearbygalaxiesaremovingawayatgreatspeed,asiftheyhadbeenlaunchedintothesky;distantgalaxiesaremovingawayatevengreaterspeed.Theuniverseisswellinglikeaballoon.Twoyearslater,theinsightisconfirmed,thankstotwoAmericanastronomers,HenriettaLeavitt

andEdwinHubble.Leavittdiscoversagoodtechniqueformeasuringthedistanceofthenebulae,confirmingthattheyareveryfaraway,outsideofourowngalaxy.UsingthistechniqueandthegreattelescopeofthePalomarObservatory,Hubblecollectsprecisedatathatconfirmthatthegalaxiesaremovingaway,ataspeedproportionaltotheirdistance.

Figure8.1HenriettaLeavitt.

ButitistheyoungBelgianpriestwhounderstands,alreadyin1927,thecrucialconsequence:ifweseeastoneflyingup,itmeansthatthestonewaspreviouslylowerdownandsomethinghasthrownitupwards.Ifweseethegalaxiesmovingawayandtheuniverseexpanding,itmeansthatthegalaxieswerepreviouslymuchcloserandtheuniversewassmaller:andsomethingcausedittostartexpanding.TheyoungBelgianpriestsuggeststhattheuniversewasoriginallyextremelysmallandcompressed,andstarteditsexpansioninagiganticexplosion.Hecallsthisinitialstatetheprimordialatom.TodayitisknownastheBigBang.HisnamewasGeorgesLemaître.InFrench,thisnamesoundslikelemaîtremeaning‘themaster’,

andfewnamesaremoreappropriateforthemanwhofirstunderstoodtheexistenceoftheBigBang.Butinspiteofthisname,Lemaître’scharacterwasreserved;heavoidedpolemics,andneverevenclaimedpriorityforthediscoveryoftheexpansionoftheuniverse,whichendedupbeingattributedtoHubble.Twoepisodesfromhislifeillustratehisprofoundintelligence.ThefirstinvolvesEinstein,thesecondapope.Asmentioned,Einsteinwasscepticalabouttheexpansionoftheuniverse.Hehadgrownup

thinkingthattheuniverseisfixed,andhadnotbeenabletoaccepttheideathatthiswasnotthecase.Eventhegreatestmakemistakesandarepreytopreconceivedideas.LemaîtremetEinstein

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andtriedtodissuadehimfromhisprejudicialview.Einsteinresisted,goingsofarastoanswerLemaître:‘Correctcalculations,abominablephysics.’Later,EinsteinwasobligedtorecognizethatLemaîtrewastheonewhowasactuallyright.Itdoesn’tfalltoeveryonetodisproveEinstein.

Figure8.2GeorgesLemaître.CopyrightArchivesGeorgesLemaître,Louvain.

Thesamethinghappenedagain.Einsteinhadintroducedthecosmologicalconstant,thesmallbutimportantmodificationofhisequationsIdescribedinChapter3,inthe(mistaken)hopeofrenderingtheequationscompatiblewithastaticuniverse.Whenhehadtoacknowledgethattheuniverseisnotstatic,heturnedagainstthecosmologicalconstant.Lemaître,forthesecondtime,triedtopersuadehimtochangehismind:thecosmologicalconstantdoesnotrendertheuniversestatic,butitisneverthelessright,andthereisnoreasontotakeitout.Onthisoccasion,too,Lemaîtrewascorrect:thecosmologicalconstantproducesanaccelerationoftheexpansionoftheuniverse,andthisaccelerationhasrecentlybeenmeasured.Onceagain,EinsteinwaswrongandLemaîtrewasright.WhentheideathattheuniversehademergedfromaBigBangbegantobeaccepted,PopePius

XIIdeclaredinapublicaddress(on22November1951)thatthetheoryconfirmedtheaccountofCreationgiveninGenesis.1Lemaîtrereactedtothispapalpositionwithgreatconcern.HegotintouchwiththescientificadvisortothepontiffandwenttogreatlengthstopersuadethePopetorefrainfrommakingreferencestolinksbetweendivinecreationandtheBigBang.Lemaîtrewasconvincedthatitwasfoolishtomixscienceandreligioninthisway:theBibleknowsnothingaboutphysics,andphysicsknowsnothingaboutGod.2PiusXIIallowedhimselftobepersuaded,andtheCatholicchurchneveragainmadepublicallusiontothesubject.ItisnotgiventoeveryonetodisprovethePope.Andofcourse,onthisalso,itwasLemaîtrewhowasright:todaythereisagreatdealoftalk

concerningthepossibilitythattheBigBangisnotarealbeginning,thattherecouldhavebeenanotheruniversebeforeit.ImagineinwhatanembarrassingpositiontheCatholicChurchwouldfinditselftoday,ifLemaîtrehadnotpreventedthePopefrommakingitofficialdoctrinethatthatBigBangandCreationwerethesamething.Fiatluxwouldhavetobechangedto‘Switchthelightbackon!’TocontestbothEinsteinandthePope,convincingboththattheyweremistaken,andtoberightin

bothcases,issurelysomethingofaresult.‘Themaster’liveduptohisname.Todayconfirmationsareoverwhelming:theuniverse,inafar-distantpast,wasextremelyhotand

extremelycompact,andhasexpandedsince.Wecanreconstructindetailthehistoryoftheuniverse,startingwithitsinitialhot,compressedstate.Weknowhowatoms,elements,galaxiesandstarsformedandhowtheuniverseasweseeittodaydeveloped.RecentextendedobservationsoftheradiationthatfillstheuniversecarriedoutmainlybythePlancksatelliteonceagainconfirmedinfullthetheoryoftheBigBang.Weknowwithareasonabledegreeofcertaintywhathappenedonalargescaletoouruniverseinthelast14billionyears,fromthetimewhenitwasaballoffire.Andtothinkthat,initially,thephrase‘theoryoftheBigBang’wascoinedbyopponentsofthe

theory,tomockanideathatseemedoutlandish…Instead,intheend,wewereallpersuaded:14billionyearsagotheuniversewasacompressedballoffire.Butwhathappenedbeforethisinitialhotandcompressedstate?Regressingintime,temperatureincreases,asdoesthedensityofmatterandenergy.Thereisa

pointatwhichtheyreachthePlanckscale:14billionyearsago.Atthatpoint,theequationsof

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generalrelativityarenolongervalid,becauseitisnolongerpossibletoignorequantummechanics.Weenterintotherealmofquantumgravity.

QuantumcosmologyTounderstandwhathappened14billionyearsago,therefore,quantumgravityisrequired.Whatdotheloopstellusaboutthesubject?Considerananalogousbutsimplersituation.Accordingtoclassicalmechanics,anelectronfalling

straightintoanatomicnucleuswouldbeswallowedbythenucleusanddisappear.Butthisisnotwhathappensinreality.Classicmechanicsisincomplete,anditisnecessarytotakequantumeffectsintoaccount.Arealelectronisaquantumobjectanddoesnotfollowaprecisetrajectory:itisn’tpossibletokeepitinsidetoosmallaregion.Themoreitisconcentrated,themoreitslipsrapidlyaway.Ifwewanttostopitaroundthenucleus,themostwecandoistoforceitintoanorbitofthesizeofthesmallestatomicorbital:itcouldnotstayanyclosertothenucleus.Quantummechanicspreventsarealelectronfromfallingintoanucleus.Aquantumrepulsionpushesawaytheelectronwhenitgetstooclosetothecentre.Thus,thankstoquantummechanics,matterisstable.Withoutit,electronswouldfallintonuclei,therewouldbenoatomsandwewouldnotexist.Thesameappliestotheuniverse.Letusimagineauniversecontractingandbecomingextremely

small,squashedbyitsownweight.AccordingtoEinstein’sequations,thisuniversewouldbesquashedadinfinitumandatacertainpointwoulddisappearaltogether,liketheelectronfallingintothenucleus.ThisistheBigBangpredictedbyEinstein’sequations,ifweignorequantumtheory.Butifwetakequantummechanicsintoaccount,theuniversecannotbeindefinitelysquashed.A

quantumrepulsionmakesitrebound.Acontractinguniversedoesnotcollapsedowntoapoint:itbouncesbackandbeginstoexpand,asifitwereemergingfromacosmicexplosion(figure8.3).

Figure8.3TheBigBounceoftheuniverseinagraphicalrepresentationbyFrancescaVidotto,theItalianscientistwhofirstusedspinfoamstocomputetheprobabilityofthisprocess.

Thepastofouruniversemaythereforewellbetheresultofjustsucharebound.AgiganticreboundknownasaBigBounceinsteadofBigBang.Thisiswhatseemstoemergefromtheequationsofloopquantumgravitywhentheyareappliedtotheexpansionoftheuniverse.Theimageofthebouncemustnotbetakenliterally.Goingbacktotheexampleoftheelectron,

recallthatifwewanttoplaceanelectronascloseaspossibletoanatom,theelectronisnolongeraparticle;wecanthinkofit,instead,asopenedupinacloudofprobabilities.Anexactpositionnolongermakessensefortheelectron.Thesamefortheuniverse:inthecrucialpassagethroughtheBigBounce,wecannolongerthinkofasingle,althoughgranular,spaceandtime,butonlyofaspread-outcloudofprobabilitiesinwhichtimeandspacewildlyfluctuate.AttheBigBounce,theworldisdissolvedintoaswarmingcloudofprobabilities,whichtheequationsstillmanagetodescribe.Ouruniversecouldthusbetheresultofthecollapseofapreviouscontractinguniversepassing

acrossaquantumphase,wherespaceandtimearedissolvedintoprobabilities.Theword‘universe’becomesambiguous.If,by‘universe’,wemean‘allthatthereis’,then,by

definition,therecannotbeaseconduniverse.Buttheword‘universe’hasassumedanothermeaningincosmology:itreferstothespacetimecontinuumthatweseedirectlyaroundus,filledwithgalaxiesthegeometryandhistoryofwhichweobserve.Thereisnoreasontobecertainthat,inthis

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sense,thisuniverseistheonlyoneinexistence.Wecanreconstructthepastuptothetimewhen,asintheimagebyJohnWheeler,thespatiotemporalcontinuumbreaksuplikeseafoamandfragmentsintoaquantumcloudofprobabilities,andthereisnoreasontodiscardthepossibilitythatbeyondthishotfoamtherecouldnotbeanotherspatiotemporalcontinuum,similartotheonewhichweperceivearoundus.TheprobabilityforauniversetocrossthephaseoftheBigBounce,passingfromcontractionto

expansion,canbecomputedusingthetechniquesdescribedintheprecedingchapter:thespacetimeboxes.Calculationsaremadeusingspinfoamsthatconnectthecontractinguniversewiththeexpandingone.Allofthisisstillatanexploratorystage,butwhatisremarkableinthisstoryisthattodaywehave

equationswithwhichtotrytodescribetheseevents.Wearebeginningtocastthefirstfewcautiousglances,forthemomentonlytheoretically,beyondtheBigBang.

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9.EmpiricalConfirmations?

TheappealofquantumcosmologygoesbeyondthefascinatingtheoreticalexplorationsofwhattheremightbebeyondtheBigBang.Thereisanotherreasonforstudyingtheapplicationofthetheorytocosmology:itmightprovidetheopportunitytofindoutwhetherornotthetheoryisactuallycorrect.Scienceworksbecause,afterhypothesesandreasoning,afterintuitionsandvisions,after

equationsandcalculations,wecancheckwhetherwehavedonewellornot:thetheorygivespredictionsaboutthingswehavenotyetobserved,andwecancheckwhetherthesearecorrect,ornot.Thisisthepowerofscience,thatwhichgroundsitsreliabilityandallowsustotrustinitwithconfidence:wecancheckwhetheratheoryisrightorwrong.Thisiswhatdistinguishessciencefromotherkindsofthinking,wheredecidingwhoisrightandwhoiswrongisusuallyamuchthornierquestion,sometimesevendevoidofmeaning.WhenLemaîtredefendstheideathattheuniverseisexpanding,andEinsteindoesnotbelieveit,

oneofthetwoiswrong;theotherright.AllofEinstein’sresults,hisfame,hisinfluenceonthescientificworld,hisimmenseauthority,countfornothing.Theobservationsprovehimwrong,andit’sgameover.AnobscureBelgianpriestisright.Itisforthisreasonthatscientificthinkinghaspower.Thesociologyofsciencehasshedlightonthecomplexityoftheprocessofscientific

understanding;likeanyotherhumanendeavour,thisprocessisbesetbyirrationality,intersectswiththegameofpowerandisaffectedbyeverysortofsocialandculturalinfluence.Nevertheless,despiteallofthis,andinoppositiontotheexaggerationsofafewpostmodernists,culturalrelativistsandthelike,noneofthisdiminishesthepracticalandtheoreticalefficacyofscientificthinking.Becauseintheend,inthemajorityofcases,itispossibletoestablishwithclaritywhoisrightandwhoiswrong.AndeventhegreatEinsteincouldgoontosay(andhedidso),‘Ah…Imadeamistake!’Scienceisthebeststrategyifwevaluereliability.Thisdoesnotmeanthatscienceisjusttheartofmakingmeasurablepredictions.Some

philosophersofscienceoverlycircumscribesciencebylimitingittoitsnumericalpredictions.Theymissthepoint,becausetheyconfusetheinstrumentswiththeobjectives.Verifiablequantitativepredictionsareinstrumentstovalidatehypotheses.Theobjectiveofscientificresearchisnotjusttoarriveatpredictions:itistounderstandhowtheworldfunctions;toconstructanddevelopanimageoftheworld,aconceptualstructuretoenableustothinkaboutit.Beforebeingtechnical,scienceisvisionary.Theverifiablepredictionsarethesharpenedtoolwhichallowsustofindoutwhenwehave

misunderstoodsomething.Atheorylackingempiricalconfirmationisatheorywhichhasnotyetpasseditsexams.Examsneverend,andatheoryisnotcompletelyconfirmedbyone,twoorthreeexperiments.Butitprogressivelyacquirescredibility,stagebystage,asitspredictionsarerevealedtobecorrect.Theoriessuchasgeneralrelativityandquantummechanics,whichinitiallyleftmanyperplexed,earnedtheircredibilitygradually,asalloftheirpredictions–eventhemostbizarre–weregraduallyconfirmedbyexperimentsandobservations.Theimportanceofexperimentalproof,ontheotherhand,doesnotmeanthat,withoutnew

experimentaldata,wecannotmakeadvances.Itisoftensaidthatsciencetakesstepsforwardonlywhenthereisnewexperimentaldata.Ifthisweretrue,wewouldhavelittlehopeoffindingthetheoryofquantumgravitybeforemeasuringsomethingnew,butthisispatentlynotthecase.WhichnewdatawereavailabletoCopernicus?None.HehadthesamedataasPtolemy.WhichnewdatadidNewtonhave?Almostnone.HisrealingredientswereKepler’slawsandGalileo’sresults.WhatnewdatadidEinsteinhavetodiscovergeneralrelativity?None.HisingredientswerespecialrelativityandNewton’stheory.Itsimplyisn’ttruethatphysicsadvancesonlywhenitisaffordednewdata.WhatCopernicus,Newton,Einsteinandmanyothersdidwastobuilduponpre-existingtheories

whichsynthesizedempiricalknowledgeacrossvastfieldsofnature,andtofindawayofcombiningandrethinkingthemtoimprovethegeneralpicture.Thisisthebasisonwhichthebestresearchonquantumgravityoperates.Theoriginof

knowledge,asalwaysinscience,isultimatelyempirical.Butthedataonwhichquantumgravityisbuiltisnotnewexperiments:itisthetheoreticaledificeswhichhavealreadystructuredourknowledgeoftheworld,informswhichareonlypartlycoherent.The‘experimentaldata’forquantumgravityaregeneralrelativityandquantummechanics.Buildingonthese,tryingto

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understandhowaworldinwhichbothquantaandcurvedspaceexistmaybemadecoherent,weattempttolooktowardstheunknown.Theenormoussuccessofthegiantswhohaveprecededusinsimilarsituations,suchasNewton,

EinsteinandDirac,givesusencouragement.Wedonotpresumetobeoftheirstature.Butwehavetheadvantageofsittingontheirshoulders,andthisallowsustolookfurtherthantheydid.Onewayoranother,wecannotbuttry.Wemustdistinguishbetweencluesandstrongevidence.CluesarewhatsetSherlockHolmeson

therighttrack,allowinghimtosolveamysteriouscase.Strongevidenceiswhatthejudgeneedstosentencetheguilty.Cluesputusontherightpathtowardsacorrecttheory.Strongevidenceisthatwhichsubsequentlyallowsustotrustwhetherthetheorywehavebuiltisagoodoneornot.Withoutclues,wesearchinthewrongdirections.Withoutevidence,atheoryisnotreliable.Thesameappliestoquantumgravity.Thetheoryisinitsinfancy.Itstheoreticalapparatusis

gainingsolidity,andthefundamentalideasarebeingclarified:thecluesaregood,andconcrete–confirmedpredictionsarestillmissing.Thetheoryhasnotyettakenitsexams.

