MOREPRAISEFORSOMETHINGDEEPLYHIDDEN

“SeanCarrollisalwayslucidandfunny,gratifyinglyreadable,whilestillexcavatingdepths.Headvocates an acceptance of quantum mechanics at its most minimal, its most austere—appealingtotheallureofthepristine.Theconsequenceisanannihilationofourconventionalnotionsofreality in favorofanutterlysurrealworldofMany-Worlds.Sean includesus in thebattlebetweenasimplerealityversusamultitudeofrealitiesthatfeelsbarelyontheperipheryof human comprehension. He includes us in the ideas, the philosophy, and the foment ofrevolution.Afascinatingandimportantbook.”

—JannaLevin,professorofphysicsandastronomyatBarnardCollegeandauthorofBlackHoleBlues

“SeanCarrollbeautifullyclarifiesthedebateaboutthefoundationsofquantummechanicsandchampions the most elegant, courageous approach: the astonishing ‘Many-Worlds’interpretation.Hisexplanationsofitsprosandconsareclear,evenhanded,andphilosophicallygob-smacking.”

—StevenStrogatz,professorofmathematicsatCornellUniversityandauthorofInfinitePowers

“Carroll gives us a front-row seat to the development of a new vision of physics: one thatconnects our everyday experiences to a dizzying hall-of-mirrors universe in which our verysenseofselfischallenged.It’safascinatingideaandonethatjustmightholdcluestoadeeperreality.”—KatieMack,theoreticalastrophysicistatNorthCarolinaStateUniversityandauthoroftheforthcomingThe

EndofEverything

“Iwasoverwhelmedbytearsofjoyatseeingsomanyfundamentalissuesexplainedaswellasthey ever have been. Something Deeply Hidden is a masterpiece, which stands along withFeynman’sQED asoneof the twobestpopularizationsofquantummechanics I’veeverseen.And ifweclassifyQED ashavinghaddifferentgoals, then it’s just thebestpopularizationofquantummechanicsI’veeverseen,fullstop.”

—ScottAaronson,professorofcomputerscienceattheUniversityofTexasatAustinanddirectorofUT’sQuantumInformationCenter

“Irresistible and an absolute treat to read. While this is a book about some of the deepestcurrentmysteriesinphysics,itisalsoabookaboutmetaphysics,asCarrolllucidlyguidesusonhowtonotonlythinkaboutthetrueandhiddennatureofrealitybutalsohowtomakesenseofit.Ilovedthisbook.”

—PriyamvadaNatarajan,theoreticalastrophysicistatYaleUniversityandauthorofMappingtheHeavens

SOMETHINGDEEPLYHIDDEN

ALSOBYSEANCARROLL

FromEternitytoHereTheParticleattheEndoftheUniverse

TheBigPicture

SOMETHINGDEEPLYHIDDENQuantumWorldsandtheEmergenceofSpacetime

SEANCARROLL

Tothinkersthroughouthistorywhostucktotheirgunsfortherightreasons

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CONTENTS

Prologue:Don’tBeAfraid

PartOneSPOOKY

What’sGoingOn:LookingattheQuantumWorldTheCourageousFormulation:AustereQuantumMechanicsWhyWouldAnybodyThinkThis?HowQuantumMechanicsCametoBeWhatCannotBeKnown,BecauseItDoesNotExist:UncertaintyandComplementarityEntangledUpinBlue:WaveFunctionsofManyParts

PartTwoSPLITTING

SplittingtheUniverse:DecoherenceandParallelWorldsOrderandRandomness:WhereProbabilityComesFromDoesThisOntologicalCommitmentMakeMeLookFat?ASocraticDialogueonQuantumPuzzlesOtherWays:AlternativestoMany-WorldsTheHumanSide:LivingandThinkinginaQuantumUniverse

PartThreeSPACETIME

WhyIsThereSpace?EmergenceandLocalityAWorldofVibrations:QuantumFieldTheoryBreathinginEmptySpace:FindingGravitywithinQuantumMechanicsBeyondSpaceandTime:Holography,BlackHoles,andtheLimitsofLocality

Epilogue:EverythingIsQuantum

Appendix:TheStoryofVirtualParticlesAcknowledgmentsFurtherReadingReferencesIndex

SOMETHINGDEEPLYHIDDEN

PROLOGUE

Don’tBeAfraid

Youdon’tneedaPhDintheoreticalphysicstobeafraidofquantummechanics.Butitdoesn’thurt.

Thatmightseemstrange.Quantummechanicsisourbesttheoryofthemicroscopicworld.Itdescribeshowatomsandparticlesinteractthroughtheforcesofnature,andmakesincrediblypreciseexperimentalpredictions.Tobesure,quantummechanicshassomethingofareputationforbeingdifficult,mysterious,justthissideofmagic.Butprofessionalphysicists,ofallpeople,should be relatively comfortable with a theory like that. They are constantly doing intricatecalculations involving quantumphenomena, and building giantmachines dedicated to testingtheresultingpredictions.Surelywe’renotsuggestingthatphysicistshavebeenfakingitallthistime?

Theyhaven’tbeen faking,but theyhaven’texactlybeenhonestwith themselveseither.Onthe one hand, quantum mechanics is the heart and soul of modern physics. Astrophysicists,particle physicists, atomic physicists, laser physicists—everyone uses quantum mechanics allthe time, and they’re very good at it. It’s not just a matter of esoteric research. Quantummechanicsisubiquitousinmoderntechnology.Semiconductors,transistors,microchips,lasers,and computermemory all rely on quantummechanics to function. For thatmatter, quantummechanics is necessary to make sense of the most basic features of the world around us.Essentiallyallofchemistryisamatterofappliedquantummechanics.Tounderstandhowthesunshines,orwhytablesaresolid,youneedquantummechanics.

Imagine closing your eyes.Hopefully things look pretty dark. Youmight think thatmakessense,becauseno light is coming in.But that’snotquite right; infrared light,witha slightlylongerwavelengththanvisiblelight,isbeingemittedallthetimebyanywarmobject,andthatincludesyourownbody.Ifoureyeswereassensitivetoinfraredlightastheyaretovisiblelight,wewouldbeblindedevenwhenourlidswereclosed,fromallthelightemittedbyoureyeballsthemselves.Buttherodsandconesthatactaslightreceptorsinoureyesarecleverlysensitivetovisiblelight,notinfrared.Howdotheymanagethat?Ultimately,theanswercomesdowntoquantummechanics.

Quantummechanics isn’tmagic. It is the deepest,most comprehensive view of realitywehave.Asfaraswecurrentlyknow,quantummechanicsisn’tjustanapproximationofthetruth;it is the truth. That’s subject to change in the face of unexpected experimental results, butwe’veseennohintsofanysuchsurprisesthusfar.Thedevelopmentofquantummechanicsinthe early years of the twentieth century, involving names like Planck, Einstein, Bohr,Heisenberg,Schrödinger,andDirac,leftusby1927withamatureunderstandingthatissurelyoneofthegreatestintellectualaccomplishmentsinhumanhistory.Wehaveeveryreasontobeproud.

On theotherhand, in thememorablewordsofRichardFeynman, “I think I cansafely saythat nobody understands quantum mechanics.” We use quantum mechanics to design newtechnologies and predict the outcomes of experiments. But honest physicists admit that wedon’ttrulyunderstandquantummechanics.Wehavearecipethatwecansafelyapplyincertainprescribed situations, and which returns mind-bogglingly precise predictions that have beentriumphantlyvindicatedbythedata.Butifyouwanttodigdeeperandaskwhatisreallygoingon,wesimplydon’tknow.Physiciststendtotreatquantummechanicslikeamindlessrobottheyrelyontoperformcertaintasks,notasabelovedfriendtheycareaboutonapersonallevel.

Thisattitudeamongtheprofessionalsseepsintohowquantummechanicsgetsexplainedtothewiderworld.WhatwewouldliketodoistopresentafullyformedpictureofNature,butwecan’tquitedothat,sincephysicistsdon’tagreeaboutwhatquantummechanicsactuallysays.Instead,populartreatmentstendtoemphasizethatquantummechanicsismysterious,baffling,impossible tounderstand.Thatmessagegoesagainst thebasicprinciples that sciencestandsfor,whichincludetheideathattheworldisfundamentallyintelligible.Wehavesomethingofamentalblockwhenitcomestoquantummechanics,andweneedabitofquantumtherapytohelpgetpastit.

Whenwe teach quantummechanics to students, they are taught a list of rules. Some of therules are of a familiar type: there’s amathematical description of quantum systems, plus anexplanationofhowsuchsystemsevolveovertime.Butthenthereareabunchofextrarulesthathavenoanalogueinanyothertheoryofphysics.Theseextrarulestelluswhathappenswhenweobserveaquantumsystem,andthatbehavior iscompletelydifferent fromhowthesystembehaveswhenwe’renotobservingit.Whatintheworldisgoingonwiththat?

There are basically two options. One is that the story we’ve been telling our students iswoefully incomplete, and in order for quantummechanics to qualify as a sensible theoryweneed tounderstandwhata“measurement”or“observation” is,andwhy it seemssodifferentfromwhatthesystemdoesotherwise.Theotheroptionisthatquantummechanicsrepresentsaviolentbreakfromthewaywehavealwaysthoughtaboutphysicsbefore,shiftingfromaviewwheretheworldexistsobjectivelyandindependentlyofhowweperceive it, toonewheretheactofobservationissomehowfundamentaltothenatureofreality.

In either case, the textbooks should by all rights spend time exploring these options, andadmit that even though quantum mechanics is extremely successful, we can’t claim to befinisheddevelopingitjustyet.Theydon’t.Forthemostpart,theypassoverthisissueinsilence,preferring to stay in the physicist’s comfort zone of writing down equations and challengingstudentstosolvethem.

That’sembarrassing.Anditgetsworse.Youmightthink,giventhissituation,thatthequesttounderstandquantummechanicswould

be the single biggest goal in all of physics.Millions of dollars of grantmoneywould flow toresearchers inquantumfoundations, thebrightestmindswould flock to theproblem,and themost important insights would be rewarded with prizes and prestige. Universities wouldcompetetohiretheleadingfiguresinthearea,danglingsuperstarsalariestolurethemawayfromrivalinstitutions.

Sadly,no.Notonlyisthequesttomakesenseofquantummechanicsnotconsideredahigh-statusspecialtywithinmodernphysics; inmanyquarters it’sconsideredbarelyrespectableatall,ifnotactivelydisparaged.Mostphysicsdepartmentshavenobodyworkingontheproblem,andthosewhochoosetodosoarelookeduponwithsuspicion.(Recentlywhilewritingagrantproposal, I was advised to concentrate on describingmywork in gravitation and cosmology,whichisconsideredlegitimate,andremainsilentaboutmyworkonthefoundationsofquantummechanics, as that would make me appear less serious.) There have been important stepsforwardoverthelastninetyyears,buttheyhavetypicallybeenmadebyheadstrongindividualswhothoughttheproblemswereimportantdespitewhatalloftheircolleaguestoldthem,orbyyoungstudentswhodidn’tknowanybetterandlaterleftthefieldentirely.

InoneofAesop’sfables,afoxseesajuicybunchofgrapesandleapstoreachit,butcan’tquitejumphighenough.Infrustrationhedeclaresthatthegrapeswereprobablysour,andhenever really wanted them anyway. The fox represents “physicists,” and the grapes are“understandingquantummechanics.”Manyresearchershavedecidedthatunderstandinghownaturereallyworkswasneverreallyimportant;allthatmattersistheabilitytomakeparticularpredictions.

Scientists are trained to value tangible results, whether they are exciting experimentalfindings or quantitative theoretical models. The idea of working to understand a theory wealreadyhave,evenifthateffortmightnotleadtoanyspecificnewtechnologiesorpredictions,canbeatoughsell.Theunderlyingtensionwas illustratedintheTVshowTheWire,whereagroupofhardworkingdetectiveslaboredformonthstometiculouslygatherevidencethatwouldbuildacaseagainstapowerfuldrugring.Theirbosses,meanwhile,hadnopatience forsuchincrementalfrivolity.Theyjustwantedtoseedrugsonthetablefortheirnextpressconference,and encouraged the police to bang heads and make splashy arrests. Funding agencies andhiring committees are like those bosses. In aworldwhere all the incentives push us towardconcrete,quantifiableoutcomes,lesspressingbig-pictureconcernscanbepushedasideasweracetowardthenextimmediategoal.

This book has three main messages. The first is that quantum mechanics should beunderstandable—even if we’re not there yet—and achieving such understanding should be ahigh-prioritygoalofmodernscience.Quantummechanicsisuniqueamongphysicaltheoriesindrawinganapparentdistinctionbetweenwhatweseeandwhatreallyis.Thatposesaspecialchallengetothemindsofscientists(andeveryoneelse),whoareusedtothinkingaboutwhatwe see as unproblematically “real,” and working to explain things accordingly. But thischallenge isn’t insuperable, and ifwe freeourminds fromcertainold-fashionedand intuitivewaysofthinking,wefindthatquantummechanicsisn’thopelesslymysticalorinexplicable.It’s

justphysics.Thesecondmessageisthatwehavemaderealprogresstowardunderstanding.Iwillfocus

on the approach I feel is clearly the most promising route, the Everett or Many-Worldsformulationofquantummechanics.Many-Worldshasbeenenthusiasticallyembracedbymanyphysicists,but ithasasketchyreputationamongpeoplewhoareputoffbyaproliferationofotherrealitiescontainingcopiesofthemselves.Ifyouareoneofthosepeople,IwanttoatleastconvinceyouthatMany-Worldsisthepurestwayofmakingsenseofquantummechanics—it’swhereweendup ifwe just follow thepathof least resistance in takingquantumphenomenaseriously. Inparticular, themultipleworldsarepredictionsof the formalismthat isalready inplace,notsomethingadded inbyhand.ButMany-Worlds isn’t theonlyrespectableapproach,andwewillmentionsomeofitsmaincompetitors.(Iwillendeavortobefair,ifnotnecessarilybalanced.)Theimportantthingisthatthevariousapproachesareallwell-constructedscientifictheories, with potentially different experimental ramifications, not just woolly-headed“interpretations”tobedebatedovercognacandcigarsafterwe’refinisheddoingrealwork.

The third message is that all this matters, and not just for the integrity of science. Thesuccess to date of the existing adequate-but-not-perfectly-coherent framework of quantummechanics shouldn’t blind us to the fact that there are circumstances under which such anapproachsimplyisn’tuptothetask.Inparticular,whenweturntounderstandingthenatureofspacetime itself, and the origin and ultimate fate of the entire universe, the foundations ofquantum mechanics are absolutely crucial. I’ll introduce some new, exciting, and admittedlytentativeproposalsthatdrawprovocativeconnectionsbetweenquantumentanglementandhowspacetime bends and curves—the phenomenon you and I know as “gravity.” For many yearsnow,thesearchforacompleteandcompellingquantumtheoryofgravityhasbeenrecognizedasan importantscientificgoal (prestige,prizes,stealingawayfaculty,andall that). Itmaybethat the secret is not to startwith gravity and “quantize” it, but to dig deeply into quantummechanicsitself,andfindthatgravitywaslurkingthereallalong.

Wedon’tknowforsure.That’stheexcitementandanxietyofcutting-edgeresearch.Butthetimehascometotakethefundamentalnatureofrealityseriously,andthatmeansconfrontingquantummechanicshead-on.

1What’sGoingOn

LookingattheQuantumWorld

AlbertEinstein,whohadawaywithwordsaswellaswithequations,was theonewhostuckquantum mechanics with the label it has been unable to shake ever since: spukhaft, usuallytranslated fromGerman toEnglish as “spooky.” If nothing else, that’s the impressionwe getfrommostpublicdiscussionsofquantummechanics.We’retoldthatit’sapartofphysicsthatisunavoidablymystifying,weird,bizarre,unknowable,strange,baffling.Spooky.

Inscrutabilitycanbealluring.Likeamysterious,sexystranger,quantummechanicstemptsusintoprojectingallsortsofqualitiesandcapacitiesontoit,whethertheyarethereornot.Abrief search for books with “quantum” in the title reveals the following list of purportedapplications:

QuantumSuccessQuantumLeadershipQuantumConsciousnessQuantumTouchQuantumYogaQuantumEatingQuantumPsychologyQuantumMindQuantumGloryQuantumForgivenessQuantumTheologyQuantumHappinessQuantumPoetryQuantumTeachingQuantumFaithQuantumLove

For a branch of physics that is often described as only being relevant to microscopicprocessesinvolvingsubatomicparticles,that’saprettyimpressiverésumé.

Tobe fair,quantummechanics—or“quantumphysics,”or“quantumtheory,” the labelsareallinterchangeable—isnotonlyrelevanttomicroscopicprocesses.Itdescribesthewholeworld,fromyouandmetostarsandgalaxies,fromthecentersofblackholestothebeginningoftheuniverse. But it is only when we look at the world in extreme close-up that the apparentweirdnessofquantumphenomenabecomesunavoidable.

Oneofthethemesinthisbookisthatquantummechanicsdoesn’tdeservetheconnotationofspookiness, in the sense of some ineffable mystery that it is beyond the human mind tocomprehend.Quantummechanicsisamazing;itisnovel,profound,mind-stretching,andaverydifferent view of reality from what we’re used to. Science is like that sometimes. But if thesubjectseemsdifficultorpuzzling,thescientificresponseistosolvethepuzzle,nottopretendit’snotthere.There’severyreasontothinkwecandothatforquantummechanicsjustlikeanyotherphysicaltheory.

Manypresentationsofquantummechanicsfollowatypicalpattern.First,theypointtosomecounterintuitive quantum phenomenon. Next, they express bafflement that the world canpossiblybethatway,anddespairofitmakingsense.Finally(ifyou’relucky),theyattemptsomesortofexplanation.

Our theme isprizingclarityovermystery, so Idon’twant toadopt thatstrategy. Iwant topresentquantummechanicsinawaythatwillmakeitmaximallyunderstandablerightfromthestart.Itwillstillseemstrange,butthat’sthenatureofthebeast.Whatitwon’tseem,hopefully,isinexplicableorunintelligible.

We will make no effort to follow historical order. In this chapter we’ll look at the basicexperimentalfactsthatforcequantummechanicsuponus,andinthenextwe’llquicklysketch

theMany-Worldsapproachtomakingsenseofthoseobservations.Onlyinthechapterafterthatwillweofferasemi-historicalaccountofthediscoveriesthatledpeopletocontemplatesuchadramatically new kind of physics in the first place. Then we’ll hammer home exactly howdramaticsomeoftheimplicationsofquantummechanicsreallyare.

Withallthatinplace,overtherestofthebookwecansetaboutthefuntaskofseeingwhereallthisleads,demystifyingthemoststrikingfeaturesofquantumreality.

Physics isoneofthemostbasicsciences, indeedoneofthemostbasichumanendeavors.Welookaroundtheworld,weseeitisfullofstuff.Whatisthatstuff,andhowdoesitbehave?

These are questions that have been asked ever since people started asking questions. Inancient Greece, physics was thought of as the general study of change and motion, of bothlivingandnonlivingmatter.Aristotlespokeavocabularyof tendencies,purposes,andcauses.How an entity moves and changes can be explained by reference to its inner nature and toexternalpowersactinguponit.Typicalobjects,forexample,mightbynaturebeatrest;inorderforthemtomove,itisnecessarythatsomethingbecausingthatmotion.

All of this changed thanks to a clever chap named Isaac Newton. In 1687 he publishedPrincipiaMathematica,themostimportantworkinthehistoryofphysics.Itwastherethathelaidthegroundworkforwhatwenowcall“classical”orsimply“Newtonian”mechanics.Newtonblew away any dusty talk of natures and purposes, revealing what lay underneath: a crisp,rigorousmathematicalformalismwithwhichteacherscontinuetotormentstudentstothisveryday.

Whatever memory you may have of high-school or college homework assignments dealingwithpendulumsand inclinedplanes, thebasic ideasofclassicalmechanicsareprettysimple.Consider an object such as a rock. Ignore everything about the rock that a geologist mightconsiderinteresting,suchasitscolorandcomposition.Putasidethepossibilitythatthebasicstructure of the rockmight change, for example, if you smashed it topieceswith ahammer.Reduceyourmentalimageoftherockdowntoitsmostabstractform:therockisanobject,andthatobjecthasalocationinspace,andthatlocationchangeswithtime.

Classicalmechanicstellsuspreciselyhowthepositionoftherockchangeswithtime.We’reveryusedtothatbynow,soit’sworthreflectingonhowimpressivethisis.Newtondoesn’thandussomevagueplatitudesaboutthegeneraltendencyofrockstomovemoreor less inthisorthatfashion.Hegivesusexact,unbreakablerulesforhoweverythingintheuniversemovesinresponsetoeverythingelse—rulesthatcanbeusedtocatchbaseballsorlandroversonMars.

Here’showitworks.Atanyonemoment,therockwillhaveapositionandalsoavelocity,arateatwhichit’smoving.AccordingtoNewton,ifnoforcesactontherock,itwillcontinuetomoveinastraightlineatconstantvelocity,foralltime.(AlreadythisisamajordeparturefromAristotle,whowouldhavetoldyouthatobjectsneedtobeconstantlypushediftheyaretobekeptinmotion.)Ifaforcedoesactontherock,itwillcauseacceleration—somechangeinthevelocityoftherock,whichmightmakeitgofaster,orslower,ormerelyalter itsdirection—indirectproportiontohowmuchforceisapplied.

That’s basically it. To figure out the entire trajectory of the rock, you need to tell me itsposition, its velocity, and what forces are acting on it. Newton’s equations tell you the rest.Forcesmightincludetheforceofgravity,ortheforceofyourhandifyoupickuptherockandthrowit,ortheforcefromthegroundwhentherockcomestoland.Theideaworksjustaswellfor billiard balls or rocket ships or planets. The project of physics, within this classicalparadigm,consistsessentiallyoffiguringoutwhatmakesupthestuffoftheuniverse(rocksandsoforth)andwhatforcesactonthem.

Classical physics provides a straightforward picture of the world, but a number of crucialmovesweremadealongthewaytosettingitup.Noticethatwehadtobeveryspecificaboutwhat information we required to figure out what would happen to the rock: its position, its

velocity,andtheforcesactingon it.Wecanthinkof thoseforcesasbeingpartof theoutsideworld,andtheimportantinformationabouttherockitselfasconsistingofjustitspositionandvelocity.Theaccelerationoftherockatanymomentintime,bycontrast,isnotsomethingweneedtospecify;that’sexactlywhatNewton’s lawsallowustocalculatefromthepositionandthevelocity.

Together,thepositionandvelocitymakeupthestateofanyobjectinclassicalmechanics.Ifwehaveasystemwithmultiplemovingparts,theclassicalstateofthatentiresystemisjustalistofthestatesofeachoftheindividualparts.Theairinanormal-sizedroomwillhaveperhaps1027moleculesofdifferent types,and thestateof thatairwouldbea listof thepositionandvelocityofeveryoneof them.(Strictlyspeaking,physicists liketousethemomentumofeachparticle, rather than its velocity, but as far as Newtonian mechanics is concerned themomentumissimplytheparticle’smasstimesitsvelocity.)Thesetofallpossiblestatesthatasystemcouldhaveisknownasthephasespaceofthesystem.

TheFrenchmathematicianPierre-SimonLaplacepointedoutaprofound implicationof theclassicalmechanicswayofthinking.Inprinciple,avastintellectcouldknowthestateofliterallyeveryobject intheuniverse, fromwhich itcoulddeduceeverythingthatwouldhappen inthefuture, as well as everything that had happened in the past. Laplace’s demon is a thoughtexperiment,notarealisticproject foranambitiouscomputerscientist,butthe implicationsofthe thought experiment are profound. Newtonian mechanics describes a deterministic,clockworkuniverse.

The machinery of classical physics is so beautiful and compelling that it seems almostinescapableonceyougraspit.ManygreatmindswhocameafterNewtonwereconvincedthatthe basic superstructure of physics had been solved, and future progress lay in figuring outexactlywhatrealizationofclassicalphysics(whichparticles,whichforces)wastherightonetodescribetheuniverseasawhole.Evenrelativity,whichwasworld-transforminginitsownway,isavarietyofclassicalmechanicsratherthanareplacementforit.

Thenalongcamequantummechanics,andeverythingchanged.

Alongside Newton’s formulation of classical mechanics, the invention of quantum mechanicsrepresentstheothergreatrevolutioninthehistoryofphysics.Unlikeanythingthathadcomebefore, quantum theory didn’t propose a particular physical model within the basic classicalframework; it discarded that framework entirely, replacing it with something profoundlydifferent.

Thefundamentalnewelementofquantummechanics,thethingthatmakesitunequivocallydistinct from its classical predecessor, centers on the question of what it means tomeasuresomethingaboutaquantumsystem.Whatexactlyameasurement is,andwhathappenswhenwe measure something, and what this all tells us about what’s really happening behind thescenes:together,thesequestionsconstitutewhat’scalledthemeasurementproblemofquantummechanics.Thereisabsolutelynoconsensuswithinphysicsorphilosophyonhowtosolvethemeasurementproblem,althoughthereareanumberofpromisingideas.

Attemptstoaddressthemeasurementproblemhaveledtotheemergenceofafieldknownasthe interpretation of quantum mechanics, although the label isn’t very accurate.“Interpretations” are things thatwemight apply to awork of literature or art,wherepeoplemighthavedifferentwaysofthinkingaboutthesamebasicobject.What’sgoingoninquantummechanics is something else: a competition between truly distinct scientific theories,incompatiblewaysofmakingsenseof thephysicalworld.For thisreason,modernworkers inthis field prefer to call it “foundations of quantum mechanics.” The subject of quantumfoundationsispartofscience,notliterarycriticism.

Nobody ever felt the need to talk about “interpretations of classical mechanics”—classicalmechanicsisperfectlytransparent.Thereisamathematicalformalismthatspeaksofpositionsandvelocitiesandtrajectories,andoh, look:thereisarockwhoseactualmotionintheworldobeysthepredictionsofthatformalism.Thereis,inparticular,nosuchthingasameasurementprobleminclassicalmechanics.Thestateofthesystemisgivenbyitspositionanditsvelocity,and ifwewant tomeasure thosequantities,wesimplydoso.Ofcourse,wecanmeasure thesystemsloppilyorcrudely,therebyobtainingimpreciseresultsoralteringthesystemitself.Butwedon’thaveto; justbybeingcareful,wecanpreciselymeasureeverythingthereistoknowabout thesystemwithoutaltering it inanynoticeableway.Classicalmechanicsoffersaclearandunambiguousrelationshipbetweenwhatweseeandwhatthetheorydescribes.

Quantummechanics, forall itssuccesses,offersnosuchthing.Theenigmaat theheartofquantumrealitycanbesummedupinasimplemotto:whatweseewhenwelookattheworldseemstobefundamentallydifferentfromwhatactuallyis.

Thinkaboutelectrons, theelementaryparticlesorbitingatomicnuclei,whose interactionsareresponsibleforallofchemistryandhencealmosteverythinginterestingaroundyourightnow.Aswedidwiththerock,wecanignoresomeoftheelectron’sspecificproperties,likeitsspinandthefactthatithasanelectricfield.(Reallywecouldjuststickwiththerockasourexample—rocks are quantum systems just as much as electrons are—but switching to a subatomicparticlehelpsus remember that the featuresdistinguishingquantummechanicsonlybecomeevidentwhenweconsiderverytinyobjectsindeed.)

Unlike inclassicalmechanics,where thestateofa system isdescribedby itspositionandvelocity,thenatureofaquantumsystemissomethingabitlessconcrete.Consideranelectroninitsnaturalhabitat,orbitingthenucleusofanatom.Youmightthink,fromtheword“orbit”aswellasfromthenumerouscartoondepictionsofatomsyouhavedoubtlessbeenexposedtooverthe years, that the orbit of an electron is more or less like the orbit of a planet in the solarsystem.Theelectron(soyoumightthink)hasa location,andavelocity,andastimepasses itzipsaroundthecentralnucleusinacircleormaybeanellipse.

Quantummechanicssuggestssomethingdifferent.Wecanmeasurevaluesofthelocationorvelocity (though not at the same time), and if we are sufficiently careful and talentedexperimenterswewillobtainsomeanswer.Butwhatwe’reseeingthroughsuchameasurementis not the actual, complete, unvarnished state of the electron. Indeed, the particularmeasurement outcome we will obtain cannot be predicted with perfect confidence, in aprofounddeparturefromtheideasofclassicalmechanics.Thebestwecandoistopredicttheprobabilityofseeingtheelectroninanyparticularlocationorwithanyparticularvelocity.

The classical notion of the state of a particle, “its location and its velocity,” is thereforereplacedinquantummechanicsbysomethingutterlyalientooureverydayexperience:acloudofprobability.Foranelectroninanatom,thiscloudismoredensetowardthecenterandthinsoutaswegetfartheraway.Wherethecloudisthickest,theprobabilityofseeingtheelectronishighest;where it isdilutedalmost to imperceptibility, theprobabilityofseeingtheelectron isvanishinglysmall.

Thiscloudisoftencalledawavefunction,becauseitcanoscillatelikeawave,asthemostprobablemeasurementoutcomechangesover time.Weusuallydenoteawave functionbyΨ,the Greek letter Psi. For every possible measurement outcome, such as the position of theparticle,thewavefunctionassignsaspecificnumber,calledtheamplitudeassociatedwiththatoutcome.Theamplitude that aparticle is at somepositionx0, for example,would bewrittenΨ(x0).

The probability of getting that outcome when we perform a measurement is given by theamplitudesquared.

Probabilityofaparticularoutcome=|Amplitudeforthatoutcome|2

1.2.

1.2.

3.

4.

ThissimplerelationiscalledtheBornrule,afterphysicistMaxBorn.*Partofourtaskwillbetofigureoutwhereintheworldsucharulecamefrom.

We’remostdefinitelynotsayingthatthereisanelectronwithsomepositionandvelocity,andwe just don’t know what those are, so the wave function encapsulates our ignorance aboutthosequantities.Inthischapterwe’renotsayinganythingatallaboutwhat“is,”onlywhatweobserve.Inchapterstocome,Iwillpoundthetableandinsistthatthewavefunctionisthesumtotalofreality,andideassuchasthepositionorthevelocityoftheelectronaremerelythingswecanmeasure.Butnoteveryoneseesthingsthatway,andforthemomentwearechoosingtodonamaskofimpartiality.

Let’s place the rules of classical and quantummechanics side by side to compare them. Thestateofaclassicalsystemisgivenbythepositionandvelocityofeachof itsmovingparts.Tofollowitsevolution,weimaginesomethinglikethefollowingprocedure:

RulesofClassicalMechanics

Setupthesystembyfixingaspecificpositionandvelocityforeachpart.EvolvethesystemusingNewton’slawsofmotion.

That’sit.Thedevilisinthedetails,ofcourse.Someclassicalsystemscanhavealotofmovingpieces.

Incontrast,therulesofstandardtextbookquantummechanicscomeintwoparts.Inthefirstpart,wehaveastructurethatexactlyparallelsthatoftheclassicalcase.Quantumsystemsaredescribedbywave functions rather thanbypositionsandvelocities. JustasNewton’s lawsofmotiongoverntheevolutionofthestateofasysteminclassicalmechanics,thereisanequationthat governs how wave functions evolve, called Schrödinger’s equation. We can expressSchrödinger’sequation inwordsas:“Therateofchangeofawavefunction isproportional totheenergyofthequantumsystem.”Slightlymorespecifically,awavefunctioncanrepresentanumberofdifferentpossibleenergies,andtheSchrödingerequationsaysthathigh-energypartsof the wave function evolve rapidly, while low-energy parts evolve very slowly. Which makessense,whenwethinkaboutit.

Whatmattersforourpurposesissimplythatthereissuchanequation,onethatpredictshowwavefunctionsevolvesmoothlythroughtime.Thatevolutionisaspredictableandinevitableasthe way objects move according to Newton’s laws in classical mechanics. Nothing weird ishappeningyet.

Thebeginningofthequantumrecipereadssomethinglikethis:

RulesofQuantumMechanics(PartOne)

SetupthesystembyfixingaspecificwavefunctionΨ.EvolvethesystemusingSchrödinger’sequation.

Sofar,sogood—thesepartsofquantummechanicsexactlyparalleltheirclassicalpredecessors.Butwhereastherulesofclassicalmechanicsstopthere,therulesofquantummechanicskeepgoing.

All theextrarulesdealwithmeasurement.Whenyouperformameasurement,suchasthepositionor spinofaparticle,quantummechanics says thereareonlycertainpossible resultsyouwill ever get. You can’t predictwhich of the results itwill be, but you can calculate theprobabilityforeachallowedoutcome.Andafteryourmeasurementisdone,thewavefunctioncollapses to a completely different function, with all of the new probability concentrated onwhateverresultyoujustgot.Soifyoumeasureaquantumsystem,ingeneralthebestyoucando is predict probabilities for various outcomes, but if you were to immediately measure thesame quantity again, you will always get the same answer—the wave function has collapsedontothatoutcome.

Let’swritethisoutingorydetail.

RulesofQuantumMechanics(PartTwo)

Therearecertainobservablequantitieswecanchoosetomeasure,suchasposition,andwhenwedomeasurethem,weobtaindefiniteresults.The probability of getting any one particular result can be calculated from the wave

5.

function. The wave function associates an amplitude with every possible measurementoutcome;theprobabilityforanyoutcomeisthesquareofthatamplitude.Uponmeasurement, thewave function collapses.However spread out itmayhavebeenpre-measurement,afterwarditisconcentratedontheresultweobtained.

In a modern university curriculum, when physics students are first exposed to quantummechanics,theyaretaughtsomeversionofthesefiverules.Theideologyassociatedwiththispresentation—treat measurements as fundamental, wave functions collapse when they areobserved,don’taskquestionsaboutwhat’sgoingonbehindthescenes—issometimescalledtheCopenhagen interpretation of quantum mechanics. But people, including the physicists fromCopenhagenwhopurportedlyinventedthisinterpretation,disagreeonpreciselywhatthatlabelshouldbetakentodescribe.Wecanjustrefertoitas“standardtextbookquantummechanics.”

Theideathattheserulesrepresenthowrealityactuallyworksis,needlesstosay,outrageous.Whatpreciselydoyoumeanbya“measurement”?Howquicklydoesithappen?Whatexactly

constitutes a measuring apparatus? Does it need to be human, or have some amount ofconsciousness, or perhaps the ability to encode information? Or maybe it just has to bemacroscopic, and if so how macroscopic does it have to be? When exactly does themeasurement occur, and how quickly? How in the world does the wave function collapse sodramatically?Ifthewavefunctionwereveryspreadout,doesthecollapsehappenfasterthanthespeedoflight?Andwhathappenstoallthepossibilitiesthatwereseeminglyallowedbythewavefunctionbutwhichwedidn’tobserve?Weretheyneverreallythere?Dotheyjustvanishintonothingness?

Toputthingsmostpointedly:Whydoquantumsystemsevolvesmoothlyanddeterministicallyaccording to the Schrödinger equation as long as we aren’t looking at them, but thendramaticallycollapsewhenwedolook?Howdotheyknow,andwhydotheycare?(Don’tworry,we’regoingtoanswerallthesequestions.)

Science, most people think, seeks to understand the natural world. We observe thingshappening,andsciencehopestoprovideanexplanationforwhatisgoingon.

Initscurrenttextbookformulation,quantummechanicshasfailedinthisambition.Wedon’tknowwhat’sreallygoingon,oratleastthecommunityofprofessionalphysicistscannotagreeonwhatitis.Whatwehaveinsteadisarecipethatweenshrineintextbooksandteachtoourstudents.IsaacNewtoncouldtellyou,startingwiththepositionandvelocityofarockthatyouhavethrownintotheairintheEarth’sgravitationalfield,justwhatthesubsequenttrajectoryofthat rock was going to be. Analogously, starting with a quantum system prepared in someparticularway,therulesofquantummechanicscantellyouhowthewavefunctionwillchangeover time,andwhat theprobabilityofvariouspossiblemeasurementoutcomeswillbeshouldyouchoosetoobserveit.

Thefactthatthequantumrecipeprovidesuswithprobabilitiesratherthatcertaintiesmightbe annoying, but we could learn to live with it. What bugs us, or should, is our lack ofunderstandingaboutwhatisactuallyhappening.

Imagine that some devious genius figured out all the laws of physics, but rather thanrevealing them to the rest of the world, they programmed a computer to answer questionsconcerning specific physics problems, and put an interface to the program on a web page.Anyone who was interested could just surf over to that site, type in a well-posed physicsquestion,andgetthecorrectanswer.

Such a program would obviously be of great use to scientists and engineers. But havingaccess to the site wouldn’t qualify as understanding the laws of physics. We would have anoracle that was in the business of providing answers to specific questions, but we ourselveswouldbecompletelylackinginanyintuitiveideaoftheunderlyingrulesofthegame.Therestoftheworld’sscientists,presentedwithsuchanoracle,wouldn’tbemovedtodeclarevictory;theywouldcontinuewiththeirworkoffiguringoutwhatthelawsofnatureactuallywere.

Quantum mechanics, in the form in which it is currently presented in physics textbooks,

represents an oracle, not a true understanding.We can set up specific problems and answerthem,butwecan’thonestlyexplainwhat’shappeningbehindthescenes.Whatwedohaveareanumberofgoodideasaboutwhatthatcouldbe,andit’spasttimethatthephysicscommunitystartedtakingtheseideasseriously.

*There’s a slight technicality,whichwe’llmentionhere and thenprettymuch forget about: the amplitude for anygivenoutcomeisactuallyacomplexnumber,notarealnumber.Realnumbersaretheonesthatappearonthenumberline, any number between minus infinity and plus infinity. Anytime you take the square of a real number, you getanotherrealnumberthat isgreaterthanorequaltozero,soasfarasrealnumbersareconcernedthere’snosuchthing as the square root of a negative number. Mathematicians long ago realized that square roots of negativenumberswouldbe reallyuseful things tohave, so theydefined the “imaginaryunit” i as the square root of -1.Animaginary number is just a real number, called “the imaginary part,” times i. Then a complex number is just acombinationofarealnumberandanimaginaryone.Thelittlebarsinthenotation|Amplitude|2intheBornrulemeanthat we actually add the squares of the real and the imaginary parts. All that is just for the sticklers out there;henceforthwe’llbehappytosay“theprobabilityistheamplitudesquared”andbedonewithit.

2TheCourageousFormulationAustereQuantumMechanics

Theattitudeinculcatedintoyoungstudentsbymodernquantummechanicstextbookshasbeencompactly summarized by physicist N. David Mermin as “Shut up and calculate!” Merminhimself wasn’t advocating such a position, but others have. Every decent physicist spends agooddealoftimecalculatingthings,whatevertheirattitudetowardquantumfoundationsmightbe.Soreallytheadmonitioncouldbeshortenedtosimply“Shutup!”*Itwasn’talwaysthus.Quantummechanicstookdecadestopiecetogether,butroundedinto

its modern form around 1927. In that year, at the Fifth International Solvay Conference inBelgium, the world’s leading physicists came together to discuss the status and meaning ofquantumtheory.Bythattimetheexperimentalevidencewasclear,andphysicistswereatlonglastinpossessionofaquantitativeformulationoftherulesofquantummechanics.Itwastimetorollupsomesleevesandfigureoutwhatthiscrazynewworldviewactuallyamountedto.The discussions at this conference help set the stage, but our goal here isn’t to get the

historyright.Wewanttounderstandthephysics.Sowe’llsketchoutalogicalpathbywhichwewill be led to a full-blown scientific theory of quantum mechanics. No vague mysticism, noseemingly ad hoc rules. Just a simple set of assumptions leading to some remarkableconclusions. With this picture in mind, many things that might otherwise have seemedominouslymysteriouswillsuddenlystarttomakeperfectsense.

TheSolvayConferencehasgonedowninhistoryasthebeginningofafamousseriesofdebatesbetweenAlbertEinsteinandNielsBohroverhowto thinkaboutquantummechanics.Bohr,aDanishphysicistbasedinCopenhagenwhoisrightfullyregardedasthegodfatherofquantumtheory,advocatedanapproachsimilartothetextbookrecipewediscussedinthelastchapter:usequantummechanicstocalculatetheprobabilitiesformeasurementoutcomes,butdon’taskof it anything more than that. Do not, in particular, worry too much about what is reallyhappening behind the scenes. Supported by his younger colleagues Werner Heisenberg andWolfgangPauli,Bohrinsistedthatquantummechanicswasaperfectlyfinetheoryasitwas.Einstein would have none of it. He was firmly convinced that the duty of physics was

preciselytoaskwhatwasgoingonbehindthescenes,andthatthestateofquantummechanicsin1927fellfarshortofprovidingasatisfactoryaccountofnature.Withhisownsympathizers,suchasErwinSchrödingerandLouisdeBroglie,Einsteinadvocatedlookingmoredeeply,andattemptingtoextendandgeneralizequantummechanicsintoasatisfactoryphysicaltheory.

Participantsinthe1927SolvayConference.Amongthemorewell-knownwere:1.MaxPlanck,2.MarieCurie,3.PaulDirac,4.ErwinSchrödinger,5.AlbertEinstein,6.LouisdeBroglie,7.WolfgangPauli,8.MaxBorn,9.Werner

Heisenberg,and10.NielsBohr.(CourtesyofWikipedia)

Einstein and his compatriots had reason to be cautiously optimistic that such a new-and-improvedtheorywasouttheretobefound.Justafewdecadesbefore,inthelateryearsofthenineteenth century, physicists had developed the theory of statistical mechanics, whichdescribedthemotionoflargenumbersofatomsandmolecules.Akeystepinthatdevelopment—whichalltookplaceundertherubricofclassicalmechanics,beforequantumtheorycameonthescene—wastheideathatwecantalkprofitablyaboutthebehaviorofalargecollectionofparticlesevenifwedon’tknowpreciselythepositionandvelocityofeachoneofthem.Allweneedtoknowisaprobabilitydistributiondescribingthelikelihoodthattheparticlesmightbebehavinginvariousways.In statistical mechanics, in other words, we think that there actually is some particular

classical state of all the particles, but we don’t know it, all we have is a distribution ofprobabilities.Happily, such a distribution is allweneed to do a great deal of useful physics,since it fixes properties such as the temperature and pressure of the system. But thedistributionisn’tacompletedescriptionofthesystem;it’ssimplyareflectionofwhatweknow(ordon’t)aboutit.Totagthisdistinctionwithphilosophicalbuzzwords,instatisticalmechanicsthe probability distribution is an epistemic notion—describing the state of our knowledge—ratherthananontologicalone—describingsomeobjectivefeatureofreality.Epistemologyisthestudyofknowledge;ontologyisthestudyofwhatisreal.It was natural, in 1927, to suspect that quantum mechanics should be thought of along

similarlines.Afterall,bythattimewehadfiguredoutthatwhatweusewavefunctionsforistocalculate the probability of any particular measurement outcome. Surely it makes sense toimaginethatnatureitselfknowspreciselywhattheoutcomeisgoingtobe,buttheformalismofquantum theory simply doesn’t completely capture that knowledge, and thus needs to beimproved.Thewave function, in this view, isn’t thewhole story; thereareadditional “hiddenvariables” that fix what the actualmeasurement outcomes are going to be, even if we don’tknow(andperhapscan’teverdetermineaheadofthemeasurement)whattheirvaluesare.Maybe.Butinsubsequentyearsanumberofresultshavebeenobtained,mostnotablybythe

physicist JohnBell in the 1960s, implying that themost simple and straightforward attemptsalongtheselinesaredoomedtofailure.Peopletried—deBroglieactuallyputforwardaspecifictheory,whichwas rediscovered and extendedbyDavidBohm in the1950s, andEinstein andSchrödingerbothbattedaroundideas.ButBell’stheoremimpliesthatanysuchtheoryrequires“action at a distance”—a measurement at one location can instantly affect the state of theuniversearbitrarily faraway.Thisseemstobe inviolationof thespirit ifnot the letterof thetheory of relativity, which says that objects and influences cannot propagate faster than thespeed of light. The hidden-variable approach is still being actively pursued, but all knownattemptsalongtheselinesareungainlyandhardtoreconcilewithmoderntheoriessuchastheStandardModelofparticlephysics,nottomentionspeculativeideasaboutquantumgravity,aswe’ll discuss later. Perhaps this is why Einstein, the pioneer of relativity, never found a

satisfactorytheoryofhisown.Inthepopularimagination,EinsteinlosttheBohr-Einsteindebates.WearetoldthatEinstein,

acreativerevolutionaryinhisyouth,hadgrownoldandconservative,andwasunabletoacceptorevenunderstand thedramatic implicationsof thenewquantum theory. (At the timeof theSolvayConference Einsteinwas forty-eight years old.) Physics subsequentlywent onwithouthim, as the great man retreated to pursue idiosyncratic attempts at finding a unified fieldtheory.Nothingcouldbefurtherfromthetruth.WhileEinsteinfailedtoputforwardacompleteand

compellinggeneralizationofquantummechanics,hisinsistencethatphysicsneedstodobetterthanshutupandcalculatewasdirectlyonpoint.Itiswildlyoffbasetothinkthathefailedtounderstandquantumtheory.Einsteinunderstood itaswellasanyone,andcontinued tomakefundamentalcontributionstothesubject, includingdemonstratingthe importanceofquantumentanglement,whichplaysacentralroleinourcurrentbestpictureofhowtheuniversereallyworks. What he failed to do was to convince his fellow physicists of the inadequacy of theCopenhagenapproach, and the importanceof tryingharder tounderstand the foundations ofquantumtheory.

If we want to follow Einstein’s ambition of a complete, unambiguous, realistic theory of thenaturalworld,butwearediscouragedbythedifficultiesoftackingnewhiddenvariablesontoquantummechanics,isthereanyremainingstrategyleft?Oneapproachistoforgetaboutnewvariables,throwawayalltheproblematic ideasabout

themeasurement process, strip quantummechanics down to its absolute essentials, and askwhathappens.What’stheleanest,meanestversionofquantumtheorywecaninvent,andstillhopetoexplaintheexperimentalresults?Every version of quantum mechanics (and there are plenty) employs a wave function or

somethingequivalent,andpositsthatthewavefunctionobeysSchrödinger’sequation,atleastmostofthetime.Thesearegoingtohavetobeingredientsinjustaboutanytheorywecantakeseriously. Let’s see if we can be stubbornly minimalist, and get away with adding little ornothingelsetotheformalism.Thisminimalist approachhas two aspects. First,we take thewave function seriously as a

direct representation of reality, not just a bookkeeping device to help us organize ourknowledge.We treat itasontological,notepistemic.That’s themostausterestrategywecanimagineadopting,sinceanythingelsewouldpositadditionalstructureoverandabovethewavefunction.But it’s also a dramatic step, sincewave functions are very different fromwhatweobserve when we look at the world. We don’t see wave functions; we see measurementoutcomes, likethepositionofaparticle.Butthetheoryseemstodemandthatwavefunctionsplayacentralrole,solet’sseehowfarwecangetbyimaginingthatrealityisexactlydescribedbyaquantumwavefunction.Second, if thewave functionusually evolves smoothly in accordancewith theSchrödinger

equation,let’ssupposethat’swhatitalwaysdoes.Inotherwords,let’seraseallofthoseextrarules aboutmeasurement in the quantum recipe entirely, and bring things back to the starksimplicity of the classical paradigm: there is a wave function, and it evolves according to adeterministicrule,andthat’sallthereistosay.Wemightcallthisproposal“austerequantummechanics,”orAQMforshort. Itstands incontrastwithtextbookquantummechanics,whereweappealtocollapsingwavefunctionsandtrytoavoidtalkingaboutthefundamentalnatureofrealityaltogether.Aboldstrategy.Butthere’sanimmediateproblemwithit:itsureseemslikewavefunctions

collapse.Whenwemakemeasurementsofaquantumsystemwithaspread-outwavefunction,wegetaspecificanswer.Evenifwethinkanelectronwavefunctionisadiffusecloudcenteredon the nucleus, when we actually look at it we don’t see such a cloud, we see a point-likeparticleatsomeparticular location.And ifwe look immediatelyagain,wesee theelectron inbasically the same location. There’s a good reason why the pioneers of quantummechanicsinventedtheideaofwavefunctionscollapsing—becausethat’swhattheyappeartodo.Butmaybethat’stooquick.Let’sturnthequestionaround.Ratherthanstartingwithwhat

weseeandtryingto inventatheorytoexplain it, let’sstartwithausterequantummechanics(wavefunctionsevolvingsmoothly,that’sit),andaskwhatpeopleinaworlddescribedbythattheorywouldactuallyexperience.Thinkaboutwhatthiscouldmean.Inthelastchapter,wewerecarefultotalkaboutthewave

functionasakindofmathematicalblackboxfromwhichpredictionsformeasurementoutcomescouldbeextracted:foranyparticularoutcome,thewavefunctionassignsanamplitude,andtheprobabilityofgettingthatoutcomeistheamplitudesquared.MaxBorn,whoproposedtheBornrule,wasoneoftheattendeesatSolvayin1927.Nowwe’resayingsomethingdeeperandmoredirect.Thewavefunctionisn’tabookkeeping

device; it’s an exact representation of the quantum system, just as a set of positions andvelocitieswouldbearepresentationofaclassicalsystem.Theworldisawavefunction,nothingmorenor less.We canuse the phrase “quantum state” as a synonym for “wave function,” indirectparallelwithcallingasetofpositionsandvelocitiesa“classicalstate.”This is adramatic claimabout thenatureof reality. Inordinary conversation, even among

grizzled veterans of quantum physics, people are always talking about concepts like “theposition of the electron.” But this wave-function-is-everything view implies that such talk iswrongheaded in an important way. There is no such thing as “the position of the electron.”There isonlytheelectron’swavefunction.Quantummechanics impliesaprofounddistinctionbetween“whatwecanobserve”and“what therereally is.”Ourobservationsaren’t revealingpre-existing facts ofwhichwewere previously ignorant; at best, they reveal a tiny slice of amuchbigger,fundamentallyelusivereality.Consider an idea you will often hear: “Atoms are mostly empty space.” Utterly wrong,

accordingto theAQMwayof thinking. Itcomes fromastubborn insistenceon thinkingofanelectron as a tiny classical dot zipping around inside of the wave function, rather than theelectronactuallybeingthewavefunction.InAQM,there’snothingzippingaround;thereisonlythequantumstate.Atomsaren’tmostlyemptyspace;theyaredescribedbywavefunctionsthatstretchthroughouttheextentoftheatom.Thewaytobreakoutofourclassicalintuitionistotrulyabandontheideathattheelectron

has some particular location. An electron is in a superposition of every possible position wecouldseeitin,anditdoesn’tsnapintoanyonespecificlocationuntilweactuallyobserveittobethere.“Superposition”isthewordphysicistsusetoemphasizethattheelectronexists inacombinationofallpositions,withaparticularamplitudeforeachone.Quantumrealityisawavefunction; classical positions and velocities aremerely what we are able to observewhenweprobethatwavefunction.

Sotherealityofaquantumsystem,accordingtoausterequantummechanics,isdescribedbyawavefunctionorquantumstate,whichcanbethoughtofasasuperpositionofeverypossibleoutcome of some observation we might want to make. How do we get from there to theannoyingrealitythatwavefunctionsappeartocollapsewhenwemakesuchmeasurements?Start by examining the statement “wemeasure the position of the electron” a littlemore

carefully. What does this measurement process actually involve? Presumably some labequipmentandabitofexperimentaldexterity,butwedon’tneedtoworryaboutspecifics.Allwe need to know is that there is some measuring apparatus (a camera or whatever) thatsomehowinteractswiththeelectron,andthenletsusreadoffwheretheelectronwasseen.In the textbookquantumrecipe, that’sasmuch insightaswewouldeverget.Someof the

peoplewhopioneeredthisapproach,includingNielsBohrandWernerHeisenberg,wouldgoalittlebitfurther,makingexplicittheideathatthemeasuringapparatusshouldbethoughtofasaclassicalobject, even if theelectron itwasobservingwasquantum-mechanical.This lineofdivisionbetweenthepartsoftheworldthatshouldbetreatedusingquantumversusclassicaldescriptions is sometimes called the Heisenberg cut. Rather than accepting that quantummechanics is fundamental and classical mechanics is just a good approximation to it inappropriate circumstances, textbook quantum mechanics puts the classical world at centerstage, as the rightway to talk about people and cameras and othermacroscopic things thatinteractwithmicroscopicquantumsystems.

Thisdoesn’tsmellright.One’s firstguessshouldbethatthequantum/classicaldivide isamatterofourpersonalconvenience,notafundamentalaspectofnature.Ifatomsobeytherulesofquantummechanicsandcamerasaremadeofatoms,presumablycamerasobeytherulesofquantum mechanics too. For that matter, you and I presumably obey the rules of quantummechanics. The fact that we are big, lumbering, macroscopic objects might make classicalphysics a good approximation to what we are, but our first guess should be that it’s reallyquantumfromtoptobottom.Ifthat’strue, it’snot justtheelectronthathasawavefunction.Thecamerashouldhavea

wavefunctionofitsown.Soshouldtheexperimenter.Everythingisquantum.That simple shift of perspective suggests a new angle on themeasurement problem. The

AQMattitudeisthatweshouldn’ttreatthemeasurementprocessasanythingmysticalorevenin need of its own set of rules; the camera and the electron simply interactwith each otheraccordingtothelawsofphysics,justlikearockandtheearthdo.A quantum state describes systems as superpositions of differentmeasurement outcomes.

Theelectronwill,ingeneral,startoutinasuperpositionofvariouslocations—alltheplaceswecould see it were we to look. The camera starts out in some wave function that might lookcomplicated,butamountstosaying“Thisisacamera,andithasn’tyetlookedattheelectron.”But then it does look at the electron, which is a physical interaction governed by theSchrödingerequation.Andafterthatinteraction,wemightexpectthatthecameraitselfisnowinasuperpositionofallthepossiblemeasurementoutcomesitmighthaveobserved:thecamerasawtheelectroninthislocation,orthecamerasawtheelectroninthatlocation,andsoon.Ifthatwerethewholestory,AQMwouldbeanuntenablemess.Electronsinsuperpositions,

camerasinsuperpositions,nothingmuchresemblingtherobustapproximatelyclassicalworldofourexperience.Fortunately we can appeal to another startling feature of quantummechanics: given two

differentobjects(likeanelectronandacamera),theyarenotdescribedbyseparate,individualwave functions. There isonly onewave function, which describes the entire systemwe careabout, all thewayup to the “wave functionof theuniverse” ifwe’re talkingabout thewholeshebang. In the case under consideration, there is a wave function describing the combinedelectron+camerasystem.Sowhatwereallyhaveisasuperpositionofallpossiblecombinationsofwheretheelectronmighthavebeen located,andwherethecameraactuallyobserved it tobe.Although such a superposition in principle includes every possibility, most of the possible

outcomesareassignedzeroweightinthequantumstate.Thecloudofprobabilityvanishesintonothingness for most possible combinations of electron location and camera image. Inparticular, there isnoprobability that theelectronwas inone locationbut thecamerasaw itsomewhereelse(aslongasyouhavearelativelyfunctionalcamera).

Thisisthequantumphenomenonknownasentanglement.Thereisasinglewavefunctionforthecombinedelectron+camerasystem,consistingofasuperpositionofvariouspossibilitiesoftheform“theelectronwasatthis location,andthecameraobserveditatthesamelocation.”Ratherthantheelectronandthecameradoingtheirownthing,thereisaconnectionbetweenthetwosystems.Now let’s take every appearance of “camera” in the above discussion and replace it with

“you.”Rather thantakingapicturewithamechanicalapparatus,we (fancifully) imagine thatyou have really good eyesight and can see where electrons are just by looking at them.Otherwise, nothing changes. According to the Schrödinger equation, an initially unentangledsituation—theelectronisinasuperpositionofvariouspossiblelocations,andyouhaven’tlookedat theelectronyet—evolvessmoothly intoanentangledone—asuperpositionofeach locationtheelectroncouldhavebeenobserved,andyouhavingseentheelectroninjustthatlocation.That’swhat therulesofquantummechanicswouldsay, ifwehadn’t tackedonallof those

extraannoyingbitsaboutthemeasurementprocess.Maybeallofthoseextraruleswerejustawaste of time. In AQM, the story we just told, about you and the electron entangling andevolving into a superposition, is the complete story. There isn’t anything special aboutmeasurement; it’s just something that happenswhen two systems interact in an appropriateway.Andafterward,youandthesystemyouinteractedwithareinasuperposition,ineachpart

ofwhichyouhaveseentheelectroninaslightlydifferentlocation.Theproblem is, this story still doesn’tmatchontowhat youactually experiencewhen you

observeaquantumsystem.Youneverfeellikeyouhaveevolvedintoasuperpositionofdifferentpossiblemeasurement outcomes; you simply think you’ve seen some specific outcome,whichcanbepredictedwith adefiniteprobability. That’swhyall of those extrameasurement ruleswere added in the first place. Otherwise you seemingly have a very pretty and elegantformalism(quantumstates,smoothevolution)thatjustdoesn’tmatchuptoreality.

Timetogeta littlephilosophical.Whatexactlydowemeanby“you”intheaboveparagraph?Constructingascientifictheoryisn’tsimplyamatterofwritingdownsomeequations;wealsoneed to indicatehow thoseequationsmaponto theworld.When it comes toyouandme,wetendtothinkthattheprocessofmatchingourselvesontosomepartofascientificformalismispretty straightforward. Certainly in the story told above, where an observer measures theposition of an electron, it definitely seems as if that observer evolves into an entangledsuperpositionofthedifferentpossiblemeasurementoutcomes.But there’s an alternative possibility. Before the measurement happened, there was one

electronandoneobserver(orcamera,ifyouprefer—itdoesn’tmatterhowwethinkaboutthething that interacts with the electron as long as it’s a big, macroscopic object). After theyinteract,however,ratherthanthinkingofthatoneobserverhavingevolvedintoasuperpositionofpossible states,wecould thinkof themashavingevolved intomultiple possible observers.Therightwaytodescribethingsafterthemeasurement,inthisview,isnotasonepersonwithmultiple ideas about where the electron was seen, but as multiple worlds, each of whichcontainsasinglepersonwithaverydefiniteideaaboutwheretheelectronwasseen.Here’s the big reveal: what we’ve described as austere quantum mechanics is more

commonlyknownastheEverett,orMany-Worlds,formulationofquantummechanics,firstputforwardbyHughEverettin1957.TheEverettviewarisesfromafundamentalannoyancewithallofthespecialrulesaboutmeasurementsthatarepresentedaspartofthestandardtextbookquantumrecipe,andsuggestsinsteadthatthereisjustasinglekindofquantumevolution.Theprice we pay for this vastly increased elegance of theoretical formalism is that the theorydescribesmanycopiesofwhatwethinkofas“theuniverse,”eachslightlydifferent,buteachtrulyrealinsomesense.Whetherthebenefitisworththecostisanissueaboutwhichpeopledisagree.(Itis.)InstumblingupontheMany-Worldsformulation,atnopointdidwetakeordinaryquantum

mechanicsandtackonabunchofuniverses.Thepotentialforsuchuniverseswasalwaysthere—theuniversehasawave function,whichcanverynaturallydescribesuperpositionsofmanydifferentwaysthingscouldbe,includingsuperpositionsofthewholeuniverse.Allwedidistopointoutthatthispotentialisnaturallyactualizedinthecourseofordinaryquantumevolution.Onceyouadmitthatanelectroncanbeinasuperpositionofdifferentlocations,itfollowsthatapersoncanbeinasuperpositionofhavingseentheelectronindifferentlocations,andindeedthatrealityasawholecanbeinasuperposition,anditbecomesnaturaltotreateveryterminthatsuperpositionasaseparate“world.”Wedidn’taddanythingtoquantummechanics,wejustfaceduptowhatwasthereallalong.We might reasonably call Everett’s approach the “courageous” formulation of quantum

mechanics. It embodies the philosophy that we should take seriously the simplest version ofunderlying reality that accounts for what we see, even if that reality differs wildly from oureverydayexperience.Dowehavethecouragetoacceptit?

ThisbriefintroductiontoMany-Worldsleavesmanyquestionsunanswered.Whenexactlydoesthewave functionsplit intomanyworlds?Whatseparates theworlds fromoneanother?Howmanyworldsarethere?Aretheotherworldsreally“real”?Howwouldweeverknow,ifwecan’tobservethem?(Orcanwe?)Howdoesthisexplaintheprobabilitythatwe’llendupinoneworldratherthananotherone?Allofthesequestionshavegoodanswers—oratleastplausibleones—andmuchofthebook

tocomewillbedevoted toanswering them.Butweshouldalsoadmit that thewholepicturemightbewrong,andsomethingverydifferentisrequired.Every versionof quantummechanics features two things: (1) awave function, and (2) the

Schrödinger equation,which governs howwave functions evolve in time. The entirety of theEverett formulation is simply the insistence that there isnothingelse, that these ingredientssuffice to provide a complete, empirically adequate account of the world. (“Empiricallyadequate”isafancywaythatphilosophersliketosay“itfitsthedata.”)Anyotherapproachtoquantummechanics consists of adding something to that bare-bones formalism, or somehow

modifyingwhatisthere.The most immediately startling implication of pure Everettian quantum mechanics is the

existence of many worlds, so it makes sense to call it Many-Worlds. But the essence of thetheoryisthatrealityisdescribedbyasmoothlyevolvingwavefunctionandnothingelse.Thereareextrachallengesassociatedwiththisphilosophy,especiallywhenitcomestomatchingtheextraordinarysimplicityoftheformalismtotherichdiversityoftheworldweobserve.Butthereare corresponding advantages of clarity and insight. Aswe’ll seewhenweultimately turn toquantumfieldtheoryandquantumgravity,takingwavefunctionsasprimaryintheirownright,free of any baggage inherited from our classical experience, is extraordinarily helpful whentacklingthedeepproblemsofmodernphysics.Giventhenecessityofthesetwoingredients(wavefunctionsandtheSchrödingerequation),

thereareafewalternativestoMany-Worldswemightalsoconsider.Oneistoimagineaddingnewphysicalentitiesoverandabovethewavefunction.Thisapproachleadstohidden-variablemodels,whichwereinthebackofthemindsofpeoplelikeEinsteinfromthestart.Thesedaysthe most popular such approach is called the de Broglie–Bohm theory, or simply Bohmianmechanics.Alternatively,wecould leavethewavefunctionby itselfbut imaginechangingtheSchrödinger equation, for example, to introduce real, random collapses. Finally, we mightimaginethatthewavefunctionisn’taphysicalthingatall,butsimplyawayofcharacterizingwhatweknowabout reality.Suchapproachesarebroadlyknownasepistemicmodels, andacurrentlypopularversionisQBism,orquantumBayesianism.All of these options—and there are many more not listed here—represent truly distinct

physical theories, not simply “interpretations” of the same underlying idea. The existence ofmultiple incompatible theories thatall lead(at least thus far) to theobservablepredictionsofquantummechanics creates a conundrum for anyonewhowants to talk aboutwhatquantumtheory really means. While the quantum recipe is agreed upon by working scientists andphilosophers,theunderlyingreality—whatanyparticularphenomenonactuallymeans—isnot.I am defending one particular view of that reality, the Many-Worlds version of quantum

mechanics,andformostofthisbookIwillsimplybeexplainingthingsinMany-Worldsterms.This shouldn’t be taken to imply that the Everettian view is unquestionably right. I hope toexplainwhatthetheorysays,andwhyit’sreasonabletoassignahighcredencetoitbeingthebestviewofrealitywehave;whatyoupersonallyendupbelievingisuptoyou.