SignalsfromnatureThemoststudiedalternativetotheresearchdirectionrecountedinthisbookisstringtheory.Themajorityofphysicistswhohaveworkedonstringtheory,orstring-relatedtheories,expectedthatassoonasthenewparticleacceleratoratCERNinGenevabegantofunction(theLHCorLargeHadronCollider),particlesofanewkindneverbeforeobserved,butanticipatedbythetheory,wouldimmediatelybecomeevident:supersymmetricparticles.Stringtheoryneedstheseparticlestobeconsistent:thatiswhythestringtheoristseagerlyexpectedthemtobefound.Loopquantumgravity,ontheotherhand,iswelldefinedevenwithoutsupersymmetricparticles.Thelooptheoristswereinclinedtothinkthattheseparticlesmightnotexist.Thesupersymmetricparticleswerenotobserved,tothegreatdisappointmentofmany.The

fanfarethatgreetedthediscoveryoftheHiggsbosonin2013alsomaskedthisdisappointment.Thesupersymmetricparticlesarenotthereattheenergywheremanystringtheoristsexpectedthemtobe.Thisisnotadefinitiveproofofanything–farfromit;butnaturehasgivenasmallclueinfavouroftheloops.Therehavebeenthreemajorexperimentalresultsinfundamentalphysicsinrecentyears.The

firstistherevelationoftheHiggsbosonatCERNinGeneva(figure9.1).ThesecondisthemeasurementsmadebythePlancksatellite(figure9.2),measurements,thedataofwhichwerealsomadepublicin2013,confirmingthestandardcosmologicalmodel.Thethirdisthefirstdetectionofgravitationalwavesannouncedinthefirstmonthsof2016.Thesearethethreesignalsthatnaturehasrecentlygivenus.

Figure9.1AneventatCERNwhichshowstheformationoftheHiggsparticle.

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Figure9.2ThePlancksatellite.

Thereissomethingincommonbetweenthesethreeresults:thecompleteabsenceofsurprise.Thisdoesnotdiminishtheirimportance:ifanything,itmakesthemevenmoremeaningful.ThediscoveryoftheHiggsbosonisarock-hardconfirmationofthevalidityoftheideasbehindthestandardmodelofelementaryparticles,basedonquantummechanics.Itistheverificationofapredictionmadethirtyyearspreviously.ThePlanckmeasurementsareasolidconfirmationofthestandardcosmologicalmodel,basedongeneralrelativitywiththecosmologicalconstant.Thedetectionofgravitationalwavesisaspectacularconfirmationofgeneralrelativity,atheoryahundredyearsold.Thethreeresults,obtainedwithstrenuoustechnologicaleffortsandextensivecollaborationsbetweenhundredsofscientists,donothingotherthanreinforcetheunderstandingthatwealreadyhadofthestructureoftheuniverse.Norealsurprises.Butsuchanabsenceofsurpriseswasinasenseitselfsurprising,becausemanyexpectedtobe

surprised,thatis,tosee‘newphysics’,notyetdescribedbyestablishedtheories.TheyexpectedsupersymmetryatCERN,nottheHiggsboson.AndmanyexpectedthatPlanckwouldmeasurediscrepanciesfromthestandardcosmologicalmodel,discrepanciesthatwouldsupportalternativecosmologicaltheoriestogeneralrelativity.Butno.Whatnatureisconfirmingissimple:generalrelativity,quantummechanicsand,within

quantummechanics,thestandardmodel.Manytheoreticalphysicistsaretodaylookingfornewtheoriesbypickingarbitraryhypotheses.

‘Letusimaginethat…’Idon’tthinkthatthiswayofdoingsciencehaseverproducedgoodresults.Ourfantasyistoolimitedto‘imagine’howtheworldmaybemade,unlesswesearchforinspirationinthetraceswehaveatourdisposal.Thetracesthatwehave–ourclues–areeitherthetheorieswhichhavebeensuccessful,ornewexperimentaldata,nothingelse.Itisinthisdataandinthesetheoriesthatwemusttrytouncoverwhatwehavebeenunableyettoimagine.ThisishowCopernicus,Newton,MaxwellandEinsteinproceeded.Theynevertriedto‘guess’anewtheory–unlike,inmyopinion,thewayinwhichtoomanytheoreticalphysicistsaretryingtodotoday.ThethreerecentexperimentalresultsImentionedspeakwiththevoiceofNatureitself:‘Stop

dreamingofnewfieldsandstrangeparticles;supplementarydimensions,othersymmetries,paralleluniverses,strings,andwhateverelse.Thepiecesofthepuzzlearesimpler:generalrelativity,quantummechanicsandthestandardmodel.Thenextstepforwardmaybe“only”aquestionofcombiningtheminthecorrectmanner.’It’sreassuringadvicefortheloopquantumgravitycommunity,becausethesearethehypothesesofthetheory:generalrelativity,quantummechanicsandcompatibilitywiththestandardmodel,nothingelse.Theradicalconceptualconsequences–thequantaofspace,thedisappearanceoftime–arenotboldhypotheses:theyaretherationalconsequencesthatfollowfromtakingthebasicinsightsofourbesttheoriesseriously.Onceagain,thesearenotdefinitiveproofs.Supersymmetricparticlesmightfinallyexist,perhaps,

atascalestillnotreached,andcouldexisteveniflooptheoryiscorrect.Supersymmetryfailedto

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showupwhereexpected,andstringtheoristsarealittledowncast,looptheoristsarebuoyant,butitisstillamatterofclues;thereisnostrongevidenceatall.Tofindmoreconcreteconfirmationofthetheory,weneedtolookelsewhere.Theprimordial

universecouldopenthewindowtopredictionscapableofconfirmingthetheory.Inanottoodistantfuture,wehope.Ortheycouldprovethetheorywrong.

AwindowontoquantumgravityIfwehavetheequationsthatdescribethetransitionoftheuniverseacrossthequantumphase,wecancomputeeffectsofquantumphenomenaupontheuniversewhichweobservetoday.Theuniverseisfilledwithcosmicradiation:aseaofphotonsremainedinthecosmossincetheearlyhotphase,theresidualglareoftheearlyhightemperature.Theelectromagneticfieldintheimmensespacebetweengalaxiestrembleslikethesurfaceofthe

seaafterabigstorm.Thisquivering,disseminatedthroughouttheuniverse,iscalledthecosmicbackgroundradiation.IthasbeenstudiedinthepastfewyearsbysatellitessuchasCOBE,WMAPand,mostrecently,Planck.Animageoftheminutefluctuationsofthisradiationisgiveninfigure9.3.Thedetailsofthestructureofthisradiationtellusthehistoryoftheuniverseand,hiddeninthefoldsofthesedetails,therecouldbefootprintsofthequantumbeginningofouruniverse.Oneofthemostactivesectorsofresearchinloopquantumgravityisstudyinghowthequantum

dynamicoftheprimordialuniverseisreflectedinthisdata.Theresultsarepreliminary,butencouraging.Withmorecalculationsandmoreprecisemeasurements,itshouldbepossibletoarriveatatestofthetheory.In2013AbhayAshtekar,IvanAgulloandWilliamNelsonpublishedanarticleinwhichthey

calculatethat,undercertainhypotheses,thestatisticaldistributionofthefluctuationsofthissourceofcosmicradiationshouldrevealtheeffectoftheinitialbounce:thewide-anglefluctuationsshouldbedifferentfromthosepredictedbythetheorythatdoesnottakequantaintoaccount.Thecurrentstateofthemeasurementisdescribedinfigure9.4,wheretheblacklinerepresentsthepredictionbyAshtekar,AgulloandNelson,andthegreydotsthemeasureddata.Fornow,thesearenotsufficienttoevaluatewhethertheupwardbendoftheblacklinepredictedbythethreeauthorsiscorrectornot.Butmeasurementsaregettingmoreprecise.Thesituationisstillfluid.Butthosewho,likemyself,havespenttheirlivesseekingtounderstandthesecretsofquantumspacearefollowingwithcloseattention,anxietyandhopethecontinuoushoningofourcapacitytomakeobservations,tomeasureandtocalculate–andareawaitingthemomentinwhichnaturewilltelluswhetherwearerightornot.

Figure9.3Thefluctuationsofthecosmicbackgroundradiation.Thisistheimageoftheoldestobjectintheuniverseavailabletous.Thesefluctuationswereproduced14billionyearsago.Inthestatisticsofsuchfluctuationswehopetofindconfirmationofthe

predictionsofquantumgravity.

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Figure9.4Possiblepredictionofthespectrumofbackgroundradiation,ofloopquantumgravity(shownbythesolidline)comparedwiththecurrentexperimentalerrors(asrepresentedbythepoints).CourtesyofA.Ashtekar,I.AgulloandW.Nelson.

Tracesofthegreatprimordialheatmustalsobeinthegravitationalfielditself.Thegravitationalfield,too,thatistosay,spaceitself,mustbetremulouslikethesurfaceofthesea.Therefore,acosmicgravitationalbackgroundradiationmustalsoexist–oldereventhantheelectromagneticone,becausethegravitationalwavesaredisturbedlessbymatterthantheelectromagneticonesandwereabletotravelundisturbedevenwhentheuniversewastoodensetolettheelectromagneticwavespass.Wehavenowobservedgravitationalwavesdirectly,withtheLIGOdetector,formedbytwoarmsof

afewmilesinlength,atarightangletoeachother,inwhichlaserbeamsmeasurethedistancebetweenthreefixedpoints.Whenagravitationalwavepasses,thespacelengthensandshortensimperceptibly,andthelasersrevealthisminusculevariation.fn46Thegravitationalwavesobservedweregeneratedbyanastrophysicalevent:collidingblackholes.Thesearephenomenadescribedbygeneralrelativitywhichdonotinvolvequantumgravity.ButamoreambitiousexperimentcalledLISAisatthestageofbeingevaluatedandiscapableofdoingthesamethingbutonamuchlargerscale:byputtingintoorbitthreesatellites,notaroundtheEarthbutaroundtheSun,asiftheywereminiatureplanetstrackingtheEarthinitsorbit.Thethreesatellitesareconnectedbylaserbeamsmeasuringthedistancebetweenthemor,betterstill,thevariationsinthedistanceswhenagravitationalwavepasses.IfLISAislaunched,itshouldbeabletoseenotonlythegravitationalwavesproducedbystarsandblackholesbutalsothediffusebackgroundofprimordialgravitationalwavesgeneratedatatimeclosetotheBigBang.Thesewavesshouldtellusaboutthequantumbounce.Inthesubtleirregularitiesofspace,weshouldbeabletofindtracesofeventswhichtookplace14

billionyearsago,attheoriginofouruniverse,andconfirmourdeductionsonthenatureofspaceandtime.

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10.QuantumBlackHoles

Blackholespopulateouruniverseingreatnumber.Theyareregionsinwhichspaceissocurvedastocollapseinonitself,andwheretimecomestoastandstill.Asmentioned,theyform,forinstance,whenastarhasburnedupalloftheavailablehydrogenandcollapses.Frequently,thecollapsedstarformedpartofapairofneighbouringstarsand,inthiscase,the

blackholeandthesurvivingcounterpartcircleonearoundtheother;theblackholesucksmatterfromtheotherstarcontinuously(asinfigure10.1).Astronomershavefoundmanyblackholeswithasize(thatis,mass)oftheorderofourSun(abit

larger,infact).Buttherearealsogiganticbackholes.Thereisoneoftheseatthecentreofalmostallofthegalaxies,includingourown.

Figure10.1Representationofacouplestar/blackhole.Thestarlosesmatter,whichispartlyabsorbedbytheblackhole,partlyprojectedbyitinjetsinthedirectionofitspoles.

Theblackholeatthecentreofourowngalaxyiscurrentlybeingstudiedindetail.IthasamassamilliontimesgreaterthanourSun.Everysooften,astargetstooclosetothismonster,isdisintegratedbythegravitationaldistortionandswallowedbythecyclopeanblackhole,likeasmallfishswallowedbyawhale.ImagineamonsterthesizeofamillionSuns,whichswallowsinaninstantourownSunanditsminiatureplanets…Thereisawonderfulongoingprojecttoconstructanetworkofradioantennaedistributedacross

theEarthfrompoletopole,withwhichastronomerswillbeabletoachievearesolutionsufficientto‘see’thegalacticblackhole.Whatweexpecttoseeisasmallblackdiscsurroundedbythelightproducedbytheradiationofthematterfallingin.Whatentersablackholedoesnotcomeoutagain,atleastifweneglectquantumtheory.The

surfaceofablackholeislikethepresent:itcanbecrossedonlyinonedirection.Fromthefuture,thereisnoreturn.Forablackhole,thepastistheoutside;thefutureistheinside.Seenfromoutside,ablackholeislikeaspherewhichcanbeenteredbutoutofwhichnothingcancome.Arocketcouldstaypositionedatafixeddistancefromthissphere,whichiscalledthehorizonoftheblackhole.Todosoitneedstokeepitsenginesfiringintensely,toresistthegravitationalpullofthehole.Thepowerfulgravityoftheholeimpliesthattimeslowsdownforthisrocket.Iftherocketstaysnearenoughtothehorizonforonehour,andthenmovesaway,itwouldthenfindthat,outside,inthemeantime,centurieshavepassed.Theclosertherocketstaystothehorizon,theslower–with

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respecttotheoutside–timerunsforit.Thus,travellingtothepastisdifficult,buttravellingtothefutureiseasy:weneedonlytogetclosetoablackholewithaspaceship,keepwithinitsvicinityforawhile,andthenmoveaway.Onthehorizonitself,timestops:ifwegetextremelyclosetoitandthenmoveawayafterafewof

ourminutes,amillionyearsmighthaveelapsedintherestoftheuniverse.Thereallysurprisingthingisthatthepropertiesofthesestrangeobjects,todaycommonly

observed,wereforeseenbyEinstein’stheory.Now,astronomersstudytheseobjectsinspace,butuntilnotlongagoblackholeswereconsideredascarcelycredibleandbizarreconsequenceofanoutlandishtheory.IremembermyuniversityprofessorintroducingthemassolutionstoEinstein’sequations,towhich‘realobjectswereunlikelyevertocorrespond’.Thisisthestupendouscapacityoftheoreticalphysicstodiscoverthingsbeforetheyareobserved.TheblackholesweobservearewelldescribedbyEinstein’stheory,andquantummechanicsisnot

neededtounderstandthem.Buttherearetwomysteriesofblackholesthatdorequirequantummechanicsinordertobeunravelledand,foreachofthese,looptheoryoffersapossiblesolution.Oneofthesecouldalsoofferanopportunitytotestthetheory.Thefirstapplicationofquantumgravitytoblackholesconcernsacuriousfactdiscoveredby

StephenHawking.Earlyinthe1970shetheoreticallydeducedthatblackholesare‘hot’.Theybehavelikehotbodies:theyemitheat.Indoingso,theyloseenergyandhencemass(sinceenergyandmassarethesamething),becomingprogressivelysmaller.They‘evaporate’.This‘evaporationofblackholes’isthemostimportantdiscoverymadebyHawking.Objectsarehotbecausetheirmicroscopicconstituentsmove.Ahotpieceofiron,forexample,isa

pieceofironwheretheatomsvibrateveryrapidlyaroundtheirequilibriumposition.Hotairisairinwhichmoleculesmovefasterthanincoldair.