*IfyoulookontheInternet,youwillfindnumerousattributionsof“Shutupandcalculate!”toRichardFeynman,aphysicistwhowasanall-timegreatatdoingdifficultcalculations.Butheneversaidanysuchthing,norwouldhehavefoundthesentimentcongenial;Feynmanthoughtcarefullyaboutquantummechanics,andnobodyeveraccusedhimofshuttingup.It’scommonforquotationstobereattributedtoplausiblespeakerswhoaremorefamousthantheactualsourceof thequote.SociologistRobertMertonhasdubbedthis theMatthewEffect,aftera linefromtheGospelofMatthew:“Foruntoeveryonethathathshallbegiven,andheshallhaveabundance:butfromhimthathathnotshallbetakenawayeventhatwhichhehath.”

3WhyWouldAnybodyThinkThis?HowQuantumMechanicsCametoBe

“Sometimes I’vebelievedasmanyassix impossible thingsbeforebreakfast,”notes theWhiteQueentoAliceinThroughtheLookingGlass.Thatcanseemlikeausefulskillasonecomestogrips with quantum mechanics in general, and Many-Worlds in particular. Fortunately, theimpossible-seeming thingswe’reasked tobelievearen’twhimsical inventionsor logic-bustingZenkoans;theyarefeaturesoftheworldthatwearenudgedtowardacceptingbecauseactualexperiments have dragged us, kicking and screaming, in that direction. We don’t choosequantummechanics;weonlychoosetofaceuptoit.

Physicsaspirestofigureoutwhatkindsofstufftheworldismadeof,howthatstuffnaturallychanges over time, and how various bits of stuff interact with one another. In my ownenvironment,Icanimmediatelyseemanydifferentkindsofstuff:papersandbooksandadeskandacomputerandacupofcoffeeandawastebasketandtwocats(oneofwhomisextremelyinterestedinwhat’s insidethewastebasket),nottomentionlesssolidthingslikeairandlightandsound.

By theendof thenineteenthcentury, scientistshadmanaged todistill every single one ofthese things down to two fundamental kinds of substances:particles and fields. Particles arepoint-like objects at a definite location in space,while fields (like the gravitational field) arespreadthroughoutspace,takingonaparticularvalueateverypoint.Whenafieldisoscillatingacrossspaceandtime,wecallthata“wave.”Sopeoplewilloftencontrastparticleswithwaves,butwhattheyreallymeanisparticlesandfields.

Quantum mechanics ultimately unified particles and fields into a single entity, the wavefunction.Theimpetustodosocamefromtwodirections:first,physicistsdiscoveredthatthingsthey thought were waves, like the electric and magnetic fields, had particle-like properties.Thentheyrealizedthatthingstheythoughtwereparticles,likeelectrons,manifestedfield-likeproperties.Thereconciliationofthesepuzzlesisthattheworldisfundamentallyfield-like(it’saquantumwavefunction),butwhenwelookatitbyperformingacarefulmeasurement,itlooksparticle-like.Ittookawhiletogetthere.

Particlesseemtobeprettystraightforwardthings:objectslocatedatparticularpointsinspace.The idea goes back to ancient Greece, where a small group of philosophers proposed thatmatterwasmadeupofpoint-like“atoms,”fortheGreekwordfor“indivisible.”InthewordsofDemocritus, the original atomist, “Sweet is by convention, bitter by convention, hot byconvention,coldbyconvention,colorbyconvention;intruththereareonlyatomsandthevoid.”

Atthetimetherewasn’tthatmuchactualevidenceinfavoroftheproposal,soitwaslargelyabandoneduntilthebeginningofthe1800s,whenexperimentershadbeguntostudychemicalreactionsinaquantitativeway.Acrucialrolewasplayedbytinoxide,acompoundmadeoftinandoxygen,whichwasdiscovered tocome in twodifferent forms.TheEnglish scientist JohnDaltonnotedthatforafixedamountoftin,theamountofoxygeninoneformoftinoxidewasexactly twice the amount in the other.We could explain this, Dalton argued in 1803, if bothelementscameintheformofdiscreteparticles,forwhichheborrowedtheword“atom”fromtheGreeks.Allwehave todo is to imagine thatone formof tinoxidewasmadeofsingle tinatomscombinedwithsingleoxygenatoms,while theother formconsistedofsingle tinatomscombined with two oxygen atoms. Every kind of chemical element, Dalton suggested, wasassociatedwithauniquekindofatom,andthetendencyof theatomstocombine indifferentways was responsible for all of chemistry. A simple summary, but one with world-alteringimplications.

Daltonjumpedthegunalittlebitwithhisnomenclature.FortheGreeks,thewholepointofatomswasthattheywereindivisible,thefundamentalbuildingblocksoutofwhicheverythingelse ismade.ButDalton’satomsarenotatall indivisible—theyconsistofa compactnucleus

surroundedbyorbitingelectrons.Ittookoverahundredyearstorealizethat,however.FirsttheEnglishphysicistJ.J.Thomsondiscoveredelectronsin1897.Theseseemedtobeanutterlynewkindofparticle,electricallychargedandonly1/1800ththemassofhydrogen,thelightestatom.In1909Thomson’sformerstudentErnestRutherford,aNewZealandphysicistwhohadmovedtotheUKforhisadvancedstudies,showedmostofthemassoftheatomwasconcentratedinacentral nucleus,while the atom’s overall sizewas set by the orbits ofmuch lighter electronstravelingaroundthatnucleus.Thestandardcartoonpictureofanatom,withelectronscirclingthe nucleus much like planets orbit the sun in our solar system, represents this Rutherfordmodelofatomicstructure.(Rutherforddidn’tknowaboutquantummechanics,sothiscartoondeviatesfromrealityinsignificantways,asweshallsee.)

Further work, initiated by Rutherford and followed up by others, revealed that nucleithemselves aren’t elementary, but consist of positively charged protons and unchargedneutrons.Theelectricchargesofelectronsandprotonsareequalinmagnitudebutoppositeinsign,soanatomwithanequalnumberofeach(andhowevermanyneutronsyou like)willbeelectricallyneutral. Itwasn’tuntil the1960sand ’70sthatphysicistsestablishedthatprotonsand neutrons are alsomade of smaller particles, called quarks, held together by new force-carryingparticlescalledgluons.

Chemicallyspeaking,electronsarewhereit’sat.Nucleigiveatomstheirheft,butoutsideofrare radioactive decays or fission/fusion reactions, they basically go along for the ride. Theorbitingelectrons,ontheotherhand,arelightandjumpy,andtheirtendencytomovearoundiswhatmakesourlivesinteresting.Twoormoreatomscanshareelectrons,leadingtochemicalbonds.Under the right conditions, electrons can change theirminds aboutwhich atoms theywanttobeassociatedwith,whichgivesuschemicalreactions.Electronscanevenescapetheiratomiccaptivityaltogetherinordertomovefreelythroughasubstance,aphenomenonwecall“electricity.”Andwhenyoushakeanelectron,itsetsupavibrationintheelectricandmagneticfieldsaroundit,leadingtolightandotherformsofelectromagneticradiation.

To emphasize the idea of being truly point-like, rather than a small object but with somedefinitenonzerosize,wesometimesdistinguishbetween“elementary”particles,whichdefineliteral points in space, and “composite” particles that are really made of even smallerconstituents.Asfarasanyonecantell,electronsaretrulyelementaryparticles.Youcanseewhydiscussions of quantummechanics are constantly referring to electronswhen they reach forexamples—they’retheeasiestfundamentalparticletomakeandmanipulate,andplayacentralroleinthebehaviorofthematterofwhichweandoursurroundingsaremade.

InbadnewsforDemocritusandhisfriends,nineteenth-centuryphysicsdidn’texplaintheworldin terms of particles alone. It suggested, instead, that two fundamental kinds of stuff wererequired:bothparticlesandfields.

Fields can be thought of as the opposite of particles, at least in the context of classicalmechanics. The defining feature of a particle is that it’s located at one point in space, andnowhereelse.Thedefiningfeatureofafieldisthatitislocatedeverywhere.Afieldissomethingthat has a value at literally every point in space. Particles need to interact with each othersomehow,andtheydosothroughtheinfluenceoffields.

Thinkof themagnetic field. It’s avector field—ateverypoint in space it looks likea littlearrow, with amagnitude (the field can be strong, or weak, or even exactly zero) and also adirection (it points along some particular axis). We can measure the direction in which themagneticfieldpointsjustbypullingoutamagneticcompassandobservingwhatdirectiontheneedlepointsin.(Itwillpointroughlynorth,ifyouarelocatedatmostplacesonEarthandnotstanding too close to anothermagnet.) The important thing is that themagnetic field exists

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invisiblyeverywherethroughoutspace,evenwhenwe’renotobservingit.That’swhatfieldsdo.There is also theelectric field,which is alsoa vectorwithamagnitudeandadirectionat

everypointinspace.Justaswecandetectamagneticfieldwithacompass,wecandetecttheelectric field by placing an electron at rest and seeing if it accelerates. The faster theacceleration,thestrongertheelectricfield.*Oneofthetriumphsofnineteenth-centuryphysicswaswhenJamesClerkMaxwellunifiedelectricityandmagnetism,showingthatbothof thesefieldscouldbethoughtofasdifferentmanifestationsofasingleunderlying“electromagnetic”field.

The other field that was well known in the nineteenth century is the gravitational field.Gravity, Isaac Newton taught us, stretches over astronomical distances. Planets in the solarsystem feel a gravitational pull toward the sun, proportional to the sun’smass and inverselyproportionaltothesquareofthedistancebetweenthem.In1783Pierre-SimonLaplaceshowedthatwecanthinkofNewtoniangravityasarisingfroma“gravitationalpotentialfield”thathasavalueateverypointinspace,justastheelectricandmagneticfieldsdo.

By the end of the 1800s, physicists could see the outlines of a complete theory of theworldcomingintofocus.Matterwasmadeofatoms,whichweremadeofsmallerparticles,interactingviavariousforcescarriedbyfields,alloperatingundertheumbrellaofclassicalmechanics.

WhattheWorldIsMadeOf(Nineteenth-CenturyEdition)

Particles(point-like,makingupmatter).Fields(pervadingspace,givingrisetoforces).

Newparticlesandforceswouldbediscoveredoverthecourseofthetwentiethcentury,butintheyear1899itwouldn’thavebeencrazytothinkthatthebasicpicturewasundercontrol.Thequantumrevolutionlurkedjustaroundthecorner,largelyunsuspected.

Ifyou’vereadanythingaboutquantummechanicsbefore,you’veprobablyheardthequestion“Isanelectronaparticle,orawave?”Theansweris:“It’sawave,butwhenwelookat(thatis,measure) that wave, it looks like a particle.” That’s the fundamental novelty of quantummechanics. There is only one kind of thing, the quantumwave function, but when observedundertherightcircumstancesitappearsparticle-liketous.

WhattheWorldIsMadeOf(TwentiethCenturyandBeyond)

Aquantumwavefunction.

Ittookanumberofconceptualbreakthroughstogofromthenineteenth-centurypictureoftheworld (classical particles and classical fields) to the twentieth-century synthesis (a singlequantumwavefunction).Thestoryofhowparticlesandfieldsaredifferentaspectsofthesameunderlying thing is one of the underappreciated successes of the quest for unification inphysics.

Togetthere,earlytwentieth-centuryphysicistsneededtoappreciatetwothings:fields(likeelectromagnetism)canbehaveinparticle-likeways,andparticles(likeelectrons)canbehaveinwave-likeways.

The particle-like behavior of fields was appreciated first. Any particle with an electricalcharge,suchasanelectron,createsanelectricfieldeverywherearoundit,fadinginmagnitudeasyougetfartherawayfromthecharge.Ifweshakeanelectron,oscillatingitupanddown,thefield oscillates along with it, in ripples that gradually spread out from its location. This iselectromagnetic radiation,or “light” for short.Every timeweheatupamaterial to sufficienttemperature, electrons in its atoms start to shake, and the material begins to glow. This isknownasblack-bodyradiation,andeveryobjectwithauniformtemperaturegivesoffaformofblackbodyradiation.

Redlightcorrespondstoslowlyoscillating, low-frequencywaves,whilebluelight israpidlyoscillating,high-frequencywaves.Givenwhatphysicistsknewaboutatomsandelectronsattheturnofthecentury,theycouldcalculatehowmuchradiationablackbodyshouldemitateverydifferent frequency, the so-called blackbody spectrum.Their calculationsworkedwell for lowfrequencies,butbecamelessand lessaccurateastheywenttohigherfrequencies,ultimatelypredicting an infinite amount of radiation coming from every material body. This was laterdubbed the “ultraviolet catastrophe,” referring to the invisible frequencies even higher thanblueorvioletlight.

Finallyin1900,GermanphysicistMaxPlanckwasabletoderiveaformulathatfitthedata

exactly.Theimportanttrickwastoproposearadicalidea:thateverytimelightwasemitted,itcame in the form of a particular amount—a “quantum”—of energy, whichwas related to thefrequency of the light. The faster the electromagnetic field oscillates, themore energy eachemissionwillhave.

Intheprocess,Planckwasforcedtoposittheexistenceofanewfundamentalparameterofnature, now known asPlanck’s constant and denoted by the letterh. The amount of energycontained in a quantumof light is proportional to its frequency, andPlanck’s constant is theconstantofproportionality:theenergyisthefrequencytimesh.Veryoftenit’smoreconvenientto use a modified version ħ, pronounced “h-bar,” which is just Planck’s original constant hdividedby2π.TheappearanceofPlanck’sconstant inanexpression isasignal thatquantummechanicsisatwork.

Planck’sdiscoveryofhisconstantsuggestedanewwayofthinkingaboutphysicalunits,suchasenergy,mass,length,ortime.Energyismeasuredinunitssuchasergsorjoulesorkilowatt-hours,whilefrequencyismeasuredinunitsof1/time,sincefrequencytellsushowmanytimessomething happens in a given amount of time. To make energy proportional to frequency,Planck’sconstantthereforehasunitsofenergytimestime.Planckhimselfrealizedthathisnewquantity could be combined with the other fundamental constants—G, Newton’s constant ofgravity,andc,thespeedoflight—toformuniversallydefinedmeasuresoflength,time,andsoforth. The Planck length is about 10-33 centimeters, while the Planck time is about 10-43seconds. The Planck length is a very short distance indeed, but presumably it has physicalrelevance, as a scale at which quantum mechanics (h), gravity (G), and relativity (c) allsimultaneouslymatter.

Amusingly, Planck’smind immediatelywent to the possibility of communicatingwith aliencivilizations. Ifwe someday start chattingwith extraterrestrial beingsusing interstellar radiosignals,theywon’tknowwhatwemeanifweweretosayhumanbeingsare“abouttwometerstall.”Butsincetheywillpresumablyknowatleastasmuchaboutphysicsaswedo,theyshouldbe aware of Planck units. This suggestion hasn’t yet been put to practical use, but Planck’sconstanthashadanimmenseimpactelsewhere.

Theideathatlightisemittedindiscretequantaofenergyrelatedtoitsfrequencyispuzzling,when you think about it. Fromwhat we intuitively know about light, it mightmake sense ifsomeonesuggestedthattheamountofenergyitcarrieddependedonhowbrightitwas,butnotonwhatcolor itwas.ButtheassumptionledPlancktoderivetherightformula,sosomethingabouttheideaseemedtobeworking.

Itwas left toAlbertEinstein, inhis singularway, tobrushawayconventionalwisdomandtake a dramatic leap into a newway of thinking. In 1905, Einstein suggested that lightwasemittedonlyatcertainenergiesbecauseitliterallyconsistedofdiscretepackets,notasmoothwave.Lightwasparticles,inotherwords—“photons,”astheyareknowntoday.Thisidea,thatlightcomesindiscrete,particle-likequantaofenergy,wasthetruebirthofquantummechanics,andwasthediscoveryforwhichEinsteinwasawardedtheNobelPrizein1921.(HedeservedtowinatleastonemoreNobelforthetheoryofrelativity,butneverdid.)Einsteinwasnodummy,andheknewthatthiswasabigdeal;ashetoldhisfriendConradHabicht,his lightquantumproposalwas“veryrevolutionary.”

NotethesubtledifferencebetweenPlanck’ssuggestionandEinstein’s.Plancksaysthatlightofa fixed frequency isemitted in certain energyamounts,whileEinstein says that’s becauselight literally is discrete particles. It’s the difference between saying that a certain coffeemachinemakesexactlyonecupatatime,andsayingthatcoffeeonlyexistsintheformofone-cup-size amounts. That might make sense when we’re talking about matter particles likeelectronsandprotons,butjustafewdecadesearlierMaxwellhadtriumphantlyexplainedthatlight was a wave, not a particle. Einstein’s proposal was threatening to undo that triumph.Planckhimselfwasreluctanttoacceptthiswildnewidea,butitdidexplainthedata.Inawildnewidea’ssearchforacceptance,that’sapowerfuladvantagetohave.

Meanwhile another problem was lurking over on the particle side of the ledger, whereRutherford’smodelexplainedatomsintermsofelectronsorbitingnuclei.

Rememberthatifyoushakeanelectron,itemitslight.By“shake”wejustmeanaccelerateinsome way. An electron that does anything other than move in a straight line at a constantvelocityshouldemitlight.

From the picture of the Rutherford atom, with electrons orbiting around the nucleus, itcertainlylookslikethoseelectronsarenotmovinginstraightlines.They’removingincirclesorellipses. In a classical world, that unambiguously means that the electrons are beingaccelerated,andequallyunambiguouslythattheyshouldbegivingofflight.Everysingleatominyourbody,andintheenvironmentaroundyou,shouldbeglowing,ifclassicalmechanicswasright.Thatmeans theelectronsshouldbe losingenergyas theyemitradiation,which in turn

implies that they should spiral downward into the central nucleus.Classically, electronorbitsshouldnotbestable.

Perhapsallofyouratomsaregivingofflight,butit’sjusttoofainttosee.Afterall,identicallogicappliestotheplanetsinthesolarsystem.Theyshouldbegivingoffgravitationalwaves—an acceleratingmass should cause ripples in the gravitational field, just like an acceleratingchargecausesripplesintheelectromagneticfield.Andindeedtheyare.Iftherewasanydoubtthat this happens, it was swept away in 2016, when researchers at the LIGO and Virgogravitational-wave observatories announced the first direct detection of gravitational waves,createdwhenblackholesoverabillionlightyearsawayspiraledintoeachother.

Buttheplanetsinthesolarsystemaremuchsmaller,andmovemoreslowly,thanthoseblackholes, which were each over thirty times the mass of the sun. As a result, the emittedgravitationalwavesfromourplanetaryneighborsareveryweakindeed.ThepoweremittedingravitationalwavesbytheorbitingEarthamountstoabout200watts—equivalenttotheoutputof a few lightbulbs, and completely insignificant compared to other influences such as solarradiationandtidalforces.IfwepretendthattheemissionofgravitationalwavesweretheonlythingaffectingtheEarth’sorbit, itwouldtakeover1023years for it tocrash intothesun.Soperhapsthesamethingistrueforatoms:maybeelectronorbitsaren’treallystable,butthey’restableenough.

This isaquantitativequestion,and it’snothardtoplug inthenumbersandseewhat fallsout.Theanswer iscatastrophic,becauseelectronsshouldmovemuchfasterthanplanetsandelectromagnetism isamuchstronger force thangravity.Theamountof time itwould takeanelectrontocrash intothenucleusof itsatomworksouttoabouttenpicoseconds.That’sone-hundred-billionth of a second. If ordinary matter made of atoms only lasted for that long,someonewouldhavenoticedbynow.

This bothered a lot of people, most notably Niels Bohr, who had briefly worked underRutherfordin1912.In1913,Bohrpublishedaseriesofthreepapers,laterknownsimplyas“thetrilogy,” in which he put forth another of those audacious, out-of-the-blue ideas thatcharacterizedtheearlyyearsofquantumtheory.Whatif,heasked,electronscan’tspiraldowninto atomic nuclei because electrons simply aren’t allowed to be in any orbit theywant, butinsteadhave tostick tocertainveryspecificorbits?Therewouldbeaminimum-energyorbit,anotheronewithsomewhathigherenergy,andsoon.Butelectronsweren’tallowedtogoanycloser to the nucleus than the lowest orbit, and they weren’t allowed to be in between theorbits.Theallowedorbitswerequantized.

Bohr’sproposalwasn’tquiteasoutlandishas itmightseemat first.Physicistshadstudiedhowlightinteractedwithdifferentelementsintheirgaseousform—hydrogen,nitrogen,oxygen,and so forth. They found that if you shined light through a cold gas, some of it would beabsorbed; likewise, ifyoupassedelectricalcurrentthroughatubeofgas,thegaswouldstart

glowing (the principle behind fluorescent lights still used today). But they only emitted andabsorbed certain very specific frequencies of light, letting other colors pass right through.Hydrogen,thesimplestelementwithjustasingleprotonandasingleelectron,inparticularhadaveryregularpatternofemissionandabsorptionfrequencies.

ForaclassicalRutherfordatom,thatwouldmakenosenseatall.ButinBohr’smodel,whereonly certain electron orbitswere allowed, therewas an immediate explanation. Even thoughelectronscouldn’tlingerinbetweentheallowedorbits,theycouldjumpfromonetoanother.Anelectroncouldfallfromahigher-energyorbittoalower-energyonebyemittinglightwithjusttherightenergytocompensate,oritcouldleapupwardinenergybyabsorbinganappropriateamountofenergyfromambientlight.Becausetheorbitsthemselveswerequantized,weshouldonly see specific energies of light interactingwith the electrons. TogetherwithPlanck’s ideathatthefrequencyoflightisrelatedtoitsenergy,thisexplainedwhyphysicistssawonlycertainfrequenciesbeingemittedorabsorbed.

Bycomparinghispredictionstotheobservedemissionoflightbyhydrogen,Bohrwasabletonot simply posit that only some electron orbits were allowed, but calculate which ones theywere. Any orbiting particle has a quantity called the angular momentum, which is easy tocalculate—it’sjustthemassoftheparticle,timesitsvelocity,timesitsdistancefromthecenterof theorbit.Bohrproposedthatanallowedelectronorbitwasonewhoseangularmomentumwasamultipleof aparticular fundamental constant.Andwhenhecompared theenergy thatelectronsshouldemitwhenjumpingbetweenorbitstowhatwasactuallyseeninlightemittedfromhydrogengas,hecouldfigureoutwhatthatconstantneededtobeinordertofitthedata.TheanswerwasPlanck’sconstant,h.Ormorespecifically,themodifiedh-barversion,ħ=h/2π.

That’sthekindofthingthatmakesyouthinkyou’reontherighttrack.Bohrwastryingtoaccountforthebehaviorofelectronsinatoms,andhepositedanadhocruleaccordingtowhichtheycouldonlymovealongcertainquantizedorbits,andinordertofitthedatahisruleendeduprequiringanewconstantofnature,andthatnewconstantwasthesameasthenewconstantthatPlanckwasforcedtoinventwhenhewastryingtoaccountforthebehaviorofphotons.Allofthismightseemramshackleandabitsketchy,buttakentogetheritappearedasifsomethingprofound was happening in the realm of atoms and particles, something that didn’t fitcomfortably with the sacred rules of classical mechanics. The ideas of this period are nowsometimes described under the rubric of “the old quantum theory,” as opposed to “the newquantumtheory”ofHeisenbergandSchrödingerthatcamealonginthelate1920s.

Asprovocativeandprovisionallysuccessfulastheoldquantumtheorywas,nobodywasreallyhappywith it. Planck and Einstein’s idea of light quanta helpedmake sense of a number ofexperimentalresults,butwashardtoreconcilewiththeenormoussuccessofMaxwell’stheoryoflightaselectromagneticwaves.Bohr’sideaofquantizedelectronorbitshelpedmakesenseofthe lightemittedandabsorbedbyhydrogen,butseemedtobepulledoutofahat,anddidn’treallyworkforelementsotherthanhydrogen.Evenbeforethe“oldquantumtheory”wasgiventhatname,itseemedclearthatthesewerejusthintsatsomethingmuchdeepergoingon.

OneoftheleastsatisfyingfeaturesofBohr’smodelwasthesuggestionthatelectronscould“jump”fromoneorbittoanother.Ifalow-energyelectronabsorbedlightwithacertainamountof energy, itmakes sense that it would have to jump up to another orbit with just the rightamountofadditionalenergy.Butwhenanelectroninahigh-energyorbitemittedlighttojumpdown,itseemedtohaveachoiceaboutexactlyhowfardowntogo,whichlowerorbittoendupin.Whatmadethatchoice?RutherfordhimselfworriedaboutthisinalettertoBohr:

Thereappearstomeonegravedifficultyinyourhypothesis,whichIhavenodoubtyoufullyrealize,namely,howdoesanelectrondecidewhatfrequencyitisgoingtovibrateatwhenitpassesfromonestationarystatetotheother?Itseemstomethatyouwouldhavetoassumethattheelectronknowsbeforehandwhereitisgoingtostop.

This business about electrons “deciding” where to go foreshadowed amuchmore drasticbreak with the paradigm of classical physics than physicists in 1913 were prepared tocontemplate.InNewtonianmechanicsonecouldimagineaLaplacedemonthatcouldpredict,atleast inprinciple,theentirefuturehistoryoftheworldfromitspresentstate.Atthispoint inthedevelopmentofquantummechanics,nobodywas reallyconfronting theprospect that thispicturewouldhavetobecompletelydiscarded.

It tookmorethantenyearsforamorecompleteframework,the“newquantumtheory,”tofinally come on the scene. In fact, two competing ideas were proposed at the time, matrixmechanics and wave mechanics, before they were ultimately shown to be mathematicallyequivalentversionsofthesamething,whichcannowsimplybecalledquantummechanics.

MatrixmechanicswasformulatedinitiallybyWernerHeisenberg,whohadworkedwithNielsBohr in Copenhagen. These two men, along with their collaborator Wolfgang Pauli, areresponsible for the Copenhagen interpretation of quantum mechanics, though who exactlybelievedwhatisatopicofongoinghistoricalandphilosophicaldebate.

Heisenberg’sapproach in1926,reflectingtheboldnessofayoungergenerationcomingonthescene,wastoputasidequestionsofwhatwasreallyhappeninginaquantumsystem,andtofocus exclusively on explaining what was observed by experimenters. Bohr had positedquantizedelectronorbitswithoutexplainingwhysomeorbitswereallowedandotherswerenot.Heisenberg dispensedwith orbits entirely. Forget aboutwhat the electron is doing; ask onlywhatyoucanobserveabout it. Inclassicalmechanics,anelectronwouldbecharacterizedbyposition and momentum. Heisenberg kept those words, but instead of thinking of them asquantities that existwhetherwe are looking at them or not, he thought of them as possibleoutcomes of measurements. For Heisenberg, the unpredictable jumps that had botheredRutherford and others became a central part of the best way of talking about the quantumworld.

Heisenbergwasonly twenty-fouryearsoldwhenhe first formulatedmatrixmechanics.Hewas clearly a prodigy, but far from an established figure in the field, and wouldn’t obtain apermanentacademicpositionuntilayearlater.InalettertoMaxBorn,anotherofhismentors,Heisenberg fretted that he “had written a crazy paper and did not dare to send it in forpublication.”But inacollaborationwithBornand theevenyoungerphysicistPascual Jordan,theywereabletoputmatrixmechanicsonaclearandmathematicallysoundfooting.

ItwouldhavebeennaturalforHeisenberg,Born,andJordantosharetheNobelPrizeforthedevelopment ofmatrixmechanics, and indeedEinsteinnominated them for the award.But itwasHeisenbergalonewhowashonoredbytheNobelcommitteein1932.IthasbeenspeculatedthatJordan’sinclusionwouldhavebeenproblematic,ashebecameknownforaggressiveright-wing political rhetoric, ultimately becoming a member of the Nazi Party and joining aSturmabteilung(Stormtrooper)unit.Atthesametime,however,hewasconsideredunreliableby his fellow Nazis, due to his support for Einstein and other Jewish scientists. In the end,Jordan never won the prize. Born was also left off the prize for matrix mechanics, but thatomissionwasmadeupforwhenhewasawardedaseparateNobelin1954forhisformulationofthe probability rule. Thatwas the last time aNobel Prize has been awarded forwork in thefoundationsofquantummechanics.

After theonsetofWorldWarII,Heisenberg ledaGermangovernmentprogramtodevelopnuclearweapons.WhatHeisenbergactuallythoughtabouttheNazis,andwhetherhetrulytriedas hard as possible to push the weapons program forward, are matters of some historicaldispute. It seems that, likeanumberofotherGermans,Heisenbergwasnot fondof theNaziParty,butpreferredaGermanvictory in theconflict to theprospectofbeingrunoverby theSoviets.There isnoevidencethatheactivelyworkedtosabotagethenuclearbombprogram,butitisclearthathisteammadeverylittleprogress.InpartthatmustbeattributedtothefactthatsomanybrilliantJewishphysicistshadfledGermanyastheNazistookpower.

As impressive as matrix mechanics was, it suffered from a severe marketing flaw: themathematical formalismwashighlyabstractanddifficult tounderstand.Einstein’sreactiontothe theory was typical: “A veritable sorcerer’s calculation. This is sufficiently ingenious andprotectedbyitsgreatcomplexity,tobeimmunetoanyproofof itsfalsity.”(Thisfromtheguywhohadproposeddescribingspacetimeintermsofnon-Euclideangeometry.)Wavemechanics,developedimmediatelythereafterbyErwinSchrödinger,wasaversionofquantumtheorythatusedconceptswithwhichphysicistswerealreadyveryfamiliar,whichgreatlyhelpedaccelerateacceptanceofthenewparadigm.

Physicists had studied waves for a long time, and with Maxwell’s formulation ofelectromagnetism as a theory of fields, they had become adept at thinking about them. Theearliest intimations of quantum mechanics, from Planck and Einstein, had been away fromwavesandtowardparticles.ButBohr’satomsuggestedthatevenparticlesweren’twhat theyseemedtobe.

In 1924, the young French physicist Louis de Broglie was thinking about Einstein’s lightquanta.Atthispointtherelationshipbetweenphotonsandclassicalelectromagneticwaveswasstillmurky.Anobvious thing tocontemplatewas that lightconsistedofbothaparticleandawave: particle-like photons could be carried along by thewell-known electromagneticwaves.Andifthat’strue,there’snoreasonwecouldn’timaginethesamethinggoingonwithelectrons—maybe there is somethingwave-like that carriesalong theelectronparticles.That’s exactlywhat de Broglie suggested in his 1924 doctoral thesis, proposing a relationship between themomentumandwavelengthofthese“matterwaves”thatwasanalogoustoPlanck’sformulaforlight,withlargermomentacorrespondingtoshorterwavelengths.

Likemanysuggestionsatthetime,deBroglie’shypothesismayhaveseemedalittleadhoc,butitsimplicationswerefar-reaching.Inparticular,itwasnaturaltoaskwhattheimplicationsof matter waves might be for electrons orbiting around a nucleus. A remarkable answersuggesteditself:forthewavetosettledownintoastationaryconfiguration,itswavelengthhadtobeanexactmultipleofthecircumferenceofacorrespondingorbit.Bohr’squantizedorbitscouldbederivedratherthansimplypostulated,simplybyassociatingwaveswiththeelectronparticlessurroundingthenucleus.

Considerastringwithitsendsheldfixed,suchasonaguitarorviolin.Eventhoughanyonepointcanmoveupordownasitlikes,theoverallbehaviorofthestringisconstrainedbybeingtieddownateitherend.Asaresult,thestringonlyvibratesatcertainspecialwavelengths,orcombinations thereof; that’s why the strings on musical instruments emit clear notes ratherthan an indistinct noise. These special vibrations are called the modes of the string. Theessentially“quantum”natureofthesubatomicworld,inthispicture,comesaboutnotbecausereality is actually subdivided into distinct chunks but because there are natural vibrationalmodesforthewavesoutofwhichphysicalsystemsaremade.

Theword“quantum,”referringtosomedefiniteamountofstuff,cangivetheimpressionthatquantummechanicsdescribesaworldthat is fundamentallydiscreteandpixelated, likewhenyouzoom in closelyona computermonitororTV screen. It’s actually theopposite;quantummechanics describes the world as a smooth wave function. But in the right circumstances,whereindividualpartsofthewavefunctionaretieddowninacertainway,thewavetakestheformofacombinationofdistinctvibrationalmodes.Whenweobservesuchasystem,weseethose discrete possibilities. That’s true for orbits of electrons, and it will also explain whyquantum fields look like sets of individual particles. In quantum mechanics, the world isfundamentallywavy; its apparent quantumdiscreteness comes from the particularway thosewavesareabletovibrate.

DeBroglie’s ideaswereintriguing,buttheyfellshortofprovidingacomprehensivetheory.ThatwaslefttoErwinSchrödinger,whoin1926putforthadynamicalunderstandingofwavefunctions, includingtheequation theyobey, laternamedafterhim.Revolutions inphysicsaregenerally a youngperson’s game, andquantummechanicswasnodifferent, butSchrödingerbuckedthetrend.AmongtheleadersofthediscussionsatSolvayin1927,Einsteinatforty-eightyearsold,Bohrat forty-two, andBornat forty-fourwere thegrandoldmen.Heisenbergwastwenty-five,Paulitwenty-seven,andDiractwenty-five.Schrödinger,attheripeoldageofthirty-eight,waslookeduponassomeonesuspiciouslylonginthetoothtoappearonthescenewithradicalnewideaslikethis.

Note the shift here from de Broglie’s “matter waves” to Schrödinger’s “wave function.”ThoughSchrödingerwasheavilyinfluencedbydeBroglie’swork,hisconceptwentquiteabitfurther, anddeservesadistinctname.Most obviously, the valueof amatterwaveat anyonepointwas some real number,while the amplitudes described bywave functions are complex

numbers—thesumofarealnumberandanimaginaryone.Moreimportant,theoriginalideawasthateachkindofparticlewouldbeassociatedwitha

matterwave.That’snothowSchrödinger’swavefunctionworks;youhavejustonefunctionthatdependsonalltheparticlesintheuniverse.It’sthatsimpleshiftthatleadstotheworld-alteringphenomenonofquantumentanglement.

Whatmade Schrödinger’s ideas an instant hit was the equation he proposed,which governshowwavefunctionschangewithtime.Toaphysicist,agoodequationmakesallthedifference.It elevates a pretty-sounding idea (“particles have wave-like properties”) to a rigorous,unforgiving framework. Unforgivingmight sound like a bad quality in a person, but it’s justwhat youwant in a scientific theory. It’s the feature that lets youmake precise predictions.Whenwesaythatquantumtextbooksspendalotoftimehavingstudentssolveequations, it’smostlytheSchrödingerequationwehaveinmind.

Schrödinger’sequationiswhataquantumversionofLaplace’sdemonwouldbesolvingasitpredicted the futureof theuniverse.Andwhile theoriginal form inwhichSchrödingerwrotedownhisequationwasmeantforsystemsofindividualparticles,it’sactuallyaverygeneralideathatappliesequallywell tospins, fields,superstrings,oranyothersystemyoumightwant todescribeusingquantummechanics.

Unlike matrix mechanics, which was expressed in terms of mathematical concepts mostphysicistsatthetimehadneverbeenexposedto,Schrödinger’swaveequationwasnotallthatdifferentinformfromMaxwell’selectromagneticequationsthatadornT-shirtswornbyphysicsstudentstothisday.Youcouldvisualizeawavefunction,oratleastyoumightconvinceyourselfthatyoucould.Thecommunitywasn’tsurewhattomakeofHeisenberg,buttheywerereadyfor Schrödinger. The Copenhagen crew—especially the youngsters, Heisenberg and Pauli—didn’treactgraciouslytothecompetingideasfromanundistinguishedoldmaninZürich.Butbeforetoolongtheywerethinkingintermsofwavefunctions,justlikeeveryoneelse.

Schrödinger’s equation involves unfamiliar symbols, but its basic message is not hard tounderstand.DeBrogliehadsuggestedthatthemomentumofawavegoesupasitswavelengthgoesdown.Schrödingerproposedasimilarthing,butforenergyandtime:therateatwhichthewave function is changing is proportional to howmuchenergy it has.Here is the celebratedequationinitsmostgeneralform:

Wedon’tneedthedetailshere,but it’snice toseetherealwaythatphysicists thinkofanequation like this. There’s some maths involved, but ultimately it’s just a translation intosymbolsoftheideawewrotedowninwords.

Ψ(theGreekletterPsi)isthewavefunction.Theleft-handsideistherateatwhichthewavefunction is changing over time. On the right-hand side we have a proportionality constantinvolvingPlanck’s constantħ, the fundamental unit of quantummechanics, and i, the squarerootofminusone.ThewavefunctionΨisactedonbysomethingcalledtheHamiltonian,orH.ThinkoftheHamiltonianasaninquisitorwhoasksthefollowingquestion:“Howmuchenergydo you have?” The concept was invented in 1833 by Irish mathematician William RowanHamilton,asawaytoreformulatethelawsofmotionofaclassicalsystem,longbeforeitgainedacentralroleinquantummechanics.

Whenphysicists startmodeling different physical systems, the first thing they try to do isworkoutamathematicalexpression for theHamiltonianof that system.Thestandardwayoffiguring out the Hamiltonian of something like a collection of particles is to start with theenergiesof theparticles themselves,and thenadd inadditional contributionsdescribinghowthe particles interactwith each other.Maybe they bump off each other like billiard balls, orperhaps they exert a mutual gravitational interaction. Each such possibility suggests aparticularkindofHamiltonian.And if youknow theHamiltonian, youknoweverything; it’s acompactwayofcapturingallthedynamicsofaphysicalsystem.

Ifaquantumwave functiondescribesasystemwithsomedefinitevalueof theenergy, theHamiltoniansimplyequalsthatvalue,andtheSchrödingerequationimpliesthatthesystemjustkeepsdoingthesamething,maintaininga fixedenergy.Moreoften,sincewavefunctionsaresuperpositionsofdifferentpossibilities,thesystemwillbeacombinationofmultipleenergies.InthatcasetheHamiltoniancapturesabitofallofthem.Thebottomlineisthattheright-handsideofSchrödinger’sequationisawayofcharacterizinghowmuchenergyiscarriedbyeachofthe contributions to a wave function in a quantum superposition; high-energy componentsevolvequickly,low-energyonesevolvemoreslowly.

Whatreallymattersisthatthereissomespecificdeterministicequation.Onceyouhavethat,

theworldisyourplayground.

Wavemechanicsmadeahugesplash,andbefore too longSchrödinger,EnglishphysicistPaulDirac,andothersdemonstratedthatitwasessentiallyequivalenttomatrixmechanics,leavinguswithaunifiedtheoryofthequantumworld.Still,allwasnotpeachesandcream.Physicistswereleftwiththequestionthatwearestillstrugglingwithtoday:What is thewave function,really?Whatphysicalthingdoesitrepresent,ifany?

IndeBroglie’sview,hismatterwavesservedtoguideparticlesaround,nottoreplacethementirely.(Helaterdevelopedthisideaintopilot-wavetheory,whichremainsaviableapproachtoquantumfoundationstoday,althoughitisnotpopularamongworkingphysicists.)Schrödinger,bycontrast,wantedtodoawaywithfundamentalparticlesentirely.Hisoriginalhopewasthathisequationwoulddescribelocalizedpacketsofvibrations,confinedtoarelativelysmallregionofspace,so thateachpacketwouldappearparticle-like toamacroscopicobserver.Thewavefunctioncouldbethoughtofasrepresentingthedensityofmassinspace.

Alas, Schrödinger’s aspirationswereundonebyhis ownequation. Ifwe startwith awavefunctiondescribingasingleparticleapproximatelylocalizedinsomeemptyregionofspace,theSchrödingerequationisclearaboutwhathappensnext:itquicklyspreadsoutallovertheplace.Lefttotheirowndevices,Schrödinger’swavefunctionsdon’tlookparticle-likeatall.*

ItwaslefttoMaxBorn,oneofHeisenberg’scollaboratorsonmatrixmechanics,toprovidethe finalmissing piece:we should think about thewave function as away of calculating theprobabilityofseeingaparticleinanygivenpositionwhenwelookforit.Inparticular,weshouldtake both the real and imaginary parts of the complex-valued amplitude, square them bothindividually,andadd the twonumbers together.The result is theprobabilityofobserving thecorresponding outcome. (The suggestion that it’s the amplitude squared, rather than theamplitudeitself,appearsinafootnoteaddedatthelastminutetoBorn’s1926paper.)Andafterwe observe it, the wave function collapses to be localized at the place where we saw theparticle.

You know who didn’t like the probability interpretation of the Schrödinger equation?Schrödinger himself. His goal, like Einstein’s, was to provide a definite mechanisticunderpinningforquantumphenomena,notjusttocreateatoolthatcouldbeusedtocalculateprobabilities.“Idon’tlikeit,andI’msorryIeverhadanythingtodowithit,”helatergroused.ThepointofthefamousSchrödinger’sCatthoughtexperiment,inwhichthewavefunctionofacatevolves(viatheSchrödingerequation)intoasuperpositionof“alive”and“dead,”wasnottomakepeoplesay,“Wow,quantummechanics is reallymysterious.” Itwas tomakepeoplesay,“Wow,thiscan’tpossiblybecorrect.”Buttothebestofourcurrentknowledge,itis.

Alotofintellectualactionwaspackedintothefirstthreedecadesofthetwentiethcentury.Overthecourseofthe1800s,physicistshadputtogetherapromisingpictureofthenatureofmatterand forces. Matter was made of particles, and forces were carried by fields, all under theumbrellaofclassicalmechanics.Butconfrontationwithexperimentaldataforcedthemtothinkbeyondthisparadigm.Inordertoexplainradiationfromhotobjects,Plancksuggestedthatlightwasemittedindiscreteamountsofenergy,andEinsteinpushedthisfurtherbysuggestingthatlightactuallycameintheformofparticle-likequanta.Meanwhile,thefactthatatomsarestableandtheobservationofhowlightwasemittedfromgasesinspiredBohrtosuggestthatelectronscouldonlymovealongcertainallowedorbits,withoccasionaljumpsbetweenthem.Heisenberg,Born, and Jordan elaborated this story of probabilistic jumps into a full theory, matrixmechanics.Fromanotherangle,deBrogliepointedoutthatifwethinkofmatterparticlessuchas electrons as actually being waves, we can derive Bohr’s quantized orbits rather thanpostulatingthem.Schrödingerdevelopedthissuggestionintoafull-blownquantumtheoryofitsown, and it was ultimately demonstrated that wave mechanics and matrix mechanics wereequivalent ways of saying the same thing. Despite initial hopes that wave mechanics couldexplain away the apparent need for probabilities as a fundamental part of the theory, Bornshowed that therightway to thinkaboutSchrödinger’swave functionwasassomething thatyousquaretogettheprobabilityofameasurementoutcome.

Whew. That’s quite a journey, taken in a remarkably short period of time, from Planck’sobservationin1900totheSolvayConferencein1927,whenthenewquantummechanicswasfleshedoutonceandforall.It’stotheenormouscreditofthephysicistsoftheearlytwentiethcenturythattheywerewillingtofaceuptothedemandsoftheexperimentaldata,andindoingsotocompletelyupendthefantasticallysuccessfulNewtonianviewoftheclassicalworld.

Theywerelesssuccessful,however,atcomingtogripswiththeimplicationsofwhattheyhadwrought.

* Annoyingly, the electron accelerates in precisely the opposite direction that the electric field points, because byhumanconventionwe’vedecidedtocallthechargeontheelectron“negative”andthatonaproton“positive.”ForthatwecanblameBenjaminFranklinintheeighteenthcentury.Hedidn’tknowaboutelectronsandprotons,buthedidfigureouttherewasaunifiedconceptcalled“electriccharge.”Whenhewenttoarbitrarily labelwhichsubstanceswerepositivelychargedandwhichwerenegativelycharged,hehadtochoosesomething,andthelabelhepickedforpositivechargecorrespondstowhatwewouldnowcall“havingfewerelectronsthanitshould.”Sobeit.* I’veemphasized that there isonlyonewave function, thewave functionof theuniverse,but thealert readerwillnoticethatIoftentalkabout“thewavefunctionofaparticle.”Thislatterconstructionisperfectlyokayif—andonlyif—theparticleisunentangledfromtherestoftheuniverse.Happily,thatisoftenthecase,butingeneralwehavetokeepourwitsaboutus.

4WhatCannotBeKnown,BecauseItDoesNotExist

UncertaintyandComplementarity

ApoliceofficerpullsoverWernerHeisenberg forspeeding. “Doyouknowhow fastyouweregoing?”asksthecop.“No,”Heisenbergreplies,“butIknowexactlywhereIam!”

I thinkwe canall agree that physics jokes are the funniest jokes there are. They are lessgood at accurately conveying physics. This particular chestnut rests on familiarity with thefamous Heisenberg uncertainty principle, often explained as saying that we cannotsimultaneouslyknowboththepositionandthevelocityofanyobject.Buttherealityisdeeperthanthat.

It’s not thatwe can’t know position andmomentum, it’s that they don’t even exist at thesametime.Onlyunderextremelyspecialcircumstancescananobjectbesaidtohavealocation—whenitswavefunction isentirelyconcentratedononepoint inspace,andzeroeverywhereelse—andsimilarlyforvelocity.Andwhenoneofthetwoispreciselydefined,theothercouldbeliterallyanything,werewe tomeasure it.Moreoften, thewave function includesa spreadofpossibilitiesforbothquantities,soneitherhasadefinitevalue.

Back in the1920s,all thiswas lessclear. Itwasstillnatural to thinkthat theprobabilisticnatureofquantummechanicssimplyindicatedthatitwasanincompletetheory,andthattherewasamoredeterministic,classical-soundingpicturewaitingtobedeveloped.Wavefunctions,inotherwords,mightbeawayofcharacterizingourignoranceofwhatwasreallygoingon,ratherthanbeing the total truth aboutwhat is going on, aswe’re advocatinghere.Oneof the firstthingspeopledidwhenlearningabouttheuncertaintyprinciplewastotrytofindloopholesinit. They failed, but in doing sowe learned a lot about howquantum reality is fundamentallydifferentfromtheclassicalworldwehadbeenusedto.

The absence of definite quantities at the heart of reality that map more or lessstraightforwardlyontowhatwecaneventuallyobserveisoneofthedeepfeaturesofquantummechanics that canbehard toacceptupon first encounter.Therearequantities thatarenotmerelyunknownbutdonotevenexist,eventhoughwecanseeminglymeasurethem.

Quantummechanics forces us to confront this yawning chasm between what we see andwhat really is. In this chapter we’ll see how that gap manifests itself in the uncertaintyprinciple, and in the next chapter we’ll see it again more forcefully in the phenomenon ofentanglement.

Theuncertaintyprincipleowes itsexistenceto the fact that therelationshipbetweenpositionandmomentum (mass times velocity) is fundamentally different in quantummechanics fromwhatitwasinclassicalmechanics.Classically,wecanimaginemeasuringthemomentumofaparticle by tracking its position over time, and seeing how fast it moves. But if all we haveaccess to isa singlemoment,positionandmomentumarecompletely independent fromeachother.IfItellyouthataparticlehasacertainpositionatoneinstant,andItellyounothingelse,youhavenoideawhatitsspeedis,andviceversa.

Physicists refer to the different numbers we use to specify something as that system’s“degrees of freedom.” In Newtonianmechanics, to tell me the complete state of a bunch ofparticles,youhavetotellmethepositionandmomentumofeveryoneofthem,sothedegreesoffreedomarethepositionsandthemomenta.Accelerationisnotadegreeoffreedom,sinceitcanbecalculatedonceweknow the forcesactingon thesystem.Theessenceofadegreeoffreedomisthatitdoesn’tdependonanythingelse.

When we switch to quantum mechanics and start thinking about Schrödinger’s wavefunctions,thingsbecomealittledifferent.Tomakeawavefunctionforasingleparticle,thinkofeverylocationwheretheparticlecouldpossiblybefound,werewetoobserveit.Thentoeachlocation assign an amplitude, a complex number with the property that the square of eachnumberistheprobabilityoffindingtheparticlethere.Thereisaconstraintthatthesquaresof

allthesenumbersadduptopreciselyone,sincethetotalprobabilitythattheparticleisfoundsomewhere must equal one. (Sometimes we speak of probabilities in terms of percentages,whicharenumerically100timestheactualprobability;a20percentchanceisthesameasa0.2probability.)

Noticewedidn’tmention“velocity”or“momentum”there.That’sbecausewedon’thavetoseparatelyspecifythemomentuminquantummechanics,aswedidinclassicalmechanics.Theprobabilityofmeasuringanyparticularvelocityiscompletelydeterminedbythewavefunctionfor all the possible positions. Velocity is not a separate degree of freedom, independent ofposition. The basic reason why is that the wave function is, you know, a wave. Unlike for aclassicalparticle,wedon’thaveasinglepositionandasinglemomentum,wehaveafunctionofall possible positions, and that function typically oscillates up and down. The rate of thoseoscillations determines what we’re likely to see if we were to measure the velocity ormomentum.

Considerasimplesinewave,oscillatingupanddowninaregularpatternthroughoutspace.PlugsuchawavefunctionintotheSchrödingerequationandaskhowitwillevolve.Wefindthata sine wave has a definite momentum, with shorter wavelengths corresponding to fastervelocity.Butasinewavehasnodefiniteposition;onthecontrary, it’sspreadouteverywhere.Andamoretypicalshape,whichisneitherlocalizedatonepointnorspreadoutinaperfectsinewave of fixed wavelength, won’t correspond to either a definite position or a definitemomentum,butsomemixtureofeach.

We see the basic dilemma. If we try to localize a wave function in space, its momentumbecomes more and more spread out, and if we try to limit it to one fixed wavelength (andthereforemomentum)itbecomesmorespreadoutinposition.That’stheuncertaintyprinciple.It’snot thatwe can’tknow bothquantities at the same time; it’s just a fact abouthowwavefunctions work that if position is concentrated near some location, momentum is completelyundetermined, and vice versa. The old-fashioned classical properties called position andmomentumaren’tquantitieswithactualvalues,they’repossiblemeasurementoutcomes.

People sometimes refer to the uncertainty principle in everyday contexts, outside of theequation-filledlanguageofphysicstexts.Soit’simportanttoemphasizewhattheprincipledoesnotsay.It’snotanassertionthat“everythingisuncertain.”Eitherpositionormomentumcouldbecertaininanappropriatequantumstate;theyjustcan’tbecertainatthesametime.

Andtheuncertaintyprincipledoesn’tsaywenecessarilydisturbasystemwhenwemeasureit.Ifaparticlehasadefinitemomentum,wecangoaheadandmeasurethatwithoutchangingitat all. The point is that there are no states for which both position and momentum aresimultaneouslydefinite.Theuncertaintyprinciple isastatementabout thenatureofquantumstatesandtheirrelationshiptoobservablequantities,notastatementaboutthephysicalactofmeasurement.

Finally,theprincipleisnotastatementaboutlimitationsonourknowledgeofthesystem.Wecan know the quantum state exactly, and that’s all there is to know about it; we still can’tpredict the results of all possible future observations with perfect certainty. The idea that“there’ssomethingwedon’tknow,”givenacertainwave function, isanoutdatedrelicofourintuitiveinsistencethatwhatweobserveiswhatreallyexists.Quantummechanicsteachesusotherwise.

You’llsometimesheartheidea,provokedbytheuncertaintyprinciple,thatquantummechanicsviolates logic itself. That’s silly. Logic deduces theorems from axioms, and the resultingtheorems are simply true. The axiomsmay ormay not apply to any given physical situation.

Pythagoras’s theorem—thesquareof thehypotenuseofaright triangleequals thesumof thesquaresoftheothertwosides—iscorrectasaformaldeductionfromtheaxiomsofEuclideangeometry,eventhoughthoseaxiomsdonotholdifwe’retalkingaboutcurvedsurfacesratherthanaflattabletop.

Theideathatquantummechanicsviolates logic lives inthesameneighborhoodofthe ideathat atoms are mostly empty space (a bad neighborhood). Both notions stem from a deepconvictionthat,despiteeverythingwe’velearned,particlesarereallypointswithsomepositionandmomentum,ratherthanbeingwavefunctionsthatarespreadout.

Consideraparticle inabox,wherewe’vedrawna linedividing thebox into leftandrightsides. Ithassomewave functionthat isspreadthroughout thebox.LetpropositionPbe“theparticleisontheleftsideofthebox,”andpropositionQbe“theparticleisontherightsideofthebox.”Wemightbetemptedtosaythatbothofthesepropositionsarefalse,sincethewavefunctionstretchesoverbothsidesofthebox.Buttheproposition“PorQ”hastobetrue,sincetheparticleisinthebox.Inclassicallogic,wecan’thavebothPandQbefalsebut“PorQ”betrue.Sosomethingfishyisgoingon.

What’sfishyisneitherlogicnorquantummechanicsbutourcasualdisregardforthenatureofquantumstateswhenassigningtruthvaluestothestatementsPandQ.Thesestatementsareneithertruenorfalse;they’rejustilldefined.Thereisnosuchthingas“thesideoftheboxtheparticle is on.” If the wave function were concentrated entirely on one side of the box andexactlyvanishedontheother,wecouldgetawaywithassigningtruthvaluestoPandQ;butinthatcaseonewouldbetrueandtheotherwouldbefalse,andclassicallogicwouldbefine.

Despite the fact that classical logic is perfectly valid whenever it is properly applied,quantummechanicshasinspiredmoregeneralapproachesknownasquantumlogic,pioneeredbyJohnvonNeumannandhiscollaboratorGarrettBirkhoff.Bystartingwithslightlydifferentlogicalaxiomsfromthestandardones,wecanderiveasetofrulesobeyedbytheprobabilitiesimpliedbytheBornruleinquantummechanics.Quantumlogicinthissenseisbothinterestinganduseful,butitsexistencedoesnotinvalidatethecorrectnessofordinarylogicinappropriatecircumstances.

Niels Bohr, in an attempt to capture what makes quantum theory so unique, proposed theconceptofcomplementarity.Theideaisthattherecanbemorethantwowaysoflookingataquantumsystem,eachofthemequallyvalid,butwiththepropertythatyoucan’temploythemsimultaneously.Wecandescribe thewave functionofaparticle in termsofeitherpositionormomentum, but not both at the same time. Similarly,we can think of electrons as exhibitingeitherparticle-likeorwave-likeproperties,justnotatthesametime.

Nowhereisthisfeaturemademoreevidentthaninthefamousdouble-slitexperiment.Thisexperimentwasn’tactuallyperformeduntilthe1970s,longafteritwasproposed.Itwasn’toneof thosesurprisingexperimental results that theoristshad to inventanewwayof thinking inordertounderstand,butratherathoughtexperiment(suggestedinitsoriginalformbyEinsteinduring his debates with Bohr, and later popularized by Richard Feynman in his lectures toCaltechundergraduates)meanttoshowthedramaticimplicationsofquantumtheory.

Theideaoftheexperimentistohomeinonthedistinctionbetweenparticlesandwaves.Westartwithasourceofclassicalparticles (maybeapelletgunthat tendstospray insomewhatunpredictabledirections),shootthemthroughasinglethinslit,thendetectthematascreenontheothersideoftheslit.Mostlytheparticleswillpassrightthrough,withperhapsveryslightdeviations if they bump up against the sides of the slit. Sowhatwe see at the detector is apatternof individualpointswherewedetect theparticles,arranged inmoreor lessaslit-likepattern.

Wecouldalsodothesamethingwithwaves,forexample,byplacingtheslitinatubofwaterandcreatingwaves thatpass through it.When thewavespass through, they spreadout inasemicircular pattern before eventually reaching the screen. Of course, we don’t observeparticle-likepointswhen thewaterwavehits the screen,but let’s imaginewehavea specialscreenthatlightsupwithabrightnessthatdependsontheamplitudethewavesreachatanyparticularpoint.Theywillbebrightestatthepointofthescreenthatisclosesttotheslit,andgraduallyfadeaswegetfartheraway.

Nowlet’sdothesamething,butwithtwoslitsinthewayratherthanjustone.Theparticlecase isn’t that much different; as long as our source of particles is sufficiently random thatparticlespassthroughbothslits,whatwe’llseeontheothersideistwolinesofpoints,oneforeachslit(oronethickline,iftheslitsthemselvesaresufficientlyclosetogether).Butthewavecaseisalteredinaninterestingway.Wavescanoscillatedownwardaswellasupward,andtwowaves oscillating in opposite directionswill cancel each other out—a phenomenon known asinterference.Sothewavespassthroughbothslitsatonce,emanatingoutwardinsemicircles,but then set up an interference pattern on the other side. As a result, if we observe the

amplitudeoftheresultantwaveatthefinalscreen,wedon’tsimplyseetwobrightlines;rather,there will be a bright line in the middle (closest to both slits), with alternating dark/brightregionsthatgraduallyfadetoeitherside.

Sofar,that’stheclassicalworldweknowandlove,whereparticlesandwavesaredifferentthingsandeveryonecaneasilydistinguishbetweenthem.Nowlet’sreplaceourpelletgunorwave machine with a source of electrons, in all their quantum-mechanical glory. There areseveraltwistsonthissetup,eachwithprovocativeconsequences.

Firstconsiderjustasingleslit.Inthiscasetheelectronsbehavejustasiftheywereclassicalparticles.Theypass through theslit, thenaredetectedby thescreenon theotherside,eachelectronleavingasingleparticle-likemark.Ifweletnumerouselectronsthrough,theirmarksarescatteredaroundacentral line inthe imageof theslit that theypassedthrough.Nothingfunnyyet.

Nowlet’sintroducetwoslits.(Theslitshavetobeveryclosetogetherforthistowork,whichis one reason it took so long for the experiment to actually be carried out.) Once again,electrons pass through the slits and leave individualmarks on the screen on the other side.However,theirmarksdonotclumpintotwolines,astheclassicalpelletsdid.Rather,theyformaseriesof lines:ahigh-densityone inthemiddle,surroundedbyparallel lineswithgraduallyfewermarks,eachseparatedbydarkregionswithalmostnomarksatall.

Inotherwords,electronsgoingthroughtwoslitsleavewhatisunmistakablyaninterferencepattern, just like waves do, even though they hit the screen with individual marks just likeparticles. This phenomenon has launched a thousand unhelpful discussions about whetherelectronsare“really”particlesorwaves,oraresometimesparticle-likeandothertimeswave-like. One way or another, it’s indisputable that something went through both slits as theelectronstraveledtothescreen.

Atthispointthisisnosurprisetous.Theelectronspassingthroughtheslitsaredescribedby

awavefunction,whichjustlikeourclassicalwavewillgothroughbothslitsandoscillateupanddown,andthereforeitmakessensethatweseeinterferencepatterns.Thenwhentheyhitthescreentheyarebeingobserved,andit’satthatpointtheyappeartousasparticles.

Let’sintroduceoneadditionalwrinkle.Imaginethatwesetuplittledetectorsateachslit,sowecantellwhetheranelectrongoesthroughit.Thatwillsettlethiscrazyideathatanelectroncantravelthroughtwoslitsonceandforall.

Youshouldbeabletofigureoutwhatwesee.Thedetectorsdon’tmeasurehalfofanelectrongoingthrougheachofthetwoslits;theymeasureafullelectrongoingthroughone,andnothingthrough the other, every time. That’s because the detector acts as a measuring device, andwhenwemeasureelectronsweseeparticles.

Butthat’snottheonlyconsequenceoflookingattheelectronasitpassesthroughtheslits.At the screen, on the other side of the slits, the interferencepatterndisappears, andwearebacktoseeingtwobandsofmarksmadebythedetectedelectrons,oneforeachslit.Withthedetectorsdoingtheirjob,thewavefunctioncollapsesastheelectrongoesthroughtheslits,sowedon’tseeinterferencefromawavepassingthroughbothslitsatonce.Whenwe’relookingatthem,electronsbehavelikeparticles.

Thedouble-slitexperimentmakesitdifficulttoclingtothebeliefthattheelectronisjustasingleclassicalpoint,andthewavefunctionsimplyrepresentsourignoranceaboutwherethatpointis.Ignorancedoesn’tcauseinterferencepatterns.Thereissomethingrealaboutthewavefunction.

Wave functionsmay be real, but they’re undeniably abstract, and once we start considering

more than one particle at a time they become hard to visualize. As we move forward withincreasinglysubtleexamplesofquantumphenomenainaction,itwillbeveryhelpfultohaveasimple, readily graspable example we can refer to over and over. The spin of a particle—adegreeoffreedominadditiontoitspositionormomentum—isjustwhatwe’relookingfor.Wehavetothinkabitaboutwhatspinmeanswithinquantummechanics,butoncewedo, itwillmakeourlivesmucheasier.

Thenotion of spin itself isn’t hard tograsp: it’s just rotation aroundanaxis, as theEarthdoeseverydayorapirouettingballetdancerdoesontheirtiptoes.Butjustliketheenergiesofanelectronorbitinganatomicnucleus, inquantummechanics thereareonlycertaindiscreteresultswecanobtainwhenwemeasureaparticle’sspin.

Foranelectron, forexample, thereare twopossiblemeasurementoutcomes forspin.Firstpick an axiswith respect towhichwemeasure the spin.We always find that the electron isspinningeitherclockwiseorcounterclockwisewhenwelookalongthataxis,andalwaysatthesame rate. These are conventionally referred to as “spin-up” and “spin-down.” Think of the“right-handrule”: ifyouwrap the fingersofyourrighthand in thedirectionof rotation,yourthumbwillbepointingalongtheappropriateup/downaxis.

A spinning electron is a tinymagnet,with north and southmagnetic poles,much like theEarth;thespinaxispointstowardthenorthpole.Onewayofmeasuringthespinofaparticularelectron is to shoot it through a magnetic field, which will deflect the electron by a bitdependingonhowitsspinisoriented.(Asatechnicality,themagneticfieldhastobefocusedintherightway—spreadoutononeside,pinchedtightlyontheother—forthistowork.)

If I told you that the electron had a certain total spin, you might make the followingprediction for such an experiment: the electron would be deflected up if its spin axis werealignedwiththeexternalfield,deflecteddownifitsspinwerealignedintheoppositedirection,anddeflectedatsomeintermediateangleifitsspinweresomewhereinbetween.Butthat’snotwhatwesee.

Thisexperimentwasfirstperformedin1922,byGermanphysicistsOttoStern(anassistanttoMaxBorn)andWalterGerlach,beforetheideaofspinhadbeenexplicitlyspelledout.Whattheysawwasremarkable.Electronsareindeeddeflectedbypassingthroughthemagneticfield,buttheyeithergoup,ortheygodown;nothinginbetween.Ifwerotatethemagneticfield,theelectrons are still deflected in the direction of the field they pass through, either along oragainst it, but no intermediate values. The measured spin, like the energy of an electronorbitinganatomicnucleus,appearstobequantized.

Thatseemssurprising.Evenifwe’veacclimatedourselvestotheideathattheenergyofanelectronorbitinganucleusonlycomesincertainquantizedvalues,at leastthatenergyseemslikeanobjectivepropertyoftheelectron.Butthisthingwecallthe“spin”oftheelectronseemsto give us different answers depending on how we measure it. No matter what particulardirectionwemeasurethespinalong,thereareonlytwopossibleoutcomeswecanobtain.

Tomake surewehaven’t lost ourminds, let’s be clever and run the electron through twomagnetsinarow.Rememberthattherulesoftextbookquantummechanicstellusthatifwegeta certain measurement outcome, then measure the same system immediately again, we willalwaysgetthesameanswer.Andindeedthat’swhathappens;ifanelectronisdeflectedupwardby onemagnet (and is therefore spin-up), it will always be deflected upward by a followingmagnetorientedinthesameway.

What ifwerotateoneof themagnetsby90degrees?Sowe’resplittingan initialbeamofelectronsintospin-upandspin-downasmeasuredbyaverticallyorientedmagnet,thentakingthespin-upelectronsandpassingthemthroughahorizontallyorientedmagnet.Whathappensthen?Dotheyholdtheirbreathandrefusetopassthrough,becausetheyareverticallyoriented

spin-upelectronsandwe’reforcingthemtobemeasuredalongahorizontalaxis?

No.Instead,thesecondmagnetsplitsthespin-upelectronsintotwobeams.Halfofthemaredeflectedtotheright(alongthedirectionofthesecondmagnet)andhalfofthemaredeflectedtotheleft.

Madness. Our classical intuition makes us think that there is something called “the axisaroundwhichtheelectronisspinning,”anditmakessense(maybe)thatthespinaroundthataxis isquantized.But theexperiments show that theaxisaroundwhich thespin isquantizedisn’t predetermined by the particle itself; you can choose any axis you like by rotating yourmagnetappropriately,andthespinwillbequantizedwithrespecttothataxis.

Whatwe’re bumping up against is anothermanifestation of the uncertainty principle. Thelessonwelearnedwasthat“position”and“momentum”aren’tpropertiesthatanelectronhas;theyarejustthingswecanmeasureaboutit.Inparticular,noparticlecanhaveadefinitevalueofbothsimultaneously.Oncewespecifytheexactwavefunctionforposition,theprobabilityofobservinganyparticularmomentumisentirelyfixed,andviceversa.

Thesameistruefor“verticalspin”and“horizontalspin.”*Thesearenotseparatepropertiesan electron can have; they are just different quantities we can measure. If we express thequantumstateintermsoftheverticalspin,theprobabilityofobservingleftorrighthorizontalspinisentirelyfixed.Themeasurementoutcomeswecangetaredeterminedbytheunderlyingquantum state, which can be expressed in different but equivalent ways. The uncertaintyprincipleexpressesthe fact that therearedifferent incompatiblemeasurementswecanmakeonanyparticularquantumstate.

Systems with two possible measurement outcomes are so common and useful in quantummechanicsthattheyaregivenacutename:qubits.Theideaisthataclassical“bit”hasjusttwopossible values, say, 0 and 1. A qubit (quantum bit) is a system that has two possiblemeasurementoutcomes, say, spin-upand spin-downalong some specifiedaxis.The stateof agenericqubit isasuperpositionofbothpossibilities,eachweightedbyacomplexnumber,theamplitude for each alternative. Quantum computers manipulate qubits in the same way thatordinarycomputersmanipulateclassicalbits.

Wecanwritethewavefunctionofaqubitas

Thesymbolsaandbarecomplexnumbers,representingtheamplitudesforspin-upandspin-down, respectively. The pieces of the wave function representing the different possiblemeasurement outcomes, in this case spin-up/-down, are the “components.” In this state, theprobabilityofobserving theparticle tobespin-upwouldbe |a|2, and theprobability for spin-downwould be |b|2. If, for example, a andb were both equal to the square root of 1/2, theprobabilityofobservingspin-uporspin-downwouldbe1/2.

Qubits can help us understand a crucial feature of wave functions: they are like thehypotenuseofarighttriangle,forwhichtheshortersidesaretheamplitudesforeachpossiblemeasurement outcome. In other words, the wave function is like a vector—an arrow with alengthandadirection.

Thevectorwe’retalkingaboutdoesn’tpointinadirectioninrealphysicalspace,like“up”or“north.”Rather,itpointsinaspacedefinedbyallpossiblemeasurementoutcomes.Forasinglespin qubit, that’s either spin-up or spin-down (once we choose some axis along which tomeasure).Whenwe say “the qubit is in a superposition of spin-up and spin-down,”whatwereallymeanis“thevectorrepresentingthequantumstatehassomecomponentinthespin-updirection,andanothercomponentinthespin-downdirection.”

It’snaturaltothinkofspin-upandspin-downaspointinginoppositedirections.Imean,justlookatthearrows.Butasquantumstates,theyareperpendiculartoeachother:aqubitthatiscompletelyspin-uphasnocomponentofspin-down,andviceversa.Eventhewavefunctionforthe position of a particle is a vector, though we normally visualize it as a smooth functionthroughoutspace.Thetrickistothinkofeverypointinspaceasdefiningadifferentcomponent,and thewave function isa superpositionofallof them.Therearean infinitenumberof suchvectors,sothespaceofallpossiblequantumstates,calledHilbertspace,isinfinite-dimensionalforthepositionofasingleparticle.That’swhyqubitsaresomucheasiertothinkabout.Twodimensionsareeasiertovisualizethaninfinitedimensions.

Whenthereareonlytwocomponentsinourquantumstate,asopposedtoinfinitelymany,itcanbehardtothinkofthestateasa“wavefunction.”It’snotverywavy,anditdoesn’tlooklikeasmoothfunctionofspace.Therightwaytothinkaboutitisactuallytheotherwayaround.Thequantum state is not a function of ordinary space, it’s a function of the abstract “space ofmeasurementoutcomes,”whichforaqubitonly includestwopossibilities.Whenthethingweobserve is the location of a single particle, the quantum state assigns an amplitude to everypossible location, which looks just like a wave in ordinary space. That’s the unusual case,however; thewave function is somethingmore abstract, andwhenmore than one particle isinvolved, it becomeshard to visualize.Butwe’re stuckwith the “wave function” terminology.Qubitsaregreatbecauseatleastthewavefunctionhasonlytwocomponents.

Thismay seem like an unnecessarymathematical detour, but there are immediate payoffs tothinkingaboutwavefunctionsasvectors.OneisexplainingtheBornrule,whichsaysthattheprobability for any particularmeasurement outcome is given by its amplitude squared.We’lldive into details later, but it’s easy to see why the ideamakes sense. As a vector, the wavefunctionhasalength.Youmightexpectthatthelengthcouldshrinkorgrowovertime,but itdoesn’t; according to Schrödinger’s equation, the wave function just changes its “direction”while maintaining a constant length. And we can compute that length using Pythagoras’stheoremfromhigh-schoolgeometry.

The numerical value of the length of the vector is irrelevant; we can just pick it to be aconvenient number, knowing that itwill remain constant. Let’s pick it to be one: everywavefunctionisavectoroflengthone.Thevectoritselfisjustlikethehypotenuseofarighttriangle,with the components forming the shorter sides. So from Pythagoras’s theorem, we have asimplerelationship:thesquaresoftheamplitudesadduptounity,|a|2+|b|2=1.

That’s the simple geometric fact underlying the Born rule for quantum probabilities.Amplitudes themselves don’t add up to one, but their squares do. That is exactly like animportantfeatureofprobability:thesumofprobabilitiesfordifferentoutcomesneedstoequalone.(Somethinghastohappen,andthetotalprobabilityofallexclusivesomethingsaddsuptounity.) Another rule is that probabilities need to be non-negative numbers. Once again,amplitudessquaredfitthebill:amplitudescanbenegative(orcomplex),buttheirsquaresarenon-negativerealnumbers.

So even before thinking too hard, we can tell that “amplitudes squared” have the rightproperties to be the probabilities of outcomes—they are a set of non-negative numbers thatalwaysadduptoone,becausethat’sthelengthofthewavefunction.Thisisattheheartofthewholematter: theBorn rule is essentiallyPythagoras’s theorem,applied to theamplitudesofdifferent branches. That’swhy it’s the amplitudes squared, not the amplitudes themselves orthesquarerootoftheamplitudesoranythingcrazylikethat.

Thevectorpicturealsoexplainstheuncertaintyprincipleinanelegantway.Rememberthatspin-upelectronssplitfifty-fiftyintoright-andleft-spinningelectronswhentheypassedthroughasubsequenthorizontalmagnet.Thatsuggeststhatanelectroninaspin-upstateisequivalenttoasuperpositionofspin-rightandspin-leftelectronstates,andlikewiseforspin-down.

Sotheideaofbeingspin-leftorspin-rightisn’tindependentfrombeingspin-uporspin-down;anyonepossibilitycanbethoughtofasasuperpositionoftheothers.Wesaythatspin-upandspin-downtogetherformabasisforthestateofaqubit—anyquantumstatecanbewrittenasasuperpositionofthosetwopossibilities.Butspin-leftandspin-rightformanotherbasis,distinctbutequallygood.Writingitonewaycompletelyfixestheotherway.

Think of this in vector terms. If we draw a two-dimensional plane with spin-up as thehorizontalaxisandspin-downas theverticalaxis, from theabove relationswe see that spin-rightandspin-leftpointat45degreeswithrespecttothem.Givenanywavefunction,wecouldexpressitintheup/downbasis,butwecouldequallywellexpressitintheright/leftbasis.Onesetofaxes isrotatedwithrespecttotheother,buttheyarebothperfectly legitimatewaysofexpressinganyvectorwelike.

Now we can see where the uncertainty principle comes from. For a single spin, theuncertainty principle says that the state can’t have a definite value for the spin along theoriginalaxes(up/down)andtherotatedaxes(right/left)atthesametime.Thisisclearfromthepicture:ifthestateispurelyspin-up,it’sautomaticallysomecombinationofspin-leftandspin-right,andviceversa.

Just as there are no quantum states that are simultaneously localized in position andmomentum, there are no states that are simultaneously localized in both vertical spin andhorizontal spin. Theuncertainty principle reflects the relationship betweenwhat really exists(quantumstates)andwhatwecanmeasure(oneobservableatatime).

*Andforthethirdperpendiculardirection,whichwemightcall“forwardspin,”thoughwedidn’tmeasurethat.

5EntangledUpinBlue

WaveFunctionsofManyParts

Popular discussions of the Einstein-Bohr debates often give the impression that Einsteincouldn’tquitehandletheuncertaintyprinciple,andspenthistimetryingtoinventcleverwaysto circumvent it. But what really bugged him about quantum mechanics was its apparentnonlocality—whathappensatonepoint inspacecanseeminglyhave immediateconsequencesfor experiments done very far away. It took him a while to codify his concerns into a well-formulatedobjection,andindoingsohehelpedilluminateoneofthemostprofoundfeaturesofthequantumworld:thephenomenonofentanglement.

Entanglement arises because there is only one wave function for the entire universe, notseparatewavefunctionsforeachpieceofit.Howdoweknowthat?Whycan’twejusthaveawavefunctionforeveryparticleorfield?

Consideranexperiment inwhichweshoot twoelectronsateachother,movingwithequalandoppositevelocities.Becausebothhaveanegativeelectriccharge,theywillrepeleachother.Classically, if we were given the initial positions and velocities of the electrons, we couldcalculatepreciselythedirectionsintowhicheachofthemwouldscatter.Quantum-mechanically,allwecandoiscalculatetheprobabilitythattheywilleachbeobservedonvariouspathsafterthey interact with each other. The wave function of each particle spreads out in a roughlysphericalpattern,untilweultimatelyobserve it andpindownadefinitedirection inwhich itwasmoving.

Whenweactuallydo thisexperiment,andobserve theelectronsafter theyhavescattered,wenoticesomethingimportant.Sincetheelectronsinitiallyhadequalandoppositevelocities,thetotalmomentumwaszero.Andmomentumisconserved,sothepost-interactionmomentumshould also be zero. This means that while the electrons might emerge moving in variousdifferentdirections,whateverdirectiononeofthemmovesin,theothermovesinpreciselytheopposite.

That’s funny, when you think about it. The first electron has a probability of scattering atvarious angles, and so does the second one. But if they each had a separate wave function,thosetwoprobabilitieswouldbecompletelyunrelated.Wecouldimaginejustobservingoneofthe electrons, and measuring the direction in which it’s moving. The other one would be

undisturbed.Howcoulditknowthatit’ssupposedtobemovingintheoppositedirectionwhenweactuallydomeasureit?

We’vealreadygivenawaytheanswer.Thetwoelectronsdon’thaveseparatewavefunctions;their behavior is described by the single wave function of the universe. In this case we canignoretherestoftheuniverse,andjustfocusinonthesetwoelectrons.Butwecan’tignoreoneof theelectronsandfocus inontheother; thepredictionswemakeforobservationsofeitheronecanbedramaticallyaffectedbytheoutcomeofobservationsoftheother.Theelectronsareentangled.

A wave function is an assignment of a complex number, the amplitude, to each possibleobservationaloutcome,and the squareof theamplitudeequals theprobability thatwewouldobservethatoutcomewerewetomakethatmeasurement.Whenwe’retalkingaboutmorethanoneparticle,thatmeansweassignanamplitudetoeverypossibleoutcomeofobservingalltheparticles at once. If what we’re observing is positions, for example, the wave function of theuniversecanbethoughtofasassigninganamplitudetoeverypossiblecombinationofpositionsforalltheparticlesintheuniverse.

Youmightwonderwhetherit’spossibletovisualizesomethinglikethat.Wecandoitforthesimple case of a single particle that we imagine only moves along one dimension, say, anelectronconfinedtoathincopperwire:wedrawalinerepresentingthepositionoftheparticle,andplota functionrepresenting theamplitude foreachposition. (Generallywecheateven inthissimplecontextbyjustplottingarealnumberratherthanacomplexnumber,butsobeit.)For two particles confined to the same one-dimensional motion, we could draw a two-dimensionalplanerepresentingthepositionsofeachofthetwoparticles,andthendoathree-dimensional contour plot for the wave function. Note that this isn’t one particle in two-dimensionalspace;it’stwoparticles,eachonaone-dimensionalspace,sothewavefunctionisdefinedonthetwo-dimensionalplanedescribingbothpositions.

BecauseofthefinitespeedoflightandafinitetimesincetheBigBang,wecanseeonlyafiniteregionofthecosmos,whichwelabel“theobservableuniverse.”Thereareapproximately1088particlesintheobservableuniverse,mostlyphotonsandneutrinos.Thatisanumbermuchgreater than two. And each particle is located in three-dimensional space, not just a one-dimensionalline.Howintheworldarewesupposedtovisualizeawavefunctionthatassignsanamplitude to every possible configuration of 1088 particles distributed through three-dimensionalspace?

We’re not. Sorry. The human imagination wasn’t designed to visualize the enormously bigmathematical spaces that are routinely used in quantum mechanics. For just one or twoparticles,wecanmuddlethrough;morethanthat,andwehavetodescribethingsinwordsandequations.Fortunately,theSchrödingerequationisstraightforwardanddefiniteinwhatitsaysabouthowthewavefunctionbehaves.Onceweunderstandwhat’sgoingonfortwoparticles,thegeneralizationto1088particlesisjustmaths.

Thefactthatwavefunctionsaresobigcanmakethinkingaboutthemalittleunwieldy.Happilywe can cast almost everything interesting to say about entanglement into the much simplercontextofjustafewqubits.

Borrowingfromawhimsicaltraditionintheliteratureoncryptography,quantumphysicistslike to consider two people named Alice and Bob who share qubits with each other. So let’simagine two electrons,A belonging to Alice andB belonging to Bob. The spins of those twoelectronsconstitutea two-qubitsystem,andaredescribedbyacorrespondingwave function.Thewave functionassignsanamplitude toeachconfigurationof thesystemasawhole,withrespect to somethingwemightobserveabout it, suchas its spin in theverticaldirection.Sotherearefourpossiblemeasurementoutcomes:bothspinsareup,bothspinsaredown,AisupandBisdown,andAisdownandBisup.Thestateofthesystemissomesuperpositionofthesefour possibilities, which are the basis states. Within each set of parentheses, the first spin is

Alice’s,andthesecondisBob’s.

Justbecausewehavetwoqubits,itdoesn’tmeantheyarenecessarilyentangled.Considerastatethatissimplyoneofthebasisstates,say,theonewherebothqubitsarespin-up.IfAlicemeasuresherqubitalongtheverticalaxis,shewillalwaysobtainspin-up,andlikewiseforBob.IfAlicemeasuresherspinalongthehorizontalaxis,shehasafifty-fiftychanceofgettingspin-rightorspin-left,andagain likewiseforBob.But ineachcase,wedon’t learnanythingaboutwhatBobwillseeby learningwhatAlicesaw.That’swhywecanoftencasuallyspeakof“thewavefunctionofaparticle,”eventhoughweknowbetter—whendifferentpartsofthesystemareunentangledwitheachother,it’sjustasiftheyhavetheirownwavefunctions.

Instead,let’sconsideranequalsuperpositionoftwobasisstates,onewithbothspinsup,andtheotherwithbothspinsdown:

IfAlicemeasuresherspinalongtheverticalaxis,shehasafifty-fiftychanceofgettingspin-uporspin-down,and likewise forBob.Thedifferencenow is that ifwe learnAlice’soutcomebeforeBobdoeshismeasurement,weknowwhatBobwillseewith100percentconfidence—he’sgoingtoseethesamethingthatAlicedid.Inthelanguageoftextbookquantummechanics,Alice’smeasurementcollapsesthewavefunctionontooneofthetwobasisstates,leavingBobwith a deterministic outcome. (In Many-Worlds language, Alice’s measurement branches thewave function, creating twodifferentBobs, eachofwhomwill get a certainoutcome.)That’sentanglementinaction.

In the aftermath of the 1927 Solvay Conference, Einstein remained convinced that quantummechanics,especiallyasinterpretedbytheCopenhagenschool,didaverygoodjobatmakingpredictionsforexperimentaloutcomes,butfellwellshortasacompletetheoryofthephysicalworld.Hisconcernswerefinallywrittenupforpublicationin1935withhiscollaboratorsBorisPodolskyandNathanRosen,inapaperthatisuniversallyknownassimplyEPR.Einsteinlatersaid that theprimary ideashadbeenhis,Rosenhaddone thecalculations,andPodolskyhaddonemuchofthewriting.

EPRconsideredthepositionandmomentumoftwoparticlesmovinginoppositedirections,but it’s easier for us to talk aboutqubits.Consider two spins that are in the entangled statewrittenabove.(It’sveryeasytocreatesuchastateinthelab.)Alicestayshomewithherqubit,butBob takeshisandembarksona long journey—say,he jumps ina rocket shipand flies toAlpha Centauri, four light-years away. The entanglement between two particles doesn’t fadeaway as they are moved apart; as long as neither Alice nor Bob measures the spins of theirqubits,theoverallquantumstatewillremainthesame.

OnceBobarrivessafelyatAlphaCentauri,Alicefinallydoesmeasurethespinofherparticle,alonganagreed-uponverticalaxis.Beforethatmeasurement,wewerecompletelyunsurewhatsuchanobservationwouldrevealforherspin,andlikewiseforBob’s.Let’ssupposethatAliceobservesspin-up.Then,bytherulesofquantummechanics,weimmediatelyknowthatBobwillalsoobservespin-up,wheneverhegetsaroundtodoingameasurement.

That’sweird.Thirtyyearsearlier,Einsteinhadestablishedtherulesofthespecialtheoryofrelativity, which says among other things that signals cannot travel faster than the speed oflight. And yet here we’re saying that according to quantum mechanics, a measurement thatAlicedoeshere andnowhas an immediate effect onBob’squbit, even though it’s four light-yearsaway.HowdoesBob’squbitknowthatAlice’shasbeenmeasured,andwhattheoutcomewas?Thisisthe“spookyactionatadistance”thatEinsteinsomemorablyfrettedabout.

It’snotnecessarilyasbadasitseems.Thefirstthingyoumightwonderabout,uponbeinginformedthatquantummechanicsapparentlysendsinfluencesfasterthanthespeedoflight,iswhetherornotwecouldtakeadvantageof thisphenomenontocommunicate instantlyacrosslargedistances.Canwebuildaquantum-entanglementphone, forwhich thespeedof light isnotalimitationatall?

No, we can’t. This is pretty clear in our simple example: if Alice measures spin-up, sheinstantlyknowsthatBobwillalsomeasurespin-upwhenhegetsaroundtoit.ButBobdoesn’tknowthat.Inorderforhimtoknowwhatthespinofhisparticleis,Alicehastosendhimhermeasurementresultbyconventionalmeans—whicharelimitedbythespeedoflight.

Youmightthinkthere’saloophole:WhatifAlicedoesn’tjustmeasureherqubitandfindoutarandomanswer,butratherforcesheranswertobespin-up?ThenBobwouldalsogetspin-up.Thatwouldseemlikeinformationhadbeentransmittedinstantaneously.

Theproblemisthatthere’snostraightforwardwaytostartwithaquantumsystemthatisina superposition and measure it in such a way that we can force a particular answer. If Alicesimplymeasuresherspin,she’llgetupordownwithequalprobabilities,noifs,ands,orbuts.WhatAlicecandoistomanipulateherspinbeforeshemeasuresit,forcingittobe100percentspin-up rather than in a superposition. For example, she can shoot a photon at her electron,withjusttherightpropertiesthatthephotonleavestheelectronaloneiftheelectronwasspin-up, and flips the electron to spin-up if it was spin-down. Now Alice’s original electron willdefinitelybemeasuredtobespin-up.ButthatelectronisalsonolongerentangledwithBob’selectron. Rather, the entanglement has been transferred to the photon, which is in asuperpositionof“leftAlice’selectronalone”and“bumpedintoAlice’selectron.”Bob’selectroniscompletelyunaffected,andhe’sgoingtogetspin-uporspin-downwithfifty-fiftyprobability,sonoinformationhasbeentransmitted.

This isageneral featureofquantumentanglement: theno-signaling theorem,accordingtowhichanentangledpairofparticlescannotactuallybeused to transmit informationbetweentwopartiesfasterthanlight.Soquantummechanicsseemstobeexploitingasubtle loophole,violatingthespiritofrelativity(nothingtravelsfasterthanthespeedoflight)whileobeyingtheletterofthelaw(actualphysicalparticles,andwhateverusefulinformationtheymightconvey,cannottravelfasterthanthespeedoflight).

Theso-calledEPRparadox (whichisn’taparadoxatall, justafeatureofquantummechanics)goesbeyondsimpleworriesaboutspookyactionatadistance.Einsteinaimedtoshownotonlythat quantum mechanics was spooky but that it couldn’t possibly be a complete theory—thatthere had to be some underlying comprehensive model for which quantum mechanics wassimplyausefulapproximation.

EPR believed in the principle of locality—the physical quantities describing nature are

defined at specific points in spacetime, not spread out all over the place, and they interactdirectlyonlywithotherquantitiesnearby,notatadistance.Saidanotherway,giventhespeed-of-lightrestrictionofspecialrelativity,localitywouldseemtoimplythatnothingwecandotoaparticleatonelocationcaninstantaneouslyaffectmeasurementswemightperformonanotherparticleveryfaraway.

On the face of it, the fact that two widely separated particles can be entangled seems toimply that locality is violated in quantum mechanics. But EPR wanted to be a little morethorough,andestablishthattherewasn’tsomecleverwork-aroundthatwouldmakeeverythingseemlocal.

Theysuggestedthefollowingprinciple:ifwehaveaphysicalsysteminaspecifiedstate,andthereisameasurementwecandoonthatsystemsuchthatweknowwith100percentcertaintywhattheoutcomewillbe,weassociateanelementofrealitywiththatmeasurementoutcome.Inclassicalmechanics,thepositionandthemomentumofeachparticlequalifyaselementsofreality.Inquantummechanics,ifwehaveaqubitinapurespin-upstate,thereisanelementofrealitycorrespondingtothespinintheverticaldirection,butthereneednotbeanelementofreality corresponding to the horizontal spin, as we don’t know what we will get when wemeasure that.A “complete” theory, in theEPR formulation, is one inwhich every element ofrealityhasadirectcounterpart in the theory itself,and theyargued thatquantummechanicscouldn’tbecompletebythiscriterion.

Let’s take Alice and Bob and their entangled qubits, and imagine that Alice has justmeasuredtheverticalspinofherparticle,findingthatitpointsupward.WenowknowthatBobwillalsomeasurespin-up,even ifBobdoesn’tknowithimself.SobyEPR’s lights, there isanelementofrealityattachedtoBob’sparticle,sayingthatthespinisup.It’snotthatthiselementof reality came into existence when Alice did her measurement, as Bob’s particle is very faraway,andlocalitysaysthattheelementofrealitymustbelocatedwheretheparticleis;itmusthavebeenthereallalong.

But now imagine that Alice didn’t do the vertical-spin measurement at all, but insteadmeasuredthespinofherparticlealongthehorizontalaxis.Let’ssayshemeasuresspin-rightfortheparticle.TheentangledquantumstatewestartedwithensuresusthatBobwillgetthesameresultthatAlicedid,nomatterwhatdirectionshechoosestomeasureherspinin.Soweknowthat Bob would also measure spin-right, and by EPR’s lights there is—and was all along—anelement of reality that says “spin-right for Bob’s qubit if it’s measured along the horizontalaxis.”

There’snowayforeitherAlice’sparticleorBob’stoknowaheadoftimewhichmeasurementAlice was going to make. Hence, Bob’s qubit must come equipped with elements of realityguaranteeingthatitsspinwouldbeupifmeasuredvertically,andrightifmeasuredhorizontally.

That’s exactly what the uncertainty principle says cannot happen. If the vertical spin isexactly determined, the horizontal spin is completely unknown, and vice versa, at leastaccording to the conventional rules of quantum mechanics. There is nothing in the quantumformalism that can determine both a vertical spin and a horizontal spin at the same time.Therefore,EPRtriumphantlyconclude,theremustbesomethingmissing—quantummechanicscannotbeacompletedescriptionofphysicalreality.

The EPR paper caused a stir that reached far beyond the community of professionalphysicists. TheNew York Times, having been tipped off by Podolsky, published a front-pagestory about the ideas. This outraged Einstein, who penned a stern letter that the Timespublished, inwhichhedecriedadvancediscussionof scientific results in the “secularpress.”It’sbeensaidthatheneverspoketoPodolskyagain.

Theresponsefromprofessionalscientistswasalsorapid.NielsBohrwroteaquickreplytothe EPR paper, which many physicists claimed resolved all the puzzles. What is less clear ispreciselyhowBohr’spaperwassupposedtohaveachievedthat;asbrilliantandcreativeashewasasathinker,Bohrwasneveranespeciallyclearcommunicator,ashehimselfadmitted.Hispaperwasfullofsentenceslike“inthisstagetherearisestheessentialproblemofaninfluenceon the precise conditions which define the possible types of prediction which regard thesubsequent behavior of the system.” Roughly, his argument was that we shouldn’t go aboutattributingelementsofrealitytosystemswithouttakingintoaccounthowtheyaregoingtobeobserved.Whatisreal,Bohrseemstosuggest,dependsnotonlyonwhatwemeasure,butonhowwechoosetomeasureit.

Einsteinandhiscollaborators laidoutwhat they took tobereasonablecriteria foraphysicaltheory—locality,andassociatingelementsofrealitytodeterministicallypredictablequantities—andshowedthatquantummechanicswasincompatiblewiththem.Buttheydidn’tconcludethatquantummechanicswaswrong, just that itwas incomplete.Thehoperemainedalive thatwewouldsomedayfindabettertheorythatbothwaslocalandrespectedreality.

ThathopewasdefinitivelysquashedbyJohnStewartBell,aphysicistfromNorthernIrelandwho worked at the CERN laboratory in Geneva, Switzerland. He became interested in thefoundations of quantum mechanics in the 1960s, at a point in physics history when it wasconsidered thoroughly disreputable to spend time thinking about such things. Today Bell’stheoremonentanglementisconsideredoneofthemostimportantresultsinphysics.

ThetheoremasksustoonceagainconsiderAliceandBobandtheirentangledqubitswithalignedspins.(SuchquantumstatesarenowknownasBellstates,althoughitwasDavidBohmwho first conceptualized the EPR puzzle in these terms.) Imagine that Alice measures theverticalspinofherparticle,andobtainstheresultthatitisspin-up.WenowknowthatifBobmeasures the vertical spin of his particle, he will also obtain spin-up. Furthermore, by theordinary rulesofquantummechanicsweknow that ifBobchooses tomeasure thehorizontalspininstead,hewillgetspin-rightandspin-leftwithfifty-fiftyprobability.WecansaythatifBobmeasures thevertical spin, thecorrelationbetweenhis resultandAlice’swillbe100percent(weknowexactlywhathe’llget),whereasifhemeasureshorizontalspin,therewillbe0percentcorrelation(wehavenoideawhathewillget).

SowhatifBob,growingboredallbyhimselfinaspaceshiporbitingAlphaCentauri,decidestomeasurethespinofhisparticlealongsomeaxisinbetweenthehorizontalandvertical?(ForconvenienceimaginethatAliceandBobactuallysharealargenumberofentangledBellpairs,so they can keep doing these measurements over and over, and we only care about whathappenswhenAliceobservesspin-up.)ThenBobwillusually,butnotalways,observethespintobepointedalongwhateverdirectionismorecloselyalignedwiththevertical“up.”Infact,wecandothemaths:ifBob’saxisisat45degrees,exactlyhalfwaybetweenverticalandhorizontal,there will be a 71 percent correlation between his results and Alice’s. (That’s one over thesquarerootoftwo,ifyou’rewonderingwherethenumbercomesfrom.)

WhatBellshowed,undercertainsuperficiallyreasonableassumptions,isthatthisquantum-mechanicalpredictionisimpossibletoreproduceinanylocaltheory.Infact,heprovedastrict

inequality:thebestyoucanpossiblydowithoutsomekindofspookyactionatadistancewouldbe to achieve a 50 percent correlation between Alice and Bob if their measurements wererotated by 45 degrees. The quantum prediction of 71 percent correlation violates Bell’sinequality. There is adistinct, undeniabledifferencebetween thedreamof simpleunderlyinglocaldynamics,andthereal-worldpredictionsofquantummechanics.

I presume you are currently thinking to yourself, “Hey, what do you mean that Bell madesuperficially reasonable assumptions? Spell them out. I’ll decide for myself what I findreasonableandwhatIdon’t.”

Fairenough.TherearetwoassumptionsbehindBell’stheoreminparticularthatonemightwanttodoubt.OneiscontainedinthesimpleideathatBob“decides”tomeasurethespinofhisqubitalongacertainaxis.Anelementofhumanchoice,orfreewill,seemstohavecrept intoour theoremaboutquantummechanics.That’shardlyunique,ofcourse;scientistsarealwaysassumingthattheycanchoosetomeasurewhatevertheywant.Butreallywethinkthat’sjustaconvenientwayoftalking,andeventhosescientistsarecomposedofparticlesandforcesthatthemselvesobeythe lawsofphysics.Sowecan imagine invokingsuperdeterminism—theideathat the true laws of physics are utterly deterministic (no randomness anywhere), andfurthermore that the initial conditionsof theuniversewere laiddownat theBigBang in justprecisely suchaway thatcertain“choices”arenevergoing tobemade. It’s conceivable thatonecouldinventaperfectlylocalsuperdeterministictheorythatwouldmimicthepredictionsofquantumentanglement, simplybecause theuniversewasprearranged tomake it appear thatway.Thisseemsunpalatabletomostphysicists;ifyoucandelicatelyarrangeyourtheorytodothat, itcanbasicallybearrangedtodoanythingyouwant,andatthatpointwhyareweevendoingphysics?Butsomesmartpeoplearepursuingtheidea.

The other potentially doubtable assumption seems uncontroversial at first glance: thatmeasurements have definite outcomes. When you observe the spin of a particle, you get anactualresult,eitherspin-uporspin-downalongwhateveraxisyouaremeasuringitwithrespectto.Seemsreasonable,doesn’tit?

But wait. We actually know about a theory where measurements don’t have definiteoutcomes—austere, Everettian quantum mechanics. There, it’s simply not true that we geteitherupordownwhenwemeasureanelectron’sspin;inonebranchofthewavefunctionwegetup,intheotherwegetdown.Theuniverseasawholedoesn’thaveanysingleoutcomeforthat measurement; it has multiple ones. That doesn’t mean that Bell’s theorem is wrong inMany-Worlds;mathematicaltheoremsareunambiguouslyright,giventheirassumptions.Itjustmeans that the theorem doesn’t apply. Bell’s result does not imply that we have to includespookyactionatadistanceinEverettianquantummechanics,asitdoesforboringoldsingle-world theories. The correlations don’t come about because of any kind of influence beingtransmitted faster than light, but because of branching of the wave function into differentworlds,inwhichcorrelatedthingshappen.

Foraresearcherinthefoundationsofquantummechanics,therelevanceofBell’stheoremtoyourworkdependsonexactlywhatitisyou’retryingtodo.Ifyouhavedevotedyourselftothetaskofinventinganewversionofquantummechanicsfromscratch,inwhichmeasurementsdohavedefiniteoutcomes,Bell’s inequality is themost importantguidepostyouhave tokeep inmind.If,ontheotherhand,you’rehappywithMany-Worldsandaretryingtopuzzleouthowtomapthetheoryontoourobservedexperience,Bell’sresultisanautomaticconsequenceoftheunderlyingequations,notanadditionalconstraintyouneedtoworryaboutmovingforward.

OneofthefantasticthingsaboutBell’stheoremisthatitturnsthesupposedspookinessofquantum entanglement into a straightforwardly experimental question—does nature exhibitintrinsically non-local correlationsbetween farawayparticles, or not?You’ll behappy to hearthat experiments have been done, and the predictions of quantum mechanics have beenspectacularlyverifiedeverytime.Thereisatraditioninpopularmediaofwritingarticleswithbreathless headlines like “Quantum Reality Is Even More Bizarre Than Previously Believed!”But when you look into the results they are actually reporting, it’s another experiment thatconfirmsexactlywhatacompetentquantummechanicwouldhavepredictedallalongusingthetheory that had been established by 1927, or at least by 1935. We understand quantummechanicsenormouslybetternowthanwedidbackthen,butthetheoryitselfhasn’tchanged.

Which isn’t to say that the experiments aren’t important or impressive; they are. TheproblemwithtestingBell’spredictions,forexample,isthatyouaretryingtomakesurethattheextra correlations predicted by quantum mechanics couldn’t have arisen due to some sneakypre-existing classical correlation. How do we know whether some hidden event in the pastsecretlyaffectedhowwechosetomeasureourspin,orwhatthemeasurementoutcomewas,orboth?

Physicistshavegonetogreatlengthstoeliminatethesepossibilities,andacottageindustry

has arisen in doing “loophole-free Bell tests.” One recent result wanted to eliminate thepossibilitythatanunknownprocessinthelaboratoryworkedtoinfluencethechoiceofhowtomeasurethespin.Soinsteadoflettingalabassistantchoosethemeasurement,orevenusingarandom-numbergeneratorsittingonanearbytable,theexperimentmadethatchoicebasedonthe polarization of photons emitted from stars many light-years away. If there were somenefariousconspiracy tomake theworld lookquantum-mechanical, ithad tohavebeensetuphundredsofyearsago,whenthelightleftthosestars.It’spossible,butdoesn’tseemlikely.

Itseemsthatquantummechanicsisrightagain.Sofar,quantummechanicshasalwaysbeenright.

6SplittingtheUniverse

DecoherenceandParallelWorlds

The 1935 Einstein-Podolsky-Rosen (EPR) paper on quantum entanglement, and Niels Bohr’sresponse to it, were the last major public salvos in the Bohr-Einstein debates over thefoundationsofquantummechanics.BohrandEinsteinhadcorrespondedaboutquantumtheorysoonafterBohrproposedhismodelofquantizedelectronorbitsin1913,andtheirdisputecametoaheadat the1927SolvayConference. In thepopular retelling,Einsteinwould raise someobjection to the rapidly coalescing Copenhagen consensus during conversations at theworkshopwithBohr,whowouldspendtheeveningfrettingaboutit,andthenatbreakfastBohrwould triumphantlypresenthis rejoinder to thechastenedEinstein.Weare told thatEinsteinsimplycouldn’tcometogripswiththefactoftheuncertaintyprincipleandthenotionthatGodplaysdicewiththeuniverse.

That’snotwhathappened.Einstein’sprimaryconcernswerenotwithrandomnessbutwithrealismandlocality.HisdeterminationtosalvagetheseprinciplesculminatedintheEPRpaperandtheirargumentthatquantummechanicsmustbeincomplete.Butbythattimethepublic-relationsbattlehadbeenlost,andtheCopenhagenapproachtoquantummechanicshadbeenadoptedbyphysicistsworldwide,whothensetaboutapplyingquantummechanicstotechnicalproblemsinatomicandnuclearphysics,aswellastheemergingfieldsofparticlephysicsandquantum field theory. The implications of the EPR paper itself were largely ignored by thecommunity.Wrestlingwiththeconfusionsattheheartofquantumtheory,ratherthanworkingonmoretangiblephysicsproblems,begantobethoughtofasasomewhateccentricendeavor.Something that could occupy the time of formerly productive physicists once they reached acertainageandwerereadytoabandonrealwork.

In1933,EinsteinleftGermanyandtookapositionatthenewInstituteforAdvancedStudyinPrinceton,NewJersey,wherehewouldremainuntilhisdeathin1955.Histechnicalworkafter1935 focused largely on classical general relativity and his search for a unified theory ofgravitation and electromagnetism, but he never stopped thinking about quantum mechanics.BohrwouldoccasionallyvisitPrinceton,whereheandEinsteinwouldcarryontheirdialogue.

John Archibald Wheeler joined the physics faculty at Princeton University, down the roadfromtheInstituteandEinstein,asanassistantprofessorin1934.InlateryearsWheelerwouldbecomeknownasoneoftheworld’sexpertsingeneralrelativity,popularizingtheterms“blackhole”and“wormhole,”but inhisearlycareerheconcentratedonquantumproblems.Hehadbriefly studied under Bohr in Copenhagen, and in 1939 he and Bohr published a pioneeringpaperonnuclearfission.WheelerhadgreatadmirationforEinstein,butheveneratedBohr;ashewouldlaterputit,“Nothinghasdonemoretoconvincemethatthereonceexistedfriendsofmankindwith the humanwisdomofConfucius andBuddha, Jesus andPericles, Erasmus andLincoln,thanwalksandtalksunderthebeechtreesofKlampenborgForestwithNielsBohr.”

Wheelermadeanimpactonphysicsinanumberofways,oneofwhichwasinthementoringoftalentedgraduatestudents, includingfutureNobel laureatessuchasRichardFeynmanandKipThorne.Oneof thosestudentswasHughEverett III,whowould introduceadramaticallynewapproachtothinkingaboutthefoundationsofquantummechanics.We’vealreadysketchedhis basic idea—the wave function represents reality, it evolves smoothly, and that evolutionleadstomultipledistinctworldswhenaquantummeasurementtakesplace—butnowwehavethetoolstodoitright.

Everett’sproposal,whicheventuallybecamehis1957PhDthesisatPrinceton,canbethoughtof as the purest incarnation of one of Wheeler’s favorite principles—that theoretical physicsshouldbe“radicallyconservative.”Theideaisthatasuccessfulphysicaltheoryisonethathasbeentestedagainstexperimentaldata,butonlyinregimesthatexperimentersareactuallyabletoreach.Oneshouldbeconservative, in thesensethatweshouldstartwiththetheoriesand

principles that are already established as successful, rather than arbitrarily introducing newapproacheswhenevernewphenomenaareencountered.Butoneshouldalsoberadical,inthesensethatthepredictionsandimplicationsofourtheoriesshouldbetakenseriouslyinregimeswelloutsidewheretheyhavebeentested.Thephrases“weshouldstart”and“shouldbetakenseriously”arecrucialhere;ofcoursenewtheoriesarewarrantedwhenoldonesareshowntoblatantlycontradict thedata,and justbecauseaprediction is takenseriouslydoesn’tmean itshouldn’tberevisedinlightofnewinformation.ButWheeler’sphilosophywasthatweshouldstart prudently, with aspects of nature we believe we understand, and then act boldly,extrapolatingourbestideastotheendsoftheuniverse.

PartofEverett’sinspirationwasthesearchforatheoryofquantumgravity,whichWheelerhadrecentlybecomeinterestedin.Therestofphysics—matter,electromagnetism,thenuclearforces—seemstofitcomfortablywithintheframeworkofquantummechanics.Butgravitywas(andremains)astubbornexception.In1915,Einsteinproposedthegeneraltheoryofrelativity,accordingtowhichspacetimeitselfisadynamicalentitywhosebendsandwarpsarewhatyouandIperceiveastheforceofgravity.Butgeneralrelativityisathoroughlyclassicaltheory,withanaloguesofpositionandmomentumforthecurvatureofspace-time,andnolimitsonhowwemight measure them. Taking that theory and “quantizing” it, constructing a theory of wavefunctionsofspace-timeratherthanparticularclassicalspacetimeshasprovendifficult.

HughEverettIII(CourtesyoftheHughEverettIIIArchiveattheUniversityofCalifornia,Irvine,andMarkEverett)

Thedifficultiesofquantumgravityarebothtechnical—calculationstendtoblowupandgiveinfinitelybiganswers—andalsoconceptual.Even inquantummechanics,whileyoumightnotbeabletosaypreciselywhereacertainparticleis,thenotionof“apointinspace”isperfectlywelldefined.Wecanspecifya locationandaskwhat is theprobabilityof finding theparticlenearby.Butifrealitydoesn’tconsistofstuffdistributedthroughspace,butratherisaquantumwavefunctiondescribingsuperpositionsofdifferentpossiblespacetimes,howdoweevenask“where”acertainparticleisobserved?

The puzzles becomeworsewhenwe turn to themeasurement problem. By the 1950s theCopenhagenschoolwasestablisheddoctrine,andphysicistshadmadetheirpeacewiththeideaofwavefunctionscollapsingwhenameasurementoccurred.Theywereevenwillingtogoalongwithtreatingthemeasurementprocessasafundamentalpartofourbestdescriptionofnature.Or,atleast,nottofrettoomuchaboutit.

But what happens when the quantum system under consideration is the entire universe?Crucial to the Copenhagen approach is the distinction between the quantum system beingmeasured and the classical observer doing themeasuring. If the system is the universe as awhole,we are all inside it; there’s no external observer towhomwe can appeal. Years later,StephenHawkingandotherswouldstudyquantumcosmologytodiscusshowaself-containeduniversecouldhaveanearliestmomentintime,presumablyidentifiedwiththeBigBang.

WhileWheelerandothersthoughtaboutthetechnicalchallengesofquantumgravity,Everettbecamefascinatedbytheseconceptualproblems,especiallyhowtohandlemeasurement.Theseeds of the Many-Worlds formulation can be traced to a late-night discussion in 1954 withfellow young physicists Charles Misner (also a student of Wheeler’s) and Aage Petersen (anassistantofBohr’s,visitingfromCopenhagen).Allpartiesagreethatcopiousamountsofsherrywereconsumedontheoccasion.

Clearly, Everett reasoned, ifwe’re going to talk about the universe in quantum terms,wecan’t carveout a separate classical realm.Everypart of theuniversewill have tobe treatedaccordingtotherulesofquantummechanics,includingtheobserverswithinit.Therewillonlybeasinglequantumstate,describedbywhatEverettcalledthe“universalwavefunction”(andwe’vebeencalling“thewavefunctionoftheuniverse”).

If everything is quantum, and the universe is described by a singlewave function, how ismeasurementsupposedtooccur?Itmustbe,Everettreasoned,whenonepartoftheuniverseinteractswithanotherpartof theuniverse insomeappropriateway.That is something that’sgoing to happen automatically, he noticed, simply due to the evolution of the universalwavefunctionaccordingtotheSchrödingerequation.Wedon’tneedto invokeanyspecialrules formeasurementatall;thingsbumpintoeachotherallthetime.

It’s for this reason that Everett titled his eventual paper on the subject “‘Relative State’Formulation ofQuantumMechanics.” As ameasurement apparatus interactswith a quantumsystem, the two become entangledwith each other. There are nowave-function collapses orclassicalrealms.Theapparatus itselfevolves intoasuperposition,entangledwiththestateofthethingbeingobserved.Theapparentlydefinitemeasurementoutcome(“theelectronisspin-up”)isonlyrelativetoaparticularstateoftheapparatus(“Imeasuredtheelectrontobespin-up”). The other possible measurement outcomes still exist and are perfectly real, just asseparateworlds.Allwehavetodoistocourageouslyfaceuptowhatquantummechanicshasbeentryingtotellusallalong.

Let’sbealittlemoreexplicitaboutwhathappenswhenameasurementismade,accordingtoEverett’stheory.

Imaginethatwehaveaspinningelectron,whichcouldbeobservedtobeinstatesofeitherspin-uporspin-downwithrespecttosomechosenaxis.Beforemeasurement,theelectronwilltypicallybeinsomesuperpositionofupanddown.Wealsohaveameasuringapparatus,whichisaquantumsysteminitsownright.Imaginethatitcanbeinsuperpositionsofthreedifferentpossibilities:itcanhavemeasuredthespintobeup,itcanhavemeasuredthespintobedown,oritmightnotyethavemeasuredthespinatall,whichwecallthe“ready”state.

Thefactthatthemeasurementapparatusdoesitsjobtellsushowthequantumstateofthecombinedspin+apparatussystemevolvesaccordingtotheSchrödingerequation.Namely,ifwestart with the apparatus in its ready state and the spin in a purely spin-up state, we areguaranteedthattheapparatusevolvestoapuremeasured-upstate,likeso:

Theinitialstateontheleftcanbereadas“thespinisintheupstate,andtheapparatusisinits readystate,”while theoneon the right,where thepointer indicates theuparrow, is “thespinisintheupstate,andtheapparatushasmeasuredittobeup.”

Likewise, the ability to successfullymeasure a pure-down spin implies that the apparatusmustevolvefrom“ready”to“measureddown”:

Whatwewant,ofcourse,istounderstandwhathappenswhentheinitialspinisnotinapureupordownstate,but insomesuperpositionofboth.Thegoodnews is thatwealreadyknoweverythingweneed. The rules of quantummechanics are clear: if you knowhow the systemevolvesstartingfromtwodifferentstates,theevolutionofasuperpositionofboththosestateswilljustbeasuperpositionofthetwoevolutions.Inotherwords,startingfromaspininsomesuperpositionandthemeasurementdeviceinitsreadystate,wehave:

Thefinalstatenowisanentangledsuperposition:thespinisupanditwasmeasuredtobe

up,plusthespinisdownanditwasmeasuredtobedown.Atthispointit’snotstrictlycorrectto say“thespin is ina superposition”or “theapparatus is ina superposition.”Entanglementprevents us from talking about the wave function of the spin, or that of the apparatus,individually,becausewhatwewillobserveaboutonecandependonwhatweobserveabouttheother.Theonlythingwecansayis“thespin+apparatussystemisinasuperposition.”

This final state is the clear, unambiguous, definitive final wave function for the combinedspin+apparatussystem,ifallwedoisevolveitaccordingtotheSchrödingerequation.Thisisthesecret toEverettianquantummechanics.TheSchrödingerequationsays thatanaccuratemeasuring apparatus will evolve into a macroscopic superposition, which we will ultimatelyinterpret as branching into separate worlds. We didn’t put the worlds in; they were alwaysthere,andtheSchrödingerequationinevitablybringsthemtolife.Theproblemisthatweneverseemtocomeacrosssuperpositionsinvolvingbigmacroscopicobjectsinourexperienceoftheworld.

The traditional remedy has been to monkey with the fundamental rules of quantummechanics in one way or another. Some approaches say that the Schrödinger equation isn’talways applicable, others say that there are additional variables over and above the wavefunction.TheCopenhagenapproachistodisallowthetreatmentofthemeasurementapparatusasaquantumsysteminthefirstplace,andtreatwavefunctioncollapseasaseparatewaythequantum state can evolve.Oneway or another, all of these approaches invoke contortions inorder to not accept superpositions like the one written above as the true and completedescription of nature. As Everett would later put it, “The Copenhagen Interpretation ishopelessly incomplete because of its a priori reliance on classical physics . . . as well as aphilosophic monstrosity with a ‘reality’ concept for the macroscopic world and denial of thesameforthemicrocosm.”

Everett’s prescriptionwas simple: stop contorting yourself. Accept the reality ofwhat theSchrödinger equationpredicts.Bothparts of the finalwave function are actually there. Theysimplydescribeseparate,never-to-interact-againworlds.

Everett didn’t introduce anything new into quantum mechanics; he removed someextraneous clunky pieces from the formalism. Every non-Everettian version of quantummechanics is,asphysicistTedBunnhasput it,a“disappearingworlds”theory.Ifthemultipleworldsbotheryou,youhavetofiddlewitheitherthenatureofquantumstatesortheirordinaryevolutioninordertogetridofthem.Isitworthit?

There’s a looming question here. We’re familiar with how wave functions representsuperpositionsofdifferentpossiblemeasurementoutcomes.Thewave functionofanelectroncanputitinasuperpositionofvariouspossiblelocations,aswellasinasuperpositionofspin-upandspin-down.Butwewerenevertemptedtosaythateachpartofthesuperpositionwasaseparate“world.”Indeed,itwouldhavebeenincoherenttodoso.Anelectronthatisinapurespin-upstatewithrespecttotheverticalaxisisinasuperpositionofspin-upandspin-downwithrespecttothehorizontalaxis.Sodoesthatdescribeoneworld,ortwo?

Everett suggested that it is logically consistent to think of superpositions involvingmacroscopicobjectsasdescribingseparateworlds.Butat the timehewaswriting,physicistshadn’t yet developed the technical tools necessary to turn this into a complete picture. Thatunderstandingonlycamelater,withtheappreciationofaphenomenonknownasdecoherence.Introduced in 1970 by the German physicist Hans Dieter Zeh, the idea of decoherence hasbecome a central part of how physicists think about quantum dynamics. To the modernEverettian, decoherence is absolutely crucial to making sense of quantum mechanics. Itexplains once and for all why wave functions seem to collapse when you measure quantumsystems—andindeedwhata“measurement”reallyis.

Weknowthereisonlyonewavefunction,thewavefunctionoftheuniverse.Butwhenwe’retalkingabout individualmicroscopicparticles, theycansettle intoquantumstateswheretheyareunentangledfromtherestoftheworld.Inthatcase,wecansensiblytalkabout“thewavefunctionof thisparticularelectron”andso forth,keeping inmindthat it’sreally justausefulshortcutwecanemploywhensystemsareunentangledwithanythingelse.

Withmacroscopicobjects,thingsaren’tthatsimple.Considerourspin-measuringapparatus,andlet’s imagineweput it inasuperpositionofhavingmeasuredspin-upandspin-down.ThedialoftheapparatusincludesapointerthatispointingeithertoUportoDown.Anapparatuslike thatdoesn’t stay separate from the restof theworld.Even if it looks like it’s just sittingthere, inrealitytheairmolecules intheroomareconstantlybumpinginto it,photonsof lightarebouncingoffof it,andsoon.Callall thatotherstuff—theentirerestof theuniverse—theenvironment. In ordinary situations, there’s no way to stop a macroscopic object frominteractingwithitsenvironment,evenifverygently.Suchinteractionswillcausetheapparatustobecomeentangledwiththeenvironment,forexample,becauseaphotonwouldreflectoffthe

dialifthepointerisinoneposition,butbeabsorbedbyitifthepointerispointingsomewhereelse.

So thewave functionwewrotedownabove,whereanapparatusbecameentangledwithaqubit,wasn’tquitethewholestory.Puttingtheenvironmentstates incurlybraces,weshouldhavewritten

Itdoesn’treallymatterwhattheenvironmentstatesactuallyare,sowe’veportrayedthemasdifferentbackgroundslabeled{E0},{E1},and{E2}.Wedon’t(andgenerallycan’t)keeptrackofexactlywhat’sgoingonintheenvironment—it’stoocomplicated.It’snotgoingtojustbeasinglephotonthatinteractsdifferentlywithdifferentpartsoftheapparatus’swavefunction,itwill be a huge number of them. Nobody can be expected to keep track of every photon orparticleinaroom.

That simple process—macroscopic objects become entangledwith the environment, whichwe cannot keep track of—is decoherence, and it comeswith universe-altering consequences.Decoherencecauses thewave function to split, orbranch, intomultipleworlds.Anyobserverbranchesintomultiplecopiesalongwiththerestoftheuniverse.Afterbranching,eachcopyoftheoriginalobserverfindsthemselvesinaworldwithsomeparticularmeasurementoutcome.To them, the wave function seems to have collapsed. We know better; the collapse is onlyapparent,duetodecoherencesplittingthewavefunction.

Wedon’tknowhowoftenbranchinghappens,orevenwhetherthat’sasensiblequestiontoask. Itdependsonwhether therearea finiteor infinitenumberofdegreesof freedom in theuniverse,whichiscurrentlyanunansweredquestioninfundamentalphysics.Butwedoknowthat there’s a lot of branching going on; it happens every time a quantum system in asuperpositionbecomesentangledwiththeenvironment. Ina typicalhumanbody,about5,000atomsundergo radioactivedecayevery second. If everydecaybranches thewave function intwo,that’s25000newbrancheseverysecond.It’salot.

Whatmakesa“world,”anyway?Wejustwrotedownasinglequantumstatedescribingaspin,anapparatus,andanenvironment.Whatmakesussaythatitdescribestwoworlds,ratherthanjustone?

One thing you would like to have in a world is that different parts of it can, at least inprinciple,affecteachother.Considerthefollowing“ghostworld”scenario(notmeantasatruedescriptionof reality, justacolorfulanalogy):when livingbeingsdie, theyallbecomeghosts.Theseghostscanseeandtalktooneanother,buttheycannotseeortalktous,norcanweseeortalktothem.TheyliveonaseparateGhostEarth,wheretheycanbuildghosthousesandgototheirghost jobs.Butneither theynor theirsurroundingscan interactwithusandthestuffaroundusinanyway.Inthiscaseitmakessensetosaythattheghostsinhabitatrulyseparateghostworld,forthefundamentalreasonthatwhathappensintheghostworldhasabsolutelynobearingonwhathappensinourworld.

Nowapplythiscriteriontoquantummechanics.We’renotinterestedinwhetherthespinanditsmeasuringapparatuscan influenceeachother—theyobviouslycan.Whatwecareabout is

whetheronecomponentof,say,theapparatuswavefunction(forexample,thepiecewherethedialispointingtoUp)canpossiblyinfluenceanotherpiece(forexample,whereit’spointingtoDown). We’ve previously come across a situation just like this, where the wave functioninfluencesitself—inthephenomenonofinterferencefromthedouble-slitexperiment.Whenwepassed electrons through two slitswithoutmeasuringwhich one theywent through,we sawinterference bands on the final screen, and attributed them to the cancellation between thecontributiontothetotalprobabilityfromeachofthetwoslits.Crucially,weimplicitlyassumedthattheelectrondidn’tinteractandbecomeentangledwithanythingalongitsjourney;itdidn’tdecohere.

When insteadwe did detect which slit the electronwent through, the interference bandswentaway.Atthetimeweattributedthistothefactthatameasurementhadbeenperformed,collapsing the electron’swave function at one slit or another. Everett gives us amuchmorecompellingstorytotell.

What actually happened was that the electron became entangled with the detector as itmovedthroughtheslits,andthenthedetectorquicklybecameentangledwiththeenvironment.The process is precisely analogous to what happened to our spin above, except that we’remeasuringwhethertheelectronwentthroughtheleftslitLortherightslitR:

No mysterious collapsing; the whole wave function is still there, evolving cheerfullyaccordingto theSchrödingerequation, leavingus inasuperpositionof twoentangledpieces.Butnotewhathappensastheelectroncontinuesontowardthescreen.Asbefore,thestateofthe electron at any given point on the screen will receive a contribution from what passedthrough slit L, and another contribution from what passed through slit R. But now thosecontributionswon’tinterferewitheachother.Inordertogetinterference,weneedtobeaddinguptwoequalandoppositequantities:

1+(-1)=0.

Butthereisnopointonthescreenwherewewillfindequalandoppositecontributionstotheelectron’swavefunctionfromtheLandRslits,becausepassingthroughthoseslitsentangledtheelectronwithdifferentstatesoftherestoftheworld.Whenwesayequalandopposite,wemean precisely equal and opposite, not “equal and opposite except for that thing we’reentangledwith.”Beingentangledwithdifferentstatesofthedetectorandenvironment—beingdecohered, in otherwords—means that the two parts of the electron’swave function can nolongerinterferewitheachother.Andthatmeanstheycan’tinteractatall.Andthatmeanstheyare, for all intents and purposes, part of separateworlds.* From the point of view of thingsentangledwithonebranchofthewavefunction,theotherbranchesmightaswellbepopulatedbyghosts.

TheMany-Worldsformulationofquantummechanicsremovesonceandforallanymysteryaboutthemeasurementprocessandcollapseofthewavefunction.Wedon’tneedspecialrulesaboutmakinganobservation:allthathappensisthatthewavefunctionkeepschuggingalongin accordance with the Schrödinger equation. And there’s nothing special about whatconstitutes“ameasurement”or“anobserver”—ameasurementisanyinteractionthatcausesaquantum system to become entangled with the environment, creating decoherence and abranchingintoseparateworlds,andanobserverisanysystemthatbringssuchaninteractionabout. Consciousness, in particular, has nothing to do with it. The “observer” could be anearthworm, a microscope, or a rock. There’s not even anything special about macroscopicsystems,other than the fact that theycan’thelpbut interactandbecomeentangledwith theenvironment.Thepricewepayforsuchpowerfulandsimpleunificationofquantumdynamicsisalargenumberofseparateworlds.

Everett himself wasn’t familiar with decoherence, so his picture wasn’t quite as robust andcomplete as the onewe’vepainted.But hisway of rethinking themeasurement problemand

offering a unified picture of quantum dynamics was compelling from the start. Even intheoretical physics, people do sometimes get lucky, hitting upon an important idea morebecause they were in the right place at the right time than because they were particularlybrilliant.That’snot the casewithHughEverett; thosewhoknewhim testifyuniformly tohisincredible intellectual gifts, and it’s clear from his writings that he had a thoroughunderstandingoftheimplicationsofhisideas.Werehestillalive,hewouldbeperfectlyathomeinmoderndiscussionsofthefoundationsofquantummechanics.

Whatwashardwasgettingotherstoappreciatethose ideas,andthat includedhisadvisor.WheelerwaspersonallyverysupportiveofEverett,buthewasalsodevotedtohisownmentor,Bohr, and was convinced of the basic soundness of the Copenhagen approach. HesimultaneouslywantedEverett’s ideas togetawidehearing,and toensure that theyweren’tinterpretedasadirectassaultonBohr’swayofthinkingaboutquantummechanics.

Yet Everett’s theorywas a direct assault on Bohr’s picture. Everett himself knew it, andenjoyedillustratingthenatureofthisassault invividlanguage.Inanearlydraftofhisthesis,Everettusedtheanalogyofanamoebadividingtoillustratethebranchingofthewavefunction:“Onecanimagineanintelligentamoebawithagoodmemory.Astimeprogressestheamoebaisconstantlysplitting,eachtimetheresultingamoebashavingthesamememoriesastheparent.Our amoeba hence does not have a life line, but a life tree.” Wheeler was put off by theblatantnessofthis(quiteaccurate)metaphor,scribblinginthemarginofthemanuscript,“Split?Better words needed.” Advisor and student were constantly tussling over the best way toexpress thenewtheory,withWheeleradvocatingcautionandprudencewhileEverett favoredboldclarity.

In1956,asEverettwasworkingon finishinghisdissertation,WheelervisitedCopenhagenand presented the new scenario to Bohr and his colleagues, including Aage Petersen. Heattempted to present it anyway; by this time the wave-functions-collapse-and-don’t-ask-embarrassing-questions-about-exactly-how school of quantum theory had hardened intoconventionalwisdom,andthosewhoaccepteditweren’tinterestedinrevisitingthefoundationswhen therewas somuch interestingappliedwork tobedone.Letters fromWheeler,Everett,and Petersen flew back and forth across the Atlantic, continuing when Wheeler returned toPrincetonandhelpedEverettcraftthefinalformofhisdissertation.Theagonyofthisprocessisreflectedintheevolutionofthepaperitself:Everett’sfirstdraftwastitled“QuantumMechanicsby the Method of the Universal Wave Function,” and a revised version was called “WaveMechanicsWithoutProbability.”Thisdocument, laterdubbedthe“longversion”of the thesis,wasn’tpublisheduntil1973.A“shortversion”was finallysubmitted forEverett’sPhDas“OntheFoundationsofQuantumMechanics,”andeventuallypublishedin1957as“‘RelativeState’Formulation of Quantum Mechanics.” It omitted many of the juicier sections Everett hadoriginally composed, includingexaminationsof the foundationsofprobabilityand informationtheoryandanoverviewofthequantummeasurementproblem,focusinginsteadonapplicationstoquantumcosmology.(Noamoebasappearinthepublishedpaper,butEverettdidmanagetoinsert the word “splitting” in a footnote added in proof while Wheeler wasn’t looking.)Furthermore,Wheeler wrote an “assessment” article that was published alongside Everett’s,which suggested that the new theory was radical and important, while at the same timeattemptingtopaperoveritsmanifestdifferenceswiththeCopenhagenapproach.