Figure10.2Thesurfaceofablackholecrossedbyloops,thatistosay,bylinksofthespinnetworkthatdescribethestateofthegravitationalfield.Eachloopcorrespondstoaquantumareaoftheblackhole’ssurface.©JohnBaez.

Whataretheelementary‘atoms’thatvibrate,makingablackholehot?Hawkingleftthisproblemunanswered.Looptheoryprovidesapossibleanswer.Theelementaryatomsofablackholethatvibrate,andarethusresponsibleforitstemperature,aretheindividualquantaofspaceonitssurface.Thus,itispossibletounderstandthepeculiarheatofblackholespredictedbyHawkingusingloop

theory:theheatistheresultofthemicroscopicvibrationsoftheindividualatomsofspace.Thesevibratebecauseintheworldofquantummechanicseverythingvibrates;nothingstaysstill.Theimpossibilityofanythingbeingentirelyandcontinuouslystillinaplaceisattheheartofquantummechanics.Black-holeheatisdirectlyconnectedtoloopquantumgravity’sfluctuationsoftheatomsofspace.Theprecisepositionoftheblackhole’shorizonisdeterminedonlyinrelationtothesemicroscopicfluctuationsofthegravitationalfield.Hence,inacertainsense,thehorizonfluctuateslikeahotbody.Thereisanotherwayofunderstandingtheoriginoftheheatofblackholes.Thequantum

fluctuationsgenerateacorrelationbetweentheinteriorandtheexteriorofahole.(IwillspeakatlengthaboutcorrelationsandtemperatureinChapter12).Quantumuncertaintyacrossthehorizonoftheblackholegeneratesfluctuationsofthehorizon’sgeometry.Butfluctuationsimplyprobability,andprobabilityimpliesthermodynamics,andthereforetemperature.Concealingfromusapartoftheuniverse,ablackholemakesitsquantumfluctuationsdetectableintheformofheat.

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Figure10.3StephenHawkingandEugenioBianchi.Ontheblackboardaretheprincipalequationsofloopquantumgravitywhichdescribeblackholes.

ItwasayoungItalianscientist,EugenioBianchi,todayaprofessorintheUnitedStates,whocompletedanelegantcalculationwhichshowshow,startingfromtheseideasandfromthebasicequationsofloopquantumgravity,itispossibletoderivetheformulafortheheatofblackholesforeseenbyHawking(figure10.3).Thesecondapplicationofloopquantumgravitytoblack-holephysicsismorespectacular.Once

collapsed,astarvanishesfromexternalview:itisinsidetheblackhole.But,insidethehole,whathappenstoit?Whatwouldyouseeifyouletyourselffallintothehole?Atfirst,nothinginparticular:youwouldcrossthesurfaceoftheblackholewithoutmajorinjuries

–thenyouwouldplummettowardsthecentre,atevergreaterspeed.Andthen?Generalrelativitypredictsthateverythingissquashedatthecentreintoaninfinitelysmallpointofinfinitedensity.Butthisis,onceagain,ifweignorequantumtheory.Ifwetakequantumgravityintoaccount,thispredictionisnolongercorrect–thereisquantum

repulsion–thesamerepulsionthatmakestheuniversebounceattheBigBang.Whatweexpectisthat,ongettingclosertothecentre,thefallingmatterissloweddownbythisquantumpressure,uptoaveryhighbutfinitedensity.Mattergetssquashed,butnotallthewaytoaninfinitelysmallpoint,becausethereisalimittohowsmallthingscanbe.Quantumgravitygeneratesahugepressurethatmakesmatterbounceout,preciselyasacollapsinguniversecanbounceoutintoanexpandinguniverse.Thebounceofacollapsingstarcanbeveryfast,ifwatchedfromdownthere.But–remember–

timepassesmuchmoreslowlytherethanoutside.Seenfromtheoutside,theprocessofthebouncecantakebillionsofyears.Afterthistime,wecanseetheblackholeexplode.Intheend,basically,thisiswhatablackholeis:ashortcuttothedistantfuture.Thus,quantumgravitymightimplythatblackholesarenoteternallystableobjects,asclassical

generalrelativitypredicted,afterall.Theyare,ultimately,unstable.Seeingtheseblack-holeexplosionswouldbeaspectacularconfirmationforthetheory.Veryold

blackholes,suchasthoseformedintheearlyuniverse,couldbeexplodingtoday.Somerecentcalculationssuggestthatthesignalsoftheirexplosioncouldbeintherangeofradiotelescopes.Ithasevenbeensuggestedthatcertainmysteriousradiopulseswhichradioastronomershavealreadymeasured,calledFastRadioBursts,couldbe,precisely,signalsgeneratedbytheexplosionofprimordialblackholes.Ifthiswasconfirmed,itwouldbefantastic:wewouldhaveadirectsignofaquantumgravitationalphenomenon.Let’swaitandsee…

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11.TheEndofInfinity

Whenwetakequantumgravityintoaccount,theinfinitecompressionoftheuniverseintoasingle,infinitelysmallpointpredictedbygeneralrelativityattheBigBangdisappears.Quantumgravityisthediscoverythatnoinfinitelysmallpointexists.Thereisalowerlimittothedivisibilityofspace.TheuniversecannotbesmallerthanthePlanckscale,becausenothingexistswhichissmallerthanthePlanckscale.Ifweignorequantummechanics,weignoretheexistenceofthislowerlimit.Thepathological

situationspredictedbygeneralrelativity,wherethetheorygivesinfinitequantities,arecalledsingularities.Quantumgravityplacesalimittoinfinity,and‘cures’thepathologicalsingularitiesofgeneralrelativity.Thesamehappensatthecentreofblackholes:thesingularitythatclassicgeneralrelativity

anticipateddisappearsassoonaswetakequantumgravityintoaccount.Thereisanothercase,ofadifferentkind,inwhichquantumgravityplacesalimittotheinfinite,

anditregardsforcessuchaselectromagnetism.Quantumfieldtheory,startedbyDiracandcompletedinthe1950sbyFeynmanandhiscolleagues,describestheseforceswellbutisfullofmathematicalabsurdities.Whenweuseittocomputephysicalprocesses,weoftenobtainresultswhichareinfinite,andmeannothing.Theyarecalleddivergences.Thedivergencesaretheneliminatedwithcalculations,usingabaroquetechnicalprocedurewhichleadstofinitefinalresults.Inpractice,itworks,andthenumbers,intheend,comeoutright;theyreproducetheexperimentalmeasurements.Butwhymustthetheorygoviatheinfinitetoarriveatreasonablenumbers?Inthelastyearsofhislife,Diracwasveryunhappywiththeinfinitiesinhistheoryandfeltthat,

allthingsconsidered,hisobjectiveoftrulyunderstandinghowthingsworkedwasnotachieved.Diraclovedconceptualclarity,evenifwhatwasclaritytohimwasnotalwaysclaritytoothers.Butinfinitiesdonotmakeforclarity.Buttheinfinitiesofquantumfieldtheoryfollowfromanassumptionatthebasisofthetheory:the

infinitedivisibilityofspace.Forexample,tocalculatetheprobabilitiesofaprocess,wesumup–asFeynmanhastaughtus–allofthewaysinwhichtheprocesscouldunfold,andtheseareinfinite,becausetheycanhappeninanyoneoftheinfinitepointsofaspatialcontinuum.Thisiswhytheresultcanbeinfinite.Whenquantumgravityistakenintoaccount,theseinfinitiesalsodisappear.Thereasonisclear:

spaceisnotinfinitelydivisible,therearenoinfinitepoints;therearenoinfinitethingstoaddup.Thegranulardiscretestructureofspaceresolvesthedifficultiesofthequantumtheoryoffields,eliminatingtheinfinitiesbywhichitisafflicted.Thisisatremendousresult:ontheonehand,takingquantummechanicsintoaccountresolvesthe

problemsgeneratedbytheinfinitiesofEinstein’stheoryofgravity,thatistosay,thesingularities.Ontheother,takinggravityintoaccountsolvestheproblemsgeneratedbyquantumfieldtheory,thatistosay,thedivergences.Farfrombeingcontradictory,astheyatfirstseemed,thetwotheorieseachofferthesolutiontotheproblemsposedbytheother!Puttingalimittoinfinityisarecurrentthemeinmodernphysics.Specialrelativitymaybe

summarizedasthediscoverythatthereexistsamaximumvelocityforallphysicalsystems.Quantummechanicscanbesummarizedasthediscoverythatthereexistsamaximumofinformationforeachphysicalsystem.TheminimumlengthisthePlancklengthLP,themaximumvelocityisthespeedoflightc,andthetotalinformationisdeterminedbythePlanckconstanth.Thisissummarizedintable11.1.Theexistenceoftheseminimumandmaximumvaluesforlength,velocityandactionfixesa

naturalsystemofunits.Insteadofmeasuringspeedinkilometresperhour,orinmetrespersecond,wecanmeasureitinfractionsofthespeedoflight.Wecanfixthevalue1forthevelocitycandwrite,forexample,v=½,forabodywhichismovingathalfthespeedoflight.Inthesameway,wecanpositbydefinitionandmeasurelengthinmultiplesofPlanck’slength.Andwecanposith=1andmeasureactionsinmultiplesofPlanck’sconstant.Inthisway,wehaveanaturalsystemoffundamentalunitiesfromwhichtheothersfollow.TheunityoftimeisthetimethatlighttakestocoverthePlancklength,andsoon.Thenaturalunitiesarecommonlyusedinresearchonquantumgravity.Theidentificationofthesethreefundamentalconstantsplacesalimittowhatseemedtobethe

infinitepossibilitiesofnature.Itsuggeststhatwhatwecallinfiniteoftenisnothingmorethansomethingwhichwehavenotyetcounted,orunderstood.Ithinkthisistrueingeneral.‘Infinite’,

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ultimately,isthenamethatwegivetowhatwedonotyetknow.Natureappearstobetellingusthatthereisnothingtrulyinfinite.Table11.1Fundamentallimitationsdiscoveredbytheoreticalphysics.

PhysicalQuantity Fundamentalconstant Theory Discovery

Velocity c Specialrelativity Amaximumvelocityexists

Information(actions) ℏ Quantummechanics Aminimumofinformationexists

Length Lp Quantumgravity Aminimumlengthexists

Thereisanotherinfinitywhichdisorientatesourthinking:theinfinitespatialextensionofthecosmos.ButasIillustratedinChapter3,Einsteinhasfoundthewayofthinkingofafinitecosmoswithoutborders.Currentmeasurementsindicatethatthesizeofthecosmosmustbelargerthan100billionlightyears.Thisistheorderofmagnitudeoftheuniversewehaveindirectaccessto.Itisaround10120timesgreaterthanthePlancklength,anumberoftimeswhichisgivenbya1followedby120zeroes.BetweenthePlanckscaleandthecosmologicalone,then,thereisthemind-blowingseparationof120ordersofmagnitude.Huge.Extraordinarilyhuge.Butfinite.Inthisspace–betweenthesizeoftheminutequantaofspace,uptoquarks,protons,atoms,

chemicalstructures,mountains,stars,galaxies(eachformedbyonehundredbillionstars),clustersofgalaxies,andrightupuntiltheseeminglyboundlessvisibleuniverseofmorethan100billiongalaxies–unfoldstheswarmingcomplexityofouruniverse;auniverseweknowonlyinafewaspects.Immense.Finite.ThecosmologicalscaleisreflectedinthevalueofthecosmologicalconstantΛ,whichentersinto

thebasicequationsofourtheories.Thefundamentaltheorycontains,therefore,averylargenumber:theratiobetweenthecosmologicalconstantandthePlancklength.Itisthislargenumberthatopensthewaytothevastcomplexityoftheworld.Butwhatweseeandunderstandoftheuniverseisnotaninfinitytodrownin.Itisawidesea,butafiniteone.ThebookofEcclesiasticus,orSirach,fn47openswithastupendousquestion:Whocannumberthesandofthesea,andthedropsofrain,andthedaysofeternity?Whocanfindouttheheightofheaven,andthebreadthoftheearth,andthedeep,andwisdom?

Notmuchlongeraftertheselineswerecomposed,anothergreattextwaswritten,withanopeningwhichstillresounds:Somethink,OKingHiero,thatthegrainsofsandcannotbecounted.

ThisistheopeningofPsammites(TheSandReckoner)byArchimedes,inwhichthegreatestscientistofantiquity…countsthegrainsofsandintheuniverse!Hedoessoinordertodemonstratethattheirnumberislargebutfinite,andcanbedetermined.

Thenumericalsystemofantiquitydidnotallowfordealingwithverylargenumberseasily.InTheSandReckoner,Archimedesdevelopsanewsystemofnumbering,similartoourexponentials,thatmakesitpossibletodealwithverylargenumbers,andshowsitspowerbycounting(certainlyplayfully)howmanygrainsofsandthereare,notjustontheseashoresbutintheentireuniverse.TheSandReckonerisplayful,butprofound.Withaflightoffancythatseemstoanticipatethe

Enlightenmentbymillennia,Archimedesrebelsagainsttheformofknowledgethatinsistsontherebeingmysterieswhichareintrinsicallyinaccessibletohumanthought.Hedoesnotclaimtoknowtheexactdimensionsoftheuniverse,ortheprecisenumberofgrainsofsand.Itisn’tthecompletenessofhisknowledgethatheisasserting.Onthecontrary,heisexplicitabouttheapproximateandprovisionalnatureofhisestimates.Hespeaksaboutpossiblealternativesregardingthetruesizeoftheuniverse–betweenwhichhedoesnotmakeadefinitechoice.Thepointatstakehereisnotthepresumptionofknowingeverything.Itistheopposite:anawarenessthatyesterday’signorancemayhavelightshedonittoday,andthattoday’smightbeilluminatedtomorrow.Thecentralpointisrebellionagainsttherenunciationofthedesiretoknow:adeclarationoffaith

inthecomprehensibilityoftheworld,aproudretaliationtothosewhoremainsatisfiedwiththeirownignorance,whocallinfinitethatwhichwedon’tunderstandanddelegateknowledgetoelsewhere.Centurieshavepassed,andthetextofEcclesiasticus,alongwiththerestoftheBiblecanbefound

incountlesshomes,whileArchimedes’textisreadonlybythefew.ArchimedeswasslaughteredbytheRomansduringthesackingofSyracuse,thelastproudremnantofMagnaGreciatofallundertheRomanyoke,duringtheexpansionofthatfutureempirewhichwouldsoonadoptEcclesiasticusasoneofthefoundationaltextsofitsofficialreligion,apositionwhichitwastooccupythereformorethanathousandyears.Duringthatmillennium,thecalculationsmadebyArchimedeslanguishedinastateofincomprehensibility:noonewasabletouse,oreventounderstandthem.