The arguments continued,withoutmuch headway beingmade. It isworth quoting from aletterEverettwrotetoPetersen,inwhichhisfrustrationcomesthrough:

Lest thediscussionofmypaperdie completely, letmeadd some fuel to the firewith . . . criticismsof the‘Copenhageninterpretation.’...IdonotthinkyoucandismissmyviewpointassimplyamisunderstandingofBohr’s position. . . . I believe that basing quantum mechanics upon classical physics was a necessaryprovisional step, but that the time has come . . . to treat [quantum mechanics] in its own right as afundamentaltheorywithoutanydependenceonclassicalphysics,andtoderiveclassicalphysicsfromit....

Let me mention a few more irritating features of the Copenhagen Interpretation. You talk of themassivenessofmacrosystemsallowingonetoneglectfurtherquantumeffects(indiscussionsofbreakingthemeasuringchain),butnevergiveanyjustificationforthisflatlyasserteddogma.[And]thereisnowheretobefound any consistent explanation for this ‘irreversibility’ of themeasuringprocess. It is again certainly notimpliedbywavemechanics,norclassicalmechanicseither.Anotherindependentpostulate?

ButEverettdecidednottocontinuetheacademicfight.BeforefinishinghisPhD,heaccepteda jobat theWeaponsSystemsEvaluationGroupfor theUSDepartmentofDefense,wherehestudiedtheeffectsofnuclearweapons.Hewouldgoontodoresearchonstrategy,gametheory,andoptimization,andplayedaroleinstartingseveralnewcompanies.It’suncleartheextenttowhich Everett’s conscious decision to not apply for professorial positions was motivated bycriticismofhisupstartnewtheory,orsimplybyimpatiencewithacademiaingeneral.

Hedid,however,maintainaninterestinquantummechanics,evenifheneverpublishedonitagain. After Everett defended his PhD and was already working for the Pentagon, WheelerpersuadedhimtovisitCopenhagenforhimselfandtalktoBohrandothers.Thevisitdidn’tgowell;afterwardEverettjudgedthatithadbeen“doomedfromthebeginning.”

Bryce DeWitt, an American physicist who had edited the journal where Everett’s thesisappeared,wrotealettertohimcomplainingthattherealworldobviouslydidn’t“branch,”sincewe never experience such things. Everett replied with a reference to Copernicus’s similarlydaringideathattheEarthmovesaroundthesun,ratherthanviceversa:“Ican’tresistasking:Do you feel themotion of the earth?”DeWitt had to admit thatwas a pretty good response.After mulling the matter over for a while, by 1970 DeWitt had become an enthusiasticEverettian. He put a great deal of effort into pushing the theory, which had languished inobscurity,towardgreaterpublicrecognition.Hisstrategiesincludedaninfluential1970articleinPhysicsToday, followedbya1973essaycollection that includedat last the longversionofEverett’sdissertation,aswellasanumberofcommentaries.ThecollectionwascalledsimplyTheMany-WorldsInterpretationofQuantumMechanics,avividnamethathasstuckeversince.

In 1976, JohnWheeler retired from Princeton and took up a position at theUniversity ofTexas,whereDeWittwasalsoonthefaculty.Togethertheyorganizedaworkshopin1977ontheMany-Worldstheory,andWheelercoaxedEverettintotakingtimeofffromhisdefenseworkinorder toattend.Theconferencewasasuccess,andEverettmadeasignificant impressionontheassembledphysicistsintheaudience.OneofthemwastheyoungresearcherDavidDeutsch,whowouldgoontobecomeamajorproponentofMany-Worlds,aswellasanearlypioneerofquantum computing. Wheeler went so far as to propose a new research institute in SantaBarbara,whereEverett could return to full-timework on quantummechanics, but ultimatelynothingcameofit.

Everett died in 1982, age fifty-one, of a sudden heart attack. He had not lived a healthylifestyle,overindulgingineating,smoking,anddrinking.Hisson,MarkEverett(whowouldgoontoformthebandEels),hassaidthathewasoriginallyupsetwithhis fatherfornottakingbettercareofhimself.Helaterchangedhismind:“Irealizethatthereisacertainvalueinmyfather’swayoflife.Heate,smokedanddrankashepleased,andonedayhejustsuddenlyandquicklydied.GivensomeoftheotherchoicesI’dwitnessed,itturnsoutthatenjoyingyourselfandthendyingquicklyisnotsuchahardwaytogo.”

*The set of all branches of thewave function is different fromwhat cosmologists often call “themultiverse.” Thecosmologicalmultiverseisreallyjustacollectionofregionsofspace,generallyfarawayfromoneanother,wherelocalconditionslookverydifferent.

7OrderandRandomness

WhereProbabilityComesFrom

One sunny day in Cambridge, England, Elizabeth Anscombe ran into her teacher, LudwigWittgenstein.“Whydopeoplesay,”Wittgensteinopened inhis inimitable fashion,“that itwasnatural to think that the sunwent round the earth, rather than that the earth turned on itsaxis?”Anscombegavetheobviousanswer,thatitjustlookslikethesungoesaroundtheEarth.“Well,”Wittgenstein replied, “what would it have looked like if the Earth had turned on itsaxis?”

Thisanecdote—recountedbyAnscombeherself,andwhichTomStoppardretold inhisplayJumpers—isafavoriteamongEverettians.PhysicistSidneyColemanusedtorelateitinlectures,andphilosopherofphysicsDavidWallaceusedittoopenhisbookTheEmergentMultiverse.ItevenbearsafamilyresemblancetoHughEverett’sremarktoBryceDeWitt.

It’seasy to seewhy theobservation is so relevant.Any reasonableperson,when first toldabouttheMany-Worldspicture,hasanimmediate,visceralobjection: it justdoesn’t feel like Ipersonally split intomultiple peoplewhenever a quantummeasurement is performed. And itcertainlydoesn’tlooklikethereareallsortsofotheruniversesexistingparalleltotheoneIfindmyselfin.

Well, the Everettian replies, channeling Wittgenstein: What would it feel and look like ifMany-Worldsweretrue?

ThehopeisthatpeoplelivinginanEverettianuniversewouldexperiencejustwhatpeopleactually do experience: a physical world that seems to obey the rules of textbook quantummechanics to a high degree of accuracy, and in many situations is well approximated byclassicalmechanics.Buttheconceptualdistancebetween“asmoothlyevolvingwavefunction”andtheexperimentaldataitismeanttoexplainisquitelarge.It’snotobviousthattheanswerwecangivetoWittgenstein’squestionistheonewewant.Everett’stheorymightbeaustereinits formulation, but there’s still a good amount of work to be done to fully flesh out itsimplications.

In this chapter we’ll confront a major puzzle for Many-Worlds: the origin and nature ofprobability.TheSchrödingerequation isperfectlydeterministic.Whydoprobabilitiesenteratall,andwhydotheyobeytheBornrule:probabilitiesequalamplitudes—thecomplexnumbersthewavefunctionassociateswitheachpossibleoutcome—squared?Doesitevenmakesensetospeakoftheprobabilityofendinguponsomeparticularbranchiftherewillbeafutureversionofmyselfoneverybranch?

InthetextbookorCopenhagenversionsofquantummechanics,there’snoneedto“derive”theBornruleforprobabilities.Wejustplopitdownthereasoneofthepostulatesofthetheory.Whycouldn’twedothesamethinginMany-Worlds?

Theansweristhateventhoughtherulewouldsoundthesameinbothcases—“probabilitiesare given by the wave function squared”—their meanings are very different. The textbookversionoftheBornrulereallyisastatementabouthowoftenthingshappen,orhowoftentheywill happen in the future. Many-Worlds has no room for such an extra postulate; we knowexactly what will happen, just from the basic rule that the wave function always obeys theSchrödinger equation. Probability in Many-Worlds is necessarily a statement about what weshouldbelieveandhowweshouldact,notabouthowoftenthingshappen.And“whatweshouldbelieve”isn’tsomethingthatreallyhasaplaceinthepostulatesofaphysicaltheory;itshouldbeimpliedbythem.

Moreover,aswewillsee,thereisneitheranyroomforanextrapostulate,noranyneedforone.Giventhebasicstructureofquantummechanics,theBornrule isnaturalandautomatic.SincewetendtoseeBornrule–likebehaviorinnature,thisshouldgiveusconfidencethatwe’reon the right track. A framework in which an important result can be derived from morefundamentalpostulatesshould,allelsebeingequal,bepreferredtoonewhere itneeds tobeseparatelyassumed.

If we successfully address this question, we will have made significant headway toward

showingtheworldwewouldexpecttoseeifMany-Worldsweretrueistheworldweactuallydosee. That is, a world that is closely approximated by classical physics, except for quantummeasurementevents,duringwhichtheprobabilityofobtaininganyparticularoutcomeisgivenbytheBornrule.

The issue of probabilities is often phrased as trying to derivewhy probabilities are given byamplitudes squared. But that’s not really the hard part. Squaring amplitudes in order to getprobabilitiesisaverynaturalthingtodo;thereweren’tanyworriesthatitmighthavebeenthewave function to the fifthpoweroranything like that.We learned thatback inChapterFive,whenweusedqubitstoexplainthatthewavefunctioncanbethoughtofasavector.Thatvectorislikethehypotenuseofarighttriangle,andtheindividualamplitudesareliketheshortersidesof thattriangle.The lengthof thevectorsequalsone,andbyPythagoras’s theoremthat’s thesum of the squares of all the amplitudes. So “amplitudes squared” naturally look likeprobabilities:they’repositivenumbersthatadduptoone.

The deeper issue is why there is anything unpredictable about Everettian quantummechanicsatall,andifso,whythereisanyspecificruleforattachingprobabilities.InMany-Worlds,ifyouknowthewavefunctionatonemomentintime,youcanfigureoutpreciselywhatit’s going to be at any other time, just by solving the Schrödinger equation. There’s nothingchancyabout it.Sohow in theworld is suchapicturesupposed to recover the realityofourobservations, where the decay of a nucleus or themeasurement of a spin seems irreduciblyrandom?

Considerour favoriteexampleofmeasuring thespinofanelectron.Let’s saywestart theelectron inanequal superpositionof spin-upand spin-downwith respect to the vertical axis,andsenditthroughaStern-Gerlachmagnet.Textbookquantummechanicssaysthatwehavea50 percent chance of thewave function collapsing to spin-up, and a 50 percent chance of itcollapsingtospin-down.Many-Worlds,ontheotherhand,saysthereisa100percentchanceofthewavefunctionoftheuniverseevolvingfromoneworldintotwo.True,inoneofthoseworldsthe experimenterwill have seen spin-up and in the other theywill have seen spin-down.Butbothworldsareindisputablythere.Ifthequestionwe’reaskingis“WhatisthechanceIwillendupbeingtheexperimenteronthespin-upbranchofthewavefunction?,”theredoesn’tseemtobeanyanswer.Youwillnotbeoneorotherexperimenters;yourcurrentsingleselfwillevolve,with certainty, into both of them.How arewe supposed to talk about probabilities in such asituation?

It’s a good question. To answer it, we have get a bit philosophical, and think aboutwhat“probability”reallymeans.

Youwillnotbesurprisedto learnthattherearecompetingschoolsof thoughtonthe issueofprobability. Consider tossing a fair coin. “Fair” means that the coin will come up heads 50percentofthetimeandtails50percentofthetime.Atleastinthelongrun;nobodyissurprisedwhenyoutossacointwiceanditcomesuptailsbothtimes.

This“inthelongrun”caveatsuggestsastrategyforwhatwemightmeanbyprobability.Forjustafewcointosses,wewouldn’tbesurprisedatalmostanyoutcome.Butaswedomoreandmore,weexpectthetotalproportionofheadstocomecloserto50percent.Soperhapswecandefinetheprobabilityofgettingheadsasthefractionoftimesweactuallywouldgetheads,ifthecoinweretossedaninfinitenumberoftimes.

This notion ofwhatwemeanby probability is sometimes called frequentism, as it definesprobability as the relative frequency of an occurrence in a very large number of trials. Itmatchesprettywellwithourintuitivenotionsofhowprobabilityfunctionswhenwetosscoins,rolldice,orplaycards.Toafrequentist,probabilityisanobjectivenotion,sinceitonlydependsonfeaturesofthecoin(orwhateverothersystemwe’retalkingabout),notonusorourstateofknowledge.

FrequentismfitscomfortablywiththetextbookpictureofquantummechanicsandtheBornrule.Maybeyoudon’tactuallysendaninfinitenumberofelectronsthroughamagneticfieldtomeasuretheirspins,butyoucouldsendaverylargenumber.(TheStern-Gerlachexperimentisafavoriteonetoreproduceinundergraduatelabcoursesforphysicsmajors,soovertheyearsquite a number of spins have been measured this way.) We can gather enough statistics toconvinceourselves that theprobability inquantummechanicsreally is just thewave functionsquared.

Many-Worldsisadifferentstory.Sayweputanelectronintoanequalsuperpositionofspin-up and spin-down, measure its spin, then repeat a large number of times. At everymeasurement, thewave function branches into aworldwith a spin-up result and onewith a

spin-down. Imagine thatwe recordour results, labeling spin-upas “0”andspin-downas “1.”Afterfiftymeasurements,therewillbeaworldwheretherecordlookslike

10101011111011001011001010100011101100011101000001.

Thatseemsrandomenough,andtoobeytheproperstatistics:therearetwenty-four0’s,andtwenty-six1’s.Notexactlyfifty-fifty,butascloseasweshouldexpect.

Buttherewillalsobeaworldwhereeverymeasurementreturnedspin-up,sothattherecordwas justa listof fifty0’s.Andaworldwhereall thespinswereobserved tobedown, so therecordwasalistoffifty1’s.Andeveryotherpossiblestringof0’sand1’s.IfEverett isright,thereisa100percentprobabilitythateachpossibilityisrealizedinsomeparticularworld.

Infact,I’llmakeaconfession:therereallyaresuchworlds.Therandom-lookingstringabovewasn’tsomethingImadeuptolookrandom,norwasitcreatedbyaclassicalrandom-numbergenerator.Itwasactuallycreatedbyaquantumrandom-numbergenerator:agizmothatmakesquantummeasurementsandusesthemtogeneraterandomsequencesof0’sand1’s.AccordingtoMany-Worlds,whenIgeneratedthatrandomnumber,theuniversesplitinto250copies(that’s1,125,899,906,842,624, or approximately 1 quadrillion), each of which carries a slightlydifferentnumber.

Ifallofthecopiesofmeinallofthosedifferentworldsstuckwiththeplanofincludingtheobtainednumber into the text of this book, thatmeans there are over a quadrillion differenttextualvariationsofSomethingDeeplyHiddenoutthere inthewavefunctionoftheuniverse.Forthemostpartthevariationswillbeminor, justrearrangingsome0’sand1’s.Butsomeofthose poor versions of me were the unlucky ones who got all 0’s or all 1’s. What are theythinking right now? Probably they thought the random-number generator was broken. Theycertainlydidn’twritepreciselythetextIamtypingatthismoment.

WhateverIortheothercopiesofmemightthinkaboutthissituation,it’squitedifferentfromthe frequentist paradigm for probabilities. It doesn’tmake toomuch sense to talk about thefrequencyinthelimitofaninfinitenumberoftrialswheneverytrialreturnseveryresult,justsomewhereelse inthewavefunction.Weneedtoturntoanotherwayofthinkingaboutwhatprobabilityissupposedtomean.

Fortunately, an alternative approach to probability exists, and long pre-dates quantummechanics.That’s thenotionofepistemicprobability,havingtodowithwhatweknowratherthansomehypotheticalinfinitenumberoftrials.

Considerthequestion“WhatistheprobabilitythatthePhiladelphia76erswillwinthe2020NBA Championship?” (I put a high value on that personally, but fans of other teams maydisagree.)Thisisn’tthekindofeventwecanimaginerepeatinganinfinitenumberoftimes;ifnothingelse,thebasketballplayerswouldgrowolder,whichwouldaffecttheirplay.The2020NBAFinalswillhappenonlyonce,and there isadefiniteanswer towhowillwin,even ifwedon’t know what it is. But professional oddsmakers have no qualms about assigning aprobabilitytosuchsituations.Nordowe,inoureverydaylives;weareconstantly judgingthelikelihood of different one-shot events, from getting a jobwe applied for to being hungry bysevenp.m.Forthatmatterwecantalkabouttheprobabilityofpastevents,eventhoughthereisa definite thing that happened, simply becausewe don’t knowwhat that thingwas—“I don’t

rememberwhat time I leftwork lastThursday,but itwasprobablybetween fivep.m.andsixp.m.,sincethat’susuallywhenIheadhome.”

Whatwe’redoinginthesecasesisassigning“credences”—degreesofbelief—tothevariouspropositionsunderconsideration.Likeanyprobability,credencesmustrangebetween0percentand100percent,andyourtotalsetofcredencesforthepossibleoutcomesofaspecifiedeventshould add up to 100 percent. Your credence in something can change as you gather newinformation;youmighthaveadegreeofbeliefthatawordisspelledacertainway,butthenyougolookitupandfindouttherightanswer.Statisticianshaveformalizedthisprocedureunderthe label ofBayesian inference, afterRev. ThomasBayes, an eighteenth-centuryPresbyterianministerandamateurmathematician.Bayesderivedanequationshowinghowweshouldupdateourcredenceswhenweobtainnewinformation,andyoucanfindhisformulaonpostersandT-shirtsinstatisticsdepartmentstheworldover.

Sothere’saperfectlygoodnotionof“probability”thatappliesevenwhensomethingisonlygoingtohappenonce,notaninfinitenumberoftimes.It’sasubjectivenotion,ratherthananobjective one; different people, in different states of knowledge, might assign differentcredences to the same outcomes for some event. That’s okay, as long as everyone agrees tofollowtherulesaboutupdatingtheircredenceswhentheylearnsomethingnew.Infact,ifyoubelieveineternalism—thefutureis justasrealasthepast;wejusthaven’tgottenthereyet—thenfrequentismissubsumedintoBayesianism.Ifyoufliparandomcoin,thestatement“Theprobabilityofthecoincomingupheadsis50percent”canbeinterpretedas“GivenwhatIknowaboutthiscoinandothercoins,thebestthingIcansayabouttheimmediatefutureofthecoinis that it isequally likely tobeheadsor tails,eventhoughthere issomedefinite thing itwillbe.”

It’sstillnotobviousthatbasingprobabilityonourknowledgeratherthanonfrequenciesisreallyastepforward.Many-Worldsisadeterministictheory,andifweknowthewavefunctionatonetimeandtheSchrödingerequation,wecanfigureouteverythingthat’sgoingtohappen.Inwhatsenseisthereanythingthatwedon’tknow,towhichwecanassignacredencegivenbytheBornrule?

There’sananswerthatistemptingbutwrong:thatwedon’tknow“whichworldwewillendupin.”Thisiswrongbecauseitimplicitlyreliesonanotionofpersonalidentitythatsimplyisn’tapplicableinaquantumuniverse.

Whatwe’reupagainsthereiswhatphilosopherscallour“folk”understandingoftheworldaroundus,andtheverydifferentviewthatissuggestedbymodernscience.Thescientificviewshouldultimatelyaccountforoureverydayexperiences.Butwehavenorighttoexpectthattheconcepts and categories that have arisen over the course of pre-scientific history shouldmaintaintheirvalidityaspartofourmostcomprehensivepictureofthephysicalworld.Agoodscientific theory should be compatible with our experience, but it might speak an entirelydifferent language. The ideas we readily deploy in our day-to-day lives emerge as usefulapproximationsofcertainaspectsofamorecompletestory.

A chair isn’t anobject thatpartakesof aPlatonic essenceof chairness; it’s a collectionofatoms arranged in a certain configuration that makes it sensible for us to include it in thecategory “chair.” We have no trouble recognizing that the boundaries of this category aresomewhat fuzzy—does a sofa count? What about a barstool? If we take something that isindubitablyachair,andremoveatoms from itonebyone, itgraduallybecomes lessand lesschairlike,butthere’snohard-and-fastthresholdthatitcrossestojumpsuddenlyfromchairtonon-chair.Andthat’sokay.Wehavenotroubleacceptingthisloosenessinoureverydayspeech.

Whenitcomestothenotionof“self,”however,we’realittlemoreprotective.Inoureverydayexperience,there’snothingveryfuzzyaboutourself.Wegrowandlearn,ourbodyages,andweinteractwiththeworldinavarietyofways.ButatanyonemomentIhavenotroubleidentifyingaspecificpersonthatisundeniably“myself.”

Quantummechanicssuggeststhatwe’regoingtohavetomodifythisstorysomewhat.Whenaspinismeasured,thewavefunctionbranchesviadecoherence,asingleworldsplitsintotwo,andtherearenowtwopeoplewhereIusedtobejustone.Itmakesnosensetoaskwhichoneis“reallyme.”Likewise,beforethebranchinghappens,itmakesnosensetowonderwhichbranch“I”willendupin.Bothofthemhaveeveryrighttothinkofthemselvesas“me.”

In a classical universe, identifying a single individual as a person aging through time isgenerallyunproblematic.Atanymomentapersonisacertainarrangementofatoms,butit’snotthe individual atoms that matter; to a large extent our atoms are replaced over time.Whatmatters is the pattern that we form, and the continuity of that pattern, especially in thememoriesofthepersonunderconsideration.

The new feature of quantummechanics is the duplication of that pattern when the wavefunction branches. That’s no reason to panic. We just have to adjust our notion of personalidentitythroughtimetoaccountforasituationthatweneverhadreasontocontemplateoverthemillenniaofpre-scientifichumanevolution.

Asstubbornasouridentityis,theconceptofasinglepersonextendingfrombirthtodeathwasalwaysjustausefulapproximation.Thepersonyouarerightnowisnotexactlythesameas

the person you were a year ago, or even a second ago. Your atoms are in slightly differentlocations,andsomeofyouratomsmighthavebeenexchangedfornewones.(Ifyou’reeatingwhilereading,youmighthavemoreatomsnowthanyouhadamomentago.)Ifwewantedtobemore precise than usual, rather than talking about “you,”we should talk about “you at 5:00p.m.,”“youat5:01p.m.,”andsoon.

Theideaofaunified“you”isusefulnotbecauseallofthesedifferentcollectionsofatomsatdifferentmomentsoftimeareliterallythesame,butbecausetheyarerelatedtooneanotherinanobviousway.Theydescribearealpattern.Youatonemomentdescendfromyouatanearliermoment,throughtheevolutionoftheindividualatomswithinyouandthepossibleadditionorsubtractionofafewofthem.Philosophershavethoughtthisthrough,ofcourse;DerekParfit,inparticular,suggestedthatidentitythroughtimeisamatterofoneinstanceinyourlife“standingin Relation R” to another instance, where Relation R says that your future self sharespsychologicalcontinuitywithyourpastself.

ThesituationinMany-Worldsquantummechanicsisexactlythesameway,exceptthatnowmore thanoneperson candescend froma singlepreviousperson. (Parfitwouldhavehadnoproblemwiththat,andinfactinvestigatedanalogoussituationsfeaturingduplicatormachines.)Rather than talkingabout“youat5:01p.m.,”weneed to talkabout“thepersonat5:01p.m.who descended from you at 5:00 p.m. andwho ended up on the spin-up branch of thewavefunction,”andlikewiseforthepersononthespin-downbranch.

Every oneof thosepeoplehas a reasonable claim tobeing “you.”Noneof them iswrong.Eachofthemisaseparateperson,allofwhomtracetheirbeginningsbacktothesameperson.In Many-Worlds, the life-span of a person should be thought of as a branching tree, withmultiple individuals at any one time, rather than as a single trajectory—much like a splittingamoeba.Andnothingabout thisdiscussionreallyhingesonwhatwe’re talkingaboutbeingaperson rather thana rock.Theworldduplicates, andeverythingwithin theworldgoes alongwithit.

We’renowsetuptoconfrontthisissueofprobabilitiesinMany-Worlds.Itmighthaveseemednaturaltothinktheproperquestionis“WhichbranchwillIendupon?”Butthat’snothowweshouldbethinkingaboutit.

Thinkinsteadaboutthemomentimmediatelyafterdecoherencehasoccurredandtheworldhasbranched.Decoherenceisanextraordinarilyrapidprocess,generallytakingatinyfractionof a second to happen. From a human perspective, the wave function branches essentiallyinstantaneously(althoughthat’sjustanapproximation).Sothebranchinghappensfirst,andweonlyfindoutaboutitslightlylater,forexample,bylookingtoseewhethertheelectronwentupordownwhenitpassedthroughthemagneticfield.

For a brief while, then, there are two copies of you, and those two copies are preciselyidentical. Each of them lives on a distinct branch of the wave function, but neither of themknowswhichoneitison.

Youcanseewhere this isgoing.There isnothingunknownabout thewave functionof theuniverse—itcontainstwobranches,andweknowtheamplitudeassociatedwitheachofthem.But there is something that the actual people on these branches don’t know: which branchthey’re on. This state of affairs, first emphasized in the quantum context by physicist LevVaidman, is called self-locatinguncertainty—you know everything there is to know about theuniverse,exceptwhereyouarewithinit.

That ignorance gives us an opening to talk about probabilities. In that moment afterbranching, both copies of you are subject to self-locating uncertainty, since they don’t know

whichbranchthey’reon.Whattheycandoisassignacredencetobeingononebranchortheother.

Whatshouldthatcredencebe?Therearetwoplausiblewaystogo.Oneisthatwecanusethestructureofquantummechanicsitselftopickoutapreferredsetofcredencesthatrationalobservers should assign to being on various branches. If you’re willing to accept that, thecredencesyou’llendupassigningareexactlythoseyouwouldgetfromtheBornrule.Thefactthattheprobabilityofaquantummeasurementoutcomeisgivenbythewavefunctionsquaredisjustwhatwewouldexpectifthatprobabilityarosefromcredencesassignedinconditionsofself-locatinguncertainty.(Andifyou’rewillingtoacceptthatanddon’twanttobebotheredwiththedetails,you’rewelcometoskiptherestofthischapter.)

Butthere’sanotherschoolof thought,whichbasicallydeniesthat itmakessensetoassignany definite credences at all. I can come up with all sorts of wacky rules for calculatingprobabilities forbeingononebranchof thewave functionoranother.Maybe Iassignhigherprobability to being on a branch where I’m happier, or where spins are always pointing up.Philosopher David Albert has (just to highlight the arbitrariness, not because he thinks it’sreasonable)suggesteda“fatnessmeasure,”wheretheprobabilityisproportionaltothenumberofatoms inyourbody.There’snoreasonable justification fordoingso,butwho’s tostopme?Theonly“rational”thingtodo,accordingtothisattitude,istoadmitthatthere’snorightwaytoassigncredences,andthereforerefusetodoso.

Thatisapositiononeisallowedtotake,butIdon’tthinkit’sthebestone.IfMany-Worldsiscorrect,wearegoingtofindourselvesinsituationsofself-locatinguncertaintywhetherwelikeitornot.Andifourgoalistocomeupwiththebestscientificunderstandingoftheworld,thatunderstandingwillnecessarilyinvolveanassignmentofcredencesinthesesituations.Afterall,partofscienceispredictingwhatwillbeobserved,evenifonlyprobabilistically.Iftherewereanarbitrarycollectionofwaystoassigncredences,andeachofthemseemedjustasreasonableastheother,wewouldbestuck.But if thestructureofthetheorypointsunmistakablytooneparticularway toassignsuchcredences,andthatway is inagreementwithourexperimentaldata, we should adopt it, congratulate ourselves on a job well done, and move on to otherproblems.

Let’ssaywebuyintotheideathattherecouldbeaclearlybestwaytoassigncredenceswhenwedon’tknowwhichbranchofthewavefunctionwe’reon.Before,wementionedthat,atheart,theBornrule is justPythagoras’stheoreminaction.Nowwecanbea littlemorecarefulandexplainwhy that’s the rationalway to think about credences in the presence of self-locatinguncertainty.

This is an important question, because ifwe didn’t already know about theBorn rule,wemight think thatamplitudesarecompletely irrelevant toprobabilities.Whenyougo fromonebranch to two, for example, why not just assign equal probability to each, since they’re twoseparateuniverses?It’seasytoshowthatthis idea,knownasbranchcounting,can’tpossiblywork. But there’s a more restricted version, which says that we should assign equalprobabilitiestobrancheswhentheyhavethesameamplitude.Andthat,wonderfully,turnsoutto be allwe need to show thatwhen branches have different amplitudes,we should use theBornrule.

Let’s first dispatch thewrong idea of branch counting before turning to the strategy thatactually works. Consider a single electron whose vertical spin has been measured by anapparatus,sothatdecoherenceandbranchinghasoccurred.Strictlyspeaking,weshouldkeeptrackofthestatesoftheapparatus,observer,andenvironment,buttheyjustgoalongfortheride,sowewon’twritethemexplicitly.Let’simaginethattheamplitudesforspin-upandspin-downaren’tequal,butratherwehaveanunbalancedstateΨ,withunequalamplitudesforthetwodirections.

Thosenumbersoutsidethedifferentbranchesarethecorrespondingamplitudes.SincetheBornrulesaystheprobabilityequalstheamplitudesquared,inthisexampleweshouldhavea1/3probabilityofseeingspin-upanda2/3probabilityofseeingspin-down.

Imaginethatwedidn’tknowabouttheBornrule,andweretemptedtoassignprobabilitiesbysimplebranchcounting.Thinkaboutthepointofviewoftheobserversonthetwobranches.Fromtheirperspective,thoseamplitudesarejustinvisiblenumbersmultiplyingtheirbranchinthewavefunctionoftheuniverse.Whyshouldtheyhaveanythingtodowithprobabilities?Bothobserversareequallyreal,andtheydon’tevenknowwhichbranchthey’reonuntil theylook.Wouldn’titbemorerational,oratleastmoredemocratic,toassignthemequalcredences?

Theobviousproblemwiththat is thatwe’reallowedtokeeponmeasuringthings. Imaginethat we agreed ahead of time that if we measured spin-up, we would stop there, but if wemeasured spin-down, an automatic mechanism would quickly measure another spin. Thissecondspinisinastateofspin-right,whichweknowcanbewrittenasasuperpositionofspin-upandspin-down.Oncewe’vemeasuredit(onlyonthebranchwherethefirstspinwasdown),wehavethreebranches:onewherethefirstspinwasup,onewherewegotdownandthenup,and one where we got down twice in a row. The rule of “assign equal probability to eachbranch”wouldtellustoassignaprobabilityof1/3toeachofthesepossibilities.

That’s silly. If we followed that rule, the probability of the original spin-up branch wouldsuddenlychangewhenwedidameasurementonthespin-downbranch,goingfrom1/2to1/3.The probability of observing spin-up in our initial experiment shouldn’t depend on whethersomeoneonanentirelyseparatebranchdecidestodoanotherexperimentlateron.Soifwe’regoing toassigncredences ina sensibleway,we’ll have tobea littlemore sophisticated thansimplebranchcounting.

Instead of simplistically saying “Assign equal probability to each branch,” let’s try somethingmorelimitedinscope:“Assignequalprobabilitytobrancheswhentheyhaveequalamplitudes.”Forexample,asinglespininaspin-rightstatecanbewrittenasanequalsuperpositionofspin-upandspin-down.

Thisnewrulesaysweshouldgive50percentcredencetobeingoneitherthespin-uporspin-downbranches,werewetoobservethespinalongtheverticalaxis.Thatseemsreasonable,asthere isa symmetrybetween the twochoices; really, any reasonable rule shouldassign themequalprobability.*

Onenicethingaboutthismoremodestproposalisthatnoinconsistencyariseswithrepeatedmeasurements.Doinganextrameasurementononebranchbutnot theotherwould leaveuswithbranchesthathaveunequalamplitudesagain,sotheruledoesn’tseemtosayanythingatall.

But in fact it’s way better than that. If we start with this simple equal-amplitudes-imply-equal-probabilitiesrule,andaskwhetherthatisaspecialcaseofamoregeneralrulethatneverleads to inconsistencies,we endupwith a unique answer.And that answer is theBorn rule:probabilityequalsamplitudesquared.

Wecanseethisbyreturningtoourunbalancedcase,withoneamplitudeequaltothesquareroot of 1/3 and the other equal to the square root of 2/3. This timewe’ll explicitly include asecondhorizontalspin-rightqubitfromthestart.Atfirst,thissecondqubitjustgoesalongfortheride.

Insisting on equal probability for equal amplitudes doesn’t tell us anything yet, since theamplitudesarenotequal.Butwecanplaythesamegamewedidbefore,measuringthesecond

spin along the vertical axis if the first spin is down. The wave function evolves into threecomponents, and we can figure out what their amplitudes are by looking back at thedecompositionofaspin-rightstateintoverticalspinsabove.Multiplyingthesquarerootof2/3bythesquarerootof1/2givesthesquarerootof1/3,sowegetthreebranches,allwithequalamplitudes.

Since the amplitudes are equal, we can now safely assign them equal probabilities. Sincetherearethreeofthem,that’s1/3each.Andifwedon’twanttheprobabilityofonebranchtosuddenly change when something happens on another branch, that means we should haveassignedprobability1/3tothespin-upbranchevenbeforewedidthesecondmeasurement.But1/3isjustthesquareoftheamplitudeofthatbranch—exactlyastheBornrulewouldpredict.

Thereareacoupleoflingeringworrieshere.Youmayobjectthatweconsideredanespeciallysimpleexample,whereoneprobabilitywasexactlytwicetheotherone.Butthesamestrategyworkswheneverwecansubdivideourstates intotherightnumberof termssothatallof theamplitudes are equal in magnitude. That works whenever the amplitudes squared are allrationalnumbers(oneintegerdividedbyanotherone),andtheansweristhesame:probabilityequalsamplitudesquared.Thereareplentyofirrationalnumbersoutthere,butasaphysicistifyou’reabletoprovethatsomethingworksforallrationalnumbers,youhandtheproblemtoamathematician,mumblesomethingabout“continuity,”anddeclarethatyourworkhereisdone.

WecanseePythagoras’stheorematwork.It’sthereasonwhyabranchthatisbiggerthananotherbranchbythesquarerootoftwocansplitintotwobranchesofequalsizetotheotherone.That’swhythehardpartisn’tderivingtheactualformula,it’sprovidingasolidgroundingforwhatprobabilitymeansinadeterministictheory.Herewe’veexploredonepossibleanswer:it comes from the credences we have for being on different branches of the wave functionimmediatelyafterthewavefunctionbranches.

Youmightworry,“But Iwant toknowwhat theprobabilityofgettingaresultwillbeevenbeforeIdothemeasurement,notjustafterward.Beforethebranching,there’snouncertaintyaboutanything—you’vealreadytoldmeit’snotrighttowonderwhichbranchI’mgoingtoendupon.SohowdoItalkaboutprobabilitiesbeforethemeasurementismade?”

Never fear. You’re right, imaginary interlocutor, it makes no sense to worry about whichbranchyou’llendupon.Rather,weknowwithcertaintythattherewillbetwodescendantsofyourpresentstate,andeachofthemwillbeonadifferentbranch.Theywillbeidentical,andthey’llbeuncertainastowhichbranchthey’reon,andtheyshouldassigncredencesgivenbytheBornrule.Butthatmeansthatallofyourdescendantswillbeinexactlythesameepistemicposition, assigning Born-rule probabilities. So it makes sense that you go ahead and assignthoseprobabilitiesrightnow.We’vebeenforcedtoshiftthemeaningofwhatprobabilityisfromasimplefrequentistmodeltoamorerobustepistemicpicture,buthowwecalculatethingsandhow we act on the basis of those calculations goes through exactly as before. That’s whyphysicistshavebeenabletodointerestingworkwhileavoidingthesesubtlequestionsallthistime.

Intuitively, this analysis suggests that the amplitudes in a quantumwave function lead todifferentbrancheshavingadifferent“weight,”whichisproportionaltotheamplitudesquared.Iwouldn’twant to take thatmental image too literally, but it provides a concrete picture thathelpsusmake sense of probabilities, aswell as of other issues like energy conservation thatwe’lltalkaboutlater.

Weightofabranch=|Amplitudeofthatbranch|2

When there are two branches with unequal amplitudes, we say that there are only twoworlds,buttheydon’thaveequalweight;theonewithhigheramplitudecountsformore.Theweightsofallthebranchesofanyparticularwavefunctionalwaysadduptoone.Andwhenonebranchsplits intotwo,wedon’tsimply“makemoreuniverse”byduplicatingtheexistingone;thetotalweightofthetwonewworldsisequaltothatofthesingleworldwestartedwith,andtheoverallweightstaysthesame.Worldsgetthinnerasbranchingproceeds.

Thisisn’ttheonlywaytoderivetheBornruleintheMany-Worldstheory.Astrategythatisevenmorepopularinthefoundations-of-physicscommunityappealstodecisiontheory—therulesbywhicha rationalagentmakeschoices inanuncertainworld.Thisapproachwaspioneered in1999byDavidDeutsch(oneofthephysicistswhohadbeenimpressedbyHughEverettattheTexasmeetingin1977),andlatermademorerigorousbyDavidWallace.

Decisiontheorypositsthatrationalagentsattachdifferentamountsofvalue,or“utility,”todifferentthingsthatmighthappen,andthenprefertomaximizetheexpectedamountofutility—theaverageofallthepossibleoutcomes,weightedbytheirprobabilities.GiventwooutcomesAandB,anagentthatassignsexactlytwicetheutilitytoBastoAshouldbeindifferentbetweenAhappeningwithcertaintyandBhappeningwith50percentprobability.Thereareabunchofreasonable-soundingaxiomsthatanygoodassignmentofutilitiesshouldobey; forexample, ifanagentprefersAtoBandalsoprefersBtoC,theyshoulddefinitelypreferAtoC.Anyonewhogoesthroughlifeviolatingtheaxiomsofdecisiontheoryisdeemedtobeirrational,andthat’sthat.

Touse this framework in thecontextofMany-Worlds,weaskhowa rational agent shouldbehave,knowingthatthewavefunctionoftheuniversewasabouttobranchandknowingwhattheamplitudesofthedifferentbranchesweregoingtobe.Forexample,anelectroninanequalsuperpositionofspin-upandspin-downisgoingtotravelthroughaStern-Gerlachmagnetandhaveitsspinbemeasured.Someoneofferstopayyou$2iftheresultisspin-up,butonlyifyoupromise topay them$1 if the result is spin-down.Should you take theoffer? Ifwe trust theBornrule,theanswerisobviouslyyes,sinceourexpectedpayoffis0.5($2)+0.5(-$1)=$0.50.Butwe’retryingtoderivetheBornrulehere;howareyousupposedtofindananswerknowingthatoneofyourfutureselveswillbe$2richerbutanotheronewillbe$1poorer?(Let’sassumeyou’resufficientlywell-offthatgainingor losingadollar issomethingyoucareabout,butnotlife-changing.)

The manipulations are trickier here than in the previous case where we were explainingprobabilities as credences in a situation of self-locating uncertainty, so we won’t go throughthemexplicitly,butthebasicideaisthesame.Firstweconsideracasewheretheamplitudesontwodifferentbranchesareequal,andweshowthatit’srationaltocalculateyourexpectedvalueasthesimpleaverageofthetwodifferentutilities.ThensupposewehaveanunbalancedstatelikeΨabove,andIaskyoutogiveme$1ifthespinismeasuredtobeupandpromisetogiveyou $1 if the spin is down. By a bit ofmathematical prestidigitation,we can show that yourexpectedutility in this situation isexactly the sameas if therewere threepossibleoutcomeswithequalamplitudes,suchthatyougiveme$1foroneoutcomeandIgiveyou$1fortheothertwo.Inthatcase,theexpectedvalueistheaverageofthethreedifferentoutcomes.

Attheendoftheday,arationalagentinanEverettianuniverseactspreciselyasiftheyliveinanondeterministicuniversewhereprobabilitiesaregivenbytheBornrule.Actingotherwisewouldbeirrational, ifweacceptthevariousplausible-seemingaxiomsaboutwhatitmeanstoberationalinthiscontext.

Onecouldstubbornlymaintainthatit’snotgoodenoughtoshowthatpeopleshouldact“asif” something is true; it needs toactuallybe true.That’smissing thepoint a littlebit.Many-Worlds quantummechanics presents uswith a dramatically different view of reality from anordinary one-world view with truly random events. It’s unsurprising that some of our mostnatural-seemingnotionsaregoingtohavetochangealongwith it. Ifwe lived intheworldoftextbookquantummechanics,wherewave-functioncollapsewastrulyrandomandobeyedtheBornrule, itwouldberational tocalculateourexpectedutility inacertainway.DeutschandWallace have shown that if we live in a deterministicMany-Worlds universe, it is rational tocalculate our expected utility in exactly the same way. From this perspective, that’s what itmeans to talk about probability: the probabilities of different events actually occurring areequivalent to theweightingwe give those eventswhenwe calculate our expected utility.Weshouldactexactlyasiftheprobabilitieswe’recalculatingapplytoasinglechancyuniverse;buttheyarestillrealprobabilities,eventhoughtheuniverseisalittlericherthanthat.

*Therearemoresophisticatedargumentsthatsucharulefollowsfromveryweakassumptions.WojciechZurekhasproposedawayof deriving suchaprinciple, andCharlesSebens and I put forwardan independent argument.Weshowedthatthisrulecanbederivedbyinsistingthattheprobabilitiesyouassignfordoinganexperimentinyourlabshouldbeindependentofthequantumstateelsewhereintheuniverse.

8DoesThisOntologicalCommitmentMakeMeLookFat?

ASocraticDialogueonQuantumPuzzles

Aliceponderedsilentlyforabitassherefilledherwineglass.“Letmegetthisstraight,”shesaidatlast.“Youactuallywanttotalkaboutthefoundationsofquantummechanics?”

“Sure,”repliedherfatherwithamischievoussmile.Hewasaphysicisthimself,onewhohadmade a successful career as a master of imposing technical calculations in particle physics.ExperimentalistswhosmashedparticlestogetherattheLargeHadronColliderwouldregularlyconsult himondifficult questions about jets of particles createdbydecaying topquarks.Butwhen it came to quantummechanics, hewas a user, not a producer. “It’s about time I got abetterunderstandingofmydaughter’sownresearch.”

“Okay,” she answered. Ingraduate schoolAlicehad initially starteddowna similar careerpathasherfather,buthadgottensidetrackedbyadoggedinsistenceonmakingsenseofwhatquantum mechanics was actually saying. It seemed to her that physicists were foolingthemselvesby ignoring the foundationsof theirmost important theory.A fewyears later,shehadaPhDintheoreticalphysicsbuthadlandedajobasanassistantprofessorinthephilosophydepartmentatamajoruniversity,andwasgainingareputationasanexpertontheMany-Worldsapproachtoquantummechanics.“Howdoyouwanttodothis?”

“Iwrotedownsomequestions,”hesaidashepulledouthisphoneandpulledsomethinguponitsscreen.

Alicefeltamixtureofcuriosityandtrepidation.“Hitme,”shesaid,sniffingfromtheglassofBordeauxshehadpoured.Itwasopeningupnicely.

“Okay,”hebegan.Hisowndrinkwasaginmartini,nottoodry,threeolives.“Let’sstartwiththeobvious. Occam’s razor. We’re all taught in kindergarten that we should prefer simpleexplanations over unnecessarily complicated ones.Now, if I follow yourwork at all—maybe Idon’t—itseemstomethatyou’recomfortablepostulatinganinfinitenumberofunseenworlds.Doesn’tthatseemabitextravagant?Directlytheoppositeofthesimplestpossibleexplanation?”

Alice nodded. “Well, it depends on how we define ‘simple,’ of course. My philosophycolleaguessometimescastthisasaworryabout‘ontologicalcommitment’—roughly,theamountofstuffweneedtoimagineiscontainedinallofreality,justtodescribeourobservedportionofit.”

“Sowouldn’t Occam’s razor suggest that having toomany ontological commitments is anunattractivefeatureinafundamentaltheory?”

“Sure, but you have to be a little careful about what that commitment actually is.Many-Worldsdoesn’tassumealargenumberofworlds.WhatitassumesisawavefunctionevolvingaccordingtotheSchrödingerequation.Theworldsarethereautomatically.”

Herfatherobjected.“Whatdoyoumeanbythat?It’sliterallycalledtheMany-Worldstheory.Ofcourseitassumesalargenumberofworlds.”

“Not really,” replied Alice, becoming more animated as she warmed to the subject. “Theingredients used in Many-Worlds are ingredients that are used by every other version ofquantum mechanics. To get rid of the other worlds, alternatives need to posit additionalassumptions:eithernewdynamicsinadditiontotheSchrödingerequation,ornewvariablesinaddition to thewave function, or an entirely separate view of reality.Ontologically speaking,Many-Worldsisasleanandmeanasyoucanpossiblyget.”

“You’rekidding.”“I’mnot!Amuchmorerespectableobjection,tobehonest, isthatMany-Worldsistoolean

andmean,andit’sthereforeanontrivialtasktomaptheformalismontothemessinessofourobservedworld.”

Herfatherseemedtocontemplatethis.Hiscocktailsattemporarilyneglected.Alice decided to press the point. “I’ll explain what I mean. If you believe that quantum

mechanics is saying something about reality, you believe that an electron can be in asuperpositionof spin-upand spin-down, forexample.Andsinceyouand I andourmeasuringapparatusesaremadeofelectronsandotherquantumparticles,thesimplestthingtoassume—thethingthatOccam’srazorwouldsuggestthatyoudo—isthatyouandIandourmeasuringapparatuses can also be in superpositions, and indeed that the whole universe can be insuperpositions. That is what is straightforwardly implied by the formalism of quantummechanics,likeitornot.It’scertainlypossibletothinkaboutcomplicatingthetheoryinvariouswaystogetridofall thosesuperpositionsorrenderthemunphysical,butyoushould imagineWilliamofOccamlookingoveryourshoulder,tut-tuttingwithdisapproval.”

“Seemslikeabitofsophistrytome,”herfathergrumbled.“Philosophizingaside,abunchofin-principle-unobservablepartsofyourtheorydoesn’tseemverysimpleatall.”

“NobodycandenythatMany-Worldsinvolves,youknow,manyworlds,”Aliceconceded.“Butthatdoesn’tcountagainstthesimplicityofthetheory.Wejudgetheoriesnotbythenumberofentitiestheycananddodescribebutbythesimplicityoftheirunderlyingideas.Theideaoftheintegers—‘-3,-2,-1,0,1,2,3...’—ismuchsimplerthantheideaof,Idon’tknow,‘-342,7,91,abillionand three, theprimenumbers less than18,and thesquarerootof3.’Therearemoreelementsintheintegers—aninfinitenumberofthem—butthereisasimplepattern,makingthisinfinitelybigseteasytodescribe.”

“Okay,” said her father. “I can see that. There are a lot of worlds, but there is a simpleprinciplethatgeneratesthem,right?Butstill,bythetimeyouactuallyhaveallthoseworlds,itmusttakeanenormousamountofmathematicalinformationtodescribeallthem.Shouldn’twebelookingforasimplertheorywheretheyjustaren’tneededatall?”

“You’rewelcometolook,”repliedAlice,“andpeoplecertainlyhave.Butbygettingridoftheworlds,youendupmakingthetheorymorecomplicated.Thinkof itthisway:thespaceofallpossiblewavefunctions,Hilbertspace, isverybig.It’snotanybiggerinMany-Worldsthaninotherversionsofquantumtheory; it’sprecisely thesamesize,and thatsize ismore thanbigenoughtodescribealargenumberofparallelrealities.Onceyoucandescribesuperpositionsofspinningelectrons,youcanjustaseasilydescribesuperpositionsofuniverses.Ifyou’redoingquantummechanics at all, the potential formanyworlds is there, and ordinary Schrödingerevolution tends tobring themabout, like itornot.Otherapproaches justchoose tosomehownotmakeuseof the fullrichnessofHilbertspace.Theydon’twanttoaccepttheexistenceofotherworlds,sotheyneedtoworkhardtogetridofthemsomehow.”

“Fine,”mutteredher father, not fully convincedbut apparently ready tomoveon to thenextquestion.Hetookasipofhisdrinkandpeeredathisphone.“Isn’t therealsoaphilosophicalproblemwith the theory? I’mnophilosophermyself, butKarlPopperand I bothknow that agoodscientifictheoryissupposedtobefalsifiable.Ifyoucan’tevenimagineanexperimentthatmightproveyourtheorywrong,it’snotreallyscience.That’sexactlythesituationwithalltheseotherworlds,isn’tit?”

“Well,yesandno.”“That’sthego-toanswertoanyphilosophyquestion.”“Thepricewepayforbeingnotorioussticklersforprecision.”Alicelaughed.“Sure,Popper

hadthisproposalthatscientifictheoriesmustbefalsifiable.Itwasanimportantidea.Butintheback of his mind he was thinking about the difference between theories such as Einstein’sgeneralrelativity,whichmadedefiniteempiricalpredictionsforthebendingoflightbythesun,andthoselikeMarxisthistoryorFreudianpsychoanalysis.Theproblemwiththelatterideas,hethought,wasthatnomatterwhatactuallyhappened,youcouldcookupastorytoexplainwhyitwasso.”

“That’swhatIthought.Ihaven’treadPoppermyself,butIappreciatethatheputhisfingeronsomethingcrucialaboutscience.”

Alicenodded.“Hedid.Buttobehonest,mostmodernphilosophersofscienceagreethatitisn’tthecompleteanswer.Scienceismessierthanthat,andwhatseparatessciencefromnon-scienceisasubtleissue.”

“Everythingisasubtleissueforyoupeople!Nowonderyounevermakeanyprogress.”“Now,now,Dad,wearegettingatsomethingsignificanthere.WhatPopperwasultimately

tryingtopinpointisthatagoodscientifictheoryhastwocharacteristics.First,itisdefinite:youcan’t just twist the theory to ‘explain’ anything at all, as Popper feared you could do withdialecticalmaterialismorpsychoanalysis.Second,itisempirical:theoriesarenotdeemedtruebysheerreasonalone.Rather,oneimaginesmanydifferentpossiblewaystheworldcouldbe,eachcorrespondingtoadifferenttheory,andthenonechoosesamongthetheoriesbygoingoutandactuallylookingattheworld.”

“Exactly.”Herfatherseemedtothinkthattheadvantagewashisonthisone.“Empirical!Butifyoucan’tactuallyobservethoseworlds,there’snothingreallyempiricalaboutyourtheoryat

all.”“Aucontraire,”Alice replied. “Many-Worldsembodiesbothof these featuresperfectly. It is

notajust-sostorythatcanbeadaptedtoanyobservedsetoffacts.Itspostulatesaresimple:theworld is described by a quantum wave function that evolves according to the Schrödingerequation. Those postulates are eminently falsifiable. Just do an experiment showing thatquantuminterferencedoesn’toccurwhenitshould,orthatentanglementreallycanbeusedforsuperluminal communication, or that a wave function really does collapse even withoutdecoherence.Many-Worldsisthemostfalsifiabletheoryeverinvented.”

“Butthosearen’t testsofMany-Worlds,”her fatherprotested,unwillingtoconcedegroundonthisone.“Thosearejusttestsofquantummechanicsgenerally.”

“Right!ButEverettianquantummechanicsisjustpure,austerequantummechanicswithoutanyadditionaladhocassumptions.Ifyoudowantto introduceextraassumptions,thenbyallmeanswecanaskwhetherthosenewassumptionsaretestable.”

“Come now. The defining feature ofMany-Worlds is the existence of all those worlds outthere.Ourworldcan’tinteractwiththem,sothatparticularaspectofthetheoryisuntestable.”

“So what? Every good theory makes some predictions that are untestable. Our currenttheoretical understanding of general relativity predicts that the force of gravity will nottomorrowsuddenlyturnoff foraperiodofonemillisecondinaparticularregionofspacetenmetersacrossandtwentymillionlight-yearsaway.That’sacompletelyuntestableprediction,ofcourse, butwemaintain a very high credence that it’s true. There’s no reason for gravity tobehaveinthatway,andimaginingthatitdidleavesuswithamuchugliertheorythantheonewehave.Theadditionalworlds inEverettianquantummechanicshaveexactly thischaracter:they are inescapable predictions of a simple theoretical formalism. We should accept themunlesswehaveaspecificreasonnotto.

“Andbesides,”Alice rushedon, “theotherworldscouldbedetected inprinciple, ifwegotincrediblylucky.Theyhaven’tgoneaway,they’restillthereinthewavefunction.Decoherencemakes it fantasticallyunlikely for oneworld to interferewithanother, butnotmetaphysicallyimpossible. Iwouldn’tsuggestapplying forgrantmoneytodosuchanexperiment, though; itwould be likemixing cream into coffee andwaiting around for them to spontaneously unmixthemselves.”

“Don’tworry,Iwasn’tplanningonit.Ijustdon’tthinkKarlPopperwouldbeveryhappywithyourapproachtothephilosophyofscience.”

“I’vegotyouthere,Dad,”saidAlice.“PopperhimselfwasaharshcriticoftheCopenhageninterpretation,which he called a ‘mistaken and even a vicious doctrine.’ In contrast, he hadgoodthingstosayaboutMany-Worlds,whichheaccuratelydescribedas‘acompletelyobjectivediscussionofquantummechanics.’”

“Seriously?PopperwasanEverettian?”“Well, no,” Alice admitted. “He ultimately parted ways with Everett because he couldn’t

understand why the wave function would branch but branches wouldn’t later fuse backtogether.Imean,that’sagoodquestion,butit’sonewecananswer.”

“I’msureyoucan.Wheredidhecomedownonthefoundationsofquantummechanics?”“Hedevelopedhisownformulationofquantummechanics,butitneverreallycaughton.”“Ha!Philosophers.”“Yeah.We’rebetterattellingyouwhyyourtheoryiswrongthanatproposingbetterones.”

Alice’sfathersighed.“Fine.I’mnotsayingyou’reconvincingmeofanything,butIdon’twanttoget bogged down in philosophical hair-splitting. Now that you mention it, Popper’s questiondoesseemkindofreasonable.Whydon’tworlds fuse togetheraswellasbranchapart? Ifwehave a spin that is an equal superposition of up anddown,we canpredict theprobability ofobservingeitheroutcome ifwedoameasurement in the future.But ifwehaveaspin that ispurelyup, andweare told that itwas justmeasured,wehaveabsolutelynowayof knowingwhat kind of superposition it was in pre-measurement (except that it wasn’t purely down).Wheredoesthedifferencecomefrom?”

Alice seemed ready for this one. “That’s just thermodynamics, really. Or at least, it’s thearrowoftime,pointingfromthepasttothefuture.Werememberyesterdaybutnottomorrow;creamandcoffeemixtogetherbuttheydon’tspontaneouslyunmix.Wavefunctionsbranch,butdon’tunbranch.”

“Sounds suspiciously circular. As I understand it, one of the purported features ofMany-WorldsisthatwavefunctionsonlyobeytheSchrödingerequation;there’snoseparatecollapsepostulate. Back when I learned quantummechanics, we knew that wave functions collapsedtoward the futureandnot toward thepast,and thatwaspartof theassumptions. Idon’t seewhy that should still be true for Everett, where the Schrödinger equation is completelyreversible.Whatdocreamandcoffeehavetodowithwavefunctions?”

Alice nodded. “Perfectly good question. Let’s set the stage a bit. The second law ofthermodynamics posits that entropy—roughly, the disorderliness or randomness of aconfiguration, as you know—never decreases in closed systems. LudwigBoltzmann explainedthisbackinthe1870s.Entropycountsthenumberofwaysthatatomscanbearrangedsothatthe system looks the same from a macroscopic perspective. The reason why it increases issimply that there are many more ways to be high-entropy than to be low-entropy, so it’simprobablethatentropywouldevergodown.Right?”

“Sure,” her father agreed. “But that’s all classical; Boltzmann didn’t know anything aboutquantummechanics.”

“Right,butthebasicideaisthesame.Boltzmannexplainedwhyentropytendstoincrease,buthedidn’tgiveareasonwhyitwaseverlowinthefirstplace.Thesedaysweappreciatethatit is a cosmological fact that the universe started out right after the Big Bang in an orderlystate,andentropyhasnaturallybeen increasingeversince,andsowehave time’sarrow.Wedon’treallyknowwhytheearlyuniversehadsuchalowentropy,thoughsomeofushaveideas.”

“Andthisisrelevantbecause...”“BecauseforEverettians,theexplanationofthequantumarrowoftimeisthesameasthatof

the entropic arrow of time: the initial conditions of the universe. Branching happens whensystemsbecomeentangledwith the environment anddecohere,whichunfolds as timemovestoward the future, not the past. The number of branches of the wave function, just like theentropy,onlyincreaseswithtime.Thatmeansthatthenumberofbrancheswasrelativelysmalltobeginwith.Inotherwords,thattherewasarelativelylowamountofentanglementbetweenvarioussystemsandtheenvironmentinthefarpast.Aswithentropy,thisisaninitialconditionweimposeonthestateoftheuniverse,andatthepresenttimewedon’tknowforsurewhyitwasthecase.”

“Okay,”saidherfather.“It’sgoodtoadmitwhatwedon’tknow.Weexplainthearrowoftime,at leastaccordingto thecurrentstateof theart,byappeal tospecial initialconditions in thepast. Is it a single condition that explains both the thermodynamic arrow and the quantumarrow,oristhatjustananalogy?”

“Ithinkit’smorethanananalogy,buttobehonest,thisisasubjectthatcouldprobablyuseabit more rigorous investigation,” Alice replied. “There certainly seems to be a connection.Entropy is related to our ignorance. If a system has low entropy, there are relatively fewmicroscopic configurations thatwould look thatway, sowe know a lot about it just from itsmacroscopicallyobservable features; if ithashighentropy,weknowrelatively little. JohnvonNeumann realized thatwe can say something similar about entangledquantum systems. If asystemiscompletelyunentangledwithanythingelse,wecansafelytalkaboutitswavefunctioninisolationfromtherestoftheworld.Butwhenitisentangled,theindividualwavefunctionisundefined,andwecanonlytalkaboutthewavefunctionforthecombinedsystem.”

Her father brightened. “Von Neumann was a brilliant guy, a real hero. There were anamazingnumberofHungarianphysicistswhoemigratedtotheUS—Szilard,Wigner,Teller—buthewasthetop.Idovaguelyrememberthathederivedaformulaforentropy.”

Alice agreed. “No question. Von Neumann realized that there was a mathematicalequivalencebetweenaclassicalsituationwhenwe’reunsureabouttheexactstateofasystem,whichgivesrisetoentropy,andthequantumsituationwheretwosubsystemsareentangled,sowecan’t talkaboutthewavefunctionofeitherpieceseparately.Hederivedaformulaforthe‘entanglemententropy’ofaquantumsystem.Themoreentangledsomethingiswiththerestoftheworld,thehigheritsentropy.”

“Aha,”exclaimedherfatherexcitedly.“Iseewhereyou’regoingwiththis.Thefactthatwavefunctionsonlybranch forward in timeandnotbackward isnotsimplyreminiscentof the factthatentropyincreases—it’sthesamefact.Thelowentropyoftheearlyuniversecorrespondstothe ideathat thereweremanyunentangledsubsystemsbackthen.Asthey interactwitheachotherandbecomeentangled,weseethatasbranchingofthewavefunction.”

“Exactly,”Alice responded,withsomething likedaughterlypride. “We’restillnot surewhythe universe is like that, but once we accept that the early universe was in a relativelyunentangled,low-entropystate,everythingelsefollows.”

“But wait a minute.” Her father seemed to have just realized something. “According toBoltzmann,entropyisonlylikelytoincrease,it’snotanabsoluterule.It’sultimatelyduetotherandom motions of atoms and molecules, so there’s a nonzero probability that entropy willspontaneously go down. Does that mean that it’s possible that decoherence will somedayreverse,andworldsactuallywillfusetogetherratherthanbranchingapart?”

“Absolutely,”saidAlicewithanod.“Butjustlikewithentropy,thechanceofthathappeningis so preposterously small that it’s irrelevant to our daily lives, or to any experiment in thehistoryofphysics.It’sextremelyunlikelythattwomacroscopicallydistinctconfigurationshaverecoheredevenonceinthelifetimeofouruniverse.”

“Soyou’resayingthere’sachance?”“I’msayingthatifyourworryaboutMany-Worldsisthatbranchesofthewavefunctionwill

someday come back together, you’ve clearly exhausted all the reasonable worries and are

graspingatstraws.”

“Well,let’snotgettoofullofourselvesjustyet,”herfathermuttered,seeminglyreturningtohisskeptical stance. He lifted the toothpick from his glass and bit off an olive. “Let me try tounderstandwhat the theory actually says. Is it right to say that the number ofworlds beingproducedateverymomentisliterallyinfinitelybig?”

“Well,”repliedAlice,somewhattentatively,“I’mafraidanhonestanswertothatquestionisgoingtorequireabitmorephilosophicalhair-splitting.”

“WhyamInotsurprised?”“We can go back to entropy as an analogy. When Boltzmann came up with his entropy

formula, he counted the number of microscopic arrangements of a system that lookedmacroscopically the same. From there, he was able to argue that entropy should naturallyincrease.”

“Sure,” said her father. “But that is real, honest physics, something we can testexperimentally.NotsurewhatithastodowithyourMany-Worldsflightsoffancy.”

“We say that now.But you have to imaginewhat peoplewere thinking back at the time.”Alice was settling comfortably into professor mode, her Bordeaux momentarily forgotten.“Boltzmannwasright,butanumberofobjectionswereraisedtohisidea.Onewasthathewasturning entropy from an objective feature of a physical system into a subjective one, whichdependedonsomenotionof‘looksthesame.’Anotherwasthathedemotedthesecondlawfromanabsolutestatementtoameretendency—itwasn’tthatentropynecessarilyincreased,itwasjustverylikelytodoso.Particlesjigglearoundrandomly,andit’sextremelyprobablethattheywill evolve towardahigher-entropy state,but it’snota lawlikecertainty.With thewisdomofaccumulated years,we can see that the subjective nature of Boltzmann’s definition does notstopitfrombeingausefulone,andthefactthatthesecondlawisareallygoodapproximationratherthananabsoluteunbreakablelawismorethangoodenoughforwhateverpurposeswemayhave.”

“Igetthat,”answeredherfather.“Entropyisanobjectivelyrealthing,butwecandefineandmeasureitonlyaftermakingafewdecisions.Butthatneverreallybotheredme—it’suseful!I’mnotsurethatextraworldsreallyare.”

“We’ll get there, but first let me elaborate on this analogy. Like entropy, the notion of a‘world’inEverettianquantummechanicsisahigher-levelconcept,notafundamentalone.It’sausefulapproximationthatprovidesgenuinephysicalinsight.Theseparatebranchesofthewavefunction aren’t put in as part of the basic architecture of the theory. It’s just extraordinarilyconvenient forushumanbeings to thinkofa superpositionofmanysuchworlds, rather thantreatingthequantumstateasanundifferentiatedabstraction.”