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NearArchimedes’SyracusethereisoneofthemostbeautifulsitesinItaly,thetheatreofTaormina,whichlooksoutattheMediterraneananduponMountEtna,thesmokingvolcano.InArchimedes’time,thetheatrewasusedtostageplaysbySophoclesandEuripides.TheRomansadapteditforgladiatorialcombat,forthepleasureofwatchinggladiatorsdie.ThesophisticatedplayfulnessofTheSandReckonerisperhapsnotonlyaboutanaudacious

mathematicalconstruction,orthevirtuosityofoneofthemostextraordinarymindsofantiquity.Itisalsoadefiantcryofreason,whichrecognizesitsownignorancebutrefusestodelegatetoothersthesourceofknowledge.Itisasmall,reservedandpowerfullyintelligentmanifestoagainstinfinity–againstobscurantism.QuantumgravityisoneofthemanylinesthatcontinuethequestofTheSandReckoner.Weare

countingthegrainsofspaceofwhichthecosmosismade.Avastcosmos,butafiniteone.Theonlytrulyinfinitethingisourignorance.

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12.Information

Weareapproachingtheconclusionofourjourney.Inthepreviousfewchapters,Ispokeabouttheconcreteapplicationsofquantumgravity:thedescriptionofwhathappenedtotheuniversearoundthetimeoftheBigBang;thedescriptionofthepropertiesoftheheatofblackholesandthesuppressionofinfinity.Beforeconcluding,Iwouldliketoreturntothetheory,butlookingatitsfuture,andtospeak

aboutinformation:aspectrethatishauntingtheoreticalphysics,arousingenthusiasmandconfusion.Thischapterisdifferentfromtheprecedingones,whereIspokeofideasandtheoriesnotyet

testedbutwelldefined;here,I’mspeakingofideasstillconfused,badlyinneedoforganization.If,dearreader,youhavefoundthejourneysofaralittlerough,thenholdontighter,becausewe’renowflyingbetweenvoidsofair.Ifthischapterseemsparticularlyopaque,it’snotbecauseyourideasareconfused.It’sbecausetheonewiththeconfusedideasisme.Manyscientistssuspecttodaythattheconceptof‘information’mayturnouttobeakeyfornew

advancesinphysics.Informationismentionedinthefoundationsofthermodynamics,thescienceofheat,thefoundationofquantummechanicsandinotherareasbesides,withthewordquiteoftenusedveryimprecisely.Ibelievethereissomethingimportantinthisidea.I’lltrytoexplainwhy,andtoshowwhatinformationhastodowithquantumgravity.Beforeanythingelse,whatisinformation?Theword‘information’isusedincommonparlanceto

meanavarietyofdifferentthings,andthisimprecisionisasourceofconfusioninscienceaswell.Thescientificnotionofinformation,however,wasdefinedwithclarityin1948,bytheAmericanmathematicianandengineerClaudeShannon,andissomethingverysimple:informationisthemeasureofthenumberofpossiblealternativesforsomething.Forexample,ifIthrowadie,itcanlandononeofsixfaces.Whenwe’veseenitfallonaparticularoneofthese,wehaveanamountofinformationN=6,becausethepossiblealternativesaresixinnumber.IfIdon’tknowwhichdayoftheyearisyourbirthday;thereare365distinctpossibilities.Ifyoutellmethedate,IhavetheinformationN=365.Andsoon.InsteadofthenumberofalternativesN,scientistsmeasureinformationintermsofaquantity

calledS,for‘Shannoninformation’.Sisdefinedasthelogarithminbase2ofN:S=log2N.TheadvantageofusingthelogarithmisthattheunitofmeasurementS=1correspondstoN=2(because1=log22),makingtheunitofinformationtheminimumnumberofalternatives:thechoicebetweentwopossibilities.Thisunitofmeasurementiscalled‘bit’.WhenIknowatroulettethatarednumberhascomeupratherthanablack,Ihaveonebitofinformation;whenIknowthatared,evennumberhaswon,Ihavetwobitsofinformation;whenanevenrednumber‘manque’(eighteenorless,inrouletteparlance)wins,Ihavethreebits.Twobitsofinformationcorrespondtofouralternatives(redeven,reduneven,blackeven,blackuneven).Threebitsofinformationcorrespondtoeightalternatives.Andsoon.fn48Akeypointisthatinformationcanbelocatedsomewhere.Imagine,forinstance,thatyouhavein

yourhandaballwhichcanbeeitherblackorwhite.ImaginethatIalsohaveaballwhichcanbeeitherblackorwhite.Therearetwopossibilitiesonmypart,andtwoonyours.Thetotalnumberofpossibilitiesisfour(2x2):white-white;white-black;black-whiteandblack-black.Now,supposethatforsomereasonwearecertainthatthetwoballsareoppositeincolour(forinstance,becausewehavetakentheballsfromaboxthatcontainedonlyonewhiteandoneblackball).Thetotalnumberofalternativesisthenonly2(white-blackorblack-white),evenifthealternativesarestilltwoonmypartandtwoonyours.Notethat,inthissituation,somethingpeculiarhappens:ifyoulookatyourball,thenyouknowthecolourofmine.Inthiscase,wesaythatthecoloursofthetwoballsarecorrelated,thatistosay,linkedtooneanother.Wesaythatmyball‘hasinformation’aboutyours(aswellasviceversa).Ifyouthinkaboutit,thisispreciselywhathappensinlifewhenwecommunicate:forexample,

whenIphoneyou,Iknowthatthephonecausesthesoundsonyoursidetobedependentonthesoundsonmine.Thesoundsonbothsidesarelinked,likethecolouroftheballs.Theexampleisnotchosenatrandom:Shannon,whoinventedthetheoryofinformation,worked

foratelephonecompany,andwaslookingforawaytomeasureaccuratelyhowmuchatelephonelinecould‘carry’.Butwhatdoesatelephonelinecarry?Itcarriesinformation.Itcarriesthecapacitytodistinguishbetweenalternatives.Forthisreason,Shannondefinedinformation.Whyisthenotionofinformationuseful,perhapsevenfundamental,tounderstandingtheworld?

Forasubtlereason:becauseitmeasurestheabilityofonephysicalsystemtocommunicatewith

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anotherphysicalsystem.Let’sreturnforafinaltimetotheatomsofDemocritus.Let’simagineaworldformedofan

interminableseaofatomswhichbounce,attractandclingtogether,andofnothingelse.Aren’twemissingsomething?PlatoandAristotleinsistedonthefactthatsomethingwasindeedmissing;theythoughtthatthe

formofthingswasthissomethingextrathathadtobeaddedtothesubstanceofwhichthingsweremadeinordertounderstandtheworld.ForPlato,formsexistbythemselves,inanetherealidealworldofforms,aworldof‘ideas’.Theideaofahorseexistspriortoandindependentlyofanyactualhorse.ForPlato,arealhorseisnothingbutapalereflectionoftheideaofahorse.Theatomswhichmakeupthehorsecountforlittle:whatcountsisthe‘horseness’,theabstractform.Aristotleisabitmorerealistic,butforhim,too,theformcannotbereducedtothesubstance.Inastatue,thereismorethanthestoneofwhichitismade.Thismore,forAristotle,istheform.ThisisthebasisofthecritiqueofDemocritus’smaterialisminantiquity.Itstillremainsacommoncritiqueofmaterialism.ButwasDemocritusreallyproposingthateverythingcanbereducedtoatoms?Let’slookatit

moreclosely.Democritussaysthatwhenatomscombinewhatcountsistheirform,theirarrangementinthestructure,aswellasthewayinwhichtheycombine.Hegivestheexampleofthelettersofthealphabet:thereareonlytwentyorsolettersbut,asheputsit,‘Itispossibleforthemtocombineindiversemodes,inordertoproducecomediesortragedies,ridiculousstoriesorepicpoems.’Therearemorethanjustatomsinthisidea:whatcountsisthewayinwhichtheyarecombined,

oneinrelationtoanother.Butwhatrelevancecanthewayinwhichtheyarecombinedhave,inaworldinwhichthereisnothingbutotheratoms?Iftheatomsarealsoanalphabet,whoisabletoreadthephraseswrittenwiththisalphabet?Theanswerissubtle:thewayinwhichtheatomsarrangethemselvesiscorrelatedwiththeway

otheratomsarrangethemselves.Therefore,asetofatomscanhaveinformation,inthetechnical,precisesensedescribedabove,aboutanothersetofatoms.This,inthephysicalworld,happenscontinuouslyandthroughout,ineverymomentandinevery

place:thelightwhicharrivesatoureyescarriesinformationabouttheobjectswhichithasplayedacross;thecolouroftheseahasinformationonthecolouroftheskyaboveit;acellhasinformationaboutthevirusthatisattackingit;anewlivingbeinghasplentyofinformationbecauseitiscorrelatedwithitsparents,andwithitsspecies;andyou,dearreader,whenreadingtheselines,receiveinformationaboutwhatIamthinkingwhilewritingthem,thatistosay,aboutwhatishappeninginmymindatthemomentinwhichIwritethistext.Whatoccursintheatomsofyourbrainisnotanymoreindependentfromwhatishappeningintheatomsofmine:wecommunicate.Theworldisn’t,then,justanetworkofcollidingatoms:itisalsoanetworkofcorrelations

betweensetsofatoms,anetworkofrealreciprocalinformationbetweenphysicalsystems.Inallofthis,thereisnothingidealisticorspiritual;it’snothingbutanapplicationofShannon’s

ideathatalternativescanbecounted.AllthisisasmuchapartoftheworldasthestonesoftheDolomites,thebuzzingofbeesandthewavesofthesea.Oncewehaveunderstoodthatthisnetworkofreciprocalinformationexistsintheuniverse,itis

naturaltoseektousethistreasuretodescribetheworld.Let’sstartwithanaspectofnaturewellunderstoodsincetheendofthenineteenthcentury:heat.Whatisheat?Whatdoesitmeantosaythatsomethingishot?Whydoesacupofscalding-hotteacoolitselfdown,ratherthanheatingitselfupfurther?ItwastheAustrianscientistLudwigBoltzmann,thefounderofstatisticalmechanics,whofirst

understoodwhy.fn49Heatistherandommicroscopicmovementofmolecules:whentheteaishotter,themovementofthemoleculesismoreagitated.Whydoesitcooldown?Boltzmannhazardedasplendidhypothesis:becausethenumberofpossiblestatesofthemoleculesinhotteaandcoldairissmallerthanthenumberincoolteaandslightlywarmerair.Thecombinedstateevolvesfromasituationwheretherearelesspossiblestatestoasituationwheretherearemorepossiblestates.Theteacan’twarmitselfup,becauseinformationcannotincreasebyitself.I’llelaborate.Themoleculesofteaareextremelynumerousandextremelysmall,andwedon’t

knowtheirprecisemovements.Therefore,welackinformation.Thislackofinformation–ormissinginformation–canbecomputed.(Boltzmanndidit:hecomputedthenumberofdistinctstatesthemoleculescanbein.Thisnumberdependsonthetemperature.)Iftheteacools,alittleofitsenergypassesintothesurroundingair;therefore,themoleculesofteamovemoreslowlyandthemoleculesofairmovemorequickly.Ifyoucomputeyourmissinginformation,youdiscoverthatithasincreased.If,instead,teaabsorbedheatfromthecolderair,thenthemissinginformationwouldbedecreased.Thatis,wewouldknowmore.Butinformationcannotfallfromthesky.Itcannotincreasebyitself,becausewhatwedon’tknow,wejustdon’tknow.Therefore,theteacannotwarmupbyitselfincontactwithcoldair.Itsoundsabitmagical,butitworks:wecanpredicthowheatbehavesjustonthebasisoftheobservationthatourinformationcannotincreaseforfree!Boltzmannwasnottakenseriously.Attheageoffifty-six,inDuino,nearTrieste,hecommitted

suicide.Today,heisconsideredoneofthegeniusesofphysics.Histombisincisedwithhisformula

S=klogW

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whichexpresses(missing)informationasthelogarithmofthenumberofalternatives,Shannon’skeyidea.Boltzmannpointedoutthatthisquantitycoincideswiththeentropyusedinthermodynamics.Entropyis‘missinginformation’,thatis,informationwithaminussign.Thetotalamountofentropycanonlyincrease,becauseinformationcanonlydiminish.fn50Today,physicistscommonlyaccepttheideathatinformationcanbeusedasaconceptualtoolto

throwlightonthenatureofheat.Moreaudacious,butdefendedtodaybyanincreasingnumberoftheorists,istheideathattheconceptofinformationcanbeusefulalsotothemysteriousaspectsofquantummechanicsillustratedinChapter5.Rememberthatakeyresultofquantummechanicsispreciselythefactthatinformationisfinite.

Thenumberofalternativeresultsthatwecanobtainmeasuringaphysicalsystemfn51isinfiniteinclassicalmechanics;but,thankstoquantumtheory,wehaveunderstoodthat,inreality,itisfinite.Quantummechanicscanbeunderstoodasthediscoverythatinformationinnatureisalwaysfinite.Infact,theentirestructureofquantummechanicscanbereadandunderstoodintermsof

information,asfollows.Aphysicalsystemmanifestsitselfonlybyinteractingwithanother.Thedescriptionofaphysicalsystem,then,isalwaysgiveninrelationtoanotherphysicalsystem,theonewithwhichitinteracts.Anydescriptionofasystemisthereforealwaysadescriptionoftheinformationwhichasystemhasaboutanothersystem,thatistosay,thecorrelationbetweenthetwosystems.Themysteriesofquantummechanicsbecomelessdenseifinterpretedinthisway,asthedescriptionoftheinformationthatphysicalsystemshaveaboutoneanother.Thedescriptionofasystem,intheend,isnothingotherthanawayofsummarizingallthepast

interactionswithit,andusingthemtopredicttheeffectoffutureinteractions.Theentireformalstructureofquantummechanicscaninlargemeasurebeexpressedintwo

simplepostulates:1

1. Therelevantinformationinanyphysicalsystemisfinite.2. Youcanalwaysobtainnewinformationonaphysicalsystem.

Here,the‘relevantinformation’istheinformationthatwehaveaboutagivensystemasaconsequenceofourpastinteractionswithit:informationallowingustopredictwhatwillbetheresultforusoffutureinteractionswiththissystem.Thefirstpostulatecharacterizesthegranularityofquantummechanics:thefactthatafinitenumberofpossibilitiesexists.Thesecondcharacterizesitsindeterminacy:thefactthatthereisalwayssomethingunpredictablewhichallowsustoobtainnewinformation.Whenweacquirenewinformationaboutasystem,thetotalrelevantinformationcannotgrowindefinitely(becauseofthefirstpostulate),andpartofthepreviousinformationbecomesirrelevant,thatistosay,itnolongerhasanyeffectuponpredictionsofthefuture.Inquantummechanicswhenweinteractwithasystem,wedon’tonlylearnsomething,wealso‘cancel’apartoftherelevantinformationaboutthesystem.fn52Theentireformalstructureofquantummechanicsfollowsinlargemeasurefromthesetwosimple

postulates.Therefore,thetheorylendsitselfinasurprisingwaytobeingexpressedintermsofinformation.Thefirsttorealizethatthenotionofinformationwasfundamentaltotheunderstandingof

quantumrealitywasJohnWheeler,thefatherofquantumgravity.Wheelercoinedthephrase‘Itfrombit’toexpressthisidea,meaningthat‘everythingisinformation’.Informationreappears,then,inthecontextofquantumgravity.Remember:theareaofany

surfaceisdeterminedbythespinsoftheloopwhichintersectthissurface.Thesespinsarediscretequantities,andeachonecontributestothearea.Asurfacewithafixedareamaybeformedfromtheseelementaryquantaofareainmanydifferent

ways,say,inanumberofwaysN.Ifyouknowtheareaofasurfacebutdon’tknowexactlyhowitsquantaofareaaredistributed,youhavemissinginformationaboutthesurface.Thisisoneofthewaysofcomputingtheheatofblackholes:thequantaofareaofablackholeenclosedinasurfaceofacertainareacanbeinNdifferentpossibledistributions.Itislikeforthecupoftea,inwhichthemoleculescanmoveinNdifferentpossibleways.Thuswecanassociateaquantityofmissinginformation,thatistosay,entropy,withablackhole.TheamountofinformationassociatedthuswithablackholedependsdirectlyupontheareaAof

thehole:thelargerthehole,thegreatertheamountofmissinginformation.Wheninformationentersintoablackhole,itisnolongerrecoverablefromoutside.Butthe

informationwhichenterstheblackholecarrieswithittheenergybywhichtheblackholebecomeslargerandincreasesitsarea.Viewedfromoutside,theinformationlostintheblackholenowappearsasentropyassociatedwiththeareaofthehole.ThefirsttosuspectsomethingsimilarwastheIsraeliphysicistJacobBekenstein.Butthesituationisanythingbutclearbecause,aswehaveseeninthelastchapter,blackholes

emitthermalradiationandveryslowlyevaporate,becomingsmallerandsmalleruntiltheyprobablydisappear,subsumedinthatoceanofmicroscopicblackholeswhichconstitutesspaceatthePlanckscale.Wheredoestheinformationthathasfallenintotheblackholeastheblackholeshrinksendup?Theoreticalphysicistsaredebatingthequestion,andnoonehasacompletelyclearanswer.

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Allofthis,Ibelieve,indicatesthatinordertograspthebasicgrammaroftheworld,weneedtomergethreebasicingredients,notjusttwo:notjustgeneralrelativityandquantummechanics,butalsothetheoryofheat,thatis,statisticalmechanicsandthermodynamics,whichwecanalsodescribeasinformationtheory.Butthethermodynamicsofgeneralrelativity,thatistosay,thestatisticalmechanicsofquantaofspace,isasyetonlyinitsfirstinfancy.Everythingisstillconfused,andthereisaverygreatdealwhichremainstobeunderstood.AllofthisbringsustothelastideaIdescribeinthisbook:thermaltime.

ThermaltimeTheproblemattherootoftheideaofthermaltimeissimple.InChapter7,Ishowedthatitisnotnecessarytousethenotionoftimetodescribephysics.Itisbettertoforgettimealtogether.Timeplaysnoroleatthefundamentallevelofphysics.Oncewehaveunderstoodthis,itiseasiertowritetheequationsofquantumgravity.Therearemanyeverydaynotionswhichnolongerhaveanyroleinthefundamentalequationsof

theuniverse;forexample,thenotionsof‘up’and‘down’,or‘hot’and‘cold’,soitisnotparticularlystrangethatsharedquotidiannotionsdisappearfromfundamentalphysics.However,oncewehaveacceptedthisidea,weobviouslyopenupasecondproblem.Howcanwerecoverthenotionsofoureverydayexperience?Howdotheyemerge,inourspecificcontext?Forexample,thenotionsof‘up’and‘down’don’tenterintoNewton’sequations,butweknow

whattheymeaninaschemawithoutabsoluteupanddown.‘Up’and‘down’aremeaningfulnearalargemass,likeaplanet.‘Down’indicatesthedirectiontowardswhichthelarge,nearmassexertsgravitationalpull;‘up’indicatestheoppositedirection.Thesamegoesfor‘hot’and‘cold’:thereareno‘hot’or‘cold’thingsatamicroscopiclevelbut,whenweputtogetheralargenumberofmicroscopicconstituentsanddescribethemintermsofaverages,thenthenotionof‘heat’appears:ahotbodyisabodywheretheaveragespeedofsingleconstituentsisraised.Weareabletounderstandthemeaningof‘up’or‘hot’incertainsituations:thepresenceofanearbymass,orthefactthatwearedealingonlywithaveragevaluesofmanymolecules,andsoon.Somethingsimilarmustapplyto‘time’.Ifthenotionoftimehasnoroletoplayatanelementary

level,itcertainlyplaysasignificantroleinourlives,justas‘up’and‘hot’do.Whatdoes‘thepassageoftime’mean,iftimeplaysnopartinthefundamentaldescriptionoftheworld?Theanswerissimple.Theoriginoftimemaybesimilartothatofheat:itcomesfromaveragesof

manymicroscopicvariables.Let’slookatthisindetail.Thatthereisalinkbetweentimeandtemperatureisanancientandrecurrentidea.Ifyouthink

aboutit,allphenomenawherewedetectthepassageoftimeareco-involvedwithtemperature.Thesalientcharacteristicoftimeisthatitmovesforwardsandnotbackwards,thatistosay,thereareirreversiblephenomena.Mechanicalphenomena–onesthatdon’tinvolveheat–arereversible.Ifwefilmthemandthenrunthefilmbackwards,weseesomethingrealistic.Ifwefilmaswingingpendulum,orastonethrownupwardsthenfalling,andthenwatchthefilminreverse,westillseeaplausiblependulumswinging,orastonerisinganddroppingtotheground.Whenthestonereachestheground,itstops,youmightobject:ifyouwatchthefilmreversed,you

seeastoneleapingupfromthegroundbyitself,andthisisimplausible.Butwhenthestonereachesthegroundandstops,wheredoesitsenergygo?Itheatstheground!Attheprecisemomentwhenheatisproduced,theprocessisirreversible:thepastdiffersfromthefuture.Itisalwaysheatandonlyheatthatdistinguishesthepastfromthefuture.Thisisuniversal.Aburningcandleistransformedintosmoke–thesmokecannottransformintoa

candle–andacandleproducesheat.Aboiling-hotcupofteacoolsdownanddoesnotheatup:itdiffusesheat.Weliveandgetold:producingheat.Ouroldbicyclewearsoutwithtime:producingheatthroughfriction.Thinkofthesolarsystem.Atfirstapproximation,itcontinuestoturnlikeanimmensemechanismalwaysequaltoitself.Itdoesn’tproduceheatand,infact,ifyouwatcheditinreverseyouwouldn’tnoticeanythingstrangeaboutit.Butlookedatmoreclosely,therearealsoirreversiblephenomena:theSunisusingupitscombustiblehydrogenandwilleventuallyexhaustitandextinguish:theSun,too,isgettingolderand,infact,producesheat.TheMoonalsoappearstoorbittheEarthunchanginglyandbealwaysequaltoitself,whereasinrealityitisslowlymovingaway.Thisisbecauseitraisestides,andthetidesheattheseaalittle,thusexchangingenergywiththeMoon.Wheneveryouconsideraphenomenoncertifyingthepassageoftime,itisthroughtheproductionofheatthatitdoesso.Thereisnopreferreddirectionoftimewithoutheat.Butheatisourwaytonameaveragesovermanyvariables.Theideaofthermaltimereversesthisobservation.Thatistosay,insteadofenquiringhowtime

producesdissipationinheat,itaskshowheatproducestime.ThankstoBoltzmann,weknowthatthenotionofheatcomesfromthefactthatweinteractwith

averages.Theideaofthermaltimeisthatthenotionoftime,too,comesfromthefactthatweinteractonlywithaveragesofmanyvariables.fn53Aslongaswehaveacompletedescriptionofasystem,allthevariablesofthesystemareonthe

samefooting;noneofthemactsasatimevariable.Thatistosay:noneiscorrelatedtoirreversiblephenomena.Butassoonaswedescribethesystembymeansofaveragesofmanyvariables,we

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haveapreferredvariablethatfunctionslikecommontime.Atimealongwhichheatisdissipated.Thetimeofoureverydayexperience.Hencetimeisnotafundamentalconstituentoftheworld,butitappearsbecausetheworldis

immense,andwearesmallsystemswithintheworld,interactingonlywithmacroscopicvariablesthataverageamonginnumerablesmall,microscopicvariables.We,inoureverydaylives,neverseeasingleelementaryparticle,orasinglequantumofspace.Weseestones,mountains,thefacesofourfriends–andeachofthesethingsweseeisformedbymyriadsofelementarycomponents.Wearealwayscorrelatedwithaverages.Averagesbehavelikeaverages:theydisperseheatand,intrinsically,generatetime.Thedifficultyofgraspingthisideacomesfromthefactthatitishardforustothinkofaworld

withouttime,andoftimeemerginginanapproximatemanner.Wearetoousedtothinkingofrealityasexistingintime.Wearebeingswholiveintime:wedwellintime,andarenourishedbyit.Weareaneffectofthistemporality,producedbyaveragevaluesofmicroscopicvariables.Butthelimitationsofourintuitionsshouldnotmisleadus.Understandingtheworldbetteroftenentailsgoingagainstintuition.Ifthiswerenotthecase,understandingwouldbeeasy.Timeisaneffectofouroverlookingofthephysicalmicrostatesofthings.Timeisinformationwe

don’thave.Timeisourignorance.

RealityandinformationWhydoesinformationplaysuchacentralroleasthis?Perhapsbecausewemustnotconfusewhatweknowaboutasystemwiththeabsolutestateofthesamesystem.Whatweknowissomethingconcerningtherelationbetweenthesystemandourselves.Knowledgeisintrinsicallyrelational;itdependsjustasmuchonitsobjectasuponitssubject.Thenotionofthe‘state’ofasystemrefers,explicitlyorimplicitly,toanothersystem.Classicalmechanicsmisledusintothinkingthatwecoulddowithouttakingaccountofthissimpletruth,andthatwecouldaccess,atleastintheory,avisionofrealityentirelyindependentoftheobserver.Butthedevelopmentofphysicshasshownthat,attheendoftheday,thisisimpossible.Careful:whenIsaythatwe‘haveinformation’aboutthetemperatureofcupoftea,orwe‘don’t

haveinformation’aboutthevelocityofeverysinglemolecule,Iamnotsayingsomethingaboutmentalstates,orabstractideas.Iamonlysayingthatthelawsofphysicsdetermineacorrelationbetweenourselvesandthetemperature(forinstance,I’velookedatathermometer),butnotbetweenourselvesandthevelocityoftheindividualmolecules.ItisthesamenotionofinformationastheoneIstartedfrominthischapter:thewhiteballinyourhand‘hasinformation’aboutthefactthattheballinmyhandisblack.We’redealingwithphysicalfacts,notmentalnotions.Aballhasinformation,inthissense,eveniftheballdoesnothavementalstates,justasaUSBstoragedevicecontainsinformation(thenumberofgigabytesprintedonthedevicetellsushowmuchinformationitcancontain),evenifaUSBstoragedevicedoesnotthink.Informationinthissense–correlationbetweenstatesofsystems–isubiquitousthroughouttheuniverse.Ibelievethatinordertounderstandrealitywehavetokeepinmindthatrealityisthisnetworkof

relations,ofreciprocalinformation,whichweavestheworld.Wesliceuptherealitysurroundingusintoobjects.Butrealityisnotmadeupofdiscreteobjects.Itisavariableflux.Thinkofanoceanwave.Wheredoesawavefinish?Wheredoesitbegin?Thinkofmountains.Wheredoesamountainstart?Wheredoesitend?HowfardoesitcontinuebeneaththeEarth’ssurface?Thesearequestionswithoutmuchsense,becauseawaveandamountainarenotobjectsinthemselves;theyarewayswhichwehaveofslicinguptheworldtoapprehendit,tospeakaboutitmoreeasily.Theselimitsarearbitrary,conventional,comfortable:theydependonus(asphysicalsystems)morethanonthewavesorthemountains.Theyarewaysoforganizingtheinformationwhichwehaveor,better,formsofinformationwhichwehave.It’sthesameforeveryobject,properlyconsidered,includinglivingorganisms.Thisiswhyit

makeslittlesensetoaskwhetherahalf-cutfingernailisstill‘me’orhasbecome‘not-me’;orifthehairsleftonmysofabythecatarestillpartofthecat,ornot;orpreciselywhenachild’slifebegins.Achildbeginstoliveonthedaywhenapersondreamsofherforthefirsttime,longbeforeherconception,orwhensheformsherfirstself-image,orwhenshebreathesforthefirsttime,orwhensherecognizeshername,orwhenweapplyanynumberofotherconventions:theyarealluseful,butarbitrary.Theyarewaystothink,andtoorientateourselveswithinthecomplexityofreality.Alivingorganismisasystemwhichcontinuallyre-formsitselfinordertoremainitself,interacting

ceaselesslywiththeexternalworld.Ofsuchorganisms,onlythosecontinuetoexistwhicharemoreefficientatdoingsoand,therefore,livingorganismsmanifestpropertieswhichhavesuitedthemforsurvival.Forthisreason,theyareinterpretable,andweinterpretthem,intermsofintentionality,ofpurpose.Thefinalisticaspectsofthebiologicalworld(thisisDarwin’smomentousdiscovery)arethereforetheresultoftheselectionofcomplexformseffectiveinpersisting.Buttheeffectivewayofcontinuingtoexistinachangingenvironmentistomanagecorrelationswiththeexternalworldbetter,thatistosay,information;tocollect,store,transmitandelaborateinformation.Forthisreason,DNAexists,togetherwithimmunesystems,senseorgans,nervoussystems,complexbrains,

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languages,books,thelibraryofAlexandria,computersandWikipedia:theymaximizetheefficiencyofinformationmanagement–themanagementofcorrelationsfavouringsurvival.ThestatuethatAristotleseesinablockofmarbleismorethantheblockofmarble:butitisnotan

abstractformthatresidesjustinthestatue.ItissomethingresidinginthecorrelationsbetweenthemindofAristotle,orours,andthemarble;somethingthatpertainstotheinformationwhichthemarbleprovidesregardingsomethingthatissignificantforAristotle,orforus.Itissomethingregardingadiscusthrower,Phidias,Aristotleandthemarble,andresidesinthecorrelateddispositionsoftheatomsofthestatue,andthecorrelationsbetweentheseandathousandothers,inourmindsorinAristotle’s.Thesespeakofadiscusthrower,justasthewhiteballinyourhandtellsyouthattheballinmineisblack.Wearestructuredtomanagepreciselythis–information–andremaininexistencethankstothis.Evenfromthisbriefoverviewitshouldbeclearthatthenotionofinformationplaysacentralrole

inourattemptstounderstandtheworld.Fromcommunicationtothebasisofgenetics,fromthermodynamicstoquantummechanicsanduptoquantumgravity,thenotionofinformationisgaininggroundasatoolforunderstanding.Theworldshouldnotbeunderstoodasanamorphousensembleofatoms–butratherasagameofmirrors,foundedonthecorrelationsbetweenthestructuresformedbycombinationsoftheseatoms.AsDemocritussaid,itisnotjustaquestionoftheseatomsbutalsooftheorderinwhichtheyare

arranged.Atomsarelikethelettersinanalphabet:anextraordinaryalphabet,sorichastobeabletoread,reflectandeventhinkaboutitself.Wearenotatoms;weareordersinwhichatomsarearranged,capableofmirroringotheratomsandmirroringourselves.Democritusgaveastrangedefinitionof‘man’:‘Maniswhatweallknow.’2Atfirstsight,this

seemsrathersillyandempty,butitisnotso.SalomonLuria,themajorscholarofDemocritus,observesthatitisnotabanalitythatDemocritus

isgivingus.Thenatureofamanisnothisinternalstructurebutthenetworkofpersonal,familialandsocialinteractionswithinwhichheexists.Itisthesewhich‘make’us,thesewhichguardus.Ashumans,wearethatwhichothersknowofus,thatwhichweknowofourselves,andthatwhichothersknowaboutourknowledge.Wearecomplexnodesinarichwebofreciprocalinformation.Allofthisisnotyetatheory.Thesearetrackswearefollowing,Ibelieve,inseekingtounderstand

theworldaroundusbetter.Therestillremainsagreatdealtounderstand.I’llspeakofthisinthefinalchapter.