Herfather’seyeswidenedabit.“ThisisworsethanIfeared.Itsoundslikeyou’regoingtotellmethata‘world’isn’tevenawell-definedconceptinMany-Worlds.”

“They’rejustaswelldefinedasentropyis.Ifwewereanineteenth-centuryLaplacedemon,whoknewthepositionandmomentumofeveryparticleintheuniverse,wewouldneverhavetostoop to defining a coarse-grainednotion like ‘entropy.’ Likewise, ifwe knew the exactwavefunctionoftheuniverse,wewouldneverhavetotalkabout‘branches.’Butinbothcaseswearepoorfinitecreatureswithdramaticallyincompleteinformation,andinvokingthesehigher-levelconceptsisextremelyuseful.”

Alicecouldtell thather fatherwas losingpatience.“I justwant toknowhowmanyworldsthereare,”hesaid.“Ifyoucan’tanswerthat,you’renotdoingaverygoodsalesjobhere.”

“Mustbethatdevotiontohonestyunderanycircumstancesthatyouinculcatedintomeatayoung age,” Alice said with a shrug. “It depends on how we divide the quantum state intoworlds.”

“Andisn’ttheresomeobviousrightway?”“Sometimes!Insimplesituationswheremeasurementshaveamanifestlydiscreteoutcome,

likemeasuring the spin of an electron,we can safely say that thewave functionbranches intwo,andthenumberofworlds(whateverthatwas)doubles.Whenwe’remeasuringaquantitythatisinprinciplecontinuous,likethepositionofaparticle,thingsarelesswelldefined.Inthatcasewecandefinea totalweightattachedtoacertainrangeofoutcomes, thewave functionsquared,butnotanabsolutenumberofbranches.Thatnumberwoulddependonhowfinelywewant to subdivide our description of themeasurement outcome,which is ultimately a choicethat’suptous.OneofmyfavoritequotesalongtheselinesisfromDavidWallace:‘Askinghowmanyworlds thereare is likeaskinghowmanyexperiencesyouhadyesterday, orhowmanyregrets a repentant criminal has had. It makes perfect sense to say that you had manyexperiences or that he had many regrets; it makes perfect sense to list the most importantcategoriesofeither;butitisanon-questiontoaskhowmany.’”

Alice’s father didn’t really seem satisfied by this. After a thoughtful pause, he responded,

“Look, I’m trying to be fair here. I’ll accept that theworlds are not fundamental, so there issomethingapproximateabouthowtheyaredefined.Butsurelyyoucantellmewhethertherearejustafinitenumberofthemorthenumberistrulyinfinite.”

“It’safairquestion,”Aliceagreed,maybeabitreluctantly.“Unfortunately,wedon’tknowtheanswer.There’sanupperlimittothenumberofworlds,whichisjustthesizeofHilbertspace,thespaceofallpossiblewavefunctions.”

“ButweknowthatHilbertspaceisinfinitelybig,”interjectedherfather.“Evenforjustoneparticle,Hilbertspace is infinite-dimensional,not tomention forquantumfield theory.So thenumberofworldssoundslikeit’sinfinite.”

“We’re not sure whether theHilbert space for our actual universe has a finite or infinitenumber of dimensions.We certainly knowof some systems forwhich the appropriateHilbertspaceisfinite-dimensional.Asinglequbitiseitherspin-uporspin-down,soitcorrespondstoatwo-dimensional Hilbert space. If we haveN qubits, the corresponding Hilbert space is 2N-dimensional—thesizeofHilbertspacegrowsexponentiallyasweincludemoreparticles.Acupof coffee contains roughly 1025 electrons, protons, and neutrons, each of whose spins isdescribedbyaqubit.SotheHilbertspaceforacupofcoffee—justincludingthespins,notyetworryingaboutthelocationsoftheparticles—hasadimensionalityofabout .

“Needlesstosay,”continuedAlice,“that’sacrazy-bignumber.Onefollowedby1025zeroes,ifyouwroteitinbinary.Whichyouwouldn’thavetimetodo,evenifyouhadbeenworkingfortheentirelifetimeofourobservableuniverse.”

“Butyou’reobviouslycheating,therealnumberismuchbiggerthanthat,”saidherfather.“You’recountingspins,butrealparticleshavelocationsinspacetoo.Andthereareaninfinitenumberofsuch locations.That’swhytheHilbertspaceforacollectionofparticles is infinite-dimensional—thenumberofdimensionsisjustthenumberofpossiblemeasurementoutcomes.”

“Right.And it’s true,HughEveretthimself thought thateveryquantummeasurementsplittheuniverseintoaninfinitenumberofworlds,andhewascomfortablewiththat.Infinitysoundslike a big number, butwe use infinite quantities in physics all the time. The number of realnumbers between 0 and 1 is infinite, as you know. IfHilbert space is infinite-dimensional, itdoesn’tmakemuchsensetotalkaboutthenumberofindividualworlds.Butwecangroupasetof similar worlds together, and talk about the total weight (amplitude-squared) they havecomparedtosomeothergroup.”

“Great.SoHilbertspaceisinfinite-dimensional,andthenumberofworldsisinfinite,butyouwanttoclaimthatweshouldonlytalkabouttherelativeweightofdifferentkindsofworlds?”

“No,I’mnotdoneyet,”Aliceinsisted.“Therealworldisn’tabunchofparticles,norisitevendescribedbyquantumfieldtheory.”

“It’snot?”saidherfatherinmockdismay.“WhathaveIbeendoingallmylife?”“You’vebeenignoringgravity,”repliedAlice,“whichisaperfectlysensiblethingtodowhile

you’rethinkingaboutparticlephysics.Butthereareindicationsfromquantumgravitythatthenumber of distinct possible quantum states is finite, not infinite. If that’s true, there is amaximumnumberofworldswecouldsensiblytalkabout,givenbythedimensionalityofHilbertspace. The kinds of estimates that get thrown around for the number of dimensions of theHilbertspaceofourobservableuniverseare things like .Abignumber,”Aliceadmitted,“butevenverybigfinitenumbersaremuchsmallerthaninfinity.”

Her father seemed to think about this. “Huh. I’m not really sure we know anything veryreliableaboutquantumgravity—”

“Maybewedon’t.That’swhyIsaidwereallydon’tknowifthenumberofworldsisfiniteorinfinite.”

“Fairenough.Butthatraisesatotallynewworry. Itseemstomethatbranchingshouldbehappeningallthetime,everytimeaquantumsystembecomesentangledwithitsenvironment.Is it conceivable that this number you just quoted, while mind-bogglingly large, isn’t largeenough?Arewe sure there’s enough room inHilbert space for all the branches of thewavefunctionthatarebeingproducedastheuniverseevolves?”

“Hmm, I never thought about that, to be honest.” Alice grabbed a napkin and startedscribblingsomenumbersonit.“Let’ssee,thereareabout1088particleswithinourobservableuniverse, mostly photons and neutrinos. For the most part these particles travel peacefullythrough space, not interacting or becoming entangled with anything. So as a generousoverestimate, let’s imagine that every particle in the universe interacts and splits the wavefunctionintwoamilliontimespersecond,andhasbeendoingsosincetheBigBang,whichwasabout1018secondsago.That’s1088×106×1018=10112splittings,producingatotalnumberofbranchesof .

“Nice!” Alice seemed pleasedwith herself. “That’s still a really big number, but it’smuchsmaller than thenumberofdimensions in theHilbert spaceof theuniverse.Pitifully smaller,really.Anditshouldbeasafeoverestimateofthenumberofbranchesrequired.Soevenifthequestion of howmany branches there are doesn’t have a definite answer, we don’t need toworrythatHilbertspaceisgoingtorunoutofroom.”

“Well,good,Iwasworriedthereforasecond.”Herfather’smartinitastedpleasantlybrinyfromthe olives. He regarded Alice, a glint in his eye. “Had you really never asked yourself thatquestionbefore?”

“IthinkmostEverettianstrainthemselvestothinkoftherelativeweightsofvariousdifferentbranches of the wave function, rather than actually counting anything. We don’t know theultimateanswer,soitdoesn’tseemtoofruitfultoworryaboutit.”

“I’llhavetoprocessthisabit,becauseIalwaysthoughtthatthereweresupposedtobeaninfinitenumberofworlds,andthatMany-Worldsimpliedthateverythinghappenedsomewhere.Thateverypossibleworldexistsoutthereinthewavefunction.Ithoughtthatwasthesellingpoint.When I was stuck on a calculation, it was comforting to think that therewas anotherworldinwhichIwasallama,orageniusbillionaireplayboyphilanthropist.”

“Wait,you’renot?”Alicefeignedsurprise.“Ialwaysthoughtyoulookedabitlikeallama.”“Imean,forthatmatter,insomeworldIshouldbeabillionairellama.”“Beforewegetofftrack,”shecontinued,“letmejustnotethatit’snot‘you’whowouldbea

llamaorabillionaire,thosewouldbeotherbeingsentirely.I’msurewe’llcomebacktothat.Butofmoredirectrelevancetotheissue,Many-Worldsdoesn’tsay‘everythingpossiblehappens’;itsays ‘the wave function evolves according to the Schrödinger equation.’ Some things don’thappen,becausetheSchrödingerequationneverleadstothemhappening.Forexample,wewillneverseeanelectronspontaneouslyconvert intoaproton.Thatwouldchange theamountofelectriccharge,andchargeisstrictlyconserved.Sobranchingwillnevercreate, forexample,universeswithmoreorlesschargethanwestartedwith.JustbecausemanythingshappeninEverettianquantummechanicsdoesn’tmeanthateverythingdoes.”

Alice’sfatherraisedhiseyebrowsinskepticism.“Dear,youaresurelynitpickingtosaveface.Maybenotstrictlyeverythinghappens,butIbelieveit’struethatagreatmanycrazy-soundingthingsdohappeninvariousworlds,no?”

“Sure,I’mhappytoadmitthat.Everytimeyourunintoawall,thewavefunctionbranchesintoanumberofworlds:somewhereyouinjureyournose,somewhereyouharmlesslytunnelrightthrough,andotherswhereyoubounceoffandarethrownacrosstheroom,forexample.”

“But that matters a lot, doesn’t it? In ordinary quantum mechanics the probability of amacroscopicobjecttunnelingthroughawallisnotzero,butit’sunimaginablytiny,andwecanjustignoreit.InMany-Worlds,theprobabilityis100percentthatithappensinsomeworld.”

Alicenodded,butherexpressionwasthatofsomeonewhohadgoneoverthisgroundmanytimesbefore.“You’reabsolutelyrightthatthisisadifference.ButIwouldarguethatitdoesn’tmatterasinglebit.IfyouaccepthowEverettiansderivetheBornrule,youshouldactasifthereisaprobabilityofyoutunnelingthroughthewall,andthatprobabilityissopreposterouslysmallthatthere’snoreasonwhatsoevertotakeitintoconsiderationasyougothroughyoureverydaylife.And if youdon’taccept thatargument, there isamuchmoreseriousworryaboutMany-Worldsforyoutofretover.”

Her fatherwasdetermined. “I think the issueof these low-probabilityworlds is important.Whatabout thoseobserverswho, in theensembleofEverettianworlds,endupseeingeventsthatseeminglydefyourBornrulepredictions? Ifwemeasureaspin fifty times, therewillbebranchesonwhichalloftheresultsreadspin-up,andothersonwhichtheyallreadspin-down.Whatarethosepoorobserverssupposedtoconcludeaboutquantummechanics?”

“Well,” said Alice, “mostlywe have to say, too bad for them. Stuff happens. But the totalweightassigned to suchobservers is so small thatweshouldn’tworryabout them toomuch.Nottomentionthat,aftertheygetfiftyspin-upsinarow,thenextfiftytrialswillstillmapontothe Born-rule predictions with overwhelming probability.Most likely they will attribute theiroriginalluckystreaktoexperimentalerror,andhaveafunstorytotelltheirlabmates.It’sjustlikeaclassicaluniversethatisjustreallybig.Ifconditionsthatweseeintheuniversearounduscontinueinfinitelyfarineverydirection,itisoverwhelminglyprobablethatthereareothercivilizations just like ours—an infinite number, in fact—doing experiments to test quantummechanics.EvenifeachofthemislikelytoseeBorn-ruleprobabilities,giventhatthereareaninfinitenumberofthem,someofthemwillseeverydifferentstatistics.Inthatcasetheymaybeledtodrawincorrectconclusionsabouthowquantummechanicsworks.Thoseobserverswouldbeunlucky,butwecantakeconsolationinthefactthattheyarealsoveryinfrequentamongthesetofallobserversintheuniverse.”

“Smallconsolationforthem!Inyourviewofphysics,therewillalwaysbeobserversouttherewhogetthelawsofnatureutterlywrong.”

“Nobodyeverpromisedthemarosegarden.Thatworryexistsinanytheorywheretherearesufficientlylargenumbersofobservers;Many-Worldsisjustoneexampleofsuchatheory.Thepoint is that inEverettianquantummechanics, there is away to compareall of thedifferentworlds: take the amplitudesof their branches, and square them.Thebranches inwhich verysurprisingthingshappenhavevery,verytinyamplitudes.Theyarerareinthesetofallworlds.We shouldn’t be anymore bothered by their existence than we are by unlucky observers in

infinitelylargeuniverses.”

“NotsureI’mconvincedhere,butlet’s justentermyworryintotherecordandmoveon.”Hesquinted at the list of questions he had brought up on his phone. “I’ve been doing a bit ofreading—evensomeofyourpapers—andonethingIdoappreciateaboutMany-Worldsisthatitremovesanylingeringmysteryaboutwhenameasurementtakesplace.There’snothingspecialabout measurement; it’s just when a quantum system that’s in a superposition becomesentangled with the larger environment, leading to decoherence and branching of the wavefunction. But there is only one wave function, the wave function of the universe, whichdescribes everything throughout space. How shouldwe think about branching from a globalperspective?Doesbranchinghappenallatonce,ordoesitgraduallyspreadoutfromthesystemwheretheinteractionoccurred?”

“Ohboy. Ihavea feeling this isgoing tobeanotherunsatisfyinganswer.”Alicepaused tosliceoffapieceofcheese.Shecarefullyarrangeditonacrackerasshethoughtaboutthebestresponse. “Basically: that’s up to you. Or, to put the point in more respectable-soundinglanguage, the very phenomenon of ‘branching’ is one that we humans invent to provide aconvenientdescriptionofacomplicatedwavefunction,andwhetherwethinkofbranchingashappeningallatonceorasspreadingoutfromapointdependsonwhat’smoreconvenientforthesituation.”

Herfathershookhishead.“Ithoughtbranchingwasthewholepoint.HowcanyouholdupMany-Worldsasarespectablescientifictheoryifnotonlycan’tyouobservetheotherbranches,and not only can’t you count them, but you don’t even have a definite criterion for how ithappens?Branchingisjust,like,youropinion,man?”Hehadalwaysbeenjustalittletoofondofmoviereferences.

“In a sense, sure. But there are better and worse opinions to have. You may prefer adescriptioninwhichnothingtravelsfasterthanthespeedoflight.Whatactuallymattersisthatyou can’t communicate or send information faster than light, and that’s true nomatterwhatdescription you choose to use.But if itmakes you feel better to limit an apparently physicaleffectlikebranchingtopropagatenofasterthanlight,youareperfectlywelcometodothat.Inthatcase,thenumberofbranchesofthewavefunctionwouldbedifferentdependingonwhereyou were in spacetime.” She took out a fresh napkin and began scribbling again, this timemaking littlediagramsoutofstraight lines.“Herewehavespacegoingfromleft torightandtime going upward. Light beams that could potentially be emitted from an event will moveupwardatforty-five-degreeangles.Ifwestartwithjustasinglebranchofthewavefunction,wecanimaginebranchinghappeningatthatevent,andthenpropagatingupwardintime,butonlygrowingatthespeedof light.Observersfartherawaywouldbedescribedbyasinglebranch,whileneareroneswouldbedescribedbytwobranches.Thisfitswellwiththeideathatdistantobservers have noway of knowing, or being influenced by, the branching event,while thosenearbydo.”

Her father studied the diagram. “I see. I guess I assumed that branching happenedsimultaneously throughout theuniverse,whichbotheredmeas someonewho isquite fondofspecial relativity. I’m sure you know as well as I do that different observers will definesimultaneitydifferently.Ikindoflikethispicturebetter,wherebranchingpropagatesoutwardatthespeedoflight.Alltheeffectslookprettylocal.”

Alicewavedherhandsbeforesheresumeddrawing.“Buttheotherwayworkstoo.Weareequallyallowedtodescribebranchingashappeningallthroughouttheuniverseallatonce.ThisviewishelpfulwhenwederivetheBornruleusingself-locatinguncertainty,aswecansensiblytalkaboutwhichbranchyouareonimmediatelyafterthebranchingoccurs,nomatterwhereithappened.Because of relativity, observersmoving at different speedswill draw thebranchesdifferently,butthere’snoobservationaldifferencecausedbydoingso.”

“Arrgh!You’vejustundoneallofyourgoodwork.Nowyou’retellingmethatbranchingcanjustaswellbethoughtofascompletelynonlocal.”

“Yeah,butwhatI’mactuallysayingisthatthequestion‘IsMany-Worldsalocaltheory?’isn’tquite the right one to ask. It would be better to ask, ‘Canwe describe branching as a localprocess,proceedingonly insidethefuture lightconeofanevent?’Theanswer is ‘Yes,butwecanequallywelldescribeitasanonlocalprocess,occurringinstantlythroughouttheuniverse.’”

Her fatherputhishandsoverhis face,butheseemedtobe trying toabsorb this,not justgivingupinfrustration.Thenhegotupandmixedhimselfanothermartini,browfurrowed.Hereturnedtohisseat,drinkinonehandandsomepeanutsintheother.“IguessthepointisthatwhetherornotIthinkapersonfarawayhasbranched,itdoesn’tmakeanydifferencetothem.Icanthinkofthemasbeingjustonecopy,orastwocopiesthatareabsolutelyidentical.It’sjustamatterofdescription.”

“Exactly!”Aliceexclaimed. “Whetherwe thinkaboutbranchingaspropagatingoutwardatthespeedoflightorhappeningallatonceisjustaquestionofwhat’smostconvenient.It’snomoreworrisomethanthefactthatwecanmeasurelengthincentimetersorinches.”

Herfatherrolledhiseyes.“Whatkindofbarbarianmeasureslengthininches?”

“Okay,let’sshiftgears,”hesaidafteramoment.“Iknowthatstringtheoristsandotherpeoplewhoaren’tverytetheredtorealityarefondoftalkingaboutextradimensions.Dothebrancheslivethere?Wherearetheseotherworldslocated,anyway?”

“Oh, come on, Robert.” Alice tended to call her father by his first name when she wasannoyedwithhim.“Youknowbetterthanthat.Thebranchesaren’t‘located’anywhere.Ifyou’restuckthinkingofthingsashavinglocationsinspace,itmightseemnaturaltoaskaboutwheretheotherworldsare.Butthereisno‘place’wherethosebranchesarehiding;theysimplyexistsimultaneously, alongwith our own, effectively out of contactwith it. I suppose they exist inHilbertspace,butthat’snotreallya‘place.’Therearemorethingsinheavenandearththanaredreamtofinyourphilosophy.”ShewasproudtokeepherreferencesShakespearean.

“Yeah,Iknow.We’reacoupleofdrinksin,IthoughtIshouldtossyouasoftball.”

Hescrolleddownthedocumentonhisphoneabit.“Allright,let’sgetmoreserioushere.Thisonehasbeenbuggingmeforever.Whataboutconservationofenergy?Wheredoesallthatstuffcomefromwhenyousuddenlycreateawholenewuniverse?”

“Well,” replied Alice, “just think about ordinary textbook quantum mechanics. Given aquantum state, we can calculate the total energy it describes. As long as the wave functionevolvesstrictlyaccordingtotheSchrödingerequation,thatenergyisexactlyconserved,right?”

“Sure.”“That’s it. In Many-Worlds, the wave function obeys the Schrödinger equation, which

conservesenergy.”“But what about the extra worlds?” her father insisted. “I could measure the energy

containedinthisworldIseearoundme,andyousayit’sbeingduplicatedallthetime.”Alice felt shewas on firm groundwith this one. “Not all worlds are created equal. Think

aboutthewavefunction.Whenitdescribesmultiplebranchedworlds,wecancalculatethetotalamount of energy by adding up the amount of energy in each world, times the weight (theamplitudesquared)forthatworld.Whenoneworlddividesintwo,theenergyineachworldisbasically the same as it previouslywas in the singleworld (as far as anyone living inside is

concerned),buttheircontributionstothetotalenergyofthewavefunctionoftheuniversehavedividedinhalf,sincetheiramplitudeshavedecreased.Eachworldgotabitthinner,althoughitsinhabitantscan’ttellanydifference.”

“Mathematically I seewhat you’re saying,” admittedher father. “But I seem tobe lackingsomeintuitionhere. Ihave,say,abowlingball,withacertainmassandpotentialenergy.Butthensomeoneinthenextroomobservesaquantumspinandbranchesthewavefunction.Nowtherearetwobowlingballs,eachofwhichhastheenergyofthepreviousone.No?”

“That ignores the amplitudes of the branches. The contribution of the bowling ball to theenergyoftheuniverseisn’tjustthemassandthepotentialenergyoftheball;it’sthat,timestheweightofitsbranchofthewavefunction.Afterthesplittingitlookslikeyouhavetwobowlingballs,but together theycontributeexactlyasmuch to theenergyof thewave functionas thesinglebowlingballdidbefore.”

Herfatherseemedtoponderthis.“I’mnotsureIagreewithyou,butIthinkyou’rewearingmedown,”hemuttered.Afteramomentheturnedbacktohislistofquestions.

“Youknow,IthinkIonlyhaveonequestionleft.”Alice’sfatherputawayhisphone,dranksomemore of his secondmartini, and leaned in a bit. “Do you really believe this?Honestly? Thatmultiplecopiesofmecomeintoexistenceeverytimesomeonemeasuresthespinofaparticle?”

Alice satback inher chair, savoredabit ofherwine, and looked thoughtful. “Youknow, Ireallydo.Atleast,IpersonallyfindEverettianquantummechanics,andallthemanyworldsthatit implies, to be by far themost plausible version of quantum theory that I know of. If thatmeansImustacceptthatmypresentselfwillevolveintoanumberofslightlydifferentfutureselveswhowillneverbeabletotalktoeachother,I’mwillingtoacceptthat.Subject,asalways,to being updated in the future if new information comes along, either in the form ofexperimentalresultsornewtheoreticalinsights.”

“Suchagoodempiricist.”Herfathersmiled.“LetmequoteDavidDeutsch,”Aliceoffered.“Heoncesaid,‘Despitetheunrivaledempirical

successofquantumtheory,theverysuggestionthatitmaybeliterallytrueasadescriptionofnatureisstillgreetedwithcynicism,incomprehension,andevenanger.’”

“What’s that supposed to mean? Every physicist thinks quantum mechanics describesnature.”

“IthinkwhenDeutschsays‘quantumtheory,’heimplicitlymeansMany-Worlds.”NowitwasAlice’sturntosmile.“WhathewasgettingatwasthatmanypeoplerejectEverettianquantummechanics more out of a visceral sense of distaste than a principled set of worries. But asphilosopherDavidLewisonceputit,‘Idonotknowhowtorefuteanincredulousstare.’”

“I hope you’re not including me there.” Alice’s father looked slightly affronted. “I’ve justbeentryingtounderstandthetheoryinaprincipledway.”

“You have!” Alice replied. “The conversation we’ve just been having—whether or not Iconvincedyouofanythingatall,thisiswhatallthoughtfulphysicistsshouldbetalkingabout.What matters to me is not that everyone become an Everettian, but that people take thechallengeofunderstandingquantummechanicsseriously.I’dmuchratherhaveadialoguewithsomeonewhoisadedicatedproponentofhiddenvariables,forexample,thantrytoengagetheinterestofsomeonewhojustdoesn’tcare.”

Her father nodded. “It’s takenme awhile, I admit. But yes, I do care.” He smiled at hisdaughter.“Ourmissionistounderstandthings,isn’tit?”

9OtherWays

AlternativestoMany-Worlds

David Albert, now a philosophy professor at Columbia and one of the world’s leadingresearchers in the foundations of quantum mechanics, had a very typical experience as agraduatestudentwhobecameinterestedinquantumfoundations.HewasinthePhDprograminthephysicsdepartmentatRockefellerUniversitywhen,afterreadingabookbyeighteenth-centuryphilosopherDavidHumeontherelationshipofknowledgeandexperience,hecametobelieve that what physics lacked was a good understanding of the quantum measurementproblem.(Humedidn’tknowaboutthemeasurementproblem,butAlbertconnecteddotsinhishead.)NobodyatRockefellerinthelate1970swasinterestedinthinkingalongthoselines,soAlbertstruckupalong-distancecollaborationwiththefamousIsraeliphysicistYakirAharonov,resultinginseveralinfluentialpapers.ButwhenhesuggestedsubmittingthatworkforhisPhDthesis,thepowersthatbeatRockefellerwereaghast.Underpenaltyofbeingkickedoutoftheprogramentirely,Albertwasforcedtowriteaseparatethesisinmathematicalphysics.Itwas,as he recalled, “clearly being assigned because it was thought it would be good for mycharacter.Therewasanexplicitlypunitiveelementthere.”

Physicists have been very bad at coming to consensus about what the foundations ofquantummechanicsactuallyare.Butinthesecondhalfofthetwentiethcentury,theydidcometoaremarkabledegreeofconsensusonarelatedissue:whateverthefoundationsofquantummechanics are,we certainly shouldn’t talk about them.Notwhile therewas realwork to bedone,doingcalculationsandconstructingnewmodelsofparticlesandfields.

Everett,ofcourse, leftacademiawithouteven trying tobecomeaphysicsprofessor.DavidBohm, who had studied and worked under Robert Oppenheimer in the 1940s, proposed aningenious way of using hidden variables to address the measurement problem. But after aseminarinwhichanotherphysicistexplainedBohm’sideas,Oppenheimerscoffedoutloud,“IfwecannotdisproveBohm, thenwemustagree to ignorehim.” JohnBell,whodidmore thananyonetoilluminatetheapparentlynonlocalnatureofquantumentanglement,purposefullyhidhis work on this subject from his colleagues at CERN, to whom he appeared as a relativelyconventionalparticletheorist.HansDieterZeh,whopioneeredtheconceptofdecoherenceasayoungresearcher inthe1970s,waswarnedbyhismentorthatworkingonthissubjectwoulddestroyhisacademiccareer.Indeed,hefounditverydifficulttopublishhisearlypapers,beingtoldbyjournalrefereesthat“thepaperiscompletelysenseless”and“quantumtheorydoesnotapply to macroscopic objects.” Dutch physicist Samuel Goudsmit, serving as the editor ofPhysicalReview,putoutamemoin1973explicitlybanningthejournalfromevenconsideringpapers on quantum foundations unless they made new experimental predictions. (Had thatpolicybeen inplaceearlier, the journalwouldhavehad to reject theEinstein-Podolsky-Rosenpaper,aswellasBohr’sreply.)

Yet,astheseverystoriesmakeclear,despiteavarietyofhurdlesputupintheirway,asubsetof physicists andphilosophers nevertheless persevered in the effort to better understand thenatureofquantumreality.TheMany-Worldstheory,especiallyoncetheprocessofwave-functionbranchinghasbeen illuminatedbydecoherence, is onepromisingapproach toanswering thepuzzlesraisedbythemeasurementproblem.Butthereareothersworthconsidering.Theyareworthwhilebothbecausetheymightactuallyberight(whichisalwaysthebestreason)andalsobecause comparing the very differentways inwhich theywork helps us to better appreciatequantummechanics,nomatterwhatourpersonalfavoriteapproachhappenstobe.

An impressive number of alternative formulations of quantum theory have been proposedover the years. (The relevantWikipedia article lists sixteen “interpretations” explicitly, alongwith a category for “other.”) Here we’ll consider three basic competitors to the Everettapproach: dynamical collapse, hidden variables, and epistemic theories. While far fromcomprehensive,theseservetoillustratethebasicstrategiesthatpeoplehavetaken.

ThevirtueofMany-Worldsisinthesimplicityofitsbasicformulation:thereisawavefunctionthat evolves according to the Schrödinger equation. All else is commentary. Some of thatcommentary,suchasthesplitintosystemsandtheirenvironment,decoherence,andbranchingof the wave function, is extremely useful, and indeed indispensable to matching the crispeleganceoftheunderlyingformalismtoourmessyexperienceoftheworld.

WhateveryourfeelingsmightbeaboutMany-Worlds,itssimplicityprovidesagoodstartingpoint for considering alternatives. If you remain profoundly skeptical that there are goodanswerstotheproblemofprobability,oraresimplyrepulsedbytheideaofallthoseworldsoutthere,thetaskyoufaceistomodifyMany-Worldsinsomeway.Given thatMany-Worldsisjust“wave functions and the Schrödinger equation,” a few plausible ways forward immediatelysuggest themselves:altering theSchrödingerequationso thatmultipleworldsneverdevelop,addingnewvariablesinadditiontothewavefunction,orreinterpretingthewavefunctionasastatement about our knowledge rather than a direct description of reality. All of these roadshavebeenenthusiasticallywalkeddown.

We turn first to the possibility of altering the Schrödinger equation. This approachwouldseemtobesquarelyinthecomfortzoneofmostphysicists;almostbeforeanysuccessfultheoryhasbeenestablished,theoristsaskhowtheycouldplayaroundwiththeunderlyingequationstomake it even better. Schrödinger himself originally hoped that his equation would describewavesthatnaturallylocalizedintoblobsthatbehavedlikeparticleswhenviewedfromfaraway.Perhaps some modification of his equation could achieve that ambition, and even provide anaturalresolutiontothemeasurementproblemwithoutpermittingmultipleworlds.

Thisisharderthanitsounds.Ifwetrythemostobviousthing,addingnewtermslikeΨ2tothe equation, we tend to ruin important features of the theory, such as the total set ofprobabilitiesaddinguptoone.Thiskindofobstaclerarelydetersphysicists.StevenWeinberg,whodevelopedthesuccessfulmodelthatunifiedtheelectromagneticandweakinteractionsinthe Standard Model of particle physics, proposed a clever modification of the Schrödingerequationthatmanagestomaintainthetotalprobabilityovertime.Itcomesatacost,however;thesimplestversionofWeinberg’stheoryallowsyoutosendsignalsfasterthanlightbetweenentangled particles, as opposed to the no-signaling theorem of ordinary quantummechanics.Thisflawcanbepatched,butthensomethingevenweirderoccurs:notonlyaretherestillotherbranchesofthewavefunction,butyoucanactuallysendsignalsbetweenthem,buildingwhatphysicistJoePolchinskidubbedan“Everettphone.”Maybethat’sagoodthing, ifyouwanttobaseyourlifechoicesontheoutcomeofaquantummeasurementandthencheckinwithyouralternateselves toseewhichone turnedout thebest.But itdoesn’tseemtobe thewaythatnatureactuallyworks.And itdoesn’t succeed insolving themeasurementproblemorgettingridofotherworlds.

Inretrospectthismakessense.Consideranelectroninapurespin-upstate.Thatcanequallywellbeexpressedasanequalsuperpositionofspin-leftandspin-right,sothatanobservationalong a horizontalmagnetic field has a 50 percent chance of observing either outcome. Butprecisely because of that equality between the two options, it’s hard to imagine how adeterministicequationcouldpredictthatwewouldseeeitheroneortheother(atleastwithouttheadditionofnewvariablescarryingadditionalinformation).Somethingwouldhavetobreakthebalancebetweenspin-leftandspin-right.

We therefore have to think a bit more dramatically. Rather than taking the Schrödingerequationandgentlytinkeringwithit,wecanbitethebulletandintroduceacompletelyseparateway for wave functions to evolve, one that squelches the appearance of multiple branches.Plentyof experimental evidenceassuresus thatwave functionsusually obey theSchrödingerequation, at least when we’re not observing them. But maybe, rarely but crucially, they dosomethingverydifferent.

Whatmightthatdifferentthingbe?Weseektoavoidtheexistentialhorrorofmultiplecopiesofthemacroscopicworldbeingdescribedinasinglewavefunction.Sowhatifweimaginedthatwave functions undergo occasional spontaneous collapse, converting suddenly from beingspreadoutoverdifferentpossibilities(say,positionsinspace)tobeingrelativelywelllocalizedaround just one point? This is the key new feature of dynamical-collapse models, the mostfamousofwhichisGRWtheory,afteritsinventorsGiancarloGhirardi,AlbertoRimini,andTullioWeber.

Envision an electron in free space, not bound to any atomic nucleus. According to theSchrödingerequation,thenaturalevolutionofsuchaparticleisforitswavefunctiontospreadoutandbecomeincreasinglydiffuse.Tothispicture,GRWaddsapostulatethatsaysateverymoment there is some probability that the wave function will change radically andinstantaneously. The peak of the new wave function is itself chosen from a probabilitydistribution,thesameonethatwewouldhaveusedtopredictthepositionwewouldmeasurefor the electron according to its original wave function. The new wave function is stronglyconcentratedaroundthiscentralpoint,sothattheparticleisnowessentiallyinonelocationasfaraswemacroscopicobserversareconcerned.WavefunctioncollapsesinGRWarerealandrandom,notinducedbymeasurements.

GRWtheoryisnotsomenebulous“interpretation”ofquantummechanics;itisabrand-newphysical theory, with different dynamics. In fact, the theory postulates two new constants ofnature:thewidthofthenewlylocalizedwavefunction,andtheprobabilitypersecondthatthedynamical collapse will occur. Realistic values for these parameters are perhaps 10-5centimetersforthewidth,and10-16fortheprobabilityofcollapsepersecond.Atypicalelectrontherefore evolves for 1016 seconds before its wave function spontaneously collapses. That’sabout 300 million years. So in the 14-billion-year lifetime of the observable universe, mostelectrons(orotherparticles)localizeonlyahandfuloftimes.

That’s a feature of the theory, not a bug. If you’re going to go messing around with theSchrödinger equation, youhadbetter do it in such away as to not ruin all of thewonderfulsuccessesof conventionalquantummechanics.Wedoquantumexperimentsall the timewithsingleparticlesorcollectionsofafewparticles.Itwouldbedisastrousifthewavefunctionsofthoseparticleskeptspontaneouslycollapsingonus. If there isa truly randomelement in theevolutionofquantumsystems,itshouldbeincrediblyrareforindividualparticles.

Then how does such a mild alteration of the theory manage to get rid of macroscopicsuperpositions?Entanglementcomestotherescue,muchasitdidwithdecoherenceinMany-Worlds.

Considermeasuringthespinofanelectron.AswepassitthroughaStern-Gerlachmagnet,the wave function of the electron evolves into a superposition of “deflected upward” and“deflecteddownward.”Wemeasurewhichwayitwent,forexample,bydetectingthedeflectedelectronona screen,which ishookedup to adialwithapointer indicatingUporDown.AnEverettian says that the pointer is a bigmacroscopic object that quickly becomes entangledwiththeenvironment, leadingtodecoherenceandbranchingofthewavefunction.GRWcan’tappealtosuchaprocess,butsomethingrelatedhappens.

It’snotthattheoriginalelectronspontaneouslycollapses;wewouldhavetowaitformillionsofyearsforthattobecomealikelyevent.Butthepointerintheapparatuscontainssomethinglike1024 electrons, protons, and neutrons. All of these particles are entangled in an obviousway: theyare indifferentpositionsdependingonwhether thepointer indicatesUporDown.Even though it’s quite unlikely that any specific particle will undergo spontaneous collapsebeforeweopenthebox,chancesareextremelygoodthatatleastoneofthemwill—thatshouldhappenroughly108timespersecond.

Youmightnotbeimpressed,thinkingthatwewouldn’tevennoticeatinysubsetofparticlesbecominglocalizedinamacroscopicpointer.Butthemagicofentanglementmeansthatifthewavefunctionofjustoneparticleisspontaneouslylocalized,therestoftheparticleswithwhichthatoneisentangledwillcomealongwithit.Ifsomehowthepointerdidmanagetoavoidanyofitsparticles localizing foracertainperiodof time,enough for it toevolve intoamacroscopicsuperpositionofUpandDown,thatsuperpositionwouldinstantlycollapseassoonasjustoneofthe particles did localize. The overall wave function goes very rapidly from describing anapparatuspointinginasuperpositionoftwoanswerstoonethatisdefinitivelyoneortheother.GRW theory manages to make operational and objective the classical/quantum split thatpartisansoftheCopenhagenapproachareforcedtoinvoke.Classicalbehaviorisseeninobjectsthatcontainsomanyparticlesthatitbecomeslikelythattheoverallwavefunctionwillundergoaseriesofrapidcollapses.

GRWtheoryhasobviousadvantagesanddisadvantages.Theprimaryadvantageisthatit’sawell-posed,specifictheorythataddressesthemeasurementprobleminastraightforwardway.Themultipleworldsof theEverettapproachareeliminatedbyaseriesof trulyunpredictablecollapses. We are left with a world that maintains the successes of quantum theory in themicroscopicrealm,whileexhibitingclassicalbehaviormacroscopically. It isaperfectly realistaccount that doesn’t invoke any fuzzy notions about consciousness in its explanation ofexperimental outcomes. GRW can be thought of as Everettian quantum mechanics plus arandomprocessthatcutsoffnewbranchesofthewavefunctionastheyappear.

Moreover,itisexperimentallytestable.Thetwoparametersgoverningthewidthoflocalizedwavefunctionsandtheprobabilityofcollapsewerenotchosenarbitrarily;iftheirvalueswereverydifferent, theyeitherwouldn’tdothe job(collapseswouldbetoorare,ornotsufficientlylocalized)ortheywouldalreadyhavebeenruledoutbyexperiment.Imaginewehaveafluidofatomsinanincrediblylow-temperaturestate,sothateveryatomismovingveryslowlyifatall.Aspontaneouscollapseofthewavefunctionofanyelectroninthefluidwouldgiveitsatomalittlejoltofenergy,whichphysicistscoulddetectasaslightincreaseinthetemperatureofthefluid.Experimentsofthisformareongoing,withtheultimategoalofeitherconfirmingGRW,orrulingitoutentirely.

Theseexperimentsareeasiersaidthandone,astheamountofenergywe’retalkingaboutisverysmall indeed.Still,GRWisagreatexampletobringupwhenyour friendscomplainthatMany-Worlds, or different approaches to quantum mechanics more generally, aren’texperimentally testable. You test theories in comparison to other theories, and these two aremanifestlydifferentintheirempiricalpredictions.

AmongGRW’s disadvantages are the fact that, well, the new spontaneous-collapse rule isutterlyadhocandoutofstepwitheverythingelseweknowaboutphysics.Itseemssuspiciousthatnaturewouldnotonlychoosetoviolateitsusuallawofmotionatrandomintervalsbutdosoinjustsuchawaythatwewouldn’tyethavebeenabletoexperimentallydetectit.

Another disadvantage, one that has prevented GRW and related theories from gainingtraction among theoretical physicists, is that it’s unclear how to construct a version of thetheorythatworksnotonlyforparticlesbutalsoforfields.Inmodernphysics,thefundamentalbuildingblocksofnaturearefields,notparticles.Weseeparticleswhenwelookcloselyenoughat vibrating fields, simply because those fields obey the rules of quantummechanics. Undersomeconditions,it’spossibletothinkofthefielddescriptionasusefulbutnotmandatory,andimaginethatfieldsarejustwaysofkeepingtrackofmanyparticlesatonce.Butthereareothercircumstances(suchas intheearlyuniverse,or insideprotonsandneutrons)wherethefield-ness is indispensable. And GRW, at least in the simple version presented here, gives usinstructions for how wave functions collapse that refers specifically to the probability perparticle. This isn’t necessarily an insurmountable obstacle—taking simple models that don’tquiteworkandgeneralizingthemuntiltheydoisthetheoreticalphysicist’sstock-in-trade—butit’sasignthattheseapproachesdon’tseemtofitnaturallywithhowwecurrentlythinkaboutthelawsofnature.

GRWdelineatesthequantum/classicalboundarybymakingspontaneouscollapsesveryrareforindividualparticles,butveryrapidforlargecollections.Analternativeapproachwouldbetomake collapse occur whenever the system reached a certain threshold, like a rubber bandbreakingwhenitisstretchedtoofar.Awell-knownexampleofanattemptalongtheselineswasput forward by mathematical physicist Roger Penrose, best known for his work in generalrelativity. Penrose’s theory uses gravity in a crucial way. He suggests that wave functionsspontaneously collapse when they begin to describe macroscopic superpositions in whichdifferent components have appreciably different gravitational fields. The criterion of“appreciablydifferent”hereturnsouttobedifficulttospecifyprecisely;singleelectronswouldnot collapse no matter how spread-out their wave functions were, while a pointer is largeenoughtocausecollapseassoonasitstartedevolvingintodifferentstates.

Mostexperts inquantummechanicshavenotwarmedtoPenrose’s theory, inpartbecausetheyareskepticalthatgravityshouldhaveanythingtodowiththefundamentalformulationofquantummechanics. Surely, they think, we can talk—and did, for most of the history of thesubject—aboutquantummechanics andwave-function collapsewithout consideringgravity atall.

It’spossible thatapreciseversionofPenrose’s criterioncouldbedeveloped inwhich it isthoughtofasdecoherenceindisguise:thegravitationalfieldofanobjectcanbethoughtofaspart of its environment, and if two different components of thewave function have differentgravitationalfields,theybecomeeffectivelydecohered.Gravityisanextremelyweakforce,andit will almost always be the case that ordinary electromagnetic interactions will causedecoherence longbeforegravitywould.But thenice thingaboutgravity is that it’suniversal(everything has a gravitational field, not everything is electrically charged), so at least thiswouldbeawaytoguaranteethatthewavefunctionwouldcollapseforanymacroscopicobject.On the other hand, branching when decoherence occurs is already part of theMany-Worldsapproach;all that thiskindofspontaneous-collapsetheorywouldsay is“It’s just likeEverett,exceptthatwhennewworldsarecreated,weerasethembyhand.”Whoknows?Thatmightbehownatureactuallyworks,butit’snotaroutethatmostworkingphysicistsareencouragedtopursue.

Sincetheverybeginningofquantummechanics,anobviouspossibilitytocontemplatehasbeenthe idea that the wave function isn’t the whole story, but that there are also other physicalvariablesinadditiontoit.Afterall,physicistswereveryusedtothinkingintermsofprobabilitydistributionsfromtheirexperiencewithstatisticalmechanics,asithadbeendevelopedinthenineteenthcentury.Wedon’tspecify theexactpositionandvelocityofeveryatom inaboxofgas,onlytheiroverallstatisticalproperties.But intheclassicalviewwetakeforgrantedthatthere is some exact position and velocity for each particle, even if we don’t know it.Maybequantum mechanics is like that—there are definite quantities associated with prospectiveobservational outcomes, but we don’t know what they are, and the wave function somehowcapturespartofthestatisticalrealitywithouttellingthewholestory.

Weknowthewave functioncan’tbeexactly likeaclassicalprobabilitydistribution.A trueprobability distribution assigns probabilities directly to outcomes, and the probability of anygiven event has to be a real number between zero and one (inclusive). A wave function,meanwhile, assigns an amplitude to every possible outcome, and amplitudes are complexnumbers. They have both a real and an imaginary part, either one of which could be either

positiveornegative.Whenwesquaresuchamplitudesweobtainaprobabilitydistribution,butif we want to explain what is experimentally observed, we can’t work directly with thatdistribution rather than keeping the wave function around. The fact that amplitudes can benegativeallowsfortheinterferencethatweseeinthedouble-slitexperiment,forexample.

There’s a simple way of addressing this problem: think of the wave function as a real,physicallyexistingthing(notjustaconvenientsummaryofourincompleteknowledge),butalsoimagine that there are additional variables, perhaps representing the positions of particles.Theseextraquantitiesareconventionallycalledhiddenvariables,althoughsomeproponentsofthisapproachdon’tlikethelabel,asit’sthesevariablesthatweactuallyobservewhenwemakeameasurement.Wecanjustcallthemparticles,sincethat’sthecasethatisusuallyconsidered.Thewave function then takeson the role of apilotwave, guiding theparticlesas theymovearound.It’slikeparticlesarelittlefloatingbarrels,andthewavefunctiondescribeswavesandcurrents in the water that push the barrels around. The wave function obeys the ordinarySchrödingerequation,whileanew“guidanceequation”governshowitinfluencestheparticles.Theparticlesareguidedtowherethewavefunctionislarge,andawayfromwhereitisnearlyzero.

ThefirstsuchtheorywaspresentedbyLouisdeBroglie,atthe1927SolvayConference.BothEinsteinandSchrödingerwerethinkingalongsimilar linesatthetime.ButdeBroglie’s ideaswere harshly criticized at Solvay, by Wolfgang Pauli in particular. From the records of theconference, itseemsas ifPauli’scriticismsweremisplaced,anddeBroglieactuallyansweredthemcorrectly.ButhewassufficientlydiscouragedbythereceptionthatdeBroglieabandonedtheidea.

In a famous book from1932,Mathematical Foundations ofQuantumMechanics, John vonNeumannproveda theoremabout thedifficulty of constructinghidden-variable theories.VonNeumannwasoneofthemostbrilliantmathematiciansandphysicistsofthetwentiethcentury,andhisnamecarriedenormouscredibilityamongresearchersinquantummechanics.Itbecamestandardpractice,wheneveranyonewouldsuggestthattheremightbeamoredefinitewaytoformulate quantum theory than the vagueness inherent in the Copenhagen approach, forsomeonetoinvokethenameofvonNeumannandtheexistenceofhisproof.Thatwouldsquelchanybuddingdiscussion.

InfactwhatvonNeumannhadprovenwassomethingabit lessthanmostpeopleassumed(often without reading his book, which wasn’t translated into English until 1955). A goodmathematicaltheoremestablishesaresultthatfollowsfromclearlystatedassumptions.Whenwewouldliketoinvokesuchatheoremtoteachussomethingabouttherealworld,however,wehavetobeverycarefulthattheassumptionsareactuallytrueinreality.VonNeumannmadeassumptions that, in retrospect,we don’t have tomake if our task is to invent a theory thatreproduces thepredictionsofquantummechanics.Heprovedsomething,butwhatheprovedwas not “hidden-variable theories can’t work.” This was pointed out by mathematician andphilosopherGreteHermann,butherworkwaslargelyignored.

Along cameDavid Bohm, an interesting and complicated figure in the history of quantummechanics. As a graduate student in the early 1940s, Bohm became interested in left-wingpolitics.HeendedupworkingontheManhattanProject,buthewas forcedtodohiswork inBerkeley,ashewasdeniedthenecessarysecurityclearancetomovetoLosAlamos.Afterthewar he became an assistant professor at Princeton, and published an influential textbook onquantummechanics. InthatbookheadheredcarefullytothereceivedCopenhagenapproach,butthinkingthroughtheissuesmadehimstartwonderingaboutalternatives.

Bohm’s interest in thesequestionswasencouragedbyoneof the few figureswhohad thestature to stand up to Bohr and his colleagues: Einstein himself. The great man had readBohm’sbook,andsummonedtheyoungprofessortohisofficetotalkaboutthefoundationsofquantum theory.Einstein explainedhis basic objections, that quantummechanics couldn’t beconsidered a complete viewof reality, and encouragedBohm to thinkmore deeply about thequestionofhiddenvariables,whichheproceededtodo.

All this took place while Bohm was under a cloud of political suspicion, at a time whenassociationwithCommunismcould ruinpeople’s careers. In1949,Bohmhad testifiedbeforetheHouseUn-AmericanActivitiesCommittee,whereherefusedtoimplicateanyofhisformercolleagues.In1950hewasarrestedinhisofficeatPrincetonforcontemptofCongress.Thoughhe was eventually cleared of all charges, the president of the university forbade him fromsettingfootoncampus,andputpressureonthephysicsdepartmenttonotrenewhiscontract.In1951,withsupportfromEinsteinandOppenheimer,BohmwaseventuallyabletofindajobattheUniversity of São Paulo, and left for Brazil. That’s why the first seminar at Princeton toexplainBohm’sideashadtobegivenbysomeoneelse.

None of this drama prevented Bohm from thinking productively about quantum mechanics.

EncouragedbyEinstein,hedevelopedatheorythatwassimilartothatofdeBroglie,inwhichparticleswereguidedbya“quantumpotential”constructedfromthewavefunction.TodaythisapproachisoftenknownasthedeBroglie–Bohmtheory,orsimplyBohmianmechanics.Bohm’spresentationofthetheorywasabitmorefleshedoutthandeBroglie’s,especiallywhenitcametodescribingthemeasurementprocess.

Even today you will sometimes hear professional physicists say that it’s impossible toconstruct a hidden-variable theory that reproduces the predictions of quantum mechanics,“because of Bell’s theorem.” But that’s exactly what Bohm did, at least for the case of non-relativistic particles. John Bell, in fact, was one of the few physicists who was extremelyimpressedbyBohm’swork,andhewasinspiredtodevelophistheorempreciselytounderstandhow to reconcile the existenceofBohmianmechanicswith thepurportedno-hidden-variablestheoremofvonNeumann.

WhatBell’stheoremactuallyprovesistheimpossibilityofreproducingquantummechanicsviaalocalhidden-variablestheory.SuchatheoryiswhatEinsteinhadlongbeenhopingfor:amodel that would attach independent reality to physical quantities associated with specificlocations in space, with effects between them propagating at or below the speed of light.Bohmianmechanicsisperfectlydeterministic,butitisresolutelynonlocal.Separatedparticlescanaffecteachotherinstantaneously.

Bohmianmechanicspositsbothasetofparticleswithdefinite(butunknowntous,untiltheyare observed) positions, and a separate wave function. The wave function evolves exactlyaccordingtotheSchrödingerequation—itdoesn’tevenseemtorecognizethattheparticlesarethere,andisunaffectedbywhattheyaredoing.Theparticles,meanwhile,arepushedaroundaccordingtoaguidanceequationthatdependsonthewavefunction.However,thewayinwhichanyoneparticleisguideddependsnotjustonthewavefunctionbutalsoonthepositionsofalltheotherparticles thatmaybe in thesystem.That’s thenonlocality; themotionofaparticlehere candepend, inprinciple, on thepositionsof otherparticles arbitrarily far away.AsBellhimselflaterputit,inBohmianmechanics“theEinstein-Podolsky-RosenparadoxisresolvedinthewaywhichEinsteinwouldhavelikedleast.”

This nonlocality plays a crucial role inunderstandinghowBohmianmechanics reproducesthe predictions of ordinary quantum mechanics. Consider the double-slit experiment, whichillustrates so vividly how quantum phenomena are simultaneously wave-like (we seeinterferencepatterns) andparticle-like (we seedots on thedetector screen, and interferencegoes away when we detect which slit the particles go through). In Bohmian mechanics thisambiguityisnotmysteriousatall:therearebothparticlesandwaves.Theparticlesarewhatweobserve;thewavefunctionaffectstheirmotion,butwehavenowayofmeasuringitdirectly.

According to Bohm, the wave function evolves through both slits just as it would inEverettianquantummechanics.Inparticular,therewillbeinterferenceeffectswherethewavefunctionaddsorcancelsonceitreachesthescreen.Butwedon’tseethewavefunctionatthescreen; we see individual particles hitting it. The particles are pushed around by the wavefunction,sothattheyaremorelikelytohitthescreenwherethewavefunctionislarge,andlesslikelytodosowhereitissmall.

TheBornruletellsusthattheprobabilityofobservingaparticleatagivenlocationisgivenbythewavefunctionsquared.Onthesurface,thisseemshardtoreconcilewiththeideathatparticle positions are completely independent variables that we can specify as we like. AndBohmianmechanics isperfectlydeterministic—therearen’tany truly randomevents,as therearewiththespontaneouscollapsesofGRWtheory.SowheredoestheBornrulecomefrom?

Theansweristhat,whileinprincipleparticlepositionscouldbeanywhereatall,inpracticethereisanaturaldistributionforthemtohave.Imaginethatwehaveawavefunctionandsomefixednumberofparticles.TorecovertheBornrule,allwehavetodoisstartwithaBornrule–likedistributionofthoseparticles.Thatis,wehavetodistributethepositionsofourparticlessothat the distribution looks like it was chosen randomly with probability given by the wavefunctionsquared.Moreparticleswheretheamplitudeislarge,fewerparticleswhereitissmall.

Suchan“equilibrium”distributionhasthenice featurethat theBornruleremainsvalidastimepassesandthesystemevolves. Ifwestartourparticles inaprobabilitydistribution thatmatches what we expect from ordinary quantum mechanics, it will continue to match thatexpectation going forward. It is believed by many Bohmians that a non-equilibrium initialdistributionwillevolvetowardequilibrium,justasagasofclassicalparticlesinaboxevolvestowardanequilibriumthermalstate;butthestatusofthisideaisnotyetsettled.Theresultingprobabilities are, of course, about our knowledge of the system rather than about objectivefrequencies;ifsomehowweknewexactlywhattheparticlepositionswere,ratherthanjusttheirdistribution,wecouldpredictexperimentaloutcomesexactlywithoutanyneedforprobabilitiesatall.

This puts Bohmian mechanics in an interesting position as an alternative formulation ofquantummechanics. GRW theorymatches traditional quantum expectations usually, but alsomakes definite predictions for new phenomena that can be tested. Like GRW, Bohmianmechanics is unambiguously a different physical theory, not simply an “interpretation.” It

doesn’t have to obey the Born rule if for some reason our particle positions are not in anequilibrium distribution. But it will obey the rule if they are. And if that’s the case, thepredictionsofBohmianmechanicsarestrictlyindistinguishablefromthoseofordinaryquantumtheory.Inparticular,wewillseemoreparticleshitthescreenwherethewavefunctionislarge,andfewerwhereitissmall.

Westillhavethequestionofwhathappenswhenwelooktoseewhichslit theparticlehasgone through. Wave functions don’t collapse in Bohmian mechanics; as with Everett, theyalwaysobeytheSchrödingerequation.Sohowarewesupposedtoexplainthedisappearanceoftheinterferencepatterninthedouble-slitexperiment?

The answer is “the same way we do in Many-Worlds.” While the wave function doesn’tcollapse, itdoesevolve. Inparticular,weshouldconsider thewave function for thedetectionapparatusaswellasfortheelectronsgoingthroughtheslits;theBohmianworldiscompletelyquantum,notstoopingtoanartificialsplitbetweenclassicalandquantumrealms.Asweknowfrom thinking about decoherence, the wave function for the detector will become entangledwith that of an electron passing through the slit, and a kind of “branching” will occur. Thedifference is that the variablesdescribing theapparatus (whicharen’t there inMany-Worlds)willbeat locationscorresponding tooneof thesebranches,andnot theother.Forall intentsand purposes, it’s just like the wave function has collapsed; or, if you prefer, it’s just likedecoherence has branched thewave function, but instead of assigning reality to each of thebranches,theparticlesofwhichwearemadeareonlylocatedononeparticularbranch.

Youwon’tbesurprisedtohearthatmanyEverettiansaredubiousaboutthiskindofstory.Ifthe wave function of the universe simply obeys the Schrödinger equation, it will undergodecoherence and branching. And you’ve already admitted that the wave function is part ofreality. The particle positions, for thatmatter, have absolutely no influence on how thewavefunctionevolves.Alltheydo,arguably,ispointtoaparticularbranchofthewavefunctionandsay, “This is the realone.”SomeEverettianshave thereforeclaimed thatBohmianmechanicsisn’t really any different from Everett, it just includes some superfluous extra variables thatservenopurposebuttoassuagesomeanxietiesaboutsplittingintomultiplecopiesofourselves.AsDeutschhasput it, “Pilot-wave theoriesareparallel-universe theories inastateofchronicdenial.”

Wewon’tadjudicate thisdispute righthere.What’s clear is thatBohmianmechanics isanexplicit construction that does what many physicists thought was impossible: to construct aprecise, deterministic theory that reproduces all of the predictions of textbook quantummechanics,withoutrequiringanymysteriousincantationsaboutthemeasurementprocessoradistinctionbetweenquantumandclassicalrealms.Thepricewepayisexplicitnonlocalityinthedynamics.

Bohmwashopefulthathisnewtheorywouldbewidelyappreciatedbyphysicists.Thiswasnottobe. Intheemotionallycharged languagethatsooftenaccompaniesdiscussionsofquantumfoundations,HeisenbergcalledBohm’stheory“asuperfluousideologicalsuperstructure,”whilePauli referred to it as “artificial metaphysics.” We’ve already heard the judgment ofOppenheimer,whohadpreviouslybeenBohm’smentorandsupporter.EinsteinseemstohaveappreciatedBohm’seffort, but thought the final constructionwasartificial andunconvincing.Unlike de Broglie, however, Bohm didn’t bow to the pressure, and continued to develop andadvocateforhistheory.Indeed,hisadvocacyinspireddeBrogliehimself,whowasstillaroundandactive(hediedin1987).InhislateryearsdeBrogliereturnedtohidden-variabletheories,developingandelaboratinghisoriginalmodel.

EvenapartfromthepresenceofexplicitnonlocalityandtheaccusationthatthetheoryisjustMany-Worlds in denial, there are other significant problems inherent in Bohmianmechanics,especiallyfromtheperspectiveofamodernfundamentalphysicist.Thelistofingredientsinthetheory is undoubtedly more complicated than in Everett, and Hilbert space, the set of allpossible wave functions, is as big as ever. The possibility of many worlds is not avoided byerasing the worlds (as in GRW), but simply by denying that they’re real. The way Bohmiandynamicsworksisfarfromelegant.Longafterclassicalmechanicswassuperseded,physicistsstill intuitivelyclingtosomething likeNewton’sthird law: ifonethingpushesonanother,thesecond thingpushesback. It therefore seemsstrange thatwehaveparticles thatarepushedaroundbyawavefunction,whilethewavefunctioniscompletelyunaffectedbytheparticles.Ofcourse, quantum mechanics inevitably forces us to confront strange things, so perhaps thisconsiderationshouldnotbeparamount.

More important, theoriginal formulationsofdeBroglieandBohmbothrelyheavilyon theideathatwhatreallyexistsare“particles.”JustaswithGRW,thiscreatesaproblemwhenwetrytounderstandthebestmodelsoftheworldthatweactuallyhave,whicharequantumfieldtheories.Peoplehaveproposedwaysof“Bohmizing”quantumfieldtheory,andtherehavebeen

somesuccesses—physicistscanbeextremelycleverwhentheywanttobe.Buttheresultsfeelforcedratherthannatural.Itdoesn’tmeantheyarenecessarilywrong,butit’sastrikeagainstBohmiantheorieswhencomparedtoMany-Worlds,whereincludingfieldsorquantumgravityisstraightforward.

InourdiscussionofBohmianmechanicswereferredtothepositionsoftheparticles,butnottotheirmomenta.ThishearkensbacktothedaysofNewton,whothoughtofparticlesashavinga position at every moment in time, and velocity (and momentum) as derived from thattrajectory,bycalculating itsrateofchange.Moremodernformulationsofclassicalmechanics(well, since1833) treatpositionandmomentumonanequal footing.Oncewego toquantummechanics, this perspective is reflected in the Heisenberg uncertainty principle, in whichpositionandmomentumappearinexactlythesameway.Bohmianmechanicsundoesthismove,treatingpositionasprimary,andmomentumassomethingthatderivedfromit.Butitturnsoutthatyoucan’tmeasureitexactly,duetounavoidableeffectsofthewavefunctionontheparticlepositionsovertime.Soattheendoftheday,theuncertaintyprincipleremainstrueinBohmianmechanicsasapracticalfactoflife,butitdoesn’thavetheautomaticnaturalnessoftheoriesinwhichthewavefunctionistheonlyrealentity.

Thereisamoregeneralprincipleatworkhere.ThesimplicityofMany-Worldsalsomakesitextremelyflexible.TheSchrödingerequationtakesthewavefunctionandfiguresouthowfastitwill evolve by applying theHamiltonian, whichmeasures the different amounts of energy indifferent components of the quantum state. You give me a Hamiltonian, and I can instantlyunderstandtheEverettianversionofitscorrespondingquantumtheory.Particles,spins,fields,superstrings,doesn’tmatter.Many-Worldsisplug-and-play.

Otherapproachesrequireagooddealmoreworkthanthat,andit’sfarfromclearthatthework is evendoable.Youhave to specifynot only aHamiltonianbut also aparticularway inwhich wave functions spontaneously collapse, or a particular new set of hidden variables tokeeptrackof.That’seasiersaidthandone.Theproblembecomesevenmorepronouncedwhenwemovefromquantumfieldtheorytoquantumgravity(which,remember,wasoneofEverett’sinitial motivations). In quantum gravity the very notion of “a location in space” becomesproblematic, as different branches of the wave function will have different spacetimegeometries.ForMany-Worldsthat’snoproblem;foralternativesit’sclosetoadisaster.

WhenBohmandEverettwere inventing their alternatives toCopenhagen in the1950s, orBellwasprovinghis theorems in the1960s,workon foundations of quantummechanicswasshunned within the physics community. That began to change somewhat with the advent ofdecoherencetheoryandquantuminformationinthe1970sand’80s;GRWtheorywasproposedin1985.Whilethissubfieldisstilllookeduponwithsuspicionbyalargemajorityofphysicists(for one thing, it tends to attract philosophers), an enormous amount of interesting andimportant work has been accomplished since the 1990s, much of it wide out in the open.However,it’salsosafetosaythatmuchcontemporaryworkonquantumfoundationsstilltakesplaceinacontextofqubitsornon-relativisticparticles.Oncewegraduatetoquantumfieldsandquantumgravity,somethingswecouldpreviouslytakeforgrantedarenolongeravailable.Justasitistimeforphysicsasafieldtotakequantumfoundationsseriously,it’stimeforquantumfoundationstotakefieldtheoryandgravityseriously.

In contemplating ways to eliminate themany worlds implied by a bare-bones version of theunderlyingquantum formalism,wehaveexploredchoppingoff theworldsbya randomevent(GRW)orreachingsomekindofthreshold(Penrose)orpickingoutparticularworldsasrealbyaddingadditionalvariables(deBroglie–Bohm).What’sleft?

Theproblemisthattheappearanceofmultiplebranchesofthewavefunctionisautomaticoncewebelieve inwave functionsandtheSchrödingerequation.Sothealternativeswehaveconsidered thus far either eliminate those branches or posit something that picks out one ofthemasspecial.

Athirdwaysuggestsitself:denytherealityofthewavefunctionentirely.By this we don’t mean to deny the central importance of wave functions in quantum

mechanics.Rather,wecanusewavefunctions,butwemightnotclaimthattheyrepresentpartof reality. They might simply characterize our knowledge; in particular, the incompleteknowledgewehaveabouttheoutcomeoffuturequantummeasurements.Thisisknownasthe“epistemic” approach to quantum mechanics, as it thinks of wave functions as capturingsomething about what we know, as opposed to “ontological” approaches that treat thewavefunctionasdescribingobjectivereality.SincewavefunctionsareusuallydenotedbytheGreekletter Ψ (Psi), advocates of epistemic approaches to quantum mechanics sometimes teaseEverettiansandotherwave-function-realistsbycallingthem“Psi-ontologists.”

We’ve already noted that an epistemic strategy cannot work in the most naïve andstraightforward way. The wave function is not a probability distribution; real probability

distributions are never negative, so they can’t lead to interference phenomena such as weobserve in thedouble-slitexperiment.Rather thangivingup,however,wecan try tobeabitmoresophisticatedinhowwethinkabouttherelationshipbetweenthewavefunctionandtherealworld.We can imagine building up a formalism that allows us to usewave functions tocalculate the probabilities associated with experimental outcomes, while not attaching anyunderlyingrealitytothem.Thisisthetasktakenupbyepistemicapproaches.

Therehavebeenmanyattempts to interpret thewave functionepistemically, just as thereare competing collapse models or hidden-variable theories. One of the most prominent isQuantum Bayesianism, developed by Christopher Fuchs, Rüdiger Schack, Carlton Caves, N.DavidMermin,andothers.ThesedaysthelabelistypicallyshortenedtoQBismandpronounced“cubism.”(Onemustadmitit’sacharmingname.)

Bayesianinferencesuggeststhatweallcarryaroundwithusasetofcredencesforvariouspropositionstobetrueorfalse,andupdatethosecredenceswhennewinformationcomesin.Allversionsofquantummechanics(andindeedallscientifictheories)useBayes’stheoreminsomeversion or another, and inmany approaches to understanding quantumprobability it plays acrucial role.QBism is distinguished bymaking our quantum credencespersonal, rather thanuniversal.According toQBism, thewave functionof an electron isn’t a once-and-for-all thingthat everyone could, in principle, agree on. Rather, everyone has their own idea of what theelectron’s wave function is, and uses that idea to make predictions about observationaloutcomes. If we do many experiments and talk to one another about what we’ve observed,QBistsclaim,wewillcometoadegreeofconsensusaboutwhatthevariouswavefunctionsare.Buttheyarefundamentallymeasuresofourpersonalbelief,notobjectivefeaturesoftheworld.WhenweseeanelectrondeflectedupwardinaStern-Gerlachmagneticfield,theworlddoesn’tchange,butwe’velearnedsomethingnewaboutit.

Thereisoneimmediateandundeniableadvantageofsuchaphilosophy:ifthewavefunctionisn’t a physical thing, there’s no need to fret about it “collapsing,” even if that collapse ispurportedlynonlocal.IfAliceandBobpossesstwoparticlesthatareentangledwitheachotherand Alicemakes ameasurement, according to the ordinary rules of quantummechanics thestate of Bob’s particle changes instantaneously. QBism reassures us that we needn’t worryabout that, as there is no such thing as “the state ofBob’s particle.”What changedwas thewave function thatAlicecarriesaroundwithher tomakepredictions: itwasupdatedusingasuitablyquantumversionofBayes’stheorem.Bob’swavefunctiondidn’tchangeatall.QBismarrangestherulesofthegamesothatwhenBobdoesgetaroundtomeasuringhisparticle,theoutcomewill agreewith the predictionwewouldmake on the basis of Alice’smeasurementoutcome.ButthereisnoneedalongthewaytoimaginethatanyphysicalquantitychangedoveratBob’slocation.Allthatchangesaredifferentpeople’sstatesofknowledge,whichafterallarelocalizedintheirheads,notspreadthroughallspace.

ThinkingaboutquantummechanicsinQBisttermshasledtointerestingdevelopmentsinthemathematicsofprobability,andoffersinsightintoquantuminformationtheory.Mostphysicists,however,will stillwant to know:What is reality supposed to be in this view? (AbrahamPaisrecalled that Einstein once asked himwhether he “really believed that themoon exists onlywhenIlookatit.”)

Theanswerisnotclear.ImaginethatwesendanelectronthroughaStern-Gerlachmagnet,but we choose not to look at whether it’s deflected up or down. For an Everettian, it isneverthelessthecasethatdecoherenceandbranchinghasoccurred,andthereisafactofthematter aboutwhich branch any particular copy of ourselves is on. TheQBist says somethingverydifferent:thereisnosuchthingaswhetherthespinwasdeflectedupordown.Allwehaveisourdegreesofbeliefaboutwhatwewillseewhenweeventuallydecidetolook.Thereisnospoon,asNeolearnedinTheMatrix.Frettingaboutthe“reality”ofwhat’sgoingonbeforewelook,inthisview,isamistakethatleadstoallsortsofconfusion.

QBists,forthemostpart,don’ttalkaboutwhattheworldreallyis.Oratleast,asanongoingresearchprogram,QBistshavechosennottodwelltoomuchonthequestionsconcerningthenatureofrealityaboutwhichtherestofuscaresomuch.Thefundamental ingredientsofthetheoryareasetofagents,whohavebeliefs,andaccumulateexperiences.Quantummechanics,in thisview, isaway foragents toorganize theirbeliefsandupdatethemin the lightofnewexperiences. The idea of an agent is absolutely central; this is in stark contrast to the otherformulationsofquantumtheorythatwe’vebeendiscussing,accordingtowhichobserversarejustphysicalsystemslikeanythingelse.

SometimesQBistswilltalkaboutrealityassomethingthatcomesintoexistenceaswemakeobservations.Merminhaswritten,“Thereisindeedacommonexternalworldinadditiontothemanydistinctindividualpersonalexternalworlds.Butthatcommonworldmustbeunderstoodatthefoundationalleveltobeamutualconstructionthatallofushaveputtogetherfromourdistinctprivateexperiences,usingourmostpowerfulhumaninvention:language.”Theideaisnot that there is no reality, but that reality is more than can be captured by any seeminglyobjectivethird-personperspective.FuchshasdubbedthisviewParticipatoryRealism:realityistheemergingtotalityofwhatdifferentobserversexperience.

QBism is relatively young as approaches to quantum foundations go, and there is muchdevelopmentyet tobedone. It’spossible that itwill run into insurmountableroadblocks,andinterest in the ideas will fizzle out. It’s also possible that the insights of QBism can beinterpreted as a sometimes-useful way of talking about the experiences of observers withinsome other, straightforwardly realist, version of quantummechanics. And finally, itmight bethatQBismorsomethingclosetoitrepresentsatrue,revolutionarywayofthinkingabouttheworld,onethatputsagentslikeyouandmeatthecenterofourbestdescriptionofreality.

Personally,as someonewho isquitecomfortablewithMany-Worlds (while recognizing thatwestillhaveopenquestions),thisallseemstomelikeanincredibleamountofeffortdevotedtosolvingproblemsthataren’treallythere.QBists,tobefair,feelasimilarlevelofexasperationwith Everett: Mermin has said that “QBism regards [branching into many simultaneouslyexistingworlds]as the reductioadabsurdum of reifying thequantumstate.”That’s quantummechanics for you, where one person’s absurdity is another person’s answer to all of life’squestions.

Thefoundations-of-physicscommunity,whichisfullofsmartpeoplewhohavethoughtlongandhard about these issues, has not reached a consensus on the best approach to quantummechanics. One reason is that people come to the problem from different backgrounds, andthereforewithdifferentconcernsforemostintheirminds.Researchersinfundamentalphysics—particle theory, general relativity, cosmology, quantum gravity—tend to favor the Everettapproach,iftheydeigntotakeapositiononquantumfoundationsatall.That’sbecauseMany-Worldsisextremelyrobusttotheunderlyingphysicalstuffitisdescribing.Yougivemeasetofparticlesandfieldsandwhathaveyou,andrulesforhowtheyinteract,andit’sstraightforwardtofitthoseelementsintoanEverettianpicture.Otherapproachestendtobemorepersnickety,demandingthatwestartfromscratchtofigureoutwhatthetheoryactuallysaysineachnewinstance.Ifyou’resomeonewhoadmitsthatwedon’treallyknowwhattheunderlyingtheoryofparticlesandfieldsandspacetimereallyis,thatsoundsexhausting,whereasMany-Worldsisanaturaleasyrestingplace.AsDavidWallacehasputit,“TheEverettinterpretation(insofarasitisphilosophicallyacceptable)istheonlyinterpretativestrategycurrentlysuitedtomakesenseofquantumphysicsaswefindit.”

Butthereisanotherreason,morebasedinpersonalstyle.Essentiallyeveryoneagreesthatsimple, elegant ideas are to be sought after as we search for scientific explanations. Beingsimple and elegant doesn’tmean an idea is correct—that’s for the data to decide—butwhenthere are multiple ideas vying for supremacy and we don’t yet have enough data to chooseamongthem,it’snaturaltogiveabitmorecredencetothesimplestandmostelegantones.

Thequestionis,whodecideswhat’ssimpleandelegant?Therearedifferentsensesoftheseterms.Everettianquantummechanicsisabsolutelysimpleandelegantfromacertainpointofview.Asmoothlyevolvingwavefunction,that’sall.Buttheresultoftheseelegantpostulates—aproliferatingtreeofmultipleuniverses—isarguablynotverysimpleatall.

Bohmianmechanics,ontheotherhand,isconstructedinakindofhaphazardway.Therearebothparticlesandwavefunctions,andtheyinteractthroughanonlocalguidanceequationthatseemsfarfromelegant.Includingbothparticlesandwavefunctionsasfundamentalingredientsis, however, a natural strategy to contemplate, oncewehave been confrontedwith the basicexperimentaldemandsofquantummechanics.Matteractssometimeslikewavesandsometimeslikeparticles,soweinvokebothwavesandparticles.GRWtheory,meanwhile,addsaweirdadhoc stochasticmodification to the Schrödinger equation. But it’s arguably the simplest,mostbrute-forcewaytophysicallyimplementthefactthatwavefunctionsappeartocollapse.

There isausefulcontrast tobedrawnbetween thesimplicityofaphysical theoryand thesimplicity with which that theory maps onto reality as we observe it. In terms of basicingredients,Many-Worldsisunquestionablyassimpleasitgets.Butthedistancebetweenwhatthetheoryitselfsays(wavefunctions,Schrödingerequation)andwhatweobserveintheworld(particles,fields,spacetime,people,chairs,stars,planets)seemsenormous.Otherapproachesmightbemorebaroqueintheirunderlyingprinciples,butit’srelativelyclearhowtheyaccountforwhatwesee.

Bothunderlyingsimplicityandclosenesstothephenomenaarevirtues intheirownrights,but it’s hard to know how to balance them against each other. This is where personal stylecomes in. All of the approaches to quantum mechanics that we’ve considered face loomingchallengesaswecontemplatedevelopingthemintorock-solidfoundationsforanunderstandingof the physicalworld. So each of us has tomake a personal judgment aboutwhich of theseproblemswilleventuallybesolved,andwhichwillprovefatalforthevariousapproaches.That’sokay;indeed,it’scrucialthatdifferentpeoplecomedowndifferentlyonthesejudgmentsabouthowtomoveforward.Thatgivesusthebestchancetokeepmultipleideasalive,maximizingtheprobabilitythatwe’lleventuallygetthingsright.

Many-Worldsoffersaperspectiveonquantummechanicsthatisnotonlysimpleandelegantatitscorebutseemsready-madeforadaptingtotheongoingquesttounderstandquantumfieldtheoryand thenatureof spacetime.That’s enough to convinceme that I should learn to livewiththeannoyanceofothercopiesofmebeingproducedallthetime.Butifitturnsoutthatanalternative approach answers our deepest questions more effectively, I’ll happily change mymind.

10TheHumanSide

LivingandThinkinginaQuantumUniverse

Inthecourseofalonglife,eachofuswilloccasionallyencounteradifficultdecisionwemustmake.Staysingleorgetmarried?Goforarunorhaveanotherdoughnut?Gotogradschoolorentertherealworld?Wouldn’t it be nice to be able to choose both sides, rather than picking one? Quantum

mechanics suggests a strategy: whenever you have a decision to make, you can do so byconsultingaquantumrandom-numbergenerator.Indeed,thereisanappavailableforiPhonescalledUniverseSplitterthatcanbeusedforthisverypurpose.(AsDaveBarrysays,IswearIamnotmakingthisup.)Let’ssayyouhaveachoicetomake:“ShouldIgetpepperoniorsausageonmypizza?”(And

let’ssayyouhavetoomuchrestrainttogivetheobviousanswerofaskingforbothonthesamepizza.)YoucanfireupUniverseSplitter,whereyouwillseetwotextboxes,intowhichyoucantype“pepperoni”and“sausage.”Thenhitthebutton,andyourphonewillsendasignalthroughthe internet to a laboratory in Switzerland, where a photon is sent toward a beam splitter(essentially a partially silvered mirror that reflects some photons and lets others through).AccordingtotheSchrödingerequation,thebeamsplitterturnsthephoton’swavefunctionintotwo components going left and right, each ofwhichheads toward a different detector.Wheneither detector notices a photon, it produces a readout that becomes entangled with theenvironment,quicklyleadingtodecoherenceandbranchingthewavefunctionintwo.Thecopyof you in the branch where the photon went left sees their phone flash with the message“pepperoni,”andintheonewhereitwentright,theysee“sausage.”Ifeachoneactuallyfollowsupwithyourplantodowhatyourphoneadvises,therewillbeoneworldinwhichaversionofyou orders pepperoni, and another inwhich a version of you orders sausage. Sadly, the twopersonshavenowayofcommunicatingwitheachothertosharetastingnotesafterward.Even for the most battle-hardened quantum physicist, one must admit that this sounds

ludicrous. But it’s the most straightforward reading of our best understanding of quantummechanics.The question naturally arises:What should we do about it? If the real world is truly this

radicallydifferent fromtheworldofoureverydayexperience,doesthishaveany implicationsforhowweliveourlives?Largely—no.Toeachindividualonsomebranchofthewavefunction,lifegoesonjustasif

they lived in a single world with truly stochastic quantum events. But the issues are worthexploring.

You are welcome to offload your hard decisions to a quantum random-number generator,thereby ensuring that there is at least one branch of the wave function in which the bestalternativewas chosen. But let’s saywe choose not to. Should the branching of our currentselves intomultiplefutureselvesaffectthechoiceswemake?Inthetextbookview,there isaprobabilitythatoneoranotheroutcomehappenswhenweobserveaquantumsystem,whileinMany-Worlds all outcomes happen,weighted by the amplitude squared of thewave function.Doestheexistenceofallthoseextraworldshaveimplicationsforhowweshouldact,personallyorethically?It’snothardtoimaginethat itmight,butuponcarefulconsiderationitturnsouttomatter

much less than youmight guess. Consider the infamous quantum suicide experiment, or therelated idea of quantum immortality. It’s an idea that has been considered ever sinceMany-Worlds came on the scene—reportedly Hugh Everett himself believed a version of quantumimmortality—buthasbeenpopularizedbyphysicistMaxTegmark.Here’sthesetup:weimagineadeadlydevicethatistriggeredbyaquantummeasurement,

suchassendingaquerytotheUniverseSplitterapp.Imaginethatthequantummeasurement

hasa50percentchanceoftriggeringagunthatshootsabullet intomyheadatcloserange,anda50percentchanceofdoingnothing.AccordingtoMany-Worlds,thatimpliestheexistenceoftwobranchesofthewavefunction,oneofwhichcontainsalivingversionofme,theotherofwhichcontainsadeadversion.Assumeforpurposesofthethoughtexperimentwebelievethatlifeitselfisapurelyphysical

phenomenon,sowecansetasideconsiderationsof lifeafterdeath.Frommyperspective, thebranch onwhich the gun fired isn’t one that any version ofme ever gets to experience—mydescendant in thatworld isdead.Butmydescendantcontinueson,unharmed,on thebranchwherethegundidn’tfire.Insomesense,then,“I”willliveforever,evenifIrepeatthismacabreprocedure over and over again. One might go so far as to argue that I shouldn’t object toactuallygoingthroughthisexperiment(puttingasidetherestoftheworld’sfeelingsaboutme,Isuppose)—inthebrancheswherethegunfired“I”don’treallyexist,whileinthesinglebranchwhere it failedto firetimeaftertimeI’mperfectlyhealthy. (Tegmark’soriginalpointwas lessgrandiose:hesimplynotedthatanexperimenterwhosurviveda largenumberoftrialswouldhavegoodreasontoaccept theEverettpicture.)Thisconclusionstands instarkcontrast toaconventionalstochasticformulationofquantummechanics,wherethereisonlyoneworld,andIwouldhaveanincreasinglytinychanceofbeingalivewithinit.Idonotrecommendthatyoutrysuchanexperimentathome.Infact,thelogicbehindnot

caringaboutthosebranchesinwhichyouarekilledismorethanalittlewonky.Considerlifeinanold-fashioned,classical,single-universepicture.Ifyouthoughtyoulivedin

suchauniverse,wouldyoumindifsomeonesneakedupbehindyouandshotyouintheheadsothatyoudiedinstantly?(Again,settingasidethepossibilitythatotherpeoplemightbeupset.)Mostofuswouldnotbeinfavorofthathappening.Butbythelogicabove,youreallyshouldn’t“mind”—afterall,onceyou’redead,there’sno“you”tobeupsetaboutwhathappened.The point beingmissed by this analysis is thatwe are upsetnow—whilewe are still very

much alive and feeling—by theprospect of beingdead in the future, especially if that futurecomessoonerratherthanlater.Andthat’savalidperspective;muchofhowwethinkaboutourcurrentlivesdependsonaprojectionintotherestofourexistence.Cuttingthatexistenceoffissomethingweareperfectlyallowedtoobjectto,evenifwewon’tbearoundtobebotheredbyitonceithappens.Andgiventhat,quantumsuicideturnsouttobejustasbleakandunpalatableasourimmediateintuitionmightsuggest.It’sokayformetoyearnforahappyandlonglifeforallthefutureversionsofmethatwillendupinvariousbranchesofthewavefunction,asmuchasitwouldbevalidformetohopeforalonglifeifIthoughttherewasjustasingleworld.This goes back to something we discussed in Chapter Seven: the importance of treating

individuals on different branches of the wave function as distinct persons, even if theydescendedfromthesameindividualinthepast.Thereisanimportantasymmetrybetweenhowwe think about “our future” versus “our past” in Many-Worlds, which ultimately can beattributedtothelow-entropyconditionofourearlyuniverse.Anyoneindividualcantracetheirlives backward in a unique person, but going forward in time we will branch into multiplepeople.Thereisnotonefutureselfthatispickedoutas“reallyyou,”andit’sequallytruethatthereisnoonepersonconstitutedbyallofthosefutureindividuals.Theyareseparate,asmuchasidenticaltwinsaredistinctpeople,despitedescendingfromasinglezygote.Wemightcareaboutwhathappenstotheversionsofourselveswholiveonotherbranches,

but it’s not sensible to think of them as “us.” Imagine that you’re just about to perform avertical-spinmeasurementonanelectronyouhavepreparedinanequalsuperpositionofspin-upandspin-down.Arandomphilanthropistentersyourlabandoffersyouthefollowingbargain:ifthespinisup,theywillgiveyouamilliondollars;ifthespinisdown,yougivethemonedollar.Youwouldbewisetotakethedeal;forallintentsandpurposes,it’sasifyouarebeingofferedabetwithequalchancesofwinningamilliondollarsorlosingjustonedollar,evenifoneofyourfutureselveswillcertainlybeoutadollar.Butnowimaginethatyouwerealittlequickerinyourexperimentalsetup,andyouobserved

aspin-downoutcomejustbeforethephilanthropistbustsin.Itturnsoutthattheyareapushydeal-maker,andtheyexplainthattheversionofyouontheotherbranchisbeinggivenamilliondollars,butyounowhavetogivethemonedollarinthisbranch.There’snoreasonforyoutobehappyaboutthis(ortogiveupthedollar),eventhoughthe

versionofyouontheotherbranchmightbehappyaboutit.Youarenotthem,andtheyarenotpart of you. Post-branching, you’re two different people. Neither your experiences nor yourrewardsshouldbethoughtofasbeingsharedbyvariouscopiesofyouondifferentbranches.Don’t play quantum Russian roulette, and don’t accept losing bargains from pushyphilanthropists.

Thatmaybeareasonablepolicywhenitcomestoyourownwell-being,butwhataboutthatofothers?Howdoesknowingabout theexistenceofotherworldsaffectournotionsofmoralor

ethicalbehavior?Therightway to thinkaboutmorality is itselfacontroversial subject,even insingle-world

versions of reality, but it’s instructive to consider two broad categories of moral theory:deontologyandconsequentialism.Deontologistsholdthatmoralbehaviorisamatterofobeyingtherightrules;actionsareinherentlyrightorwrong,whatevertheirconsequencesmightturnout to be. Consequentialists, unsurprisingly, have the alternative view: we should work tomaximize the beneficent consequences of our actions. Utilitarians, who advocatemaximizingsome measure of overall well-being, are paradigmatic consequentialists. There are otheroptions,buttheseillustratethebasicpoint.Deontology would seem to be unaffected by the possible presence of other worlds. If the

wholepointof your theory is thatactionsare intrinsically rightorwrong, regardlessofwhatoutcomestheyleadto,theexistenceofmoreworldsinwhichthoseoutcomescanoccurdoesn’treallymatter.AtypicaldeontologicalruleisKant’scategoricalimperative:“Actonlyaccordingtothatmaximwherebyyoucan,atthesametime,willthatitshouldbecomeauniversallaw.”Itseemslikeitwouldbesafeheretoreplace“auniversallaw”by“alawholdinginallbranchesofthewavefunction,”withoutalteringanysubstantivejudgmentaboutwhatkindofactionsmightqualify.Consequentialismisanothermatterentirely.Imaginethatyouareano-nonsenseutilitarian,

whobelievesthereisaquantitycalledutilitythatmeasurestheamountofwell-beingassociatedwithconsciouscreatures,andthatthisquantitycanbeaddedamongallcreaturestoobtainatotal utility, and that the morally right course of action is the one that maximizes this totalutility.Imaginefurtherthatyoujudgethetotalutilityintheentireuniversetobesomepositivenumber.(Ifyoudidn’t,you’dbeinfavoroftryingsomehowtodestroytheuniverse,whichmakesforagoodsupervillainoriginstorybutnotforgoodneighbors.)Itwould follow that, if theuniversehaspositiveutility andourgoal is tomaximizeutility,

creatinganewcopyofthewholeuniversewouldbeoneof themostmorallyvalorousactionsyoucouldpossiblytake.Therightthingtodowouldthenbetobranchthewavefunctionoftheuniverseasoftenaspossible.Wecould imaginebuildingaquantumutilitymaximizingdevice(QUMaD), perhaps an apparatus that continually bounces electrons through a device thatmeasuresfirsttheirverticalspin,thentheirhorizontalspin.Everytimeanelectronundergoeseithermeasurement, the universe branches in two, doubling the total utility of all universes.HavingbuiltQUMaDandturnediton,youwouldbethemostmoralpersonevertolive!Somethingaboutthissmellsfishy,however.TurningonQUMaDhasnoimpactwhatsoeveron

thelivesofpeopleinthisuniverseoranyother.Theydon’tevenknowthemachineexists.Arewereallysureithassuchamorallypraiseworthyeffect?Happilythereareacoupleofwaysoutofthispuzzle.Oneistodenytheassumptions:maybe

thiskindofno-nonsenseutilitarianismisn’tthebestmoraltheory.Thereisalongandhonorabletraditionofpeopleinventingthingsthatwouldnominallyincreasetheutilityoftheuniverse,butdon’tresembleourmoralintuitionswhatsoever.(RobertNozickimagineda“utilitymonster,”ahypotheticalbeingthatwassogoodatexperiencingpleasurethatthemostmoralthinganyonecoulddowouldbetokeepthemonsterashappyaspossible,nomatterwhoelsemightsufferthereby.) QUMaD is just another example along these lines. The simple idea of adding uputilities among different people doesn’t always lead to the results we might initially haveimagined.But there’s another solution, one that comports more directly with the Many-Worlds

philosophy. When we talked about deriving the Born rule, we discussed how to apportioncredencesinconditionsofself-locatinguncertainty:youknowthewavefunctionoftheuniverse,butyoudon’tknowwhichbranchyouareon.Theanswerwas thatyourcredencesshouldbeproportionaltotheweightofthebranch—thecorrespondingamplitude,squared.This“weight”isacrucially importantaspectofhowwethinkaboutworlds inanEverettianpicture. It’snotjustprobability thatgoes thatway; conservationofenergyalsoonlyworks ifwemultiply theenergyofeachbranchbyitsassociatedweight.Itmakes sense, then, thatwe should do the samewith utility. Ifwe have a universewith

somegiven totalutility,andwemeasureaspin tobranch it in two, thepost-branchingutilityshouldbethesumoftheweightsofeachbranchtimestheirutilities.Then,inthelikelyeventthatourspinmeasurementdidn’taffectanyone’sutilityinasubstantialway,thetotalutilityiscompletely unchanged by ourmeasurement. That’s just what our intuitionmight expect. It’salsowhatwewould directly conclude from the decision-theoretic approach to probabilitywementioned in Chapter Six. From this perspective, Many-Worlds shouldn’t change our ideasaboutmoralactioninanynoticeableway.It’sneverthelesspossibletocookupasysteminwhichthedifferencebetweenMany-Worlds

andcollapsetheoriesreallywouldbemorallyrelevant.ImaginethatsomequantumexperimentwillleadtoequallylikelyoutcomesAorB,withAbeingextremelygoodandBbeingjustalittlebitgood,andthattheseeffectsapplytoeveryoneintheworldwithequalmeasure.Inasingle-worldview,autilitarian(oranycommonsensicalperson,really)wouldbeinfavorofrunningtheexperiment,sinceeitherthevastgoodofAortheminorgoodofBwouldraisethenetutilityof

theworld. But imagine that your ethical code is entirely devoted to equality: you don’t carewhathappens,aslongasithappenstoeveryoneequally.Onthecollapsetheory,youdon’tknowwhichoutcomewillhappen,buteitheronemaintainsequality,soit’sstillagoodideatoruntheexperiment. But inMany-Worlds, people in one branchwill experience A while those on theotherbranchwillexperienceB.Evenifthebranchescan’tcommunicateorotherwiseinteract,thiscouldconceivablyoffendyourmoralsensibilities,soyou’dbeagainstdoingtheexperimentat all. Personally I don’t think that inequality between people who literally live in differentworldsshouldmatterthatmuchtous,butthelogicalpossibilityisthere.Excluding such artificial constructions, Many-Worlds doesn’t seem to have many moral

implications.Thepictureofbranchingas“creating”anentirelynewcopyof theuniverse isavivid one, but not quite right. It’s better to think of it as dividing the existing universe intoalmost-identicalslices,eachoneofwhichhasasmallerweight than theoriginal. Ifwe followthatpicturecarefully,weconclude that it’s correct to thinkaboutour futureexactlyas ifwelived in a single stochastic universe that obeyed the Born rule. As counterintuitive asMany-Worldsmightseem,attheendofthedayitdoesn’treallychangehowweshouldgothroughourlives.

Sofarwe’vetreatedbranchingofthewavefunctionassomethingthathappensindependentlyofourselves,sothatwesimplyhavetogoalongfortheride.It’sworthaskingwhetherthat’stheproperperspective.WheneverImakeadecision,aredifferentworldscreatedwhereIchosedifferent things? Are there realities out there corresponding to every series of alternativechoicesIcouldhavemade,universesthatactualizeallthepossibilitiesofmylife?Theideaof“makingadecision”isn’tsomethinginscribedinthefundamentallawsofphysics.

It’soneofthoseuseful,approximate,emergentnotionsthatwefindconvenienttoinvokewhendescribing human-scale phenomena. What you and I label “making a decision” is a set ofneurochemical processes happening in our brain. It’s perfectly okay to talk about makingdecisions, but it’s not something over and above ordinarymaterial stuff obeying the laws ofphysics.So the question is, do the physical processes going on in your brain when you make a

decisioncausethewavefunctionoftheuniversetobranch,withdifferentdecisionsbeingmadeineachbranch?IfI’mplayingpokerandloseallmychipsaftermakinganill-timedbluff,canItakesolaceintheideathatthereisanotherbranchwhereIplayedmoreconservatively?No,youdonotcausethewavefunctiontobranchbymakingadecision.Inlargepartthat’s

justduetowhatwemean(oroughttomean)bysomething“causing”somethingelse.Branchingistheresultofamicroscopicprocessamplifiedtomacroscopicscales:asysteminaquantumsuperpositionbecomesentangledwithalargersystem,whichthenbecomesentangledwiththeenvironment, leading todecoherence.Adecision, on the otherhand, is a purelymacroscopicphenomenon.Therearenodecisionsbeingmadebytheelectronsandatomsinsideyourbrain;they’rejustobeyingthelawsofphysics.Decisionsandchoicesandtheirconsequencesareusefulconceptswhenwearetalkingabout

things at the macroscopic, human-size level. It’s perfectly okay to think of choices as reallyexisting and having influences, as long as we confine such talk to the regime in which theyapply.Wecanchoose, inotherwords, to talkaboutapersonasabunchofparticlesobeyingSchrödinger’sequation,orwecanequallywell talkaboutthemasanagentwithvolitionwhomakes decisions that affect the world. But we can’t use both descriptions at once. Yourdecisionsdon’tcausethewavefunctiontobranch,because“thewavefunctionbranching”isarelevantconceptattheleveloffundamentalphysics,and“yourdecisions”isarelevantconceptattheeverydaymacroscopiclevelofpeople.Sothereisnosenseinwhichyourdecisionscausebranching.Butwecanstillaskwhether

thereareotherbrancheswhereyoumadedifferentdecisions.Andindeedtheremightbe,buttherightwaytothinkaboutthecausalityis“somemicroscopicprocesshappenedthatcausedbranching, and on different branches you ended upmaking different decisions,” rather than“youmadeadecision,whichcausedthewavefunctionoftheuniversetobranch.”Forthemostpart,however,whenyoudomakeadecision—evenonethatseemslikeaclosecallatthetime—almostalloftheweightwillbeconcentratedonasinglebranch,notspreadequallyovermanyalternatives.Theneuronsinourbrainsarecellsconsistingofacentralbodyandanumberofappendages.

Mostofthoseappendagesaredendrites,whichtakeinsignalsfromsurroundingneurons,butone of them is the axon, a longer fiber down which outgoing signals are sent. Chargedmolecules(ions)buildupintheneuronuntiltheyreachapointwhereanelectrochemicalpulseis triggered, travelingdown the axon and across synapses to thedendrites of other neurons.Combinemanysuchevents,andwehavethemakingsofa“thought.”(We’reglossingoversomecomplicationshere;hopefullyneuroscientistswillforgiveme.)

For themostpart, theseprocessescanbe thoughtof asbeingpurely classical, orat leastdeterministic.Quantummechanicsplaysaroleatsomelevelinanychemicalreaction,sinceit’squantum mechanics that sets the rules for how electrons want to jump from one atom toanother or bind two atoms together. Butwhen you get enough atoms together in one place,their net behavior can be described without any reference to quantum concepts likeentanglement or the Born rule—otherwise you wouldn’t have been able to take a chemistryclass in high school without first learning the Schrödinger equation and worrying about themeasurementproblem.So“decisions”arebestthoughtofasclassicalevents,notquantumones.Whileyoumightbe

personallyunsurewhatchoiceyouwilleventuallymake,theoutcomeisencodedinyourbrain.We’renotabsolutelysureabouttheextenttowhichthisistrue,sincethere’sstillalotwedon’tknowaboutthephysicalprocessesbehindthinking.It’spossiblethattheratesofneurologicallyimportant chemical reactions can vary slightly depending on the entanglement between thedifferentatomsinvolved.Ifthatturnsouttobetrue,therewouldbeasenseinwhichyourbrainisaquantumcomputer,albeitalimitedone.Atthesametime,anhonestEverettianadmitsthattherewillalwaysbebranchesofthewave

function on which quantum systems appear to have done very unlikely things. As AlicementionedinChapterEight,therewillbebrancheswhereIrunintoawallandhappentotunnelthroughit,ratherthanbouncingoff.Likewise,eveniftheclassicalapproximationtomybrainimpliesthatI’mgoingtobetallmychipsatthepokertable,thereissometinyamplitudeforabunch of neurons to do unlikely things and cause me to make a snug fold. But it’s not mydecision that’s causing the branching; it’s the branching that I interpret as leading to mydecision.Underthemoststraightforwardunderstandingofthechemistrygoingoninourbrains,most

ofour thinkinghasnothing todowithentanglementandbranchingof thewave function.Weshouldn’t imagine thatmaking a difficult decision splits theworld intomultiple copies, eachcontaining a version of you that chose differently. Unless, of course, you don’t want to takeresponsibility,andturnyourdecision-makingovertoaquantumrandom-numbergenerator.

Similarly, quantummechanicshasnothing todowith thequestionof freewill. It’s natural tothinkthatitmight,asfreewillisoftencontrastedwithdeterminism,theideathatthefutureiscompletelydeterminedbythepresentstateoftheuniverse.Afterall,ifthefutureisdetermined,whatroomisthereformetomakechoices?Inthetextbookpresentationofquantummechanics,measurementoutcomesaretrulyrandom,sophysicsisnotdeterministic.Maybethatopensthedooracrackforfreewill tosneakbackin,after itwasbanishedbytheNewtonianclockworkparadigmofclassicalmechanics?There’s so much wrong with this that it’s hard to know where to start. First, “free will”

versus “determinism” isn’t the right distinction to draw. Determinism should be opposed to“indeterminism,” and free will should be opposed to “no free will.” Determinism isstraightforward to define: given the exact current state of the system, the laws of physicsdetermine precisely the state at later times. Freewill is trickier. One usually hears freewilldefinedassomethinglike“theabilitytohavechosenotherwise.”Thatmeanswe’recomparingwhatreallyhappened(wewereinasituation,wemadeadecision,andweactedaccordingly)toadifferenthypotheticalscenario(wewindtheclockbackwardtotheoriginalsituation,andaskwhetherwe “could have” decideddifferently).Whenplaying this game, it’s crucial to specifyexactly what is kept fixed between the real and hypothetical situations. Is it absolutelyeverything, down to the last microscopic detail? Or do we just imagine fixing our availablemacroscopicinformation,allowingforvariationwithininvisiblemicroscopicdetails?Let’s say we’re hard-core about this question, and compare what actually happened to a

hypotheticalre-runningoftheuniversestartingfromexactlythesameinitialcondition,downtothe precise state of every last elementary particle. In a classical deterministic universe theoutcomewouldbepreciselythesame,sothere’snopossibilityyoucouldhave“madeadifferentdecision.”Bycontrast,accordingtotextbookquantummechanics,anelementofrandomnessisintroduced, so we can’t confidently predict exactly the same future outcome from the sameinitialconditions.Butthathasnothingtodowith freewill.Adifferentoutcomedoesn’tmeanwemanifested

somekindofpersonal,supra-physicalvolitionalinfluenceoverthelawsofnature.Itjustmeansthatsomeunpredictablequantumrandomnumberscameupdifferently.Whatmatters for thetraditional “strong”notion of freewill is notwhetherweare subject to deterministic lawsofnature, but whether we are subject to impersonal laws of any sort. The fact that we can’tpredictthefutureisn’tthesameastheideathatwearefreetobringitabout.Evenintextbookquantummechanics,humanbeingsarestillcollectionsofparticlesandfieldsobeyingthelawsofphysics.

1.2.

For that matter, quantum mechanics is not necessarily indeterministic. Many-Worlds is acounterexample.Youevolve,perfectlydeterministically,fromasinglepersonnowintomultiplepersonsatafuturetime.Nochoicescomeintothematteranywhere.Ontheotherhand,wecanalsocontemplateaweakernotionoffreewill,onethatrefersto

themacroscopicallyavailableknowledgeweactuallyhaveabouttheworld,ratherthanrunningthoughtexperimentsbasedonmicroscopicallyperfectknowledge.Inthatcase,adifferentformofunpredictability arises.Givenapersonandwhatwe (or they, or anyone) knowabout theircurrentmentalstate,therewilltypicallybemanydifferentspecificarrangementsofatomsandmolecules in theirbodiesandbrains thatarecompatiblewith thatknowledge.Someof thosearrangementsmay leadtosufficientlydifferentneuralprocessesthatwewouldendupactingverydifferently,ifthosearrangementshadbeentrue.Inthatcase,thebestwecanrealisticallydo to describe theway human beings (or other conscious agents) act in the realworld is toattributevolitiontothem—theabilitytochoosedifferently.Attributing volition to people is what every one of us actually does aswe go through life

talkingaboutourselvesandothers.Forpracticalpurposesitdoesn’tmatterwhetherwecouldpredict the future from perfect knowledge of the present, because we don’t have suchknowledge,norwillweever.Thishasledphilosophers,goingbackasfarasThomasHobbes,toproposecompatibilismbetweenunderlyingdeterministiclawsandtherealityofhumanchoice-making.Mostmodernphilosophersarecompatibilistsabout freewill (whichdoesn’tmeanit’sright,ofcourse).Freewill is real, just like tablesand temperatureandbranchesof thewavefunction.Asfarasquantummechanicsisconcerned,itdoesn’tmatterwhetheryouareacompatibilist

or an incompatibilist concerning freewill. In neither case should quantumuncertainty affectyour stance;even if youcan’tpredict theoutcomeofaquantummeasurement, thatoutcomestems from the laws of physics, not any personal choicesmade by you.We don’t create theworldbyouractions,ouractionsarepartoftheworld.

I would be remiss to talk about the human side of Many-Worlds without confronting thequestion of consciousness. There is a long history of claiming that human consciousness isnecessarytounderstandquantummechanics,orthatquantummechanicsmaybenecessarytounderstand consciousness. Much of this can be attributed to the impression that quantummechanicsismysterious,andconsciousnessismysterious,somaybetheyhavesomethingtodowitheachother.That’s not wrong, as far as it goes. Maybe quantum mechanics and consciousness are

somehow interconnected; it’s a hypothesis we’re welcome to contemplate. But according toeverythingwecurrentlyknow,thereisnogoodevidencethisisactuallythecase.Let’sfirstexaminewhetherquantummechanicsmighthelpusunderstandconsciousness.It’s

conceivable—thoughfarfromcertain—thattheratesofvariousneuralprocessesinyourbraindependonquantumentanglementinaninterestingway,sothattheycannotbeunderstoodbyclassicalreasoningalone.Butaccountingforconsciousness,aswetraditionallythinkabout it,isn’t a straightforward matter of the rates of neural processes. Philosophers distinguishbetweenthe“easyproblem”ofconsciousness—figuringouthowwesensethings,reacttothem,thinkaboutthem—andthe“hardproblem”—oursubjective,first-personexperienceoftheworld;whatitisliketobeus,ratherthansomeoneelse.Quantummechanicsdoesn’tseemtohaveanythingtodowiththehardproblem.Peoplehave

tried:RogerPenrose,forexample,hasteamedwithanesthesiologistStuartHamerofftodevelopa theory inwhich objective collapse of thewave functions ofmicrotubules in thebrainhelpsexplainwhyweexperienceconsciousness.Thisproposalhasnotgainedmuchacceptanceintheneurosciencecommunity.Moreimportant,it’sunclearwhyitshouldmatterforconsciousnessatall. It’s perfectly conceivable that some subtle quantum processes in the brain, involvingmicrotubulesorsomethingcompletelydifferent,affecttherateatwhichourneuronsfire.Butthisisofnohelpwhatsoeverinbridgingthegapbetween“thefiringofourneurons”and“oursubjective,self-awareexperience.”Manyscientistsandphilosophers,myself included,havenotroublebelievingthatthisgapisverybridgeable.Butatinychangeintherateofthisorthatneurochemical process doesn’t seem to be relevant to understanding how. (And if it were,there’snoreasontheeffectcouldn’tberepeatedinnonhumancomputers.)Everettian quantum mechanics has nothing specific to say about the hard problem of

consciousnessthatwouldn’tbesharedbyanyotherviewinwhichtheworldisentirelyphysical.Insuchaview,therelevantfactsaboutconsciousnessincludethese:

Consciousnessarisesfrombrains.Brainsarecoherentphysicalsystems.

That’sall. (“Coherent”heremeans“madeofmutually interactingparts”; twocollectionsofneuronsontwonon-interactingbranchesofthewavefunctionaretwodistinctbrains.)Youcanextend“brains”to“nervoussystems”or“organisms”or“information-processingsystems”ifyoulike. The point is that we aren’t making extra assumptions about consciousness or personalidentity in order to discuss Many-Worlds quantum mechanics; it is a quintessentiallymechanistic theory, with no special role for observers or experiences. Conscious observersbranch along with the rest of the wave function, of course, but so do rocks and rivers andclouds. The challenge of understanding consciousness is as difficult, nomore and no less, inMany-Worldsasitwouldhavebeenwithoutquantummechanicsatall.There are many important aspects of consciousness that scientists don’t currently

understand. That is precisely what we should expect; the human mind generally, andconsciousness in particular, are extremely complex phenomena. The fact that we don’t fullyunderstandthemshouldn’ttemptusintoproposingentirelynewlawsoffundamentalphysicstohelp ourselves out. The laws of physics are enormously better understood, and thatunderstandinghasbeenmuchbetterverifiedbyexperiment,thanthefunctioningofourbrainsandtheirrelationshiptoourminds.Wemightsomedayhavetocontemplatemodifyingthelawsofphysicstosuccessfullyaccountforconsciousness,butthatshouldbeamoveoflastresort.

We can also flip the question on its head: If quantum mechanics doesn’t help account forconsciousness,isitneverthelesspossiblethatconsciousnessplaysacentralroleinaccountingforquantummechanics?Many things are possible. But there’s a bit more to it than that. Given the prominence

afforded to the act of measurement in the rules of standard textbook quantum theory, it’snatural to wonder whether there isn’t something special about the interaction between aconsciousmindandaquantumsystem.Couldthecollapseofthewavefunctionbecausedbytheconsciousperceptionofcertainaspectsofphysicalobjects?Accordingtothetextbookview,wavefunctionscollapsewhentheyaremeasured,butwhat

preciselyconstitutes“measurement”isleftalittlevague.TheCopenhageninterpretationpositsadistinctionbetweenquantumandclassicalrealms,andtreatsmeasurementasaninteractionbetweenaclassicalobserverandaquantumsystem.Whereweshoulddrawthelineishardtospecify.IfwehaveaGeigercounterobservingemissionfromaradioactivesource,forexample,itwouldbenaturaltotreatthecounteraspartoftheclassicalworld.Butwedon’thaveto;eveninCopenhagen,wecould imaginetreatingGeigercountersasquantumsystemsthatobeytheSchrödingerequation. It’sonlywhentheoutcomeofameasurement isperceivedbyahumanbeing that (in thisway of thinking) thewave function absolutely has to collapse, because nohumanbeinghaseverreportedbeinginasuperpositionofdifferentmeasurementoutcomes.Sothelastpossibleplacewecandrawthecutisbetween“observerswhocantestifyastowhetherthey are in a superposition” and “everything else.” Since the perception of not being in asuperposition is part of our consciousness, it’s not crazy to ask whether it’s actuallyconsciousnessthatcausesthecollapse.This ideawasput forwardasearlyas1939,byFritzLondonandEdmondBauer,and later

gained favor with EugeneWigner, who won the Nobel Prize for his work on symmetries. InWigner’swords:

All that quantum mechanics purports to provide are probability connections between subsequentimpressions (also called “apperceptions”) of the consciousness, and even though the dividing linebetweentheobserver,whoseconsciousnessisbeingaffected,andtheobservedphysicalobjectcanbeshifted towards the one or the other to a considerable degree, it cannot be eliminated. Itmay bepremature to believe that the present philosophy of quantummechanics will remain a permanentfeatureof futurephysical theories; itwill remainremarkable, inwhateverwayour futureconceptsmaydevelop,thattheverystudyoftheexternalworldledtotheconclusionthatthecontentoftheconsciousnessisanultimatereality.

Wignerhimself laterchangedhismindabout the roleofconsciousness inquantumtheory,butothershavetakenupthetorch.It’snotgenerallyaviewyouwillhearspokenofapprovinglyatphysicsconferences,buttherearesomescientistsouttherewhocontinuetotakeitseriously.If consciousnessdidplaya role in thequantummeasurementprocess,whatexactlywould

that mean? The most straightforward approach would be to posit a dualist theory ofconsciousness,accordingtowhich“mind”and“matter”aretwodistinct,interactingcategories.Thegeneralideawouldbethatourphysicalbodiesaremadeofparticleswithawavefunctionthatobeys theSchrödingerequation,but thatconsciousness resides in a separate immaterialmind,whose influence causeswave functions to collapse upon being perceived. Dualism haswanedinpopularitysinceitsheydayinthetimeofRenéDescartes.Thebasicconundrumisthe

“interaction problem”: How do mind and matter interact with each other? In the presentcontext,howisanimmaterialmind,lackingextentinspaceandtime,supposedtocausewavefunctionstocollapse?There is another strategy, however, that seems at once less clunky and considerablymore

dramatic. This is idealism, in the philosophical sense of theword. It doesn’tmean “pursuingloftyideals,”butratherthatthefundamentalessenceofrealityismental,ratherthanphysical,in character. Idealism can be contrastedwith physicalismormaterialism,which suggest thatrealityisfundamentallymadeofphysicalstuff,andmindsandconsciousnessariseoutofthatascollectivephenomena.Ifphysicalismclaimsthatthere isonlythephysicalworld,anddualismclaims that there arebothphysical andmental realms, idealismclaims that there is only thementalrealm.(Thereisnotalotofsupportonthegroundfortheremaininglogicalpossibility,thatneitherthephysicalnorthementalexists.)For an idealist,mind comes first, andwhat we think of as “matter” is a reflection of our

thoughts about the world. In some versions of the story, reality emerges from the collectiveeffortofalltheindividualminds,whereasinothers,asingleconceptof“themental”underliesbothindividualmindsandtherealitytheybringtobe.Someofhistory’sgreatestphilosophicalminds, including many in various Eastern traditions but also Westerners such as ImmanuelKant,havebeensympathetictosomeversionofidealism.It’s not hard to see how quantum mechanics and idealism might seem like a good fit.

Idealism says thatmind is the ultimate foundation of reality, and quantummechanics (in itstextbookformulation)saysthatpropertieslikepositionandmomentumdon’texistuntiltheyareobserved,presumablybysomeonewithamind.Allvarietiesofidealismarechallengedbythefactthat,asidefromthecontentiousexception

ofquantummeasurement,therealworldseemstomovealongquitewellwithoutanyparticularhelpfromconsciousminds.Ourmindsdiscoverthingsabouttheworldthroughtheprocessofobservationandexperiment,anddifferentmindsendupdiscoveringaspectsoftheworldthatalwaysendupbeingwholly consistentwithoneanother.Wehaveassembledquiteadetailedandsuccessfulaccountofthefirstfewminutesofthehistoryoftheuniverse,atimewhentherewere no known minds around to think about it. Meanwhile, progress in neuroscience hasincreasinglybeenabletoidentifyparticularthoughtprocesseswithspecificbiochemicaleventstakingplaceinthematerialthatmakesupourbrains.Ifitweren’tforquantummechanicsandthemeasurementproblem,allofourexperienceofrealitywouldspeaktothewisdomofputtingmatterfirstandmindemergentfromit,ratherthantheotherwayaround.So, is theweirdness of the quantummeasurement process sufficiently intractable thatwe

should discard physicalism itself, in favor of an idealistic philosophy that takes mind as theprimary ground of reality? Does quantum mechanics necessarily imply the centrality of themental?No. We don’t need to invoke any special role for consciousness in order to address the

quantum measurement problem. We’ve seen several counterexamples. Many-Worlds is anexplicit example, accounting for the apparent collapse of thewave function using the purelymechanistic process of decoherence and branching. We’re allowed to contemplate thepossibilitythatconsciousnessissomehowinvolved,butit’sjustascertainlynotforcedonusbyanythingwecurrentlyunderstand.Ofcourse,wewilloftentalkaboutconsciousexperiencesinour attempts tomap the quantum formalism onto theworld aswe see it, but onlywhen thethingswe’retryingtoexplainarethoseexperiencesthemselves.Otherwise,mindshavenothingtodowithit.These are difficult, subtle issues, and this isn’t the place for a completely fair and

comprehensive adjudication of the debate between idealism and physicalism. Idealism isn’tsomething that’s easy to disprove; if someone is convinced it’s right, it’s hard to point toanythingthatwouldobviouslychangetheirmind(orMind).Butwhattheycan’tdoisclaimthatquantummechanicsforcesusintosuchaposition.Wehaveverystraightforwardandcompellingmodels of theworld inwhich reality exists independently of us; there’s no need to thinkwebringrealityintoexistencebyobservingorthinkingaboutit.

11WhyIsThereSpace?EmergenceandLocality

Okay,atlonglastwe’rereadytothinkabouttheactualworld.Waitaminute,Ihearyouthinking.Ithoughtweweretalkingabouttheactualworldalready.

Isn’tquantummechanicssupposedtodescribetheactualworld?Well,sure.Butquantummechanicscanalsodescribeplentyofworldsotherthanouractual

one.Quantummechanicsitselfisn’tasingletheory,inthesenseofbeingamodelofonespecificphysicalsystem.It’sa framework, just likeclassicalmechanics is, inwhichwecantalkaboutmanydifferentphysicalsystems.Wecantalkaboutthequantumtheoryofasingleparticle,oroftheelectromagneticfield,orofasetofspins,oroftheentireuniverse.Nowit’stimetofocusinonwhatthequantumtheoryofouractualworldmightlooklike.

This goal—finding the right quantum theory of the actual world—has been pursued bygenerationsofphysicistssincetheearlytwentiethcentury.Byanypossiblemeasure,theyhavebeenextraordinarilysuccessful.Oneimportantinsightwastothinkofthebasicbuildingblocksofnaturenotasparticlesbutasfieldspervadingspace,thusleadingtoquantumfieldtheory.

Backinthenineteenthcentury,physicistsseemedtobehominginonaviewoftheworldinwhichbothparticlesandfieldsplayedarole:matterwasmadeofparticles,andtheforcesbywhichtheyinteractedweredescribedbyfields.Thesedaysweknowbetter;eventheparticlesthatweknowandloveareactuallyvibrationsinfieldsthatsuffusethespacearoundus.Whenweseeparticle-liketracksinaphysicsexperiment,that’sareflectionofthefactthatwhatwesee is not what there really is. Under the right circumstances we see particles, but our bestcurrenttheoriessaythatfieldsaremorefundamental.

Gravity istheonepartofphysicsthatdoesn’t fitcomfortably intothequantum-field-theoryparadigm.Youwilloftenhearthat“wedon’thaveaquantumtheoryofgravity,”butthat’sabittoostrong.Wehaveanextremelygoodclassicaltheoryofgravity:Einstein’sgeneralrelativity,which describes the curvature of spacetime. General relativity is itself a field theory—itdescribesa fieldpervadingallofspace, in thiscase thegravitational field.Andwehaveverywell understood procedures for taking a classical field theory and quantizing it, yielding aquantumfieldtheory.Applythoseprocedurestotheknownfieldsoffundamentalphysics,andweendupwithsomethingcalled theCoreTheory. TheCoreTheoryaccuratelydescribesnotonlyparticlephysicsbutalsogravity,as longasthestrengthof thegravitational fielddoesn’tgrow too large. It is sufficient to describe every phenomenon that happens in your everydayexperience,andquiteabitbeyond—tablesandchairs,amoebasandkittens,planetsandstars.

The problem is that the Core Theory doesn’t cover a number of situations beyond theeveryday,includingplaceswheregravitybecomesextreme,likeblackholesandtheBigBang.In other words, we have a theory of quantum gravity that is adequate when gravity is fairlyweak,one that isperfectlycapableofdescribingwhyapples fall fromtreesorhowthemoonorbits the Earth. But it’s limited; once gravity becomes very strong, or we try to push ourcalculations too far, our theoretical apparatus fails us. As far as we can tell, this situation isuniquetogravity.Foralltheotherparticlesandforces,quantumfieldtheoriesseemtobeabletohandleanysituationwecanimagine.

Facedwiththedifficultyofquantizinggeneralrelativityaswewouldanyotherfieldtheory,thereareanumberofstrategiesthatwemighttry.Oneissimplytothinkharder;maybethereisagoodwaytodirectlyquantizegeneralrelativity,butitinvolvesnewtechniquesthatwehaven’tneeded forother field theories.Adifferentapproach is to imagine thatgeneralrelativity isn’ttherighttheorytoquantize;maybeweshouldstartwithadistinctclassicalprecursor,suchasstring theory, and thenquantize that, hoping tobuild aquantum theory that includesgravityalong with everything else. Physicists have been trying both of these approaches for somedecadesnow,withsomesuccessesbutstillalotofpuzzlesleftunanswered.

Herewe’regoingtoconsideradifferentstrategy,onethatfacesuptothequantumnatureofrealityfromthestart.Everyphysicistunderstandsthattheworldisfundamentallyquantum,butas we actually do physics we can’t help but be influenced by our experience and intuitions,

whichhavelongbeentrainedonclassicalprinciples.Thereareparticles,therearefields,theydo things, we can observe them. Even when we explicitly move to quantum mechanics,physicistsgenerallystartbytakingaclassical theoryandquantizing it.Butnaturedoesn’tdothat. Nature simply is quantum from the start; classical physics, as Everett insisted, is anapproximationthatisusefulintherightcircumstances.

Thisiswherewereachthepayoffforallofourhardworkoverthepreviouschapters.Many-Worlds is uniquely suited to the task of throwing away all of our classical intuition, beingquantumfromtheget-go,anddetermininghowtheapproximatelyclassicalworldthatweseearoundusultimatelyemergesfromthewavefunctionoftheuniverse,spacetimeandall.

In alternatives to Many-Worlds, one often needs additional variables (such as in Bohmianmechanics)orrulesabouthowwavefunctionsspontaneouslycollapse(suchasinGRW).Theseare typically derived from our experience with the classical limit of the theory underconsideration, and it’s exactly that experience that has failed us so far for quantum gravity.Many-Worlds, by contrast, doesn’t rely on any additional superstructure.Ultimately it’s not atheory of particular kinds of “stuff,” just quantum states evolving under the Schrödingerequation.Thatcreatesextraworkforusunderordinarycircumstances,aswehavetoexplainwhyweseeaworldofparticlesandfieldsatall.Butinthisuniquequantum-gravitycontext,it’san advantage, since we have to do that work anyway. Many-Worlds, with its quantum-firstperspective, is the right approach if you feel that we don’t know of any classical theory thatcouldserveastherightstartingpointforconstructingaquantumtheoryofgravity.

Beforediggingintoquantumgravityproper,weneedtolaysomegroundwork.Generalrelativityisatheoryofthedynamicsofspace-time,sointhischapterwe’llaskwhytheconceptof“space”issoimportantinthefirstplace.Theanswerresidesintheconceptoflocality—thingsinteractwithoneanotherwhen theyarenearby in space. In thenext chapterwe’ll seehowquantumfields propagating through space embody this principle of locality, and teach us somethingaboutthenatureofemptyspace.Inthechapterafterthatwe’llinvestigatehowtoextractspaceitself from the quantum wave function. And in the final chapter we’ll see that when gravitybecomesstrong,localityitselfwillhavetobeabandonedasacentralprinciple.Themysteryofquantumgravityseemstobeintimatelyconnectedwiththevirtuesandtheshortcomingsoftheideaoflocality.

It’sworthbeingcarefulabout“locality,”asitisusedintwosomewhatdifferentsenses:whatwemightcallmeasurementlocalityanddynamicallocality.TheEPRthoughtexperimentshowsthatthereissomethingthatseemsnonlocalaboutquantummeasurement.Alicemeasuresherspin,andwhatBobwillmeasureforhisspinfarawayisimmediatelyaffected,evenifhedoesn’tknowit.Bell’stheoremimpliesthatanytheoryinwhichmeasurementshavedefiniteoutcomes—basically,everyapproachtoquantummechanicsotherthanMany-Worlds—isgoingtofeaturethiskindofmeasurementnonlocality.WhetherMany-Worlds isnonlocal in thissensedependsonhowwechoosetodefineourbranchesof thewave function;we’reallowedtomakeeitherlocalornonlocalchoices,wherebranchinghappensonlynearbyorimmediatelyallthroughoutspace.

Dynamical locality,on theotherhand, refers to thesmoothevolutionof thequantumstatewhennomeasurementorbranchingishappening.That’sthecontextinwhichphysicistsexpecteverything to be perfectly local, with disturbances at one location only immediately affectingthingsrightnearby.Thiskindoflocalityisenforcedbytheruleinspecialrelativitythatnothingcan travel faster than light. And it’s this dynamical locality that we’re concerned with at themomentaswestudythenatureandemergenceofspaceitself.

With that in mind, we can roll up our sleeves a bit and dig into the question of how thestructure of our observed reality—we live in a world that looks like a collection of objectslocated in space, behaving approximately classically except for occasional quantum jumps—emerges from the quantum wave function. Everettian quantum mechanics purports to tell astory about many such worlds, but the postulates of the theory (wave functions, smoothevolution) don’t even mention “worlds” at all. Where do the worlds come from, and why doworldslookapproximatelyclassical?

Inourdiscussionofdecoherence,wepointedoutthatyoucanthinkofaquantumsystemashaving split into multiple separate copies once it becomes entangled with the largerenvironment around it, since whatever happens to each copy won’t be able to interfere withwhateverhappenstotheothers.Ifwewanttobesticklers,however,that’stellingusthatwe’reallowedtothinkaboutthedecoheredwavefunctionasdescribingseparateworlds—notthatweshouldthinkofitthatway,muchlessthatweneedtothinkofitthatway.Canwedobetter?

Thetruth is,nothingforcesustothinkof thewavefunctionasdescribingmultipleworlds,evenafter decoherencehas occurred.We could just talk about the entirewave function as awhole.It’sjustreallyhelpfultosplititupintoworlds.

Many-Worldsdescribestheuniverseusingasinglemathematicalobject,thewavefunction.Therearemanywaysoftalkingaboutthewavefunctionthatgiveusphysicalinsightintowhatisgoingon.Itmaybeusefulinsomecasestotalkintermsofposition,forexample,andinothercases in termsofmomentum.Likewise, it is oftenhelpful to talk about thepost-decoherencewavefunctionasdescribingasetofdistinctworlds;that’s justified,becausewhathappensoneach branch doesn’t affect what happens on the others. But ultimately, that language is aconvenience forus,not something that the theory itself insistson.Fundamentally, the theoryjustcaresaboutthewavefunctionasawhole.

Bywayofananalogy, thinkofall thematter in theroomaroundyourightnow.Youcoulddescribe it—helping ourselves to the classical approximation for the moment—by listing theposition and velocity of every atom in the room. But that would be crazy. You neither haveaccesstoallthatinformation,norcouldyouputittouseifyoudid,nordoyoureallyneedit.Instead,youchunkupthestuffaroundyouintoasetofusefulconcepts:chairs,tables,lights,floors,andsoon.That’sanenormouslymorecompactdescriptionthanlistingeveryatomwouldbe,butstillgivesusagreatdealofinsightintowhat’sgoingon.

Similarly, characterizing the quantum state in terms of multiple worlds isn’t necessary—itjustgivesusanenormouslyusefulhandleonanincrediblycomplexsituation.AsAliceinsistedinChapterEight,theworldsaren’tfundamental.Rather,they’reemergent.

Emergenceinthissensedoesnotrefertoeventsunfoldingovertime,aswhenababybirdemerges fromitsegg. It’sawayofdescribingtheworldthat isn’tcompletelycomprehensive,butdividesuprealityintomoremanageablechunks.Notionslikeroomsandfloorsarenowheretobefoundinthefundamentallawsofphysics—they’reemergent.Theyarewaysofeffectivelydescribing what’s going on even if we lack perfect knowledge of each and every atom andmoleculearoundus.Tosaythatsomethingisemergentistosaythatit’spartofanapproximatedescription of reality that is valid at a certain (usually macroscopic) level, and is to becontrastedwith“fundamental”things,whicharepartofanexactdescriptionatthemicroscopiclevel.

IntheLaplace’sdemonthoughtexperiment,weimagineavastintelligencethatwouldknowall the laws of physics and the exact state of the world, as well as having unlimitedcomputational capacity.To thedemon,everything that is,was, andeverwillbe is completelyknown.ButnoneofusisLaplace’sdemon.Inreality,wehaveatbestpartialinformationaboutthestateof theworld,andquite limitedcomputationalcapacity.Noneofus looksatacupofcoffeeandseeseveryparticleineveryatom;weseesomecoarsemacroscopicfeaturesoftheliquidandthecup.Butthatcanbealltheinformationweneedtohaveausefuldiscussionaboutthe coffee, and to predict its behavior in a variety of circumstances. A cup of coffee is anemergentphenomenon.

The same thing can be said for worlds in Everettian quantum mechanics. For a quantumversionofLaplace’sdemon,withexactknowledgeofthequantumstateoftheuniverse,therewould never be any need to divide the wave function into a set of branches describing acollectionofworlds.Butitisenormouslyconvenientandhelpfultodoso,andwe’reallowedtotake advantage of this convenience because the individual worlds don’t interact with oneanother.

That doesn’t mean that the worlds aren’t “real.” Fundamental versus emergent is onedistinction,andrealversusnot-realisacompletelyseparateone.Chairsandtablesandcupsofcoffeeare indubitablyreal,as theydescribe truepatterns in theuniverse,ones thatorganizetheworldinwaysthatreflecttheunderlyingreality.ThesamegoesforEverettianworlds.Wechoosetoinvokethemwhencarvingupthewavefunctionforourconvenience,butwedon’tdothat carving randomly. There are right and wrong ways to divide the wave function intobranches, and the right ways leave us with independent worlds that obey approximatelyclassicallawsofphysics.Whichwaysactuallyworkisultimatelydeterminedbythefundamentallawsofnature,notbyhumanwhimsy.

Emergenceisnotagenericfeatureofphysicalsystems.Ithappenswhenthere’saspecialwayofdescribingthesystemthatinvolvesmuchlessinformationthanacompletedescriptionwould,butneverthelessgivesusausefulhandleonwhat’sgoingon.That’swhyitmakessenseforusto carveup reality in thewaywedo, describing tables and chairs andbranches of thewavefunction.