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13.Mystery

Thetruthisinthedepths.–Democritus1

I’vedescribedwhatIthinkisthenatureofthingsinthelightofwhatwehavelearnedtodate.I’vesummarizedthedevelopmentofsomekeyideasoffundamentalphysics,andIhaveillustratedthegreatdiscoveriesmadebyphysicsinthetwentiethcenturyandtheimageoftheworldemergingfromtheresearchintoaquantumtheoryofgravity.

AmIsureaboutallthis?Iamnot.OneoftheveryfirstandmostbeautifulpagesinthehistoryofscienceisthepassageinPlato’s

PhaedoinwhichSocratesexplainstheshapeoftheEarth.Socratessayshe‘believes’theEarthisasphere,withgreatvalleyswheremenlive.He’sbasically

right,ifabitconfused.Headds,‘I’mnotsure.’Thispageisworthmuchmorethanallofthenonsenseontheimmortalityofthesoulwhichfillstherestofthedialogue.ItisnotjusttheoldesttexttocomedowntouswhichspeaksexplicitlyofthefactthattheEarthmustbespherical.Moreimportantly,itshineswiththecrystallineclaritywithwhichPlatoacknowledgesthelimitsoftheknowledgeofhistime.‘I’mnotsure,’saysSocrates.

Thisacuteawarenessofourignoranceistheheartofscientificthinking.Itisthankstothisawarenessofthelimitsofourknowledgethatwehavelearnedsomuch.Wearenotcertainofallwhichwesuspect,justasSocrateswasnotsureofthesphericalnatureoftheEarth.Weareexploringatthebordersofourknowledge.

Awarenessofthelimitsofourknowledgeisalsoawarenessofthefactthatwhatweknowmayturnouttobewrong,orinexact.Onlybykeepinginmindthatourbeliefsmayturnouttobewrongisitpossibletofreeourselvesfromwrongideas,andtolearn.Tolearnsomething,itisnecessarytohavethecouragetoacceptthatwhatwethinkweknow,includingourmostrootedconvictions,maybewrong,oratleastnaïve:shadowsonthewallsofPlato’scave.

Scienceisbornfromthisactofhumility:nottrustingblindlyinourpastknowledgeandourintuition.Notbelievingwhateveryonesays.Nothavingfaithintheaccumulatedknowledgeofourfathersandgrandfathers.Welearnnothingifwethinkthatwealreadyknowtheessentials,ifweassumethattheywerewritteninabookorknownbytheeldersofthetribe.Thecenturiesinwhichpeoplehadfaithinwhattheybelievedwerethecenturiesinwhichlittlenewwaslearned.Hadtheytrustedtheknowledgeoftheirfathers,Einstein,NewtonandCopernicuswouldneverhavecalledthingsintoquestionandwouldhaveneverbeenabletomoveourknowledgeforwards.Ifnoonehadraiseddoubts,wewouldbestillworshippingpharaohsandthinkingthattheEarthissupportedonthebackofagiantturtle.Evenourmostefficaciousknowledge,suchasthatfoundbyNewton,mayeventuallyturnout,asEinsteinshowed,tobesimplistic.

Scienceissometimescriticizedforpretendingtoexplaineverything,forthinkingthatithasananswertoeveryquestion.It’sacuriousaccusation.Aseveryresearcherworkingineverylaboratorythroughouttheworldknows,doingsciencemeanscominguphardagainstthelimitsofyourignoranceonadailybasis–theinnumerablethingswhichyoudon’tknow,andcan’tdo.Thisisquitedifferentfromclaimingtoknoweverything.Wedon’tknowwhichparticleswemightseenextyearatCERN,orwhatournexttelescopeswillreveal,orwhichequationstrulydescribetheworld;wedon’tknowhowtosolvetheequationswehave,andsometimeswedon’tunderstandwhattheysignify;wedon’tknowifthebeautifultheoryonwhichweareworkingisright.Wedon’tknowwhatthereisbeyondtheBigBang;wedon’tknowhowastormworks,orabacterium,oraneye–orthecellsinourownbodies,orourthoughtprocesses.Ascientistissomeonewholivesimmersedintheawarenessofourdeepignorance,indirectcontactwithourowninnumerablelimits,withthelimitsofourunderstanding.

Butifwearecertainofnothing,howcanwepossiblyrelyonwhatsciencetellsus?Theanswerissimple.Scienceisnotreliablebecauseitprovidescertainty.Itisreliablebecauseitprovidesuswiththebestanswerswehaveatpresent.Scienceisthemostweknowsofarabouttheproblemsconfrontingus.Itispreciselyitsopenness,thefactthatitconstantlycallscurrentknowledgeintoquestion,whichguaranteesthattheanswersitoffersarethebestsofaravailable:ifyoufindbetteranswers,thesenewanswersbecomescience.WhenEinsteinfoundbetteranswersthanNewton,hedidn’tquestionthecapacityofsciencetogivethebestpossibleanswers–onthecontrary,heconfirmedit.

Theanswersgivenbyscience,then,arenotreliablebecausetheyaredefinitive.Theyarereliablebecausetheyarenotdefinitive.Theyarereliablebecausetheyarethebestavailabletoday.Andthey

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arethebestwehavebecausewedon’tconsiderthemtobedefinitive,butseethemasopentoimprovement.It’stheawarenessofourignorancethatgivesscienceitsreliability.

Anditisreliabilitythatweneed,notcertainty.Wedon’thaveabsolutecertainty,andneverwillhaveit–unlessweacceptblindbelief.Themostcredibleanswersaretheonesgivenbyscience,becausescienceisthesearchforthemostcredibleanswersavailable,notforanswerspretendingtocertainty.

Thoughrootedinpreviousknowledge,scienceisanadventurebasedoncontinuouschange.ThestoryIhavetoldreachesbackovermillennia,tracinganarrativeofsciencethathastreasuredgoodideasbuthasn’thesitatedtothrowideasawaywhensomethingwhichworksbetterwasfound.Thenatureofscientificthinkingiscritical,rebelliousanddissatisfiedwithaprioriconceptions,withreverenceandsacredoruntouchabletruth.Thesearchforknowledgeisnotnourishedbycertainty:itisnourishedbyaradicaldistrustincertainty.

Thismeansnotgivingcredencetothosewhosaytheyareinpossessionofthetruth.Forthisreason,scienceandreligionfrequentlyfindthemselvesonacollisioncourse.Notbecausesciencepretendstoknowultimateanswersbutpreciselyfortheoppositereason:becausethescientificspiritdistrustswhoeverclaimstobetheonehavingultimateanswers,orprivilegedaccesstoTruth.Thisdistrustisfoundtobedisturbinginsomereligiousquarters.Itisnotsciencewhichisdisturbedbyreligion:therearecertainreligionsthataredisturbedbyscientificthinking.

Toacceptthesubstantialuncertaintyofourknowledgeistoacceptlivingimmersedinignoranceand,therefore,inmystery,toacceptlivingwithquestionstowhichwedonotknowtheanswers.Perhapswedon’tknowthemyetor–whoknows?–weneverwill.

Tolivewithuncertaintymaybedifficult.Therearethosewhopreferanycertainty,evenifunfounded,totheuncertaintywhichcomesfromrecognizingourownlimits.Therearesomewhoprefertobelieveinastoryjustbecauseitwasbelievedbythetribe’sancestorsratherthanbravelytoacceptuncertainty.

Ignorancecanbescary.Outoffear,wecantellourselvescalmingstories:uptherebeyondthestars,thereisanenchantedgarden,withagentlefatherwhowillwelcomeusintohisarms.Itdoesn’tmatterifthisistrue,itisreassuring.

Thereisalways,inthisworld,someonewhopretendstotellustheultimateanswers.TheworldisfullofpeoplewhosaythattheyhaveTheTruth.Becausetheyhavegotitfromthefathers;theyhavereaditinaGreatBook;theyhavereceiveditdirectlyfromagod;theyhavefounditinthedepthsofthemselves.ThereisalwayssomeonewhohasthepresumptiontobethedepositoryofTruth,neglectingtonoticethattheworldisfullofotherdepositoriesofTruth,eachonewithhisownrealTruth,differentfromthatoftheothers.Thereisalwayssomeprophetdressedinwhite,utteringthewords,‘Followme,Iamthetrueway.’

Idon’tcriticizethosewhoprefertobelieveinthis:weareallfreetobelieveinwhateverwewant.Maybe,afterall,thereisagrainoftruthinthejokereportedbyStAugustine:WhatwasGoddoingbeforecreatingtheworld?HewaspreparingHellforthosewhoseektoscrutinizedeepmysteries.2Butthesedeepmysteriesarepreciselythe‘depths’inwhichDemocritus,inthequotethatopensthischapter,invitesustoseekthetruth.

Formypart,Iprefertolookourignoranceintheface,acceptitandseektolookjustabitfurther:totrytounderstandthatwhichweareabletounderstand.Notjustbecauseacceptingthisignoranceisthewaytoavoidbeingentangledinsuperstitionsandprejudices–butbecausetoacceptourignoranceinthefirstplaceseemstometobethetruest,themostbeautifuland,aboveall,themosthonestway.

Toseektolookfurther,togofurther,seemstometobeoneofthesplendidthingswhichgivessensetolife.Likeloving,orlookingatthesky.Thecuriositytolearn,todiscover,tolookoverthenexthill,thedesiretotastetheapple:thesearethethingswhichmakeushuman.AsDante’sUlyssesremindshiscompanions,wearenotmade‘tolivelikebrutes,buttoseekvirtueandknowledge’.

Theworldismoreextraordinaryandprofoundthananyofthefablestoldbyourforefathers.Iwanttogoandseeit.Toacceptuncertaintydoesn’tdetractfromoursenseofmystery.Onthecontrary:weareimmersedinthemysteryandthebeautyoftheworld.Theworldrevealedbyquantumgravityisanewandstrangeone–stillfullofmystery,butcoherentwithitssimpleandclearbeauty.

Itisaworldwhichdoesnotexistinspaceanddoesnotdevelopintime.Aworldmadeupsolelyofinteractingquantumfieldstheswarmingofwhichgenerates–throughadensenetworkofreciprocalinteractions–space,time,particles,wavesandlight(figure13.1)

Itcontinues,itcontinues,teeminglife,anddeathTenderandhostile,clearandunknowable.

Andthepoetgoeson:Somuchtheeyecansee,fromthiswatchingtower.3

Aworldwithoutinfinity,wheretheinfinitelysmalldoesnotexist,becausethereisaminimumscaletothisteeming,beneathwhichthereisnothing.Quantaofspaceminglewiththefoamofspacetime,andthestructureofthingsisbornfromreciprocalinformationwhichweavesthe

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correlationsbetweentheregionsoftheworld.Aworldwhichweknowhowtodescribewithasetofequations.Perhaps,tobecorrected.

Figure13.1Anintuitiverepresentationofquantumgravity.

It’savastworld,withmuchstilltoclarifyandexplore.It’smyfondestdreamthatsomeone–oneoftheyoungerreadersofthisbook,Ihope–willbeabletovoyageacrossitandilluminateitbetter.Beyondthenexthillthereareworldsstillmorevast,stilltobediscovered.

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Notes

1.Grains1OnAnaximanderandtheMilesians,seeCarloRovelli,TheFirstScientist:AnaximanderandHisLegacy,trans.MarionLignanaRosenberg(Yardley,Westholme,2007).

2TheMilesianoriginofLeucippusisgiven,forinstance,bySimplicius(seeM.Andolfo,Atomistiantichi.Frammentietestimonianze(AncientAtomism.FragmentsandTestimonies),(Milan,Rusconi,1999),p.103.Butitisnotcertain.ThereferencetoMiletusandtoEleaissignificantinrelationtohisculturalroots;thedebtLeucippusowedtoZenoofEleaisdiscussedinthefollowingpages.

3Seneca,Naturalesquestiones,VII,3,2d.4Cicero,Academicapriora,II,23,73.5SextusEmpiricus,Adversusmathematicos,VIII,135(trans.R.G.BuryAgainsttheProfessors),(LoebClassicalLibrary,1989).

6SeeAristotle,OnGenerationandCorruption,A1,315b6,inTheCompleteWorksofAristotle,Vol.I,ed.JonathanBarnes(Princeton,PrincetonUniversityPress,1984).

7AcollectionofancientfragmentsandtestimonieswhichspeakofatomismisgiveninM.Andolfo’sAncientAtomists.AcompleteanthologyoffragmentsandtestimoniesconcerningDemocritushasbeenpublishedbySolomonLuria.(Seeentry‘Democritus’inthebibliographyforanEnglishalternative.)

8ForabriefandinterestingrecentworkonthethoughtofDemocritus,placingitinthecontextofhumanism,seeS.Martini,Democrito:filosofodellanaturaofilosofodell’uomo?(Democritus:PhilosopherofNatureorPhilosopherofMan?),(Rome,Armando,2002.)

9Plato,Phaedo,ed.DavidGallup(Oxford,OxfordUniversityPress,2009),XLVI.10RichardFeynman,TheFeynmanLecturesonPhysics,Vol.1,eds.RobertLeightonandMatthewSands(London,BasicBooks,2011).

11SeeAristotle,OnGenerationandDecay,A2,316a,inTheCompleteWorksofAristotle,Vol.I,ed.JonathanBarnes(Princeton,PrincetonUniversityPress,1984).

12AgoodrecentaccountofZeno’sparadoxes,andoftheirphilosophicalandmathematicalrelevance,isgivenbyVincenzoFanoinIparadossidiZenone(Zeno’sParadoxes),(Rome,Carocci,2012).

13Amores(LovePoems),I,15,23–4.14Lucretius,OntheNatureoftheUniverse,trans.E.A.Latham,(Harmondsworth,Penguin,1951),p.173

15Ibid.,p.89.16Ibid.,p.2717Ibid.,p.60.18GuidoCavalcanti,Rime.,trans.LeonardCottrell.19ForanaccountoftherediscoveryofLucretius’stextanditsimpactuponEuropeanculture,seeStephenGreenblatt,TheSwerve:HowtheWorldBecameModern(NewYork,Norton,2011).

20SeeM.Camarota,‘Galileo,LucretiusandAtomism’,inF.CittiandM.Beretta(eds.),Lucrezio,lanaturaelascienza(Lucretius,NatureandScience),(Florence,LeoS.Olshki,2008),pp.141–75.

21SeeR.Kargon,AtomisminEnglandfromHariottoNewton(Oxford,OxfordUniversityPress,1966).

22WilliamShakespeare,RomeoandJuliet,1.4,55–60,CompleteWorks,eds.JonathanBateandEricRassmussen(London,Macmillan/TheRoyalShakespeareCompany,2007),p.1690.

23OnTheNatureoftheUniverse,pp.63–4.24PiergiorgioOdifreddihaspublishedafinetranslationofandcommentaryonLucretius’stext,designedforuseinschools.(Comestannolecose.IlmioLucrezio,lamiaVenere(TheNatureofThings.MyLucretius,MyVenus),(Milan,Rizzoli,2013).Itwouldbewonderfulifschoolsadoptedthisbookandifthisextraordinarytextwasmorewidelyknown.Areadingofthetext,andofitsauthor,diametricallyopposedtoOdifreddi’sisgivenbyV.E.AlfieriinLucrezio(Lucretius),(Florence,LeMonnier,1929)andemphasizesthepoignant,poeticqualitiesofthework,derivingfromthemanoblebutbitterinterpretationofthecharacterofLucretius.