Thinkofaplanetorbitingthesun.AplanetliketheEarthcontainsroughly1050particles.Todescribe the state of the Earth exactly, even at the classical level, would require listing theposition and momentum of every one of those particles, something that is beyond even ourwildestimaginationofsupercomputingpower.Happily,ifwhatwecareaboutisjusttheorbitofthe planet, the vast majority of that information is completely unnecessary. We can insteadidealize theEarthasa singlepoint, locatedat theEarth’s centerofmassandwith the same

totalmomentum.Thestateofthisidealizedpointisspecifiedbyapositionandmomentum,andthatverytinyamountof information(sixnumbers, threeeachforpositionandmomentum,asopposed to 6×1050 numbers, positions and momenta for each particle) is all we need tocalculate its trajectory. That’s emergence: a way of capturing important features of a systemusingfarlessinformationthananexhaustivedescriptionwouldentail.*

Weoftentalkaboutemergentdescriptions in termsofhow“convenient” theyare forus touse, but don’t be tricked into thinking there’s anythinganthropocentric goingon. Tables andchairs and planets would still exist even if there were no human beings to talk about them.“Convenience”isashorthandforindicatinganobjectivephysicalproperty:theexistenceofanaccurate model of the system that requires only a tiny fraction of the full informationcharacterizingit.

Emergence is not automatic. It’s a special, precious thing, and provides an enormoussimplificationwhenitoccurs.Imagineweknowthepositionofeveryoneofthe1050particlesintheEarth,butwedon’tknowthemomentumofanyofthem.Wepossessanenormousamountofinformation—fullyhalfof thetotal informationavailable—butwehavepreciselyzeroability topredictwheretheEarthwouldbegoingnext.Strictlyspeaking,evenifweknowthemomentumof all but one of the particles in the Earth, but have no knowledge at all of exactly onemomentum,wecan’tsaywhattheEarthwilldonext;it’spossiblethatthissingleparticlehasasmuchmomentumasalloftheotherscombined.

That’sthegenericsituationinphysics.Inordertoaccuratelypredictwhatasystemmadeofmanypartswilldonext,youneedtokeeptrackoftheinformationofalltheparts.Losejustalittle bit, and you know nothing. Emergence happens when the opposite is possible: we canthrowawayalmostalltheinformation,keepingjustalittlebit(aslongasyoucorrectlyidentifywhichbit),andstillsayquitealotaboutwhatwillhappen.

Inthecaseofthecenterofmassofanobjectmadeofmanyparticles,thekindofinformationintheemergentdescriptionwehaveisexactlythesameasthekindwestartedwith(positionandmomentum),justalotlessofit.Butemergencecanbemoresubtlethanthat;theemergentdescriptionmaybeofanentirelydifferentthingfromwhatwestartedwith.

Consider the air in our room. Imagine that we divide space into tiny boxes, perhaps onemillimeter on each side. Each box still contains a huge number of molecules. But instead ofkeepingtrackofthestateofeachoneofthem,wekeeptrackofaveragequantitiessuchasthedensity,pressure,andtemperatureineachbox.Itturnsoutthatthisisalltheinformationweneedtomakeaccuratepredictionsforhowtheairwillbehave.Theemergenttheorydescribesadifferentkindof thing,a fluidrather thanacollectionofmolecules,but that fluiddescriptionsuffices todescribe theair to ahighdegreeof precision.Treating theair as a fluid requiresmuchlessdatathantreatingitasacollectionofparticles;thefluiddescriptionisemergent.

Everettianworldsarethesameway.Wedon’tneedtokeeptrackoftheentirewavefunctiontomakeusefulpredictions,justwhathappensinanindividualworld.Toagoodapproximationwe can treatwhat happens in eachworldusing classicalmechanics,with just the occasionalquantuminterventionwhenweentanglewithmicroscopicsystemsinsuperposition.That’swhyNewton’s laws of gravitation and motion are sufficient to fly rockets to the moon withoutknowingthecompletequantumstateoftheuniverse;ourindividualbranchofthewavefunctiondescribesanemergentalmost-classicalworld.

Branches of the wave function, describing separate worlds, are not mentioned in thepostulatesofMany-Worlds.NoraretablesandchairsandairmentionedintheCoreTheoryofparticles and forces. As the philosopher Daniel Dennett has put it, in terms that were thenported into the quantum context by David Wallace, each world is an emergent feature thatcaptures “real patterns” within the underlying dynamics. A real pattern gives us an accuratewayoftalkingabouttheworld,withoutappealingtoacomprehensivemicroscopicdescription.That’s what makes emergent patterns in general, and Everettian worlds in particular,indisputablyreal.

Once youbelieve that branches of thewave function canusefully be thought of as emergentworlds,youmightstartwonderingwhyit’sthissetofworldsinparticular.Whydoweendupseeing macroscopic objects with pretty well-defined locations in space, rather than being insuperpositionsofdifferentlocations?Whyis“space”apparentlysuchacentralconceptatall?Textbooks in introductory quantum mechanics sometimes give the impression that classicalbehaviorisinevitableonceobjectsbecomeverybig,butthat’snonsense.Wehavenotroubleatall imagining a wave function that describes macroscopic objects in all sorts of weirdsuperpositions.Therealanswerismoreinteresting.

We can begin to get a handle on the special nature of space by comparing how we thinkabout position to how we think about momentum. When Isaac Newton first wrote down theequationsofclassicalmechanics,positionclearlyplayedaprivilegedrole,whereasvelocityand

momentumwerederivedquantities.Positionis“whereyouareinspace,”whilevelocityis“howfastyouaremovingthroughspace,”andmomentumismasstimesvelocity.Spacewouldappeartobethemainthing.

But adeeper look reveals that the conceptsofpositionandmomentumareonmoreof anequalfootingthantheyfirstappear.Perhapsweshouldn’tbesurprised;afterall,positionandmomentumarethetwoquantitiesthattogetherdefinethestateofaclassicalsystem.Indeed,intheHamiltonianformulationofclassicalmechanics,positionandmomentumareexplicitlyonanequalfooting.Isthisareflectionofsomeunderlyingsymmetrythatisn’tobviousonthesurface?

Inoureveryday lives,positionandmomentumseemquitedifferent.Whatamathematicianwouldcall“thespaceofallpossiblepositions”iswhattherestofus justcall“space”; it’sthethree-dimensionalworldinwhichwelive.The“spaceofallpossiblemomenta,”or“momentumspace,”isalsothree-dimensional,butit’saseeminglyabstractconcept.Nobodybelieveswelivethere.Whynot?

The feature that makes space special is locality. Interactions between different objectshappenwhentheyarenearbyinspace.Twobilliardballsbounceoffeachotherwhentheycometogetheratthesamespatialposition.Nothingofthesorthappenswhenparticleshavethesame(oropposite)momenta;ifthey’renotinthesamelocation,theyjustkeepgoingtheirmerryway.That’snotanecessaryfeatureofthelawsofphysics—wecouldimagineotherpossibleworldswhereitwasn’tthecase—butit’sonethatseemstoholdprettywellinourworld.

Ricochetingbilliardballsareclassical,butthesamediscussioncouldbehadaboutquantummechanics.Thebasicquantum formalismalso treatspositionandmomentumequally.Wecanexpress the wave function by attaching a complex amplitude to every possible location theparticlecanbein,orwecouldjustaswellexpressitbyattachingacomplexnumbertoeverypossiblemomentumtheparticlecouldhave.The twowaysofdescribing thesameunderlyingquantum state are equivalent, expressing the same information in different ways, as we sawwhendiscussingtheuncertaintyprinciple.

Thisiskindofprofound.We’vesaidthatawavefunctionofdefinitemomentumlookslikeasinewave.Butthat’swhatitlookslikeintermsofposition,whichisthelanguagewenaturallytend to speak. Expressed in terms of momentum, the same quantum state would look like aspikelocatedatthatparticularmomentum.Astatewithdefinitepositionwouldlooklikeasinewavespreadoverallpossiblemomenta.Thisbeginstosuggestthatwhatreallymattersistheabstractnotionof“thequantumstate,”notitsspecificrealizationasawavefunctionintermsofeitherpositionormomentum.

The symmetry is broken, once again, by the fact that in our particular world, interactionshappenwhen systemsarenearby in space. This is dynamical locality atwork.FromaMany-Worldsperspectivethattreatsquantumstatesasfundamentalandeverythingelseasemergent,thissuggeststhatweshouldreallyturnthingsaround:“positionsinspace”arethevariablesinwhichinteractionslooklocal.Spaceisn’tfundamental;it’sjustawaytoorganizewhat’sgoingonintheunderlyingquantumwavefunction.

This point of view helps us understand why the Everettian wave function can naturally bedividedintoasetofapproximatelyclassicalworlds.Thisissueisknownasthepreferred-basisproblem.Many-Worldsisbasedonthefactthatthewavefunctionoftheuniversewillgenerallydescribe all sorts of superpositions, including states where macroscopic objects are insuperpositionsofbeinginverydifferentlocations.Butweneverseechairsorbowlingballsorplanets in superpositions; as far as our experience is concerned, they always seem to havedefinite locations, and their motion obeys the rules of classical mechanics to a very goodapproximation.Whydon’t thestatesweseeever involvemacroscopicsuperpositions?Wecanwrite the wave function as a combination of many distinct worlds, but why divide it up intotheseworldsinparticular?

The answer was essentially figured out in the 1980s, using decoherence, althoughresearchers are still hammering out the details. To get there, it’s useful to turn to that oldthought-experiment standby,Schrödinger’sCat.Wehave a sealedbox containing a cat andacontainerofsleepinggas.Schrödinger’soriginalscenarioinvolvedpoison,butthere’snoreasonwehavetoimaginekillingthecat.(HisdaughterRuthoncemused,“Ithinkmyfatherjustdidn’tlikecats.”)

Our experimenter has rigged a spring to pull open the container, releasing the gas andputting the cat to sleep, but only when a detector such as a Geiger counter clicks upondetectingaparticleofradiation.Nexttothedetectorisaradioactivesource.Weknowtherateat which particles are emitted from the source, so we can calculate the probability that thecounterwillclickandreleasethehammerafteranygivenperiodoftime.

Radioactiveemission is a fundamentallyquantumprocess.Whatwe informallydescribeastheoccasional,randomemissionofaparticleisactuallyasmoothevolutionofthewavefunction

oftheatomicnucleiwithinthesource.Eachnucleusevolvesfromastateofpurelyun-decayedtoasuperpositionof(un-decayed)+(decayed),withthelatterpartgraduallygrowingovertime.Theemissionappearsrandombecausethedetectordoesn’tmeasurethewavefunctiondirectly;itonlyseeseither(un-decayed)or(decayed),justasaverticalStern-Gerlachmagnetonlyeverseesspin-uporspin-down.

The point of the thought experiment is to take a microscopic quantum superposition andmagnifyittoamanifestlymacroscopicsituation.Thathappensassoonasthedetectorclicks.Allthebusinesswiththesleepinggasandthecat is just tomaketheamplificationofaquantumsuperposition to themacroscopicworldmore vivid. (Theword “entanglement,” or inGermanVerschränkung,wasfirstappliedtoquantummechanicsbySchrödingerinthediscussionofhiscat,whicharoseoutofcorrespondencewithEinstein.)

Schrödinger’s experiment was posed in the context of the textbook approach to themeasurementproblem,wherewavefunctionscollapsewhentheyareliterallyobserved.So,hesays,imaginethatwekeeptheboxclosed—notobservingwhat’sinside—untilthewavefunctionevolves to an even superposition of “at least one nucleus has decayed” and “no nuclei havedecayed.” In that case, the wave functions of the detector, the gas, and the cat will all alsoevolveintoanequalsuperposition,of“thedetectorclicked,thegaswasreleased,andthecatisasleep”and“thedetectordidn’tyetclick,thegasisstillinthecontainer,andthecatisawake.”Surely,asksSchrödinger,youdon’tseriouslybelievethattheboxcontainsasuperpositionofanawakecatandanasleepcatuntilweopenit?

As far as that goes, he was right. Once we have an Everettian perspective on quantumdynamics,weacceptthatthewavefunctionsmoothlyevolvesintoanequalsuperpositionoftwopossibilities,oneinwhichthecatisasleepandtheotherinwhichitisawake.Butdecoherencetellsus that thecat isalsoentangledwith itsenvironment,consistingofall theairmoleculesandphotonswithinthebox.Theeffectivebranchingintoseparateworldshappensalmostrightawayafter thedetectorclicks.By the time theexperimentergetsaround toopening thebox,there are two branches of the wave function, each of which has a single cat and a singleexperimenter,notasuperposition.

ThissolvesSchrödinger’soriginalworry,butraisesanotherone.Whyisitthatwhenweopenthebox,theparticulardecoheredquantumstatesweseeareeitherthatofanawakecat,oranasleep cat? Why don’t we see some superposition of both? “Awake” and “Asleep” togetherrepresentjustonepossiblebasisforthecatsystem,justas“spin-up”and“spin-down”dofortheelectron.Whyisthatbasispreferredoveranyotherone?

The physical process that matters is stuff in the environment—gas molecules, photons—interactingwiththephysicalsystemunderconsideration.Whetheraparticularparticleactuallydoesinteractwiththecatwilldependonwherethecat is.Agivenphotonmightverywellbeabsorbedbyacatthatisawakeandprowlingaroundthebox,butcompletelymissacatthatissleepingonthefloor.

What’sspecialaboutthe“Awake”/“Asleep”basis,inotherwords,isthattheindividualstatesdescribewell-definedconfigurationsinspace.Andspaceisthequantitywithrespecttowhichphysical interactionsarelocal.Aparticlecanbumpintoacatiftheparticleandthecatcomeintophysicalcontact.Thetwopartsofthecatwavefunction,“Awake”and“Asleep,”comeintocontactwithdifferentparticlesintheenvironment,andthereforebranchintodifferentworlds.

Thisisthebasicanswertothequestionofwhyweseetheparticularworldsthatwedo:thepreferred-basisstatesarethosethatdescribecoherentobjects inspace,becausesuchobjectsinteractconsistentlywiththeirenvironments.Theseareoftencalledpointerstates,astheyarethestatesinwhichthepointerofamacroscopicmeasuringdevicewillindicateadefinitevalue,rather than being in a superposition. The pointer basis is where a well-behaved classicalapproximationmakessense,andthereforeit’sthatkindofbasisthatdefinesemergentworlds.Decoherence is the phenomenon that ultimately links the austere simplicity of Everettianquantummechanicstothemessyparticularityoftheworldwesee.

* Sadly there are competing definitions of the word “emergence,” some of which mean almost the opposite of thesense used here. Our definition is sometimes called “weak emergence” in the literature, as opposed to “strongemergence,”inwhichthewholeisirreducibletothesumofitsparts.

12AWorldofVibrationsQuantumFieldTheory

The phrase “action at a distance,” usuallymodified by Einstein’s adjective “spooky,” is ofteninvokedindiscussionsofquantumentanglementandtheEPRpuzzle.Buttheideaismucholderthanthat—itgoesbackatleasttoIsaacNewtonandhistheoryofgravity.

If Newton had done nothing more than put together the basic structure of classicalmechanics,hewouldbealeadingcandidateforthegreatestphysicistofalltime.Whatclincheshisclaimtothecrownisthathedidmuchmorethanthat,includinglittlethingslikeinventingcalculus.Still,whenmostpeopleseeapictureofNewtoninhismagnificentwig,theythinkofhistheoryofgravity.

Newtonian gravity can be summed up in the famous inverse-square law: the gravitationalforce between two objects is proportional to the mass of each of them, and inverselyproportionaltothesquareofthedistancebetweenthem.SoifyoumovedthemoontobetwiceasfarawayfromtheEarth,thegravitationalforcebetweenthemwouldbeonlyone-fourthaslarge.Using this simple rule,Newtonwasable to show thatplanetswouldnaturallymove inellipses around the sun, confirming the empirical relationship that had been posited byJohannesKepleryearsbefore.

But Newton was never really satisfied with his own theory, precisely because it featuredactionatadistance.Theforcebetweentwoobjectsdependsonwhereeachofthemislocated,andwhen an objectmoves, the direction of its gravitational pull changes instantaneously allthroughout theuniverse.Therewasnothing inbetweenthatwouldmediatesuchachange; itsimply happened. This bugged Newton—not because it was illogical or incompatible withobservation,butjustbecauseitseemedwrong.Spooky,onemightsay.

Itisinconceivablethatinanimatebrutemattershould,withouttheMediationofsomethingelsewhichisnotmaterial,operateuponandaffectothermatterwithoutmutualcontact. . . .Gravitymustbecausedbyanagentactingconstantlyaccordingtocertainlaws;butwhetherthisagentbematerialorimmaterial,Ihavelefttotheconsiderationofmyreaders.

There is indeed an “agent” that causes gravity to act the way it does, and that agent isperfectlymaterial—it’sthegravitationalfield.ThisconceptwasfirstintroducedbyPierre-SimonLaplace,whowasabletorewriteNewton’stheoryofgravitysothattheforcewascarriedbyagravitationalpotential field,ratherthansimplyhoppingmysteriouslyacross infinitedistances.But a change in the force still happened instantaneously through all of space. Itwasn’t untilEinstein came along with general relativity that changes in the gravitational field, just likechangesintheelectromagneticfield,wereshowntotravelthroughspaceatthespeedoflight.General relativity replaces Laplace’s potential with the “metric” field, a mathematicallysophisticated way of characterizing the curvature of spacetime, but the general idea of agravitationalfieldpervadingallofspacehasremainedintact.

Theideaofafieldcarryingaforceisconceptuallyappealingbecauseitinstantiatestheideaof locality.As theEarthmoves, thedirectionof itsgravitationalpull doesn’t change instantlythroughouttheuniverse.Rather,itchangesrightwheretheEarthislocated,andthenthefieldatthatpointtugsonthefieldnearby,whichtugsonthefieldalittlefartheraway,andsooninawavemovingoutwardatthespeedoflight.

Modernphysicsextendsthisideatoliterallyeverythingintheuniverse.TheCoreTheoryisconstructed by starting with a set of fields and then quantizing them. Even particles likeelectrons and quarks are really vibrations in quantum fields. That’s a wonderful story all byitself, but our aim in this chapter is slightly more modest: to understand the “vacuum” inquantumfieldtheory,thequantumstatecorrespondingtoemptyspace.(I’verelegatedabriefdiscussion of states with actual particles in them to the Appendix.) Later we’ll tackle thequantumemergenceofspaceitself,butfornowwe’llbedrearilyconventionalandthinkabout

quantumfieldtheoryaswhatyougetwhenyouquantizeaclassicalfieldtheoryinapreexistingspace.

One of the lessonswewill learn is that entanglement plays an evenmore central role inquantum field theory than it does in quantum particle theories. When particles were ourprimary concern, entanglement was something that may or may not have been important,dependingonthephysicalcircumstances.Youcancreateastateoftwoentangledelectrons,butthereareplentyof interestingstatesof twoelectronswhere theparticlesaren’tentangledatall.Infieldtheory,bycontrast,essentiallyeveryphysicallyinterestingstateisonethatfeaturesanenormousamountofentanglement.Evenemptyspace,whichyoumight thinkofasprettystraightforward, is described in quantum field theory as an intricate collection of entangledvibrations.

QuantummechanicsfirstbeganwhenPlanckandEinsteinarguedthatelectromagneticwaveshadparticle-likeproperties,andthenBohr,deBroglie,andSchrödingersuggestedthatparticlescouldhavewave-likeaspects.Buttherearetwodifferentkindsof“waviness”atworkhere,andit’s worth being careful to distinguish between them. One kind of waviness arises when wemake the transition from a classical theory of particles to a quantum version, obtaining thequantumwave functionofa setofparticles.Theotherkind iswhenwehaveaclassical fieldtheorytostartwith,evenbeforequantummechanicsbecomes involvedatall.That’s thecasewithclassicalelectromagnetism,orwithEinstein’stheoryofgravity.Classicalelectromagnetismandgeneralrelativityareboth theoriesof fields (andthereforeofwaves),butare themselvesperfectlyclassical.

Inquantumfield theory,westartwithaclassical theoryof fieldsandconstructaquantumversionof that. Insteadofawave function that tellsus theprobabilityof seeingaparticleatsome location, we have a wave function that tells us the probability of seeing a particularconfigurationofafieldthroughoutspace.Awavefunctionofawave,ifyoulike.

Therearemanywaystoquantizeaclassicaltheory,butthemostdirectoneistheroutewehavealreadytaken.Thinkingofacollectionofparticles,wecanask,“Wherecantheparticlesbe?”Theanswerforeachindividualparticleissimply“Atanypointinspace.”Iftherewerejustoneparticle,thewavefunctionwouldthereforeassignanamplitudetoeverypointinspace.Butwhenwehaveseveralparticles,thereisn’taseparatewavefunctionforeachparticle.Thereisonebigwavefunction,assigningadifferentamplitudetoeverypossiblesetoflocationsthatalltheparticlescouldbeinatonce.That’showentanglementcanhappen;foreveryconfigurationoftheparticles,thereisanamplitudewecouldsquaretogettheprobabilityofobservingthemthereallatthesametime.

It’s the same thing for fields, with “possible configuration of the particles” replaced by“possibleconfigurationsofthefield,”whereby“configuration”wenowmeanthevaluesofthefield at each point throughout all of space. Thiswave function considers every possible fieldconfiguration, and assigns an amplitude to each. If we could imagine observing the fieldeverywhereatonce,theprobabilityofgettinganyparticularshapeofthefieldwillbeequaltothesquareoftheamplitudeassignedtothatconfiguration.

Thisisthedifferencebetweenaclassicalfieldandaquantumwavefunction.Aclassicalfieldisafunctionofspace,andaclassicaltheorywithmanyfieldswoulddescribemultiplefunctionsof space overlapping with one another. The wave function in quantum field theory is not afunctionofspace,it’safunctionofthesetofallconfigurationsofalltheclassicalfields.(IntheCoreTheory,thatwouldincludethegravitationalfield,theelectromagneticfield,thefieldsforthe various subatomic particles, and so on.) An intimidating beast, but something physicistshavelearnedtounderstandandevencherish.

AllofthisimplicitlyassumestheMany-Worldsversionofquantummechanics.Wedidn’tsayanything about decoherence andbranching, butwehavebeen taking for granted that allwereallyneedisaquantumwavefunctionandanappropriateversionoftheSchrödingerequation,and the restwill take careof itself.That’s exactly theEverettian situation. (Sometimeswhenpeoplesay“theSchrödingerequation”theyarereferringspecificallytotheversionSchrödingeroriginallywrotedown,whichisonlyappropriatefornon-relativisticpointparticles,butthere’sno difficulty in finding a version of the equation for relativistic quantum fields or any othersystem with a Hamiltonian.) In other theories, one often needs additional variables or rulesabout how wave functions spontaneously collapse. When we move to field theory, it’s notimmediatelyclearwhatthoseextraingredientsshouldbe.

Ifquantumfieldtheorydescribestheworldasawavefunctionofaclassicalfieldconfiguration,that seems to be waviness on top of waviness. If we asked how much wavier things couldpossiblyget,theanswer(toparaphraseNigelTufnelofSpinalTap)mightbe“nonemorewavy.”Andyet,whenwemakeobservationsofquantumfields,forexample,inadetectorattheLargeHadronColliderinGeneva,whatweseeareindividualtracksrepresentingthepathsofpoint-likeobjects,notdiffusewavyclouds.Somehowwehavecircledbacktoparticles,despitebeingaswavyascanbe.

The reason for this goes back to the same reason why we see discrete energy levels forelectrons inatoms.Anelectronmovingthroughspaceallby itselfcanhaveanyenergyatall,butinthevicinityoftheattractiveforceexertedbyanatomicnucleus,it’sasiftheelectronistrappedinabox.Thewavefunctionfallstozerofarawayfromtheatom;wecanthinkofitasbeingtieddown, justas forastring tieddownonbothendsand free tomove inbetween. Insuchcircumstances,thetied-downstringcanonlyperformadiscretesetofvibrations;likewise,thewavefunctionoftheelectronhasadiscretesetofenergylevels.Anytimethewavefunctionof a system is “tied down” by going to zero for large/faraway/extreme configurations, it willexhibitasetofdiscreteenergylevels.

Returningtofieldtheory,consideraverysimplefieldconfiguration,asinewavestretchingthroughoutallofspace.Wecallsuchaconfigurationamodeofthefield;it’saconvenientwayof thinking, since any field configuration at all can be thought of as a combination of manymodes of different wavelengths. That sine wave contains energy, and the energy increasesrapidlyasweimaginewavesofgreaterandgreaterheight.Wewanttoconstructthequantumwavefunctionofthatfield.Becausetheenergyofthefieldriseswiththeheightofthewave,thewavefunctionneedstodecreaserapidlyastheheightofthewaveincreases,soastonotgivetoomuchprobabilitytoveryhigh-energywaves.Forallintentsandpurposes,thewavefunctionistieddown(itgoestozero)atlargeenergies.

Asaresult, just likeavibratingstringoranelectron inanatom, there isadiscretesetofenergylevelsforthevibrationsofaquantumfield.Infact,everymodeofthefieldcanbeinits

lowest-energystate,oritsnext-highest,ornext-highest,andsoon.Theoverallminimum-energywavefunctionisoneinwhicheverysinglemodehasthelowestpossibleenergy.That’sauniquestate, whichwe call the vacuum.When quantum field theorists talk about the vacuum, theydon’tmeanamachine that liftsdustoff your floors, orevena regionof interplanetary spacedevoidofmatter.Whattheymeanis“thelowest-energystateofyourquantumfieldtheory.”

Youmightthinkthatthequantumvacuumwouldbeemptyandboring,butit’sactuallyawildplace.Anelectroninanatomhasalowest-energystateitcanbein,butifwethinkaboutitasawavefunctionofthepositionoftheelectron,thatfunctioncanstillhaveaninterestingshape.Likewise, thevacuumstate in field theorycanstillhave interestingstructure ifweaskaboutindividualpartsofthefield.

The next energy level has a bit more going on, since wemake it out of the next-highestenergiesofeachmode.Thatgivesusabitoffreedom;therecanbestatesthataremostlyshort-wavelengthmodes,orstatesthataremostlylong-wavelengthmodes,oranymixture.Whattheyhaveincommoniseachmodeisinits“firstexcitedstate,”withjustabitmoreenergythantheminimum.

Puttingthattogether,thewavefunctionforthefirstexcitedstateofaquantumfieldtheorylooks exactly like that of a singleparticle, expressedas a function ofmomentum rather thanposition.Therewillgenerallybecontributions fromdifferentwavelengths,whichwe interpretasdifferentmomentaintheparticlewavefunction.Mostimportant,thiskindofstatebehavesina particle-like way when we observe it: if we measure a bit of energy in one location(interpretedas“Ijustsawaparticlethere”),itbecomesoverwhelminglyprobablethatyouwillobservethesameamountofenergynearbyifyoulookamomentlater,evenifthewavefunctionwasoriginallyallspreadout.Whatyouendupseeingisalocalizedvibrationpropagatinginthefield,leavingatrackinanexperimentaldetectorjustlikeaparticleissupposedtodo.Ifitlookslikeaparticleandquackslikeaparticle,itmakessensetocallitaparticle.

Can we have a quantum-field-theory wave function that combines some modes in theirlowest-energy states and some others in their first excited states? Sure—that would be asuperpositionofazero-particlestateandaone-particlestate,givingastatewithoutadefinitenumberofparticles.

As youmight be prepared to guess, the next-highest energywave functions of a quantumfield theory look like thewave functionof twoparticles.The storygoeson forquantum fieldstatesrepresentingthreeparticles,orfour,orwhatever.JustasweobserveSchrödinger’scattobeeitherawakeorasleep,andnotanysuperpositionthereof,collectionsofparticlesarewhatweobservewhenwemakemeasurementsofgentlyvibratingquantumfields.Inthelanguageofthepreviouschapter,as longas the fieldsaren’t fluctuatingtoowildly, the“pointerstates”ofquantumfieldtheorylooklikecollectionsofdefinitenumbersofparticles.Thosearethekindsofstatesweseewhenweactuallylookattheworld.

Even better, quantum field theory can describe transitions between states with differentnumbersofparticles,justasanelectroncanhopupordowninenergyinanatom.Inordinaryparticle-basedquantummechanics, thenumberofparticles is fixed,butquantum field theoryhas no problem describing particles decaying or annihilating or being created in collisions.Whichisgood,becausethingslikethathappenallthetime.

Quantum field theory represents one of the great triumphs of unification in the history ofphysics, tying together the seemingly opposed ideas of particles andwaves.Oncewe realizethatquantizingtheelectromagneticfieldleadstoparticle-likephotons,perhapsitshouldn’tbesurprising that other particles such as electrons and quarks also arise fromquantized fields.Electronsarevibrationsintheelectronfield,varioustypesofquarksarevibrationsinvarioustypesofquarkfields,andsoon.

Introductionstoquantummechanicssometimescontrastparticlesandwavesas if theyaretwoequalsidesofthesamecoin,butultimatelythebattlebetweenparticlesandfieldsisnotafair fight. Fields aremore fundamental; it’s fields that provide the best picturewe currentlyhaveofwhattheuniverseismadeof.Particlesaresimplywhatweseewhenweobservefieldsunder the right circumstances. Sometimes the circumstances aren’t right; inside a proton orneutron,eventhoughweoftenspeakaboutquarksandgluonsasifthey’reindividualparticles,it’smoreaccuratetothinkofthemasdiffusefields.AsphysicistPaulDaviesoncetitledapaper,withonlyabitofrhetoricalexaggeration,“ParticlesDoNotExist.”

Our interest here is in the basic paradigm of quantum reality, not in the specific pattern ofparticlesand theirmassesand interactions.Wecareaboutentanglementandemergenceandhowtheclassicalworldarisesfromthebranchingwavefunction.Happily,forthesepurposeswecanconcentrateourattentiononthequantumfieldtheoryvacuum—thephysicsofemptyspace,withoutanyparticlesflyingaround.

Tobringhometheinterestingnessofthefield-theoryvacuum,let’sfocusononeofitsmost

obviousaspects,itsenergy.It’stemptingtothinkthattheenergyiszerobydefinition.Butwe’vebeencarefulnottosaythat: thevacuumisthe“lowest-energystate,”notnecessarilya“zero-energystate.”Infact,itsenergycanbeanythingatall;it’saconstantofnature,aparameterofthe universe that is not determined by any other set of measurable parameters. As far asquantumfieldtheoryisconcerned,youhavetojustgooutandmeasurewhattheenergyofthevacuumactuallyis.

Andwehavemeasuredthevacuumenergy,oratleastwethinkwehave.It’snoteasytodo;youcan’tsimplyputacupfulofemptyspaceonascaleandaskhowmuchitweighs.Thewaytodo it is to look for the gravitational influence of the vacuum energy. According to generalrelativity, energy is the source of the curvature of spacetime, and therefore of gravity. Theenergyofempty space takesaparticular form: there isaprecisely constantamount ineverycubic centimeter of space, unchanging through the universe, even as spacetime expands orwarps.Einsteinreferredtothevacuumenergyasthecosmologicalconstant,andcosmologistslongdebatedwhetheritsvaluewasexactlyzeroorsomeothernumber.

That debate seems to have been settled in 1998, when astronomers discovered that theuniverseisnotonlyexpandingbutalsoaccelerating.Ifyoulookatadistantgalaxyandmeasurethe velocity with which it is receding, that velocity is increasing with time. That would beextremelysurprising ifall theuniversecontainedwereordinarymatterandradiation,bothofwhichhavethegravitationaleffectofpullingthingstogetherandslowingdowntheexpansionrate.Apositivevacuumenergyhastheoppositeeffect:itpushestheuniverseapart,leadingtoaccelerated expansion. Two teams of astronomers measured the distances and velocities ofextragalactic supernovae, expecting to measure the deceleration of the universe. What theyactually found was that it is speeding up. The discomfiting surprise at obtaining such anunexpected result was partly ameliorated by winning the Nobel Prize in 2011. (The debate“seems to”havebeensettled,because it’sstillanopenpossibility thatcosmicacceleration iscausedbysomethingother thanvacuumenergy.But that’sby far the leadingexplanation,onboththeoreticalandobservationalgrounds.)

Youmight think thatwould be the end of it. Empty space has energy,we’vemeasured it,cocoaandcupcakesallaround.

But there’s another question we’re allowed to ask: What should we expect the vacuumenergytobe?That’safunnyquestion;sinceit’sjustaconstantofnature,maybewedon’thavetherighttoexpectthatit’sanyparticularvalueatall.Whatwecando,however,isaquick-and-dirtyestimateofhowbigwemightguessthevacuumenergyshouldbe.Theresultissobering.

The traditional way to estimate the vacuum energy is to distinguish between what theclassical cosmological constantwouldbe, andhowquantumeffects change that value.That’snot really right; nature doesn’t care that human beings like to start classically and buildquantummechanicsontopofthat.Natureisquantumfromthestart.Butsinceallwe’retryingtodoisgetaveryroughestimate,maybethisprocedureisokay.

Asitturnsout,it’snotokay.Thequantumcontributiontothevacuumenergyisinfinitelybig.Thiskindofproblemisendemictoquantumfieldtheory;manycalculationsthatwetrytodobygraduallyincludingquantumeffectsendupgivingusnonsensical,infinitelybiganswers.

Butweshouldn’ttakethoseinfinitiestooseriously.Theycanultimatelybetracedtothefactthat a quantum field can be thought of as a combination of vibratingmodes at all differentwavelengths,fromincrediblylongallthewaydowntozero.Ifweassume(fornoespeciallygoodreason) that the classicalminimumenergyof eachmode is zero, then the real-world vacuumenergy is just the sum of all the additional quantum energies for eachmode. Adding up thequantum energies for all those modes is what gives us an infinite vacuum energy. That’sprobablynotphysicallyrealistic.Afterall,atveryshortdistancesweshouldexpectspacetimeitself tobreakdownasausefulconcept,asquantumgravitybecomesimpossibleto ignore.ItmightmakemoresensetoonlyincludecontributionswithwavelengthslargerthanthePlancklength, forexample.Wecall this imposingacutoff—looking at quantum field theory, but onlyincludingmodeswithwavelengthslongerthanacertaindistance.

Unfortunatelythisdoesn’tquitefixtheproblem.Ifweestimatethequantumcontributiontothe vacuum energy by imposing a Planckscale cutoff on the allowed modes, we get a finiteanswer rather than an infinite one, but that answer is 10122 times larger than the value weactuallyobserve.Thismismatch,knownasthecosmologicalconstantproblem,hasoftenbeencalledthebiggestdiscrepancybetweentheoryandobservationinallofphysics.

Thecosmologicalconstantproblemisnotreallyaconflictbetweentheoryandobservationinthe strict sense. We don’t have anything like a reliable theoretical prediction for what thevacuum energy should be. Our very wrong estimate comes from making two dubiousassumptions:thattheclassicalcontributiontothevacuumenergyiszero,andthatweimposeacutoff at thePlanck scale. It’s alwayspossible that the classical contributionwe should startwithisalmostexactlyaslargeasthequantumpiece,butwiththeoppositesign,sothatwhenweaddthemtogetherwegetanobserved“physical”vacuumenergywitharelativelytinyvalue.Wejusthavenoideawhythatshouldbetrue.

Theproblemisnotthattheoryconflictswithobservation;it’sthatourroughexpectationsare

wayoff,whichmostpeopletakeasacluethatsomethingmysteriousandunknownisatwork.Since the energywe estimatedwas a purely quantum-mechanical effect, andwemeasure itsexistenceusing itsgravitationaleffect, it’splausiblethatwewon’tsolvetheproblemuntilwehaveafullyworkingquantumtheoryofgravity.

Populardiscussionsofquantumfieldtheorywilloftendescribethevacuumasfullof“quantumfluctuations,” or even “particles popping in and out of existence in empty space.” That’s anevocativepicture,butit’smorefalsethantrue.

Inemptyspacedescribedbythequantum-field-theoryvacuum,nothingisfluctuatingatall;the quantum state is absolutely stationary. The picture of particles popping in and out ofexistenceisentirelydifferentfromthereality,inwhichthestateispreciselythesamefromonemomenttoanother.Thereisundoubtedlyanintrinsicallyquantumcontributiontotheenergyofemptyspace,but it’smisleading tospeakof thatenergyascoming from“fluctuations,”whennothing is actually fluctuating. The system is sitting peacefully in its lowest-energy quantumstate.

Why, then, are physicists constantly talking about quantum fluctuations? It’s the samephenomenonwe have noted in other contexts:we human beings have an irresistible urge tothink ofwhatwe see as being real, even though quantummechanics keeps telling us to dobetter.Hidden-variable theoriesgive in to thisurgebymakingsomething realother than thesmoothlyevolvingwavefunction.

Everettianquantummechanicsisclear:emptyspaceisdescribedbyastationary,unchangingquantumstate,wherenothing ishappening frommoment tomoment.But ifwewere to looksufficientlycarefully,measuringthevaluesofaquantumfieldinsomesmallregion,wewouldseewhat looked likearandommess.And ifwe lookedagainamoment later,wewouldseeadifferent-looking random mess. The temptation to conclude that there is something movingaround in empty space, evenwhenwe’renot looking, is overwhelming.But that’s notwhat’sgoingon.Rather,we’re seeingamanifestationofwhatwe talkedabout in the context of theuncertainty principle: when we observe a quantum state, we typically see something quitedifferentfromwhatthestatewasbeforewelooked.

Todrivethispointhome,imaginethatwedoamoreexperimentallyfeasiblemeasurement.Ratherthanmeasuringthevalueofafieldateverypoint,let’sjustmeasurethetotalnumberofparticlesinthevacuumstateofaquantumfieldtheory.Inanidealthought-experimentworld,we can imagine doing that measurement throughout all of space all at once. Since byconstructionwe’reinthelowest-energystate,youwon’tbesurprisedtohearthatwewill,withperfectconfidence,detectnoparticlesanywhere. It’s justemptyspace.But in therealworld,wewillbeconfinedtodoinganexperimentinsomespecificregionofspace,suchastheinteriorofourlaboratory,andaskinghowmanyparticlesthereare.Whatshouldweexpecttosee?

Thisdoesn’tsoundlikeahardquestion.Iftherearenoparticlesanywhere,thencertainlywewon’tseeanyparticlesinourlab,right?Alas,no.That’snothowquantumfieldtheoryworks.Eveninthevacuumstate,ifourexperimentalprobeisconfinedtosomefiniteregion,therewillalwaysbeasmallprobabilityofobservingoneormoreparticles.Generallytheprobabilitywillbereally,reallysmall—notsomethingwehavetoworryaboutinrealisticexperimentalsetups—butitwillbethere.Theconverseisalsotrue:therewillbequantumstatesforwhichourlocalexperiment will never see particles, but such states will have more energy overall than thevacuumstate.

You might be tempted to ask: But are the particles really there? How can there be zeroparticles in the universe as a whole, and yet we might see particles when we look in anyparticularlocation?

Butwe’renotdealingwithatheoryofparticles;it’satheoryoffields.Particlesarewhatwesee when we observe the theory in particular ways. We shouldn’t be asking, “How manyparticles are there, really?” We should be asking, “What are the possible measurementoutcomeswhenweobserveaquantumstateinthisspecificway?”Ameasurementoftheform“Howmanyparticlesarethere intheentireuniverse?” is fundamentallydifferent fromoneoftheform“Howmanyparticlesarethereinthisroom?”Sodifferentthat,justasforpositionandmomentum,noquantumstatewillgivedefiniteanswers forbothquestionsat thesame time.Thenumberofparticlesweseeisn’tanabsolutereality,itdependsonhowwelookatthestate.

This leads us directly to an important property of quantum field theory: the entanglementbetweenpartsofthefieldindifferentregionsofspace.

Imaginedividingtheuniverseintotworegionsbydrawinganimaginaryplanesomewhereinspace. Call the regions “left” and “right” for convenience. Classically, since fields live

everywhere, toconstructanyparticular fieldconfigurationwewouldhavetospecifywhatthefieldisdoingbothintheleftregionandintherightregion.Ifthereisamismatchofthevalueofthefieldsacrosstheboundary,thatwillcorrespondtoasharpdiscontinuityintheprofileofthefieldoverall.That’sconceivable,butitcostsenergyforthefieldtochangefrompointtopoint,soadiscontinuous jump impliesa largeamountof energyat thatpoint.This iswhyordinaryfieldconfigurationstendtovarysmoothly,ratherthansuddenly.

At the quantum level, the classical statement “The field value tends to match across theboundary”turnsinto“Thefieldsintheleftandrightregionstendtobehighlyentangledwitheachother.”Wecanconsiderquantumstateswherethetworegionsareunentangled,buttherewouldbeaninfiniteamountofenergyattheboundary.

This reasoning extends further. Imagine dividing up all of space into equal-sized boxes.Classically,thefieldwouldbedoingsomethingineachbox,buttoavoidinfiniteenergydensitiesthe valuesmustmatch at the boundaries between boxes. In quantum field theory, therefore,what’shappening inoneboxmustbehighlyentangledwithwhat’shappening inneighboringboxes.

That’s not all. If a box is entangled with its neighbors, and those neighboring boxes areentangledwiththeirneighbors,itstandstoreasonthatthefieldsinouroriginalboxshouldbeentangled not onlywith its neighbors, butwith the fields one box away. (That’s not logicallynecessary,butitseemsreasonableinthiscase,andacarefulcalculationaffirmsthatitistrue.)Therewillbealotlessentanglementwiththefieldsoneboxawaythanfordirectneighbors,buttherewillstillbesomethere.Andindeedthispatterncontinuesallthroughoutspace:thefieldsinanyoneboxareentangledwith the fields ineveryotherbox in theuniverse,although theamount of entanglement becomes less and less as we consider boxes that are farther andfartherapart.

Thatmay seem like a stretch, since after all there are an infinite number of boxes in aninfinitelybiguniverse.Canthefieldsinonelittleregion,say,asinglecubiccentimeter,reallybeentangledwithfieldsineveryothercubiccentimeteroftheuniverse?

Yes, they can. In field theory, even a single cubic centimeter (or a box of any other size)contains an infinite number of degrees of freedom. Remember that we defined a degree offreedominChapterFourasanumberneededtospecifythestateofasystem,suchas“position”or“spin.”Infieldtheory,thereareaninfinitenumberofdegreesoffreedominanyfiniteregion:ateverypointinspace,thevalueofthefieldatthatpointisaseparatedegreeoffreedom.Andthereareaninfinitenumberofpointsinspace,eveninjustasmallregion.

Quantum-mechanically, the space of all the possible wave functions for a system is thatsystem’sHilbertspace.So theHilbertspacedescribinganyregion inquantumfield theory isinfinite-dimensional,becausethereareaninfinitenumberofdegreesoffreedom.Aswe’llsee,thatmightnotcontinuetoholdtrueinthecorrecttheoryofreality;therearereasonstothinkthat quantum gravity features only a finite number of degrees of freedom in a region. Butquantumfieldtheory,withoutgravity,allowsforinfinitepossibilitiesinanytinybox.

Those degrees of freedom share a lot of entanglement with the degrees of freedomelsewhere in space. To drive home just howmuch, imagine starting with the vacuum state,taking one of those one-cubic-centimeter boxes, and poking the quantum fields inside. By“poking”wemeananywaywecouldconceivably imagineaffecting the field just in that localregion, bymeasuring it or otherwise interactingwith it.Weknow thatmeasuringaquantumstatechangesitintoanotherstate(indeed,todifferentstatesoneachbranchofthenewwavefunction). Do you think that by poking the state strictly inside a given box, it’s possible toinstantlychangethestateoutsidethebox?

Ifyouknowalittlerelativity,youmightbetemptedtoanswer“no”—itshouldtaketimeforany effects to propagate to faraway regions. But then you remember the EPR thoughtexperiment,whereAlice’smeasurementonaspincanaffectthequantumstateofBob’sspin,nomatterhowfarawaytheyarefromeachother.Entanglementisthesecretingredient.Andwe

justsaidthatthevacuumstateinquantumfieldtheoryishighlyentangled,suchthateveryboxisentangledwitheveryotherbox.Graduallyyouwillbegintowonderwhetherpokingthefieldinoneboxmightbeabletocausedrasticchangesintherestofthestate,evenveryfaraway.

Indeeditcan.Bypokingaquantumfieldinonetinyregionofspace,it’spossibletoturnthequantum state of the whole universe into literally any state at all. Technically this result isknownastheReeh-Schliedertheorem,butithasalsobeencalledtheTajMahaltheorem.That’sbecause it implies thatwithout leavingmyroom, Icandoanexperimentandgetanoutcomethatimpliesthereisnow,suddenly,acopyoftheTajMahalonthemoon.(Oranyotherbuilding,atanyotherlocationintheuniverse.)

Don’t get too excited.We can’t purposefully force theTajMahal to be created, or reliablybringanythingparticular intoexistence. In theEPRexampleAlicecanmeasureherspin,butshe can’t guarantee what outcome that measurement is going to get. The Reeh-Schliedertheoremimpliesthatifwemeasurequantumfieldslocally,thereissomemeasurementoutcomewecouldget thatwouldbeassociatedwithaTajMahal suddenlybeingon themoon.Butnomatterhowhardwe try, theprobabilityofactuallygetting thatoutcomewillbe really, really,really tiny. Almost all the time, a localmeasurement leaves distant parts of theworld prettymuchunaltered.Likemanyremarkableresultsinquantummechanics,it’snotapracticalworry.

Apopular after-dinnerdiscussion among certain circles is “Shouldwebe surprisedby theReeh-Schliedertheorem,ornot?”Itcertainlyseemssurprisingthatwecandoameasurementinourbasementthatturnsthestateoftheuniverseintoliterallyanything.Assurprisingthingsgo,that’s up there. But the other side argues that once you understand entanglement, andappreciatethatthingscantechnicallybepossiblebutaresoincrediblyimprobablethatitreallydoesn’tmatter,weshouldn’tbeverysurprisedafterall.Lookedatintherightway,thepotentialforaTajMahalonthemoonwasthereallalong, insometinypartof thequantumstate.Ourexperimentsimplylifteditoutofthevacuumbybranchingthewavefunctioninanappropriateway.

Ithinkit’sokaytobesurprised.Butmoreimportant,weshouldappreciatetherichnessandcomplexityofthevacuum.Inquantumfieldtheory,evenemptyspaceisanexcitingplacetobe.

13BreathinginEmptySpace

FindingGravitywithinQuantumMechanics

Quantum field theory is able to successfully account for everyexperiment everperformedbyhumanbeings.Whenitcomestodescribingreality,it’sthebestapproachwehave.It’sthereforeextremely tempting to imagine that future physical theories will be set within the broadparadigmofquantumfieldtheory,orperhapssmallvariationsthereof.Butgravity,atleastwhenitbecomesstrong,doesn’tseemtobewelldescribedbyquantum

field theory. So in this chapter we’ll ask whether we can make progress by attacking theproblemfromadifferentangle.Following Feynman, physicists love to remind one another that nobody really understands

quantummechanics.Meanwhile, they have long lamented that nobody understands quantumgravity.Maybethesetwolacksofunderstandingarerelated.Gravity,whichdescribesthestateofspacetimeitselfratherthanjustparticlesorfieldsmovingwithinspacetime,presentsspecialchallengeswhenwetrytodescribeitinquantumterms.Perhapsthatshouldn’tbesurprising,ifwedon’tthinkwefullyunderstandquantummechanicsitself.It’spossiblethatthinkingaboutthefoundationsofquantumtheory—inparticular,theMany-Worldsperspectivethattheworldisjust a wave function, and everything else emerges out of that—will shed new light on howcurvedspacetimeemergesfromquantumunderpinnings.Ourself-appointed task isoneof reverseengineering.Rather than takingclassicalgeneral

relativityandquantizing it,wewill try to findgravitywithinquantummechanics.That is,wewill take the basic ingredients of quantum theory—wave functions, Schrödinger’s equation,entanglement—and ask under what circumstances we can obtain emergent branches of thewavefunctionthatlooklikequantumfieldspropagatinginacurvedspacetime.Uptothispointinthebook,basicallyeverythingwe’vetalkedaboutiseitherwellunderstood

andestablisheddoctrine(suchastheessentialsofquantummechanics),oratleastaplausibleandrespectablehypothesis(theMany-Worldsapproach).Nowwe’vereachedtheedgeofwhatis safely understood, and will be venturing out into uncharted territory. We’ll be looking atspeculativeideasthatmightbeimportanttounderstandingquantumspacetimeandcosmology.But they might not be. Only years, possibly decades, of further investigation will reveal theanswerwithanyconfidence.Byallmeanstaketheseideasasprovocationstofurtherthinking,andkeepaneyeonwherethediscussiongoesintimestocome,butkeepinmindtheintrinsicuncertainty that comes with wrestling with hard problems at the bleeding edge of ourunderstanding.

AlbertEinsteinoncemusedtoacolleague,“OnquantumtheoryIusemorebraingreasethanrelativity.”Butitwashiscontributionstorelativitythatmadehimanintellectualsuperstar.Like“quantummechanics,”“relativity”doesnotrefertoaspecificphysicaltheory,butrather

aframeworkwithinwhichtheoriescanbeconstructed.Theoriesthatare“relativistic”shareacommonpictureofthenatureofspaceandtime,oneinwhichthephysicalworldisdescribedbyeventshappeninginasingleunified“spacetime.”Evenbeforerelativity,itwasstillpossibletotalk about spacetime in Newtonian physics: there is three-dimensional space, and onedimensionof time,and to locateanevent in theuniverseyouhave tospecifybothwhere theeventisinspaceandwhenitoccursintime.ButbeforeEinstein,therewasn’tmuchmotivationfor combining them into a single four-dimensional concept. Once relativity came along, thatbecameanaturalstep.There are two big ideas that go under the name of “the theory of relativity,” the special

theoryandthegeneraltheory.Specialrelativity,whichcametogetherin1905,isbasedontheideathateveryonemeasureslighttotravelatthesamespeedinemptyspace.Combiningthatinsightwithaninsistencethatthereisnoabsoluteframeofmotionleadsusdirectlytotheideathat timeand space are “relative.”Spacetime is universal and agreeduponby everyone, but

howwedivvyitupinto“space”and“time”willbedifferentfordifferentobservers.Specialrelativity isa frameworkthat includesmanyspecificphysical theories,allofwhich

aredubbed“relativistic.”Classicalelectromagnetism,puttogetherbyJamesClerkMaxwell inthe 1860s, is a relativistic theory even though it was invented before relativity; the need tobetterunderstandthesymmetriesofelectromagnetismwasadrivingforcebehindwhyrelativitywasinventedinthefirstplace.(Sometimespeoplemisusetheword“classical”toinclude“non-relativistic,” but it’s better to reserve it to mean “non-quantum.”) Quantum mechanics andspecialrelativityare100percentcompatiblewitheachother.Thequantumfieldtheoriesusedinmodernparticlephysicsarerelativistictotheircores.The other big idea in relativity came ten years later, when Einstein proposed general

relativity, his theory of gravity and curved spacetime. The crucial insight was that four-dimensionalspacetimeisn’t justastaticbackgroundonwhichtheinterestingpartsofphysicstakeplace;ithasalifeofitsown.Spacetimecanbendandwarp,anddoessoinresponsetothepresence of matter and energy. We grow up learning about the flat geometry described byEuclid, inwhich initiallyparallel linesremainparallel foreverandtheangles insideatrianglealwaysaddupto180degrees.Spacetime,Einsteinrealized,hasanon-Euclideangeometry, inwhichthesevenerablefactsarenolongerthecase.Initiallyparallelraysoflight,forexample,can be focused together while moving through empty space. The effects of this warping ofgeometry are what we recognize as “gravity.” General relativity came with numerous mind-stretchingconsequences,suchastheexpansionoftheuniverseandtheexistenceofblackholes,thoughithastakenphysicistsalongtimetoappreciatewhatthoseconsequencesare.Specialrelativityisaframework,butgeneralrelativityisaspecifictheory.JustlikeNewton’s

laws govern the evolution of a classical system or the Schrödinger equation governs theevolutionofaquantumwavefunction,Einsteinderivedanequationthatgovernsthecurvatureofspacetime.AswithSchrödinger’sequation,it’sfuntoactuallyseeEinstein’sequationwrittenout,evenifwedon’tbotherwithallthedetails:

Rμν—(½)Rgμν=8πGTμν

Themaths behindEinstein’s equation is formidable, but the basic idea is simple, andwaspithilysummarizedbyJohnWheeler:mattertellsspacetimehowtocurve,andspacetimetellsmatterhowtomove.Theleft-handsidemeasuresthecurvatureofspacetime,whiletheright-handsidecharacterizesenergy-likequantities,includingmomentum,pressure,andmass.Generalrelativityisclassical.Thegeometryofspacetimeisunique,evolvesdeterministically,

and can in principle bemeasured to arbitrary precisionwithout disturbing it.Once quantummechanicscamealong,itwasperfectlynaturaltotryto“quantize”generalrelativity,obtainingaquantumtheoryofgravity.Easiersaidthandone.Whatmakesrelativityspecialisthatit’satheory of spacetime rather than a theory of stuff within spacetime. Other quantum theoriesdescribewave functions that assign probabilities to observing things at definite,well-definedlocations in space and moments in time. Quantum gravity, by contrast, will have to be aquantumtheoryofspacetimeitself.Thatraisessomeissues.Einstein,naturally,wasoneofthefirsttoappreciatetheproblem.In1936,hemusedonthe

difficultyofevenimagininghowtoapplytheprinciplesofquantummechanicstothenatureofspacetime:

PerhapsthesuccessoftheHeisenbergmethodpointstoapurelyalgebraicalmethodofdescriptionofnature,thatistotheeliminationofcontinuousfunctionsfromphysics.Then,however,wemustalsogive up, by principle, the space-time continuum. It is not unimaginable that human ingenuity willsomedayfindmethodswhichwillmakeitpossibletoproceedalongsuchapath.Atthepresenttime,however,suchaprogramlookslikeanattempttobreatheinemptyspace.

Here Einstein is contemplating Heisenberg’s approach to quantum theory, which you’llrememberprovidedadescriptionintermsofexplicitquantumjumpswithouttryingtofillinthedetails about microscopic processes happening along the way. Similar worries persist if weswitchtoamoreSchrödingerianpointofviewwithwavefunctions.Presumablywewouldneedawavefunctionthatassignsamplitudestodifferentpossiblegeometriesofspacetime.Butifweimagine, forexample,twobranchesofsuchawavefunctionthatdescribedifferentspacetimegeometries, there is no unique way of specifying that two events in the two branchescorrespondtothe“same”pointinspace-time.Thereisnouniquemap,inotherwords,betweentwodifferentgeometries.Consideratwo-dimensionalsphereandtorus.Imaginethatafriendofyourspicksoutapoint

onasphere,andthenasksyoutopickout“thesame”pointonthetorus.You’dbestymied,andforgoodreason;there’snowaytodoit.

Apparently,spacetimecan’tplaythesamecentralroleinquantumgravitythatitdoesintherest of physics. There isn’t a single spacetime, there’s a superposition of many differentspacetime geometries. We can’t ask what the probability might be to find an electron at acertainpointinspace,sincethere’snoobjectivewaytospecifywhichpointwe’retalkingabout.Quantumgravity, then,comeswithasetofconceptual issuesthatdistinguish it fromother

quantum-mechanicaltheories.Theseissuescanhaveimportantramificationsforthenatureofour universe, including the question of what happened at the beginning, or if there was abeginningat all.Wecanevenaskwhether spaceand timeare themselves fundamental, or iftheyemergeoutofsomethingdeeper.

Just like the foundations of quantummechanics, the field of quantum gravity was relativelyignoredfordecadesasphysicistsconcentratedonotherthings.Notcompletely;HughEverettwas inspired to propose the Many-Worlds approach in part by thinking about the quantumtheory of the entire universe, where gravity plays an important role, and his mentor, JohnWheeler,worried about the problem for years. But even putting aside the conceptual issues,otherobstaclesgotinthewayofmakingseriousprogressonquantizinggravity.Amajorroadblockisthedifficultyofgettingdirectexperimentaldata.Gravityisaveryweak

force; the electric repulsion between two electrons is about 1043 times stronger than theirgravitational attraction. In any realistic experiment involving just a few particles, where wemightexpectquantumeffectstobevisible,theforceofgravityisutterlynegligiblecomparedtoother influences. We can imagine building a particle accelerator powerful enough to smashparticles together at the Planck energy, where quantum gravity should become important.Unfortunately, if we simply scale up the technology in current machines, the resultingacceleratorwouldhavetobelight-yearsindiameter.It’snotafeasibleconstructionprojectatthistime.Therearealsotechnicalproblemswiththetheory itself, inadditiontotheconceptualones

just mentioned. General relativity is a classical field theory. The field involved is called themetric. (The symbol gμν in the middle of Einstein’s equation represents the metric, and theother quantities depend on it.) Theword “metric” ultimately derives from theGreekmetron,“somethingusedtomeasure,”andthat’sexactlywhatthemetricfieldallowsustodo.Givenapaththroughspace-time,themetrictellsusthedistancealongthatpath.Themetricessentiallyupdates Pythagoras’s theorem, which works in flat Euclidean geometry but has to begeneralizedwhen spacetime is curved. Knowing the length of every curve suffices to fix thegeometryofspacetimeateverypoint.Spacetimehasametric even in special relativity, or for thatmatter inNewtonianphysics.

Butthatmetricisrigid,unchanging,andflat—thecurvatureofspacetimeiszeroateverypoint.The big insight of general relativity was to make the metric field into something that isdynamicalandaffectedbymatterandenergy.Wecanattempttoquantizethatfieldjustaswewould any other. Small ripples in the quantized gravitational field look like particles calledgravitons, just like ripples in the electromagnetic field look like photons. Nobody has everdetectedagraviton,and it’spossible thatnobodyeverwill, since thegravitational force issoincredibly weak. But if we accept the basic principles of general relativity and quantummechanics,theexistenceofgravitonsisinevitable.Wecanthenaskwhathappenswhengravitonsscatteroffeachotheroroffotherparticles.

Sadly,whatwefindisthatthetheorypredictsnonsense,ifitpredictsanythingatall.Aninfinitenumberofinputparametersareneededtocalculateanyparticularquantityofinterest,sothetheoryhasnopredictivepower.Wecan restrictourattention toan “effective” field theoryofgravity,wherebyfiatwelimitourattentiontolongwavelengthsandlowenergies.That’swhatallowsustocalculatethegravitationalfieldinthesolarsystem,eveninquantumgravity.Butifwe want a theory of everything, or at least a theory of gravity that is valid at all possibleenergies,we’restuck.Somethingdramaticiscalledfor.The most popular contemporary approach to quantum gravity is string theory, which

replacesparticlesby little loopsorsegmentsofone-dimensional“string.” (Don’taskwhat thestringsaremadeof—stringstuffiswhateverythingelseismadeof.)Thestringsthemselvesare

incredibly small, so much so that they appear like particles when we observe them from adistance.String theorywas initially proposed to help understand the strong nuclear force, but that

didn’tworkout.Oneof theproblemswas that the theory inevitablypredicts theexistenceofparticles that look and behave exactly like gravitons. That was initially perceived as anannoyance, but pretty soon physicists thought to themselves, “Hmm, gravity actually exists.Maybestringtheoryisaquantumtheoryofgravity?”Thatturnsouttobetrue,andevenbetterthereisabonus:thetheorymakesfinitepredictionsforallphysicalquantities,withoutneedingan infinite number of input parameters. The popularity of strings exploded in 1984 whenMichaelGreenandJohnSchwarzshowedthatthetheoryismathematicallyconsistent.Today, string theory is themostpursuedapproach toexploringquantumgravitybyawide

margin, although other ideasmaintain their adherents. The second-most-popular approach isloopquantumgravity,whichbeganasawayofdirectlyquantizinggeneralrelativitybyusingacleverchoiceofvariables—ratherthanlookingatthecurvatureofspacetimeateachpoint,weconsiderhowvectorsarerotatedwhentheytravelaroundclosedloopsinspace.(Ifspaceisflat,theydon’trotateatall,whileifspaceiscurved,theycanrotatebyalot.)Stringtheoryaspiresto be a theory of all the forces andmatter at once,while loopquantumgravity only aims atgravityitself.Unfortunately,theobstaclestogatheringexperimentaldatarelevanttoquantumgravityareequallyformidableforallthealternatives,sowe’restucknotreallyknowingwhichapproach(ifany)isontherighttrack.Whilestringtheoryhasbeensomewhatsuccessfulindealingwiththetechnicalproblemsof

quantum gravity, it hasn’t shed much light on the conceptual problems. Indeed, one way ofthinking about different approacheswithin the quantum-gravity community is to ask howweshouldthinkabouttheconceptualsideofthings.Astringtheoristislikelytobelievethatifwetakecareofallthetechnicalissues,theconceptualproblemswilleventuallyresolvethemselves.Someone who thinks otherwise might be nudged toward loop quantum gravity or anotheralternativeapproach.Whenthedatadon’tpointonewayortheother,opinionstendtobecomedeeplyentrenched.Stringtheory,loopquantumgravity,andotherideasshareacommonpattern:theystartwith

a set of classical variables, then quantize. From the perspectivewe’ve been following in thisbook,that’salittlebackward.Natureisquantumfromthestart,describedbyawavefunctionevolvingaccordingtoanappropriateversionoftheSchrödingerequation.Thingslike“space”and “fields” and “particles” are useful ways of talking about that wave function in anappropriateclassicallimit.Wedon’twanttostartwithspaceandfieldsandquantizethem;wewanttoextractthemfromanintrinsicallyquantumwavefunction.