25H.DielsandW.Kranz,eds.,DieFragmentederVorsokratiker(ThePresocratics:Fragments),(Berlin,Weidmann,1903),68b247.

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2.TheClassics1OnthevalueofAristotle’sphysics,seeC.Rovelli,‘Aristotle’sPhysics:APhysicist’sLook’,inJournaloftheAmericanPhilosophicalAssociation,1(2015),pp.23–40.

2IamblichusofChalchis,Summapitagorica(Milan,Bompiani,2006).ForanEnglishtranslation,seeIamblichus,LifeofPythagoras,trans.ThomasTaylor(Vermont,InnerTraditionsInternational,1986).

3IsaacNewton,Opticks(1704),(NewYork,Dover,1979),p.400.4GiacomoLeopardi,TheCanti,withaSelectionofHisProse,trans.J.G.Nichols(NewYork,Routledge,2003),p.53.

5IsaacNewton,LetterstoBentley(Montana,Kessinger,2010).CitedinH.S.Thayer,Newton’sPhilosophyofNature(NewYork,Hafener,1953),p.54.

6Ibid.7MichaelFaraday,ExperimentalResearchesinElectricity(London,BernardQuaritch,1839–55),3vols.,pp.436–7.

3.Albert1Simplicius,AristotelisPhysica(Aristotle’sPhysics),28,15.2A.Calaprice,DearProfessorEinstein.AlbertEinstein’sLetterstoandfromChildren(NewYork,PrometheusBooks,2002),p.140.

3TheletterisquotedinA.Fölsing,Einstein:ABiography(London,Penguin,1998),p.337.4F.P.DeCeglia(ed.)ScienziatidiPuglia:secoliVaC.-XXI,dC,Parte3(ScientistsofPuglia:FromtheFifthCenturyBCtotheTwentiethCenturies,Part3)(Bari,Adda,2007),p.18.

5A.Calaprice,DearProfessorEinstein.AlbertEinstein’sLetterstoandfromChildren(NewYork,PrometheusBooks,2002),p.208.

4.Quanta1AlbertEinstein,‘ÜbereinendieErzeugungundVerwandlungdesLichtesbetreffendenGesichtspunkt’,AnnalenderPhysik,17,pp.132–48.

2ForabiographyofDiracwhichillustrateshisdisconcertingpersonality,seeGrahamFarmelo,TheStrangestMan:TheHiddenLifeofPaulDirac,QuantumGenius(London,Faber,2009).

3Lucretius,OntheNatureoftheUniverse,trans.E.A.Latham,(Harmondsworth,Penguin,1951),II,218.

4Foradiscussionofthisrelationalinterpretationofquantummechanics,see‘RelationalQuantumMechanics’intheonlineStanfordEncyclopediaofPhilosophy,http://plato.stanford.edu/archives/win2003/entries/rovelli,orCarloRovelli,‘RelationalQuantumMechanics’,inInternationalJournalofTheoreticalPhysics,35,1637,1996,http://arxiv.org/abs/quant-ph/9609002.

5BasvanFrassen,‘Rovelli’sWorld’,inFoundationsofPhysics,40,2010,390–417;MichelBitbol,PhysicalRelationsorFunctionalRelations?ANon-metaphysicalConstrualofRovelli’sRelationalQuantumMechanics,PhilosophyofScienceArchives,2007,http://philsci-archive.pitt.edu/3506/;MauroDorato,Rovelli’sRelationalQuantumMechanics,MonismandQuantumBecoming,PhilosophyofScienceArchives,2013,http://philsci-archive.pitt.edu/9964/,andChecos’èiltempo?Einstein,Gödelel’esperienzacommune(WhatisTime?Einstein,GödelandSharedExperience),(Rome,Carocci,2013).

5.SpacetimeisQuantum1TheworkonthemeasurabilityoffieldsbyNielsBohrandLeonRosenfeldis‘DetKongelikeDanskeVidenskabernesSelskabs’,inMathematiks-fysikeMeddelelser,12,1933.

2SeeMatveiBronštejn,‘QuantentheorieschwacherGravitationsfelder’,inPhysikalischeZeitschriftderSowjetunion,9,1936,140–57;and‘Kvantovaniegravitatsionnykhvoln’,inPi’smavZhurnalEksperimental’noiITeoreticheskoiFiziki,6,1936,195–236.

3SeeF.GorelikandV.Frenkel,MatveiPetrovichBronsteinandSovietTheoreticalPhysics(Boston,BirkhauserVerlag,1994).‘Bronstein’wasalsotherealsurnameofTrotsky.

4TheepisodeisrecalledbyBryceDeWitt,http://www.aip.org/history/ohilist/23199.html.

7.TimeDoesNotExist1Lucretius,OntheNatureoftheUniverse,trans.E.A.Latham,(Harmondsworth,Penguin,1951),p.41.

2WilliamShakespeare,AMidsummerNight’sDream,5.1,23–7,RSCedition,p.403.

8.BeyondtheBigBang

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1ThespeechcanbefoundontheVaticanwebsite:http://www.vatican.va/holy_father/pius_xii/speeches/1951/documents/hf_p-xii_spe_19511122_di-serena_it.html#top.

2SeeS.Singh,BigBang(London,HarperCollins,2010),p.362.

12.Information1AdetaileddiscussionofthesetwopostulatescanbefoundinCarloRovelli,‘RelationalQuantumMechanics’,inInternationalJournalofTheoreticalPhysics,35,1637,1996,http://arxiv.org/abs/quant-ph/9609002.

2Cicero,Academicapriora,II,23,73.

13.Mystery1CitedinDiogenesLaertius,LivesofEminentPhilosophers,2vols.,(NewYork,Loeb,1989).2StAugustine,Confessions,XI,12(Harmondsworth,Penguin,2002).3MarioLuzi,Dallatorre,inDalfondodellecampagne(Turin,Einaudi)p.214.

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AnnotatedBibliography

Andolfo,Matteo,Atomistiantichi.Frammentietestimonianze(AncientAtomism.FragmentsandTestominies),Milan,Rusconi,1999.

Aristotle,OnGenerationandCorruption,inTheCompleteWorksofAristotle,Vol.I.Ed.JonathanBarnes,Princeton,PrincetonUniversityPress,1984.TheprincipalworkbyAristotlewhichprovidesinformationonthethoughtofDemocritus.

Bitbol,Michel,PhysicalRelationsorFunctionalRelations?ANon-metaphysicalConstrualofRovelli’sQuantumMechanics.PhilosophyofScienceArchives,2007,http://philsci-archive.pitt.edu/3506/.CommentaryonandKantianinterpretationofrelationalquantummechanics.

Baggott,Jim,TheQuantumStory:AHistoryinFortyMoments.NewYork,OxfordUniversityPress,2011.Agoodandcompletereconstructionofthemainstagesinthedevelopmentofquantummechanics,fromitsorigintothepresentday.

Bojowald,Martin,OnceBeforeTime:AWholeStoryoftheUniverse.NewYork,AlfredA.Knopf,2010.Anexplanatorydescriptionoftheapplicationofloopquantumgravitytotheoriginoftheuniverse–byoneofthefirstscientiststouseitinthisway.Includesanaccountoftheso-called‘BigBounce’oftheuniverse,whichmayhavehappenedbeforetheBigBang.

Calaprice,Alice(ed.),DearProfessorEinstein.AlbertEinstein’sLetterstoandfromChildren.NewYork,PrometheusBooks,2002.AdelightfulcollectionoflettersexchangedbetweenEinsteinandseveralchildren.

Democritus,Raccoltadeiframmenti(CollectedFragments).InterpretationandCommentarybyS.Luria.Trans.It.Milan,Bompiani,2007.ThereisnoEnglishtranslationofthisclassicedition.

Democritus,TheAtomists:LeucippusandDemocritus.Fragments.AtextandtranslationwithcommentarybyC.C.W.Taylor.Toronto,TorontoUniversityPress,1999.Diehls,H.andKranz,W.(eds),DieFragmentederVorsokratiker.Berlin,Weidmann,1903.Theclassiccollection,inthreevolumes,ofthe‘fragments’ofthePresocratics.

Dorato,Mauro,Checos’èiltempo?Einstein,Gödelel’esperienzacommune.Rome,Carocci,2013.Preciseandcompletediscussion,centredaroundspecialrelativity,oftheEinsteinianmodificationoftheconceptoftime.

—,Rovelli’sRelationalQuantumMechanics,MonismandQuantumBecoming.PhilosophyofScienceArchives,2013,http://philsci-archive.pitt.edu/9964/.DiscussionbyanItalianphilosopherofinterpretationsofquantummechanics.

Fano,Vincenzo,IparadossidiZenone.Trans.It.Rome,Carocci,2012.AnexcellentstudywhichhighlightsthetopicalityoftheproblemsposedbyZeno’sparadoxes.

Farmelo,Graham,TheStrangestMan:TheHiddenLifeofPaulDirac,QuantumGenius.London,Faber,2009.AnextensivebuthighlyreadableaccountofthelifeandbafflingcharacterofthegreatestphysicistafterEinstein.

Feynman,Richard,TheFeynmanLecturesonPhysics,eds.RichardB.LeightonandMatthewSands(3vols.).London,BasicBooks,2011.EssentialphysicstextbookdrawnfromthelecturesofthegreatestAmericanphysicist.Theyarebrilliant,original,lively,theproductofanoutstandingintelligence.Nostudentofphysicswhoistrulyinterestedinscienceshouldfailtoreadthem.

Fölsing,Albrecht,AlbertEinstein:ABiography.NewYork,Penguin,1998.ExtensiveandcompletebiographyofEinstein.

Gorelik,GennalyE.andFrenkel,Victor,MatveiPetrovichBronsteinandSovietTheoreticalPhysicsintheThirties.Boston,BirkhauserVerlag,1994.HistoricalstudyofBronštejn,theyoungRussianwhoinitiatedresearchinquantumgravityandwasexecutedbyStalin.

Greenblatt,Stephen,TheSwerve:HowtheWorldbecameModern.NewYork,W.W.Norton,2011.AbookwhichreconstructstheinfluenceoftherediscoveryofLucretiusuponthebirthofthemodernworld.

Heisenberg,Werner,PhysicsandPhilosophy:TheRevolutioninModernScience.NewYork,Harper&Row,1962.Thetrueoriginatorofquantummechanicsreflectsongeneralproblemsofphilosophyandscience.

Kumar,Manjit,Quantum:Einstein,BohrandtheGreatDebateabouttheNatureofReality.London,IconBooks,2009.Afineexplanatory,detailedreconstructionofthebirthofquantummechanicsand,aboveall,ofthelongdialoguebetweenBohrandEinsteinonthemeaningofthenewtheory.

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Lucretius,OnTheNatureoftheUniverse(Dererumnatura),trans.R.E.Latham.Harmondsworth,Penguin,1951.Thewonderfulpoemfromwhichweknowtheideasandspiritofancientatomism.

Newton,Isaac,ATreatiseoftheSystemoftheWorld.London,F.Fayram,1731.Alittle-knownworkbyNewtoninwhichhegiveshistheoryofuniversalgravityinamuchlesstechnicalformthaninhisgreattreatise(Principia).

Odifreddi,P.,Comestannolecose.IlmioLucrezio,lamiaVenere.Milan,Rizzoli,2013.TranslationwithextensivecommentaryofLucretius’spoem.Anidealschooltextbook.

Plato,Phaedo,ed.DavidGallop.Oxford,OxfordUniversityPress,2009.Theoldestextanttexttospeakexplicitlyaboutthesphericalnatureoftheearth.

Rovelli,Carlo,‘Aristotle’sPhysics:APhysicist’sLook’inJournaloftheAmericanPhilosophicalAssociation,1(2015)23–40.

—,SevenBriefLessonsonPhysics,trans.SimonCarnellandEricaSegre.London,AllenLane,2015.Aconciseguidethroughthescientificrevolutionthatshookphysicsinthetwentiethcenturyandstillcontinuestoshakeustoday.

—,TheFirstScientist:AnaximanderandHisLegacy,trans.MarionLignanaRosenberg.Yardley,Westholme,2007.AreconstructionoftheideasofAnaximander–oneofthegreatestscientistsofalltime–andtheirinfluenceonthedevelopmentofscience.Thebookisalsoareflectiononthebirthandnatureofscientificthinking:itsdistinctivecharacter,itsdifferencefrommythicalthinking,itslimitationsandstrength.

—,‘QuantumGravity’,inButterfield,J.andEarman,J.(eds.),HandbookofThePhilosophyofScience,PhilosophyofPhysics.Amsterdam,Elsevier/North-Holland,2007,pp.1287–330.Alongarticleaimedatphilosophers,withadetaileddiscussionofthecurrentstateofquantumgravity,ofitsunresolvedproblemsandthevariousapproachestothem.

—,QuantumGravity.Cambridge,CambridgeUniversityPress,2004.Technicalmanualonquantumgravity.Definitelynotrecommendedforanyonewithoutabackgroundinphysics.

—,‘RelationalQuantumMechanics’inTheStanfordEncyclopaediaofPhilosophy,http://plato.stanford.edu/archives/win2003/entries/rovelli/.Synthesisoftherelationalinterpretationofquantummechanics.

—,‘RelationalQuantumMechanics’inInternationalJournalofTheoreticalPhysics,35(1996),1637,http://arxiv.org/abs/quant-ph/9609002.Thearticlewhichfirstintroducesthetheoryofrelationalquantummechanics.

Smolin,Lee,ThreeRoadstoQuantumGravity.NewYork,BasicBooks,2002.Anintroductiontoquantumgravityanditsopenquestions.

VanFraassen,Bas.‘Rovelli’sWorld’inFoundationsofPhysics,40(2010),390–417.Adiscussionofrelationalquantummechanics,byanimportantanalyticphilosopher.

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ALLENLANEUK|USA|Canada|Ireland|AustraliaIndia|NewZealand|SouthAfricaAllenLaneispartofthePenguinRandomHousegroupofcompanieswhoseaddressescanbefoundatglobal.penguinrandomhouse.com

FirstpublishedinItalianunderthetitleLarealtànonècomeciapparebyRaffaelloCortinaEditoreSpA2014ThistranslationfirstpublishedinGreatBritainbyAllenLane2016Copyright©RaffaelloCortinaEditoreSpA,2014Translationcopyright©SimonCarnellandEricaSegre,2016ThemoralrightoftheauthorandtranslatorshasbeenassertedCovercredit:CoralieBickford-SmithISBN:978-0-241-25797-5

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fn1Intechnicalterms,thereareconverginginfinitesums.Fortheexampleofthestring,theinfinitesum½+¼+1⁄8+1⁄16…convergesto1.InfiniteconvergentsumswerenotunderstoodinZeno’stime.Archimedesunderstoodthemafewcenturieslater,andusedthemtocalculateareas.Newtonusedthemheavily,butnotuntilthenineteenthcentury,withBolzanoandWeierstrass,wasconceptualclarityonthesemathematicalobjectsachieved.Aristotle,however,hadalreadyunderstoodthatthiswasapossiblewaytoanswerZeno;theAristoteliandistinctionbetweenactualinfinityandpotentialinfinityalreadycontainsthekeyidea:thedifferencebetweentheabsenceofalimittodivisibility,andthepossibilityofhavingalreadydividedsomethinganinfinitenumberoftimes.

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fn2HereisthelistofalloftheworksofDemocritus,withtheirtitlesasgivenbyDiogenesLaertius:GreatCosmology;LittleCosmology;Cosmography;OnthePlanets;OnNature;OnHumanNature;OnIntelligence;OntheSenses;OntheSoul;OnFlavours;OnColour;OnDiverseMovementsoftheAtoms;OfChangesinShape;TheCausesofCelestialPhenomena;TheCausesofAtmosphericPhenomena;OnFireandOnThingsinFire;TheCausesofAcousticPhenomena;ConcerningtheMagnet;TheCausesofSeeds,PlantsandFruits;OnAnimals;ADescriptionoftheSky;Geography;ADescriptionofthePole;OnGeometry;GeometricalReality;OntheTangentsoftheCircleandtheSphere;Numbers;OnIrrationalLinesandSolids;Projections;Astronomy;AstronomicalTable;OnRaysofLight;OnReflectedImages;OnRhythmandHarmony;OnPoetry;OntheBeautyofSong;OnEuphonyandCacophony;ConcerningHomer,oronCorrectEpicDiction;TheScienceofMedicine;OnAgriculture;OnWords;OnNames;OnValues,orOnVirtue;OntheDispositionwhichCharacterizestheWise;OnPainting;ATreatiseonTactics;CircumnavigationoftheOcean;OnHistory;TheThoughtofChaldea;TheThoughtofthePhrygians;OntheSacredWritingsofBabylon;OntheSacredWritingsofMeroe;OnFeversandtheCoughsDerivingfromIllness;OnAporiae;LegalQuestions;Pythagoras;OnLogic,orCriterionofThought;Confirmations;PointsofEthics;OnWell-being.Alllost…

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fn3ThebadreputationofAristotelianphysicsdatesbacktothepolemicsofGalileo.Galileohadtomoveforwardandthereforeneededtobecritical.HeattackedAristotleviciously,withscornandsarcasm.ButhetookAristotle’sphysicsveryseriously.fn4(x=½at²).fn5Thesquareoftheperiodofrevolutionisproportionaltothecubeoftheradiusoftheorbit.Thislawwasshowntobecorrectnotonlyfortheplanetsorbitingthesun(Kepler),butalsoforthemoonsofJupiter(Huygens).Newtonassumes,byinduction,thatitshouldalsoholdforthehypotheticallittlemoonorbitingtheEarth.Theconstantofproportionalitydependsonthebodyaroundwhichtheorbitismade:thisiswhydataonthelunarorbitallowustocomputetheperiodofthelittlemoon.fn6a=v²/r,wherevisthespeedandrtheradiusoftheorbit.fn7Theenergyreleasedbycombustionenginesischemicalandtherefore,ultimately,electromagnetic.fn8TheequationsfillapageinMaxwell’soriginaltreatise.Todaythesameequationscanbewritteninhalfaline:dF=0,d*F=J.We’llsoonseewhy.fn9Ifyouvisualizethefieldasavector(anarrow)ateachpointofspace,thepointofthearrowindicatesthedirectionoftheFaradaylines,thatistosay,thetangentoftheFaradaylines,andthelengthofthearrowisproportionaltothedensityoftheFaradaylines.

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fn10Thesetofeventsataspace-likedistancefromareferenceevent.fn11Theastutereaderwillobjectthatthehalfwaymomentofmyquarterofanhourcanbeconsideredsimultaneoustoyourreply.Thereaderwhohasstudiedphysicswillrecognizethatthisis‘Einstein’sconvention’fordefiningsimultaneity.ThisdefinitionofsimultaneitydependsonhowImove,andconsequentlydoesnotdefinesimultaneitybetweentwoeventsbutonlyasimultaneityrelativetothestateofmovementofparticularbodies.Infigure3.3adotishalfwaybetweenaandb,thepointsatwhichIexitfromthepastoftheobserverandenterhisfuture.Theotherdotishalfwaybetweeneandd,thepointsatwhichIexitfromthepastoftheobserverandenterintohisfutureifImovealongadifferenttrajectory.Bothdotsaresimultaneousasregardsthereader,accordingtothisdefinitionof‘simultaneity’,buttheyoccurinsuccessivetimes.Thetwodotsareeachsimultaneoustothereader,butrelativetotwodifferentmotionsofmine.Hencetheterm‘relativity’.fn12Airplaneandballfollowageodesicinacurvedspace.Inthecaseoftheball,thegeometryisapproximatelygivenbythemetricds²=(1−2Φ(x))dt²−dx²,whereΦ(x)istheNewtonianpotential.Theeffectofthegravitationalfieldisreducedtothedilationoftimewithaltitude.(Thereaderfamiliarwiththetheorywillnoticethecurioussigninversion:thephysicaltrajectorymaximizespropertime.)fn13ObservationsofthebinarysystemPSRB193+16showthatthetwostarswhichrevolvearoundoneanotherradiategravitationalwaves.TheseobservationsbroughtaNobelPrizeforRussellHulseandJosephTaylorin1993.fn14Plutarch,Adversuscolotem,4,1108.Thewordϕύσινmeans‘nature’,andincludesthesense‘thenatureofsomething’.fn15Thistermiscalled‘cosmological’becauseitseffectsoccuronlyatanextremelylarge,or‘cosmological’distances.TheconstantΛiscalledthe‘cosmologicalconstant’,anditsvaluewasmeasuredattheendofthe1990s,bringingaNobelPrizein2011fortheastronomersSaulPerlmutter,BrianP.SchmidtandAdamG.Riess.fn16Göttingen,whereHilbertworked,wasatthistimetheseatofthemostimportantschoolofgeometry.fn17AsphereisthesetofpointsinR3determinedbytheequationx2+y2+z2=1.The3-sphereisthesetofpointsinR4determinedbytheequationx2+y2+z2+u2=1.fn18IthasbeenobjectedthatDantespeaksof‘circles’andnotof‘spheres’.Buttheobjectionisinvalid.BrunettoLatiniwritesof‘acircle,liketheshellofanegg’.Theword‘circle’,forDante,asforhisteacherandmentor,designateseverythingwhichiscircular,includingspheres.fn19OnthesurfaceoftheEarth,forinstance,theNorthPoleandtwopointsontheequatorcanmakeatrianglewiththreesidesofequallengthandthreerightangles–somethingwhichclearlycannotbedoneonaplane.

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fn20AHilbertspace.fn21Thesearetheeigenvaluesoftheoperatorassociatedwiththephysicalvariableinquestion.Thekeyequationistheeigenvalueequation.fn22Thiscloudisdescribedbyamathematicalobjectcalledwavefunction.TheAustrianphysicistErwinSchrödingerhaswrittenanequationdescribingitsevolutionintime.Quantummechanicsisoftenmistakenlyidentifiedwiththisequation.Schrödingerhadhopesthatthe‘wave’couldbeusedtoexplaintheodditiesofquantumtheory:fromthoseoftheseatoelectromagneticones,wavesaresomethingweunderstandwell.Eventoday,somephysiciststrytounderstandquantummechanicsbythinkingthatrealityistheSchrödingerwave.ButHeisenbergandDiracunderstoodatoncethatthiswouldnotdo.ToviewSchrödinger’swaveassomethingrealistogiveittoomuchweight–itdoesn’thelpustounderstandthetheory;onthecontrary,itleadstogreaterconfusion.Exceptforspecialcases,theSchrödingerwaveisnotinphysicalspace,andthisdivestsitofallitsintuitivecharacter.ButthemainreasonwhySchrödinger’swaveisabadimageofrealityisthefactthat,whenaparticlecollideswithsomethingelse,itisalwaysatapoint:itisneverspreadoutinspacelikeawave.Ifweconceiveanelectronasawave,wegetintroubleexplaininghowthiswaveinstantlyconcentratestoapointateachcollision.Schrödinger’swaveisnotausefulrepresentationofreality:itisanaidtocalculationwhichpermitsustopredictwithsomedegreeofprecisionwheretheelectronwillreappear.Therealityoftheelectronisnotawave:itishowitmanifestsitselfininteractions,likethemanwhoappearedinthepoolsoflamplightwhiletheyoungHeisenbergwanderedpensivelyintheCopenhagennight.fn23Dirac’sequation.fn24Thereisaphenomenonwhichseemsnottobereducibletothestandardmodel:‘darkmatter’.Astrophysicistsandcosmologistsobserveintheuniverseeffectsofmatterwhichseemsnottobethetypeofmatterdescribedbythestandardmodel.Outthere,therearestillmanythingsthatwedon’tknow.fn25IfindtheclaimthattheHiggsboson‘explainsmass’exaggerated.TheHiggsbosondoesnot‘explain’anythingabouttheoriginofmass.Whatwould‘explain’themassoftheHiggs?Thepointistechnical:thestandardmodelreliesoncertainsymmetries,andthesesymmetriesseemedtopermitonlyparticlesdevoidofmass.ButHiggsandothersrealizedthatitispossibletohavebothsymmetriesandmass,aslongasthelatterentersindirectlyviatheinteractionwiththefieldknowntodayastheHiggsfield.fn26Afiniteregionofthephasespace–thespaceofthepossiblestatesofasystem–containsaninfinitenumberofdistinguishableclassicstates,butalwaysonlyafinitenumberoforthogonalquantumstates.Thisnumberisgivenbythevolumeoftheregion,dividedbythePlanckconstant,raisedtothenumberofdegreesoffreedom.Thisresultisgeneral.fn27OrFeynman’sintegral.TheprobabilityofgoingfromAtoBisthesquaremoduleoftheintegraloverallthepathsoftheexponentialoftheclassicalactionofthetrajectory,multipliedbytheimaginaryunitanddividedbyPlanck’sconstant.fn28Amechanismintheboxopensthesmallwindowontherightforaninstant,allowingaphotontoescapeatsomeprecisetime.Byweighingthebox,itispossibletodeducetheenergyofthereleasedphoton.Einsteinhopedthatthiswouldcreatedifficultiesforquantummechanics,whichpredictsthattimeandenergycannotbebothpreciselydetermined.Bohrreplied,mistakenly,thatthewayoutofthedifficultyrequiredEinstein’sgeneralrelativity,andEinstein,mistakenly,acceptedBohr’sreply.ThecorrectresponsetoEinstein,whichBohrwasunabletofindbutthatiscleartoday,isthatthepositionoftheescapingphotonandtheweightoftheboxremaintiedtoeachother(‘correlated’),evenifthephotonisalreadyfaraway.

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fn29ThemarkonthehofPlanck’sconstantservesonlytoindicatethatPlanck’sconstantisinthisequationdividedby2π,aratheruselessandidiosyncraticadditionbytheoreticalphysicists:placingthesmall,hard,angularmarkontheh‘makesitelegant’.fn30Tohearthismetaphordirectlyinhisownvoice,gotothesitehttp://www.webofstories.com/play/9542?o=MS.fn31DeWittreplacesderivativeswithderivativeoperatorsintheHamilton–Jacobiequationforgeneralrelativity(writtenalittlewhileearlierbyPeres).Thatis,hedoeswhatSchrödingerhaddonetowritehisequation,inhisfirstwork:replacingderivativeswithderivativeoperatorsintheHamilton–Jacobiequationofaparticle.fn32Orthe‘Einstein–Schrödinger’equation.fn33Thebestknownalternativetoloopquantumgravityisstringtheory,whosemainconcernisnotsomuchstudyingthequantumpropertiesofspaceandtime,butratherwritingaunifiedtheoryofallknownfields,anobjectivethatmightbeprematuregivencurrentknowledge.

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fn34Theeigenvalueequationforthevolumeoperator.fn35Hencethequantumstatesofgravityareindicatedwith|jl,vn>,wherenindicatesthenodesandlthelinksofthegraph.fn36ImaginewhatanonsensicalhotchpotchtheideasofAristotleandPlatowouldseemifweonlyhadthecommentariesonthemwrittenbyothersandwereunabletoaccessthelucidityandcomplexityoftheoriginaltexts!fn37ThequantumnumberofthestatesofphotonsinFock’sspaceisthemomentum,Fourier’stransformationofposition.fn38Theoperatorassociatedwiththegeometryofgranularspaceistheholonomyofthegravitationalconnections,orrather,inphysicalterms,a‘Wilsonloop’ofgeneralrelativity.

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fn39Thegravitationalpotential.fn40x(t)=½at².fn41Especiallysincehehadbecomeexcited…fn42Theactualstructureoftheverticesofthespinfoamisabitmorecomplexthantheoneinfigure7.2,andresemblesmorecloselytheoneshowninfigure7.4.fn43It’saFeynmandiagrambecauseitisahistoryofquanta,asintheFeynmandiagrams.Exceptthatnow,thequantaarenotquantamovinginspace,butratherquantaofspace.Thegraphtheydrawintheirinteractionsisnotarepresentationofthemovementofparticlesinspace,butrepresentstheplotofspaceitself.Buttheresultingpictureisalsopreciselyalatticeliketheoneusedinthelatticeapproximation,becauseitrepresentsadiscretizedspacetime.Withthedifferencethatitisnolongeranapproximation,buttherealdiscretestructureofspaceatasmallscale.fn44ThefirstdefinestheHilbertspaceofthetheory.Theseconddescribesthealgebraoftheoperators.Thethirddescribesthesizeoftransitionofeachvertex,suchastheoneshowninfigure7.4.fn45‘[…]allthedifferentelementaryparticlescouldbereducedtooneuniversalsubstancewhichcouldequallybecalledenergyormatter,andnoneoftheparticlesshouldbeprivilegedandconsideredmorefundamental.ThispointofviewcorrespondstoAnaximander’sdoctrine,andIamconvincedthatinmodernphysicsthisisthecorrectpointofview.’WernerHeisenberg,PhysicsandPhilosophy:TheRevolutioninModernScience(NewYork,Harper&Row,1962).

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fn46Thisisaninterferometer:itusestheinterferencebetweenthelaserswhichrunalongthetwoarmstorevealtheminutevariationsinlengthofthesearms.

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fn47TheEcclesiasticusisconsideredpartoftheBiblebyCatholics,mostoftheOrientalOrthodoxChurchandsomeJews.TheLutheranchurchesincludeitintheirlectionaries,andasabookproperforreading,devotionandprayer,butnotintheBible.FormostJewsand.theAnglicanChurchthesituationissimilar.

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fn48Asubtlepoint:informationdoesn’tmeasurewhatIknowbutthenumberofpossiblealternatives.TheinformationIamgivenwhenthenumber3comesupinrouletteisN=37,becausethereare37numbers;buttheinformationIhavewhennumber3winsonredisN=18,becausethereare18rednumbers.HowmuchinformationdowehaveifwelearnwhichofthebrothersKaramazovmurderedtheirfather?TheanswerdependsonhowmanyKaramazovbrothersthereare.fn49Boltzmanndidnotusetheconceptofinformation,buthisworkcanbereadinthisway.fn50Entropyisproportionaltothelogarithmofthevolumeofthephasespace.Theconstantofproportionality,k,isBoltzmann’sconstant,whichtransformstheunitsofmeasurementforinformation(bits)intotheunitsofmeasurementforentropy(Kelvin’sjoules).fn51Inafiniteregionofitsphasespace.fn52Thisiswhatcametobecalled,inappropriately,the‘collapse’ofthewavefunction.fn53Hereishowitworkstechnically:aBoltzmannstatisticalstateisdescribedbyafunctiononphasespacegivenbytheexponentialoftheHamiltonian.TheHamiltonianisthegeneratoroftimeevolution.Inasysteminwhichtimeisnotdefined,thereisnoHamiltonian.Butifwehaveastatisticalstate,wejusttakeitslogarithmandthisdefinesaHamiltonian,andhenceanotionoftime.