How can we find “space” within a wave function?We want to identify features of the wavefunctionthatresemblespaceasweknowit,andinparticularsomethingthatwouldcorrespondto a metric that defines distances. So let’s think about how distances show up in ordinaryquantum field theory. For simplicity, let’s just think about distances in space;we’ll talk laterabouthowtimemightenterintothegame.There’soneobviousplacethatdistancesshowupinquantumfieldtheory,whichwe’veseen

inthelastchapter:inemptyspace,fieldsindifferentregionsareentangledwitheachother,andregions that are far away are less entangled than ones that are nearby. Unlike “space,” theconceptof“entanglement”isalwaysavailabletousinanyabstractquantumwavefunction.Soperhapswecangetsomepurchasehere, lookingat theentanglementstructureofstatesandusing that to define distances. What we need is a quantitative measure of how entangled aquantumsubsystemactuallyis.Happily,suchameasureexists:it’stheentropy.John von Neumann showed how quantum mechanics introduces a notion of entropy that

parallels the classical definition. As explained by Ludwig Boltzmann, we start with a set ofconstituents that canmix together in variousways, like atoms andmolecules in a fluid. Theentropy is then a way of counting the number of ways those constituents can be arrangedwithoutchangingthemacroscopicappearanceof thesystem.Entropy isrelated to ignorance:high-entropystatesarethoseforwhichwedon’tknowmuchaboutthemicroscopicdetailsofasystemjustfromknowingitsobservablefeatures.VonNeumannentropy,meanwhile,ispurelyquantummechanicalinnature,andarisesfrom

entanglement. Consider a quantum system that is divided into two parts. It could be twoelectrons, or the quantum fields in two different regions of space. The system as awhole isdescribedbyawavefunction,asusual.Ithassomedefinitequantumstate,evenifwecanonlypredictmeasurement outcomes probabilistically. But as long as the two parts are entangled,thereisonlytheonewavefunctionforthewholething,notaseparatewavefunctionforeachpart.Theparts,inotherwords,arenotindefinitequantumstatesoftheirown.Von Neumann showed that, for many purposes, the fact that entangled subsystems don’t

havedefinitewavefunctionsof theirownisanalogoustohavingawavefunction,butwe justdon’t know what it is. Quantum subsystems, in other words, closely resemble the classicalsituationwhere there aremany possible states that lookmacroscopically the same. And thisuncertaintycanbequantifiedintowhatwenowcalltheentanglemententropy.Thehighertheentropyofaquantumsubsystem,themoreit’sentangledwiththeoutsideworld.Thinkabouttwoqubits,onebelongingtoAliceandtheothertoBob.Itmightbethattheyare

unentangled,soeachqubithas itsownwave function, forexample,anequalsuperpositionofspin-upandspin-down.Inthatcase,theentanglemententropyofeachqubitiszero.Evenifwecanonlypredictmeasurement outcomesprobabilistically, each subsystem is still in adefinitequantumstate.Butimaginethatthetwoqubitsareentangled,inanequalsuperpositionof“bothqubitsare

spin-up” and “both qubits are spin-down.” Alice’s qubit doesn’t have its own wave function,because it’s entangled with Bob’s. Indeed, Bob could perform a measurement of his spin,branchingthewavefunction,sothatnowtherearetwocopiesofAlice,eachofwhomhasaspinin a definite state. But neither copy of Alice knows which state that is; she’s in a state ofignorance,wherethebestshecandoissaythatthereisafifty-fiftychanceherqubitisspin-uporspin-down.Notethesubtledifference:Alice’squbitisnotinaquantumsuperpositionwhereshedoesn’tknowwhatthemeasurementoutcomewillbe;it’sinastateoneachbranchthatwillgive a definite measurement outcome, but she doesn’t know which state it is. We thereforedescribeherqubitashavinganonzeroentropy.VonNeumann’sideawasthatweshouldascribeanonzeroentropytoAlice’squbitevenbeforeBobmeasureshis,becauseafterallshedoesn’tevenknowwhetherhe’sdoneameasurement.That’stheentanglemententropy.

Let’sseehowentanglemententropyappearsinquantumfieldtheory.Forgettingaboutgravityfor a second, consider a regionof empty space in the vacuumstate, specifiedby aboundaryseparating inside the region from outside. Empty space is a richly textured place, full ofquantumdegreesoffreedomthatwecanthinkofasmodesofvibratingfields.Themodesinsidetheregionwillbeentangledwith themodesoutside,so theregionhasanentropyassociatedwithit,eveniftheoverallstateissimplythevacuum.We can even calculate what that entropy is. The answer is: infinity. This is a common

complication with quantum field theory, that many questions of apparent physical relevancehave seemingly infinite answers because there are an infinite number of possibleways for afieldtovibrate.Butjustaswedidforthevacuumenergyinthelastchapter,wecanaskwhathappenswhenweimposeacutoff,allowingonlymodeslongerthanacertainwavelength.Theresulting entropy is finite, and it turns out to be naturally proportional to the area of theregion’sboundary.Thereasonisn’thardtounderstand:fieldvibrationsinonepartofspaceareentangled with regions all over, but most of the entanglement is concentrated on nearbyregions.Thetotalentropyofaregionofemptyspacedependsontheamountofentanglementacrosstheboundary,whichisproportionaltohowbigthatboundaryis—itsarea.That’san intriguing featureofquantumfield theory.Pickoutaregionwithinemptyspace,

andtheentropyof thatregion isproportional to theareaof itsboundary.Thatrelateson theone hand a geometric quantity, the area of a region, to a “matter” quantity, the entropycontainedinside.ItallsoundsvaguelyreminiscentofEinstein’sequation,whichalsoconnectsgeometry(thecurvatureofspacetime)toamatterquantity(energy).Aretheysomehowrelated?

Theycouldbe,aswaspointedoutinaprovocative1995paperbyTedJacobson,aningeniousphysicist at the University of Maryland. In ordinary quantum field theory without gravity,entropyisproportionaltoareainthevacuumstate,butinhigher-energystatesitdoesn’thaveto be. Jacobson postulated that there’s something special about gravity: when gravity isincluded,theentropyofaregionisalwaysproportionalto itsboundaryarea.That’snotatallwhatwewouldexpect inquantum field theory,butmaybe ithappensoncegravityenters thegame.Wecanimaginethatitmightbethecase,andseewhathappens.Whathappensisprettywonderful.Jacobsonpositedthattheareaofasurfaceisproportional

totheentropyoftheregionitencloses.Areaisageometricquantity;wecan’tcalculatetheareaofasurfacewithoutknowingsomethingaboutthegeometryofthespaceitisapartof.Jacobsonnotedthatwecouldrelatetheareaofaverysmallsurfacetothesamegeometricquantitythatappears on the left-hand side of Einstein’s equation. Meanwhile, entropy tells us somethingabout“matter,”broadlyconstrued;aboutthestuffthatislivingwithinspacetime.Theconceptofentropyoriginallyarosewithinthermodynamics,where itwasrelatedto theheat leavingasystem.Andheatisaformofenergy.Jacobsonalsoarguedthatthisentropycouldbedirectlyrelated to the energy term appearing on the right-hand side of Einstein’s equation. Throughthesemaneuvers hewas able toderive Einstein’s equation for general relativity, rather thandirectlypostulatingit,asEinsteindid.Tosaythesamethingmoredirectly,weconsiderasmallregioninflatspacetime.Ithassome

entropy,because themodes inside the regionareentangledwith thoseoutside.Now imaginechangingthequantumstatealittlebit,sothatwedecreasetheamountbywhichthatregionisentangled, and therefore decrease its entropy. In Jacobson’s picture, the area bounding ourregionchangesinresponse,shrinkingbyabit.Andheshowsthatthisresponseofthegeometryof spacetime toachange in thequantumstate isequivalent toEinstein’sequationofgeneralrelativity,relatingcurvaturetoenergy.This was the beginning of a surge of interest in what is now called “entropic” or

“thermodynamic” gravity; other important contributions were made by Thanu Padmanabhan(2009)andErikVerlinde(2010).Thebehaviorofspacetimeingeneralrelativitycanbethoughtofassimplythenaturaltendencyofsystemstomovetowardconfigurationsofhigherentropy.Thisisafairlyradicalchangeofperspective.Einsteinthoughtintermsofenergy,adefinite

quantityassociatedwithparticularconfigurationsofstuffintheuniverse.Jacobsonandothershave argued thatwe can reach the same conclusions by thinking about entropy, a collectivephenomenonthatemergesfromthemutualinteractionofmanysmallconstituentsofasystem.This simple shift in focus might offer a crucial way forward in our quest to discover afundamentallyquantumtheoryofgravity.

Jacobsonwasn’thimselfproposingatheoryofquantumgravity;hewaspointingtoanewwaytoderive Einstein’s equation for classical general relativity, with quantum fields acting as thesourceofenergy.Theappearanceofwordslike“area”and“regionofspace”shouldindicatetous that the above discussion treated spacetime as a tangible, classical thing. But given thecentral role thatentanglemententropyplays inhisderivation, it’snatural toaskwhetherwemightadapt thebasic ideas toanapproach that ismore intrinsicallyquantumfromthestart,wherespaceitselfemergesfromthewavefunction.In Many-Worlds, a wave function is just an abstract vector living within the super-high-

dimensional mathematical construct of Hilbert space. Usually we make wave functions bystartingwithsomethingclassicalandquantizingit,whichgivesusanimmediatehandleonwhatthewave function is supposed to represent, thebasicparts fromwhich it is constructed.Butherewedon’thaveanysuchluxury.AllwehaveisthestateitselfandSchrödinger’sequation.We speak abstractly of “degrees of freedom,” but they aren’t the quantized version of anyreadily identifiable classical stuff—they are the quantum-mechanical essence out of whichspacetime,andeverythingelse,emerges.JohnWheelerusedtotalkabouttheideaof“ItfromBit,”suggestingthatthephysicalworldarose(somehow)outofinformation.Thesedays,whenentanglementofquantumdegreesoffreedomisthemainfocus,weliketotalkabout“ItfromQubit.”IfwelookbackattheSchrödingerequation,itsaysthattherateatwhichthewavefunction

changeswithtimeisgovernedbytheHamiltonian.RememberthattheHamiltonianisawayofdescribinghowmuchenergythesystemcontains,andit’sacompactwayofcapturingallofthesystem’sdynamics.AstandardfeatureofHamiltoniansintherealworldisdynamicallocality—subsystems interactwith other subsystems onlywhen they are next to each other, notwhentheyarefaraway.Influencescantravelthroughspace,butonlyatspeedslessthanorequaltothespeedoflight.Soaneventatoneparticularmomentonlyimmediatelyaffectswhat’sgoingonatitspresentlocation.With the problemwe’ve assigned to ourselves—how does space emerge from an abstract

quantumwavefunction?—wedon’thavetheconvenienceofstartingwith individualpartsandaskinghow they interact.Weknowwhat “time”means in this context—it’s right there in theSchrödingerequation,thelettert—butwedon’thaveparticles,orfields,orevenlocationsinathree-dimensionalworld.We’recaughtbreathinginemptyspace,andneedto lookforoxygenwherewecanfindit.Happily,thisisacasewherereverseengineeringworksquitewell.Ratherthanstartingwith

individualpiecesofa systemandaskinghow they interact,wecango theotherwayaround:Giventhesystemasawhole(theabstractquantumwavefunction)anditsHamiltonian,isthere

asensiblewaytobreakitupintosubsystems?It’slikebuyingslicedbreadallyourlife,andthenbeinghandedanun-slicedloaf.Therearemanywayswecouldimagineslicingit; isthereoneparticularwaythat’sclearlythebest?Yes, there is, if we believe that locality is an important feature of the realworld.We can

tackletheproblembitbybit,orqubitbyqubit,atanyrate.Agenericquantumstatecanbe thoughtofasasuperpositionofasetofbasis stateswith

definite fixed energy. (Just like a generic state of a spinning electron can be thought of as asuperpositionofanelectronthatisdefinitelyspin-upandonethatisdefinitelyspin-down.)TheHamiltoniantellsuswhattheactualenergyisforeachpossibledefinite-energystate.Giventhatlistofpossibleenergies,wecanaskwhetheranyparticularwayofdividingthewavefunctioninto subsystems implies that those subsystems interact “locally.” In fact, for a random list ofenergies,therewon’tbeanywayofdividingthewavefunctionintolocalsubsystems,butfortherightkindofHamiltonian,therewillbeexactlyonesuchway.Demandingthatphysicslooklocaltellsushowtodecomposeourquantumsystemintoacollectionofdegreesoffreedom.Inotherwords,wedon’tneedtostartwithasetof fundamentalbuildingblocksofreality,

thenstickthemtogethertomaketheworld.Wecanstartwiththeworld,andaskifthereisaway to think about it as a collection of fundamental building blocks. With the right kind ofHamiltonian,therewillbe,andallofourdataandexperienceoftheworldsuggeststhatwedohave the right kind of Hamiltonian. It’s easy to imagine possible worlds where the laws ofphysicsweren’tlocalatall.Butit’shardtoimaginewhatlifewouldbelikeinsuchaworld,orevenwhetherlifewouldbepossible;thelocalityofphysicalinteractionshelpsbringordertotheuniverse.

Wecanbegintoseehowspaceitselfemergesfromthewavefunction.Whenwesaythatthere’sauniquewayofdividingupoursystemintodegreesoffreedomthatinteractlocallywiththeirneighbors,allwereallymeanisthateachdegreeoffreedominteractswithonlyasmallnumberofotherdegreesoffreedom.Thenotionsof“local”and“nearest”aren’timposedfromthestart—theypopoutfromthefactthattheseinteractionsareveryspecial.Thewaytothinkaboutitisn’t “degrees of freedom interact only when they are nearby,” but rather “we define twodegreesof freedomtobe ‘nearby’whentheydirectly interactwitheachother,and ‘faraway’when they don’t.” A long list of abstract degrees of freedom has been knit together into anetwork, inwhicheachdegreeoffreedomisconnectedtoasmallnumberofotherones.Thisnetworkformstheskeletononwhichspaceitselfisconstructed.That’s a start, butwewant to do evenmore.When someone asks you how far apart two

differentcitiesare,they’relookingforsomethingabitmorespecificthan“near”or“far.”Theywantanactualdistance,andthat’swhatthemetriconspacetimeordinarilyletsuscalculate.Inourabstractwavefunctiondividedupintodegreesoffreedom,wehaven’tyetconstructedafullgeometry,justanotionofnearandfar.Wecandobetter.Remembertheintuitionfromvacuumstatesinquantumfieldtheorythat

JacobsonusedtoderiveEinstein’sequation: theentanglemententropyofaregionofspace isproportionaltotheareaofitsboundary.Inourcurrentcontextofaquantumstatedescribedintermsofabstractdegreesoffreedom,wedon’tknowwhat“area”issupposedtomean.Butwedohaveentanglementbetweenthedegreesoffreedom,andforanycollectionofthemwecancomputetheirentropy.Soonceagain followingour reverse-engineeringphilosophy,wecandefine the “area” of a

collectionofdegreesoffreedomtobeproportionaltoitsentanglemententropy.Infact,wecanassertthisforeverypossiblesubsetofdegreesoffreedom,assigningareastoeverysurfacewecan imagine drawingwithin our network. Happily, mathematicians long ago figured out thatknowing the area of every possible surface in a region is enough to fully determine thegeometryofthatregion;it’scompletelyequivalenttoknowingthemetriceverywhere.Inotherwords, the combination of (1) knowing how our degrees of freedom are entangled, and (2)postulating that the entropy of any collection of degrees of freedom defines an area of theboundary around that collection, suffices to fully determine the geometry of our emergentspace.

Wecandescribethisconstructioninequivalentbutslightlylessformalterms.Pickouttwoofourspacetimedegreesoffreedom.Theywillgenerallyhavesomeentanglementbetweenthem.Iftheyweremodesofvibratingquantumfieldsinthevacuumstate,weknowexactlywhatthatdegreeofentanglementwouldbe:itwouldbehighiftheywerenearby,andlowiftheywerefaraway.Nowwearesimplythinkingtheotherwayaround.Ifthedegreesoffreedomarehighlyentangled,wedefine themtobenearby,and the fartherand fartheraway, the lessentangledtheyare.Ametriconspacehasemergedfromtheentanglementstructureofthequantumstate.Thinking thisway is a bit unusual, even for physicists, becausewe’re used to thinking of

particlesmovingthroughspace,whiletakingspaceitselfforgranted.AsweknowfromtheEPRthought experiment, twoparticles canbe completely entanglednomatter how far away theyare; there’s no necessary relationship between entanglement and distance. Here, however,we’renottalkingaboutparticlesbutaboutthefundamentalbuilding-blockdegreesoffreedomthatmakeupspaceitself.Thosearen’tentangledinanyoldway;theyarestrungtogetherinaveryspecificstructure.*

NowwecanuseJacobson’strickwithentropyandarea.Knowingtheareaofeverysurfaceinournetworkgivesusageometry,andknowing theentropyofeach region tellsus somethingabout theenergy in that region. I’vebeen involvedwith thisapproachmyself, inpapers from2016and2018withmycollaboratorsChunJun(Charles)CaoandSpyridonMichalakis.Closelyrelated ideashavebeen investigatedbyTomBanks,WillyFischler,SteveGiddings,andotherphysicists who are willing to contemplate the idea that spacetime isn’t fundamental, butemergesfromthewavefunction.Wearen’tquiteatthepointwherewecansimplysay,“Yes,thisemergentgeometryonspace

evolveswithtimeinexactlytherightwaytodescribeaspacetimethatobeysEinstein’sequationofgeneral relativity.”That’s theultimategoal, butwe’renot there yet.Whatwecando is tospecify a list of requirements under which that’s exactly what does happen. The individualrequirements seem reasonable—things like “at long distances, physics looks like an effectivequantumfieldtheory”—butmanyofthemremainunprovenasyet,andsofarthemostrigorousresultsareavailableonlyinsituationswherethegravitationalfieldisrelativelyweak.Wedon’tyet have away of describing black holes or theBig Bang, though there are some promisingideas.That’slifeasatheoreticalphysicist.Wedon’thavealltheanswers,butlet’snotlosesightof

the overall ambition: starting from an abstract quantumwave function,we have a roadmapdescribing how space emerges, with a geometry fixed by quantum entanglement, and thatgeometry seems toobey thedynamical rulesofgeneral relativity.Thereare somanycaveatsandassumptionsgoingintothisproposalthatit’shardtoknowwheretostartlistingthem.Butthereseemstobeaveryrealprospectthattheroutetounderstandingtheuniverseliesnotinquantizinggravity,butinfindinggravitywithinquantummechanics.

Youmayhavenoticedatinyimbalanceinthisdiscussion.We’vebeenaskinghowspacetimecanemergefromentanglementinquantumgravity.Butifwe’rehonest,we’vereallyonlylookedat

howspaceemerges;we’vetakentimeforgrantedassomethingthatcomesalongfortheride.Andit’spossiblethatthisapproachiscompletelyfair.Althoughrelativitytreatsspaceandtimeas if theywereonanequal footing,quantummechanicsgenerallydoesnot.TheSchrödingerequation,inparticular,treatsthemverydifferently:itliterallydescribeshowthequantumstateevolveswithtime.“Space”mayormaynotbepartofthatequation,dependingonwhatsystemwe’relookingat,buttimeisfundamental.It’splausiblethatthesymmetrybetweenspaceandtimethatwe’refamiliarwithfromrelativityisn’tbuiltintoquantumgravity,butemergesintheclassicalapproximation.It is nevertheless overwhelmingly tempting to wonderwhether time, like space,might be

emergentratherthanfundamental,andwhetherentanglementmighthaveanythingtodowithit.Theanswerisyesonbothcounts,althoughthedetailsremainalittlesketchy.If we take the Schrödinger equation at face value, time seems to be right there in a

fundamentalway. Indeed, it immediately follows that theuniverse lastseternally towardboththepastandfuture,foralmostallquantumstates.Youmightthinkthatthisconflictswiththeoft-repeated fact that theBigBangwas the beginning of our universe, butwe don’t actuallyknowthatoft-repeatedfacttobetrue.That’sapredictionofclassicalgeneralrelativity,notofquantum gravity. If quantum gravity operates according to some version of the Schrödingerequation,thenforalmostallquantumstates,timerunsfromminusinfinity inthepasttoplusinfinityinthefuture.TheBigBangmightbesimplyatransitionalphase,withaninfinitelyolduniverseprecedingit.We have to say “almost all” in these statements because there is one loophole. The

Schrödingerequationsaysthattherateofchangeofthewavefunctionisdrivenbyhowmuchenergythequantumsystemhas.Whatifweconsidersystemswhoseenergyispreciselyzero?Thenall theequationsays isthatthesystemdoesn’tevolveatall; timehasdisappearedfromthestory.You might think it’s extremely implausible that the universe has exactly zero energy, but

general relativity suggests you shouldn’t be so sure. Of course there seem to be energy-containingthingsallaroundus—stars,planets,interstellarradiation,darkmatter,darkenergy,andsoon.Butwhenyougothroughthemaths,thereisalsoacontributiontotheenergyoftheuniverse fromthegravitational field itself,which isgenerallynegative. Inacloseduniverse—onethatwrapsaroundonitselftoformacompactgeometry,likeathree-dimensionalsphereortorus,ratherthanstretchingtoinfinity—thatgravitationalenergypreciselycancelsthepositiveenergy from everything else. A closed universe has exactly zero energy, regardless ofwhat’sinside.That’saclassicalstatement,butthere’saquantum-mechanicalanaloguethatwasdeveloped

byJohnWheelerandBryceDeWitt.TheWheeler-DeWittequationsimplysaysthatthequantumstateoftheuniversedoesn’tevolveatallasafunctionoftime.Thisseemscrazy,orat least in flagrantcontradiction toourobservationalexperience.The

universecertainlyseemstoevolve.Thispuzzlehasbeencleverlylabeledtheproblemoftimeinquantumgravity,anditiswherethepossibilityofemergenttimemightcometotherescue.Ifthequantumstateof theuniverseobeystheWheeler-DeWittequation(which isplausible,butfarfromcertain),timehastobeemergentratherthanfundamental.OnewaythatmightworkwassuggestedbyDonPageandWilliamWoottersin1983.Imagine

aquantumsystemconsistingoftwoparts:aclock,andeverythingelseintheuniverse.Imaginethatboththeclockandtherestofthesystemevolveintimeasusual.Nowtakesnapshotsofthequantumstateat regular intervals,perhapsonceper secondoronceperPlanck time. Inanyparticular snapshot, thequantumstatedescribes theclock readingsomeparticular time,andtherestofthesysteminwhateverconfigurationitwasinatthattime.Thatgivesusacollectionofinstantaneousquantumstatesofthesystem.Thegreatthingaboutquantumstatesisthatwecansimplyaddthemtogether(superposing

them) tomakeanewstate.So let’smakeanewquantumstatebyadding togetherallofoursnapshots.Thisnewquantumstatedoesn’tevolveovertime;itjustexists,asweconstructeditby hand. And there is no specific time reading on the clock; the clock subsystem is in asuperpositionofallthetimesatwhichwetooksnapshots.Itdoesn’tsoundmuchlikeourworld.Buthere’sthething:withinthatsuperpositionofallthesnapshots,thestateoftheclockis

entangledwiththestateoftherestofthesystem.Ifwemeasuretheclockandseethatitreadssomeparticular time, then the rest of the universe is inwhatever state our original evolvingsystemwascaughtinatpreciselythattime.

In other words, there’s not “really” time in the superposition state, which is completely

static.Butentanglementgeneratesarelationshipbetweenwhattheclockreadsandwhattherestoftheuniverseisdoing.Andthestateoftherestoftheuniverseispreciselywhatitwouldbe if it were evolving as the original state did over time. We have replaced “time” as afundamental notion with “what the clock reads in this part of the overall quantumsuperposition.” In that way, time has emerged from a static state, thanks to the magic ofentanglement.The juryremainsoutonwhethertheenergyof theuniverseactually iszero,andtherefore

timeisemergent,oritisanyothernumber,suchthattimeisfundamental.Atthecurrentstateoftheart,itmakessensetokeepouroptionsopenandinvestigatebothpossibilities.

* In 2013, JuanMaldacena and Leonard Susskind suggested that we should think of entangled particles as beingconnected by a microscopic (and impossible-to-travel-through) wormhole in spacetime. This has been dubbed the“ER=EPRconjecture,”aftertwofamouspapersfrom1935:onebyEinsteinandNathanRosen,wheretheyintroducedthe concept of wormholes; and the other of course by Einstein, Rosen, and Boris Podolsky, where they discussedentanglement.Howfarsuchasuggestioncanbetakenisstillunclear.

14BeyondSpaceandTime

Holography,BlackHoles,andtheLimitsofLocality

BeforeStephenHawking’sdeathin2018,hewasthemostfamouslivingscientistintheworldby a comfortable margin. That celebrity was entirely deserved; not only was Hawking acharismatic and influential public figure, and not only did he have an inspirational personalstory,buthisscientificcontributionswereincrediblysignificantintheirownright.

Hawking’sgreatestachievementwasshowingthat,onceweincludetheeffectsofquantummechanics,blackholes “ain’t soblack,”ashe liked to say.Blackholesactuallyemita steadystreamofparticlesoutintospace,andthoseparticlescarryenergyawayfromtheblackhole,causing it to shrink in size. This realization led both to profound insights (black holes haveentropy)aswellasunexpectedpuzzles(wheredoestheinformationgowhenblackholesformandthenevaporateaway?).

Thefactthatblackholesradiate,andtheimplicationsofthatsurprisingidea,arethesinglebest cluewe have about the nature of quantumgravity.Hawking didn’t first construct a fulltheoryofquantumgravityandthenuseittoshowthatblackholesradiate.Instead,heusedareasonableapproximation,treatingspacetimeitselfasclassical,withdynamicalquantumfieldslivingon topof it.Wehope that this isa reasonableapproximation,anyway;but someof thepuzzling aspects of Hawking’s insight have given us second thoughts. Forty-five years afterHawking’soriginalpaperonthesubject,tryingtounderstandblack-holeradiationisstilloneofthehottesttopicsincontemporarytheoreticalphysics.

While that task is far from complete, one implication seems clear: the simple picturesketched in the last chapter,where space emerges from a set of entangled nearest-neighbordegreesof freedom, isprobablynot theentire story. It’s a verygood story, andmightbe therightstartingpointforconstructingatheoryofquantumgravity.Butitreliesheavilyontheideaof locality—what happens at one point in space can have an immediate effect only on pointsrightnextdoor.Blackholes,totheextentthatweunderstandthem,seemtobeindicatingthatnature is more subtle than that. In some circumstances the world looks like a collection ofdegreesoffreedominteractingwiththeirnearestneighbors,butwhengravitybecomesstrong,that simplepicturebreaksdown.Rather thanbeingdistributed throughout space,degreesoffreedomsqueezetogetheronasurface,and“space” ismerelyaholographicprojectionof theinformationcontainedtherein.

Locality undoubtedly plays an important role in our everyday lives, but it seems like thefundamentalnatureof realitycan’tquitebecapturedbya setof thingshappeningatpreciselocations in space.Once again,whatwehavehere is a job for theMany-Worlds approach toquantummechanics.Other formulations take space as a given andworkwithin it; thewave-function-first Everettian philosophy allows us to accept that space can appear fundamentallydifferent depending on how we look at it, if it’s a useful concept at all. Physicists are stillwrestling with the implications of this idea, but it’s already led us to some very interestingplacesindeed.

Ingeneral relativity, ablackhole is a regionof spacetime that is curved sodramatically thatnothingcanescape from it,noteven light itself.Theedgeof theblackhole,demarcating theinside from theoutside, is theeventhorizon.According to classical relativity, theareaof theeventhorizoncanonlygrow,notshrink;blackholes increase insizewhenmatterandenergyfallin,butcannotlosemasstotheoutsideworld.

Everyone thought that was true in nature until 1974, when Hawking announced thatquantum mechanics changes everything. In the presence of quantum fields, black holesnaturallyradiateparticlesintotheirsurroundings.Thoseparticleshaveablackbodyspectrum,soeveryblackholehasa temperature;moremassiveblackholesarecooler,whileverysmallblackholes are incredibly hot. The formula for the temperature of a blackhole’s radiation is

engravedonHawking’sgravestoneinWest-minsterAbbey.Particles radiated by a black hole carry away energy, causing the hole to lose mass and

eventuallyevaporateawaycompletely.WhileitwouldbenicetoobserveHawkingradiationinatelescope, it’s not going to happen for any of the black holes we know about. The Hawkingtemperatureofablackholethemassofthesunwouldbeabout0.00000006Kelvin.Anysuchsignalwouldbeswampedbyothersources,suchastheleftovermicrowaveradiationfromtheBigBang,whichhasatemperatureofabout2.7Kelvin.Evenifsuchablackholenevergrewbyaccretingmatterandradiation,itwilltakeover1067yearsforittoevaporateawaycompletely.

Thereisastandardstorythat istoldtoexplainwhyblackholesemitradiation.I’vetoldit,Hawkinghastold it,everyonetells it. Itgoes likethis:accordingtoquantumfieldtheory, thevacuum is a bubbling stew of particles popping in and out of existence, typically in pairsconsistingofoneparticleandoneanti-particle.Ordinarilywedon’tnotice,butinthevicinityofablackholeeventhorizon,oneoftheparticlescanfallinsidetheholeandthennevergetout,whiletheotherescapestotheoutsideworld.Fromtheperspectiveofsomeonewatchingfromafar, the escaping particle has positive energy, so to balance the books the infalling particlemust have negative energy, and the black hole shrinks inmass as it absorbs these negative-energyparticles.

Given our wave-functions-first Everettian perspective, there’s a more accurate way todescribe what’s happening. The particles-appearing-and-disappearing story is a colorfulmetaphor that often provides physical intuition, and this is definitely one of those cases.Butwhatwereallyhaveisaquantumwavefunctionofthefieldsneartheblackhole.Andthatwavefunctionisnotstatic;itevolvesintosomethingelse,inthiscaseasmallerblackholeplussomeparticles traveling away from it in all directions. It’s not that different from an atomwhoseelectronshaveabitofextraenergy,andwhichthereforedropdowntolower-energystatesbyemittingphotons.Thedifferenceisthattheatomeventuallyreachesastateoflowestpossibleenergyandstaysthere,whiletheblackhole(asfarasweunderstand)justdecaysawayentirely,explodingatthelastsecondinaflashofhigh-energyparticles.

The story of how black holes radiate and evaporate was derived by Hawking using thetechniquesofconventionalquantumfieldtheory,justinacurvedspacetimeofgeneralrelativityratherthanaparticlephysicist’susualno-gravitycontext.It’snotagenuinelyquantum-gravityresult; spacetime itself is treated classically, not as part of the quantum wave function. Butnothingaboutthescenarioactuallyseemstorequiredeepknowledgeofquantumgravity.Asfarasphysicistscantell,Hawkingradiation isarobustphenomenon.Wheneverwedo figureoutquantumgravity,inotherwords,itshouldreproduceHawking’sresult.

Thatraisesaproblem,onethathasbecomenotoriouswithintheoreticalphysicsastheblackhole informationpuzzle.Remember thatquantummechanics, in itsMany-Worldsversion, isadeterministictheory.Randomnessisonlyapparent,arisingfromself-locatinguncertaintywhenthe wave function branches and we don’t know which branch we’re on. But in Hawking’scalculation,black-holeradiationseemsnottobedeterministic;it’strulyrandom,evenwithoutanybranching.Startingfromaprecisequantumstatedescribingmatterthatcollapsestomakeablackhole,thereisnowayofcomputingtheprecisequantumstateoftheradiationintowhichitevaporates.Theinformationspecifyingtheoriginalstateseemstobelost.

Imaginetakingabook—maybetheveryoneyouarereadingrightnow—andthrowingitintoafire,lettingitburncompletelyaway.(Don’tworry,youcanalwaysbuymorecopies.)Itmightappear that the informationcontained in thebook is lost in the flames.But ifwe turnonourphysicist’spowersofthought-experimentingenuity,werealizethatthislossisonlyapparent.Inprinciple, ifwe captured everybit of light andheat anddust and ash from the fire, andhadperfectknowledgeofthelawsofphysics,wecouldreconstructexactlywhatwentintothefire,including all the words on the pages of the book. It’ll never happen in the real world, butphysicssaysit’sconceivable.

Most physicists think that black holes should be just like that: throw a book in, and theinformation contained in its pages shouldbe secretly encoded in the radiation that theblackhole emits. But this is not what happens, according to Hawking’s derivation of black-holeradiation;rather,theinformationinthebookappearstobetrulydestroyed.

It’s possible, of course, that this implication is correct, that the information really isdestroyed,andthatblack-holeevaporationisnothinglikeanordinaryfire.It’snotlikewehaveany experimental input oneway or the other. Butmost physicists believe that information isconserved,andthatitreallydoesgetoutsomehow.Andtheysuspectthatthesecrettogettingitoutliesinabetterunderstandingofquantumgravity.

That’seasiersaidthandone.Onewayofthinkingaboutwhyblackholesaresupposedtobeblackinthefirstplaceisthatinordertoescape,youwouldhavetobeabletotravelfasterthanlight.Hawking radiation avoids that difficulty because it actually originates right outside theeventhorizon,notdeepintheinterior.Butanybookwethrowinsidedoesindeedplungeintothe interior, with all its information intact. You might wonder whether the information issomehowcopiedontotheoutgoingradiationasthebookfallsthroughthehorizon,andcarriedout that way. Unfortunately that’s in contradiction with the basic principles of quantum

mechanics;thereisaresultcalledtheno-cloningtheoremthatsayswecan’tduplicatequantuminformationwithoutdestroyingtheoriginalcopy.

The other possibility seems to be that the book falls all the way in, but as it hits thesingularity inside the black hole, its information is somehow transferred to the outgoingradiation at the horizon. Unfortunately, that would seemingly require faster-than-lightcommunication.Or,equivalently,dynamicalnonlocality—occurrencesatonepointinspacetimeimmediatelyinfluencingwhathappenssomedistanceaway.Thiskindofnonlocalityispreciselywhatcannothappen,accordingtotheordinaryrulesofquantumfieldtheory.Thisisacluethatthoserulesmighthavetobedramaticallyrevisedoncequantumgravitybecomesimportant.*

Hawking’sproposalthatblackholesradiatedidn’tcomeoutoftheblue.ItcameinresponsetoasuggestionfromJacobBekenstein—whoatthetimewasyetanothergraduatestudentofJohnWheeler’satPrinceton—thatblackholesshouldhaveentropy.

One of themotivations behind Bekenstein’s idea was the fact that, according to classicalgeneral relativity, the area of a black hole’s event horizon can never decrease. That soundssuspiciouslylikethesecondlawofthermodynamics,accordingtowhichtheentropyofaclosedsystem can never decrease. Inspired by this similarity, physicists constructed an elaborateanalogy between the laws of thermodynamics and the behavior of black holes, according towhichthemassoftheblackholeisliketheenergyofathermodynamicsystem,andtheareaoftheeventhorizonisliketheentropy.

Bekensteinsuggestedthatitwasmorethanananalogy.Theareaoftheeventhorizonisn’tjustliketheentropy,it istheentropyoftheblackhole,oratleastproportionaltoit.Hawkingand others scoffed at the suggestion at first—if black holes have entropy like conventionalthermodynamic systems, they should also have a temperature, and then they should give offradiation!Motivated to disprove this ridiculous-sounding notion, Hawking ended up showingthat it was all true. These days we refer to the entropy of a black hole as the Bekenstein-Hawkingentropy.

Onereasonwhy this issuchaprovocativeresult is thatclassically,blackholesdon’t seemlikethingsthatshouldhaveentropyatall.They’rejustregionsofemptyspace.Yougetentropywhenyoursystemismadeofatomsorothertinyconstituents,whichcanbearrangedinmanydifferent ways while maintaining the same macroscopic appearance. What are theseconstituentssupposedtobeforablackhole?Theanswerhastocomefromquantummechanics.

It’s natural to presume that the Bekenstein-Hawking entropy of a black hole is a kind ofentanglemententropy.Therearesomedegreesoffreedominsidetheblackhole,andtheyareentangledwiththeoutsideworld.Whatarethey?

We might first guess that the degrees of freedom are simply vibrational modes of thequantumfieldsinsidetheblackhole.Thereareacoupleofproblemswiththat.Foronething,the real answer for the entropy of a region in quantum field theorywas “infinity.”We couldwrestle thatdowntoa finitenumberbychoosingto ignorevery-small-wavelengthmodes,butthat involved introducing an arbitrary cutoff on the energies of the field vibrations wewereconsidering. TheBekenstein-Hawking entropy, on the other hand, is just a finite number, fullstop.Foranotherthing,theentanglemententropyinfieldtheoryshoulddependonexactlyhowmanyfieldsareinvolved—theelectrons,quarks,neutrinos,andsoforth.Theformulaforblack-holeentropythatHawkingderivedmakesnomentionofsuchthingsatall.

Ifwecan’tsimplyattributeblack-holeentropytothequantumfieldsinside,thealternativeisto imagine that spacetime itself is made of some quantum degrees of freedom, and theBekenstein-Hawkingformulameasurestheentanglementofthedegreesoffreedominsidetheblackholewiththedegreesoffreedomoutside.Ifthatsoundsprettyvague,that’sbecauseitis.We’re not precisely surewhat these spacetime degrees of freedom are, or how they interactwithoneanother.Butthegeneralprinciplesofquantummechanicsshouldstillberespected.Ifthere’sentropy,andthatentropycomesfromentanglement,theremustbedegreesoffreedom

thatcanentanglewiththerestoftheworldinmanydifferentways,evenifclassicalblackholesareallfeatureless.

Ifthisstoryisright,thenumberofdegreesoffreedominablackholeisn’tinfinite,butitisvery large indeed. Our Milky Way galaxy contains a supermassive black hole at its center,associatedwitharadiosourcecalledSagittariusA*.Fromobservinghowstarsorbitaroundthehole,wecanmeasureitsmasstobe4milliontimesthemassofthesun.Thatcorrespondstoanentropy of 1090, which is greater than the entropy of all the known particles in the entireobservableuniverse.Thenumberofdegreesoffreedominaquantumsystemhastobeatleastaslargeasitsentropy,sincethatentropycomespreciselyfromthosedegreesoffreedombeingentangledwiththeoutsideworld.Sotheremustbeatleast1090degreesoffreedomintheblackhole.

Whilewetendtopayattentiontothestuffweseeintheuniverse—matter,radiation,andsoon—almostalloftheuniverse’squantumdegreesoffreedomareinvisible,doingnothingmorethan stitching spacetime together. In a volume of space roughly the size of an adult human,theremustbeatleast1070degreesoffreedom;weknowthatbecausethat’stheentropyofablackholethatwouldfillsuchavolume.Butthereareonlyabout1028particlesinaperson.Wecan thinkofaparticleasadegreeof freedom thathasbeen“turnedon,”whileall theotherdegrees of freedom are peacefully “turned off” in the vacuum state. As far as quantum fieldtheory isconcerned,ahumanbeingor thecenterofastar isn’tall thatdifferent fromemptyspace.

Maybethefactthattheentropyofablackholeisproportionaltoitsareaisjustwhatweshouldexpect.Inquantumfieldtheoryit’snaturalforregionsofspacetohaveanentropyproportionaltotheirboundaryarea,andablackholeisjustaregionofspace.Butaproblemlurksbeneaththe surface. It’s natural for a region of space in the vacuum state to have an entropyproportionaltoitsboundaryarea.Butablackholeisn’tpartofthevacuumstate;there’sablackholethere,andspacetimeisnoticeablycurved.

Blackholeshaveaveryspecialproperty: theyrepresent thehighest-entropystateswecanhave inanygivensize regionof space.Thisprovocative factwas firstnoticedbyBekenstein,andlaterrefinedbyRaphaelBousso.Ifyoustartfromaregionwithinthevacuumstateandtryto increase its entropy, youmust also increase its energy. (Since you started in the vacuum,there’snowherefortheenergytogobutup.)Asyoukeepthrowinginentropy,theenergyalsoincreases.Eventuallyyouhavesomuchenergyinafixedregionthatthewholethingcan’thelpbutcollapseintoablackhole.That’sthelimit;youcan’tfitanymoreentropyintoaregionthanyouwouldhaveifablackholewerethere.

Thatconclusionisprofoundlydifferentfromwhatwewouldexpectinanordinaryquantumfieldtheorywithoutgravity.There,thereisnolimitonhowmuchentropywecanfitinaregion,becausethere’salsonolimitonhowmuchenergytherecanbe.Thisreflectsthefactthatthereare an infinite number of degrees of freedom in quantum field theory, even in a finite-sizedregion.

Gravityappearstobedifferent.Thereisamaximumamountofenergyandentropythatcanfitintoagivenregion,whichseemstoimplythatthereareonlyafinitenumberofdegreesoffreedomthere.Somehowthesedegreesoffreedombecomeentangledintherightwaytostitchtogetherintothegeometryofspacetime.It’snotjustblackholes:everyregionofspacetimehasamaximumentropywecouldimaginefittingintoit(theentropythatablackholeofthatsizewouldhave),andthereforeafinitenumberofdegreesoffreedom.It’seventruefortheuniverseasawhole;because there isvacuumenergy, theaccelerationofspace isexpanding,and thatmeansthereisahorizonallaroundusthatdelineatestheextentoftheobservablepartofourcosmos.Thatobservablepatchofspacehasafinitemaximumentropy,sothereareonlyafinitenumberofdegreesoffreedomneededtodescribeeverythingweseeoreverwillsee.

Ifthisstoryisontherighttrack,ithasanimmediate,profoundconsequencefortheMany-Worldspictureofquantummechanics.Afinitenumberofquantumdegreesoffreedomimpliesafinite-dimensionalHilbert space for thesystemasawhole (in thiscase,anychosenregionofspace).Thatinturnimpliesthatthereissomefinitenumberofbranchesofthewavefunction,notaninfinitenumber.That’swhyAlicewascageybackinChapterEightaboutwhetherthereare an infinite number of “worlds” in thewave function. Inmany simplemodels of quantummechanics, including that of a fixed set of particles moving smoothly through space or anyordinaryquantumfieldtheory,Hilbertspaceisinfinite-dimensionalandtherecouldpotentiallybeaninfinitenumberofworlds.Butgravityseemstochangethingsaroundinanimportantway.Itpreventsmostofthoseworldsfromexisting,becausetheywoulddescribetoomuchenergybeingpackedintoalocalregion.

Somaybeintherealuniverse,wheregravitycertainlyexists,Everettianquantummechanicsonlydescribesafinitenumberofworlds.ThenumberAlicementionedforthedimensionalityof

Hilbertspacewas .Nowwecanrevealwherethatnumbercamefrom:it’sfromcalculatingtheentropythatour

observableuniversewillhaveonceitreachesmaximumentropy,andworkingbackwardtofindout how big Hilbert space needs to be to accommodate that much entropy. (The size of theobservableuniverseissetbythevacuumenergy,sotheexponent10122istheratioofthePlanckscale to the cosmological constant, familiar from our discussion in Chapter Twelve.) Ourconfidenceinthebasicprinciplesofquantumgravityisn’tstrongenoughtobeabsolutelysurethat there are only a finite number of Everettian worlds, but it seems reasonable, and itcertainlywouldmakethingsmuchsimpler.

Themaximum-entropy nature of black holes also has an important consequence for quantumgravity.Inclassicalgeneralrelativity,there’snothingspecialabouttheinteriorregionofablackhole,inbetweentheeventhorizonandthesingularity.There’sagravitationalfieldthere,buttoaninfallingobserveritotherwiselookslikeemptyspace.Accordingtothestorywetoldinthelastchapter,thequantumversionof“emptyspace”issomethinglike“acollectionofspacetimedegreesoffreedomentangledtogetherinsuchawayastoformanemergentthree-dimensionalgeometry.”Implicitinthatdescriptionisthatthedegreesoffreedomarescatteredmoreorlessuniformlythroughoutthevolumeofspacewe’relookingat.Andifthatweretrue,themaximum-entropy state of that form would have all of those degrees of freedom entangled with theoutsideworld.Theentropywouldthusbeproportionaltothevolumeoftheregion,nottheareaofitsboundary.What’sup?

Thereisacluefromtheblackholeinformationpuzzle.Theissuetherewasthatthereisnoobvious way to transmit information from a book that has fallen into the black hole to theHawking radiationemitted from theeventhorizon, at leastnotwithout signalsmoving fasterthan light.Sowhatabout this crazy idea:maybeall of the informationabout the stateof theblackhole—the“inside”aswellasthehorizon—canbethoughtofaslivingonthehorizonitself,not buried in the interior. The black-hole state “lives,” in some sense, on a two-dimensionalsurface,ratherthanbeingstretchedacrossathree-dimensionalvolume.

FirstdevelopedbyGerard ’tHooftandLeonardSusskind in the1990s,based inpartonapaper by Charles Thorn from 1978, this idea is known as the holographic principle. In anordinary hologram, shining light on a two-dimensional surface reveals an apparently three-dimensional image. According to the holographic principle, the apparently three-dimensionalinteriorofablackholereflectsinformationencodedonthetwo-dimensionalsurfaceofitseventhorizon. If this is true, maybe it’s not so hard to get information from the black hole to itsoutgoingradiation,becausetheinformationwasalwaysonthehorizontostartwith.

Physicists still haven’t settled on the precise meaning of holography for real-world blackholes. Is it justawayofcounting thenumberofdegreesof freedom,orshouldwethink thatthereisanactualtwo-dimensionaltheorylivingontheeventhorizonthatdescribesthephysicsoftheblackhole?Wedon’tknow,butthere isadifferentcontext inwhichholography isveryprecise: the so-called AdS/CFT correspondence, proposed by Juan Maldacena in 1997. The“AdS” in the label stands for “anti–deSitter space,” a hypothetical spacetimewith nomattersourcesotherthananegativevacuumenergy(asopposedtothepositivevacuumenergyofourrealworld).“CFT”standsforconformalfieldtheory,aparticularkindofquantumfieldtheorythatcanbedefinedonan infinitely farawayboundaryofAdS.According toMaldacena, thesetwotheoriesaresecretlyequivalenttoeachother.That’sextremelyprovocative,foracoupleofreasons.First, theAdS theory includesgravity,while theCFT isanordinary field theory thathasnogravityatall.Second,theboundaryofaspacetimehasonefewerdimensionsthanthespacetimeitself.Ifweconsiderfour-dimensionalAdS,forexample,thatisequivalenttoathree-dimensionalconformalfieldtheory.Youcouldn’taskforamoreexplicitexampleofholographyinaction.

Going into the details of AdS/CFT would require another book entirely. But it is worthmentioning that it is in this context that most modern research on the connection betweenspacetimegeometryandquantumentanglementisbeingcarriedout.AsnotedbyShinseiRyu,TadashiTakayanagi,MarkVanRaamsdonk,BrianSwingle,andothersintheearly2000s,thereisadirectconnectionbetweenentanglementintheboundaryCFTandtheresultinggeometryintheAdSinterior.BecauseAdS/CFTisrelativelywelldefinedasmodelsofquantumgravitygo,

understandingthisconnectionhasbeenthetargetofaveryintenseeffortoverthepastseveralyears.

Alas, it’s not the real world. All of the fun of AdS/CFT comes from relating things in theinterior, where gravity happens, to things on the boundary, where gravity is absent. But theexistence of the boundary is very special to anti–de Sitter space, which relies on a negativevacuumenergy.Ouruniverseappearstohaveapositivevacuumenergy,notanegativeone.

There’sanoldjokeaboutthedrunkwhoislookingunderalamppostforhislostkeys.Whensomeoneasks ifhe’ssurehelostthemthere,hereplies,“Ohno,I lostthemsomewhereelse,butthe light ismuchbetteroverhere.” Inthequantum-gravitygame,AdS/CFTis theworld’sbrightestlamppost.Bystudyingitwe’veuncoveredalargenumberoffascinatingconceptsthatare useful to theoretical physicists, but there is no direct route to using that knowledge tounderstandwhyapplesfall fromtrees,orotheraspectsofgravity inthespacearoundus.It’sworthcontinuing thepursuit,but important tokeepoureyeson theprize:understanding theworldinwhichweactuallylive.

The implicationsofholography for real-worldblackholesare less clear than theyare for theimaginary world of AdS/CFT. Are we saying that classical general relativity was completelywrongabouttheinteriorofablackholeappearingempty,andthatinfactaninfallingobserverwouldsmack intoaholographicsurfaceuponencounteringtheeventhorizon?Wearenot—atleast,most adherents of holography aren’t saying that. Rather, they appeal to a related andequally startling idea, black-hole complementarity. It was proposed by Susskind and others,usingterminologythatintentionallyrecallsBohr’sphilosophyofquantummeasurement.

Theblack-hole versionof complementarity says that thingsarea littlemorenuanced thansimply “the interior of a black hole looks like ordinary empty space” or “all the informationabouttheblackholeisencodedontheeventhorizon.”Infactbotharetrue,butwecan’tspeakboth languages at the same time. Or, as physicists are more likely to put it, they don’tsimultaneously appear true to any single observer. To an observer falling through the eventhorizon,everythinglookslikenormalemptyspace,whiletoanobserverlookingattheholefromfaraway,alloftheinformationisspreadacrossthehorizon.

Even though this behavior is fundamentally quantum-mechanical, it does have a classicalprecursor.Thinkaboutwhathappenstoabook(orastar,orwhatever)whenwethrowitintoablackholeinclassicalgeneralrelativity.Fromthebook’spointofview,itjustpassesrightintotheinterior.Buttheeffectofspacetimewarpingisstrongneartheeventhorizon,sothat’snotwhat an external observer would see. They would see the book appear to slow down as itapproachedthehorizon,becomingredderanddimmeralongtheway.Theywouldn’teverseeitcross;tosomeonefaraway,objectsappeartobefrozenintimeastheyapproachthehorizon,rather than plunging in. This led astrophysicists to develop a picture called themembraneparadigm,accordingtowhichwecanmodelthephysicalpropertiesofablackholebyimaginingthat there is a physicalmembrane at the horizon,with certain calculable properties such astemperatureandelectricalconductivity.Themembraneparadigmwasoriginallythoughtofasaconvenient shortcut throughwhich astrophysicists could simplify calculations involving blackholes,butcomplementarityclaims thatexternalobserversreallydoseeblackholesas if theywerevibratingquantummembraneswheretheclassicaleventhorizonwouldbe.

If you tend to thinkof spacetimeasa fundamental thing, thismightmakenosenseatall.Spacetime has some geometry, there’s nothing else to it. But quantum-mechanically it’sperfectly plausible; there’s a wave function of the universe, and different observations canreveal different things about it. It’s not that much different from saying that the number ofparticlesinastatedependsonhowweobserveit.

Theworld isaquantumstateevolving inHilbertspace,andphysicalspaceemergesoutofthat.Itshouldn’tcomeasasurprisethatasinglequantumstatemightexhibitdifferentnotionsofpositionandlocalitydependingonwhatkindofobservationsweperformonit.Accordingtoblack-holecomplementarity, there’sno such thingas “what thegeometryof spacetime is,”or“wherethedegreesoffreedomare”;youaskeitherwhatthequantumstateis,orwhatisseenbysomeparticularobserver.

This sounds different from the picture we explored in the last chapter, where degrees offreedom were distributed in a network filling space, and became entangled to define anemergentgeometry.Butthatpicturewasonlymeanttoapplywhengravitywasweak,andblackholes definitely do not qualify asweak. In the view presented in this chapter, there are stillabstractdegreesoffreedomcomingtogethertoformspacetime,but“wheretheyarelocated”dependsonhowtheyarebeingobserved.Spaceitselfisnotfundamental;it’sjustausefulwayoftalkingfromcertainpointsofview.

Hopefully these last chapters have successfully conveyed the way in which Many-Worldsquantum mechanics might have significant implications for the long-standing problem ofquantum gravity. To be honest, many physicists working on these problems don’t think ofthemselvesasusingMany-Worlds, though theyare implicitlydoingso.Theycertainlyarenotusing hidden variables, or dynamical collapses, or an epistemic approach to quantummechanics.When itcomes tounderstandinghowtoquantize theuniverse itself,Many-Worldsseemstobethemostdirectpathtotake,ifnothingelse.

Is the picture we’ve sketched, where the entanglement between degrees of freedomsomehow comes together to define the geometry of our approximately classical spacetime,actuallyontherighttrack?Nobodyknowsforsure.Whatseemsclear,giventhecurrentstateofourknowledge,isthatbothspaceandtimecouldemergefromanabstractquantumstateinthedesiredway—all the ingredients are there, and it’s not out of place to hope that a fewmoreyearsofworkwillbringamuchsharperpictureintofocus.Ifwetrainourselvestodiscardourclassical prejudices, and take the lessons of quantum mechanics at face value, we mayeventuallylearnhowtoextractouruniversefromthewavefunction.

*It’snotcompletelyagreeduponthatinfallingobjectsactuallydotraveldeepintotheinteriorofablackhole.In2012agroupofphysicistsarguedthat,ifinformationisgoingtoescapefromevaporatingblackholeswithoutviolatingthebasic tenets of quantum mechanics, something dramatic has to happen at the event horizon: not quiet, emptyspacetime,asisusuallyassumed,butablastofhigh-energyparticlesknownasafirewall.Opinionsaboutthefirewallproposalaredivided,astheoristscontinuetoarguebackandforthabouttheissue.

EPILOGUE

EverythingIsQuantum

WhatwouldEinsteinhavethoughtofMany-Worldsquantumtheory?Likelyhewouldhavebeenrepulsed,at leastat firstexposure.Buthewouldhave toadmit that thereareaspectsof theideathatfitverywellwithhispictureofhownatureshouldoperate.

EinsteindiedinPrincetonin1955,justasEverettwaswranglinghisideaintoshape.Hewasfirmlycommittedtotheprincipleoflocality,andwasenormouslybotheredbythespookyactionatadistance impliedbyquantumentanglement. In that sense,hemightverywellhavebeenhorrified by Many-Worlds and the holographic principle, ideas that treat space itself asemergent rather than fundamental.The suggestion that reality isdescribedasavector inanenormous Hilbert space, rather than as matter and energy in good old four-dimensionalspacetime,isnotonehewouldhavefoundcongenial.Butthere’sagoodchancethathewouldhave been pleased that Everett returns our best description of the universe to one featuringdefinite,deterministicevolution—andreaffirmstheprinciplethatrealityisultimatelyknowable.

Lateinlife,Einsteinrelatedastoryfromhischildhood.

AwonderofthiskindIexperiencedasachildoffourorfiveyearswhenmyfathershowedmeacompass.Thatthisneedlebehaved insuchadeterminedwaydidnotatall fit in thekindofoccurrences thatcould findaplaceintheunconsciousworldofconcepts(efficacyproducedbydirect“touch”).Icanstillremember—oratleastbelieveIcanremember—thatthisexperiencemadeadeepandlastingimpressionuponme.

Somethingdeeplyhiddenhadtobebehindthings.

ItseemstomethatthisimpulseliesattheheartofallofEinstein’sworriesaboutquantummechanics.Hemighthavefrettedoutloudaboutindeterminismandnonlocality,butwhatreallybugged him was his sense that Copenhagen quantum mechanics replaced the crisp rigor ofgoodscientifictheorieswithafuzzyparadigminwhichanill-definednotionof“measurement”playeda central role.Hewas always on the lookout for thedeeplyhidden thingbeneath thesurface, theprinciple thatwould restore intelligibility to thatwhichhaddrifted intomystery.Littledidhesuspectthatwhatwashiddenmightbeotherbranchesofthewavefunction.

It doesn’t really matter what Einstein would have actually thought, of course; scientifictheoriesriseorfallontheirmerits,notbecausewecanconjureuphypotheticalghostsofgreatmindsfromthepasttonodtheirapproval.

Butit’susefultopayattentiontothosegreatminds,ifonlytoberemindedoftheconnectionsbetweendebatesofthepastandresearchinthepresent.TheissuesdiscussedinthisbookstemdirectlyfromthediscussionsbetweenEinsteinandBohrandothersinthe1920s.InthewakeoftheSolvayConference,popularopinionwithin thephysicscommunity swungBohr’sway,andtheCopenhagenapproachtoquantummechanicssettledinasentrencheddogma.It’sproventobe an amazingly successful tool at making predictions for experiments and designing newtechnologies.Butasafundamentaltheoryoftheworld,itfallswoefullyshort.

I’ve laid out the case for why Many-Worlds is the most promising formulation of quantummechanics.ButIhaveenormousrespectfor,andhavefrequentproductiveconversationswith,partisans for other approaches. What makes me melancholy are professional physicists whodismiss foundationalworkanddon’t thinkthe issuesareworthtakingseriously.Afterreadingthis book, whether or not you would describe yourself as an Everettian, I hope you areconvincedoftheimportanceofgettingquantummechanicsrightonceandforall.

I’moptimisticabouthowthingsareprogressing.Themodernstudyofquantumfoundationsisn’tjustabunchofelderlyphysicistschattingaboutfantasticalideasovertumblersofscotchafter the real work is done for the day. Much of the recent progress in developing ourunderstanding of quantum theory has been spurred, directly or indirectly, by technologicalinnovations: quantum computing, quantum cryptography, and quantum information moregenerally.We’vereachedapointwhereit isnolongerpracticaltodrawabrightlinebetweenthe quantum and classical realms. Everything is quantum. This state of affairs has forcedphysicists to take the foundationsof quantummechanics abitmore seriously, andhas led to

newinsightsthatmighthelpexplaintheemergenceofspaceandtimethemselves.Ithinkwe’llbemakingsignificantprogressonthesedifficultpuzzlesinthenearfuture.And

I like tobelievemostof theotherversionsofmeonotherbranchesof thewave function feellikewise.

APPENDIX

TheStoryofVirtualParticles

OurdiscussionofquantumfieldtheoryinChapterTwelvewouldseemamusinglyidiosyncratictomostworkingquantumfieldtheorists.Whatwecaredaboutwasjustthevacuumstate,thelowest-energyconfigurationofasetofquantumfieldsfillingspace.Butthat’sjustonestateoutofaninfinitenumber.Whatmostphysicistscareaboutarealltheotherstates—thosethatlooklikeparticlesmovingandinteractingwithoneanother.

Justasit’snaturaltospeakabout“thepositionoftheelectron”whenwereallyknowbetterandshouldspeakabouttheelectron’swavefunction,physicistswhounderstandperfectlywellthat the world is made of fields tend to talk about particles all the time. They even callthemselves “particle physicists” without discernible embarrassment. It’s an understandableimpulse:particlesarewhatwesee,regardlessofwhat’sgoingonbeneaththesurface.

Thegoodnewsis,that’sokay,aslongasweknowwhatwe’redoing.Formanypurposes,wecantalkasifwhatreallyexistsisacollectionofparticlestravelingthroughspace,bumpingintoone another, being created and destroyed, and occasionally popping into or out of existence.Thebehaviorofquantumfieldscan,under therightcircumstances,beaccuratelymodeledasthe repeated interactionofmanyparticles.Thatmight seemnaturalwhen thequantumstatedescribes some fixed number of particle-like field vibrations, far away from one another andblissfully unaware of the others’ existence.But ifwe follow the rules,we can calculatewhathappensusingparticlelanguageevenwhenabunchoffieldsarevibratingrightontopofoneanother,exactlywhenyoumightexpecttheirfield-nesstobemostimportant.

That’s the essential insight from Richard Feynman and his well-known tool of Feynmandiagrams. When he first invented his diagrams, Feynman held out the hope that he wassuggestingaparticle-basedalternativetoquantumfieldtheory,butthatturnsoutnottobethecase. What they are is both a wonderfully vivid metaphorical device and an incrediblyconvenientcomputationalmethod,withintheoverarchingparadigmofquantumfieldtheory.

A Feynman diagram is simply a stick-figure cartoon representing particles moving andinteracting with one another.With time running from left to right, an initial set of particlescomes in, they jumbleupwithvariousparticlesappearingordisappearing, thena finalsetofparticles emerges. Physicists use these diagrams not only to describe what processes areallowedtohappenbuttopreciselycalculatethelikelihoodthattheyactuallywill.Ifyouwanttoask,forexample,whatparticlesaHiggsbosonmightdecayintoandhowrapidly,youwoulddoa calculation involving a boatload of Feynman diagrams, each representing a certaincontributiontothefinalanswer.Likewiseifyouwanttoknowhowlikelyit isthatanelectronandapositronwillscatteroffeachother.

Here isasimpleFeynmandiagram.Thewaytothinkaboutthispicture is thatanelectronand a positron (straight lines) come in from the left, meet each other, and annihilate into aphoton (wavy line),which travels forawhilebeforeconvertingback intoanelectron/positronpair. There are specific rules that allow physicists to attach precise numbers to every suchdiagram, indicating the contribution that this picture makes to the overall process of “anelectronandapositronscatteroffeachother.”

ThestorywetellbasedontheFeynmandiagramsis justthat,astory.It’snot literallytruethatanelectronandapositronchangeintoaphotonandthenchangeback.Foronething,realphotonsmoveatthespeedoflight,whileelectron/positronpairs(eithertheindividualparticlesorthecenterofmassofapairofthem)donot.

What actually happens is that both the electron field and thepositron field are constantlyinteractingwiththeelectromagneticfield;oscillationsinanyelectricallychargedfield,suchasthe electron or positron, are necessarily accompanied by subtle oscillations in theelectromagneticfieldaswell.Whentheoscillationsintwosuchfields(whichweinterpretastheelectronandpositron)comeclosetoeachotheroroverlap,allofthefieldspushandpullononeanother,causingouroriginalparticlestoscatteroffinsomedirection.Feynman’sinsightisthatwecancalculatewhat’sgoingon in the field theorybypretending that thereareabunchofparticlesflyingaroundincertainways.

This represents an enormous computational convenience; working particle physicists useFeynmandiagramsall the time,andoccasionallydreamabout themwhilesleeping.But therearecertainconceptualcompromisesthatneedtobemadealongtheway.TheparticlesconfinedtotheinterioroftheFeynmandiagrams,whichdon’teithercomeinfromtheleftorexittotheright,don’tobeytheusualrulesforordinaryparticles.Theydon’t,forexample,havethesameenergyormassthataregularparticlehas.Theyobeytheirownsetofrules,justnottheusualones.

Thatshouldn’tbesurprising,asthe“particles”insideFeynmandiagramsarenotparticlesatall; they’re a convenientmathematical fairy tale. To remind ourselves of that, we label them“virtual”particles.Virtualparticlesarejustawaytocalculatethebehaviorofquantumfields,by pretending that ordinary particles are changing into weird particles with impossibleenergies, and tossing such particles back and forth between themselves. A real photon hasexactlyzeromass,butthemassofavirtualphotoncanbeabsolutelyanything.Whatwemeanby “virtual particles” are subtle distortions in the wave function of a collection of quantumfields.Sometimestheyarecalled“fluctuations”orsimply“modes”(referringtoavibrationinafield with a particular wavelength). But everyone calls them particles, and they can besuccessfullyrepresentedaslineswithinFeynmandiagrams,sowecancallthemthat.

Thediagramwedrewforanelectronandapositronscatteringoffeachotherisn’ttheonlyonewecouldpossiblydraw;infact,it’sjustoneofaninfinitenumber.Therulesofthegametellusthat we should sum up all of the possible diagrams with the same incoming and outgoingparticles. We can list such diagrams in order of increasing complexity, with subsequentdiagramscontainingmoreandmorevirtualparticles.

Thefinalnumberweobtainisanamplitude,sowesquareittogettheprobabilityofsuchaprocesshappening.UsingFeynmandiagrams,wecancalculatetheprobabilityoftwoparticlesscatteringoffeachother,ofoneparticledecayingintoseveral,orforparticlesturningintootherkindsofparticles.

An obviousworry pops up: If there are an infinite number of diagrams, how can you addthem all up and get a sensible result? The answer is that diagrams contribute smaller andsmalleramountsastheybecomemorecomplicated.Eventhoughthereareaninfinitenumberofthem, the sum total of all the very complicated ones can be a tiny number. In practice, as amatteroffact,weoftengetquiteaccurateanswersbycalculatingonlythefirstfewdiagramsintheinfiniteseries.

Thereisonesubtletyalongthewaytothisniceresult,however.Consideradiagramthathasaloopinit—thatis,wherewecantracearoundsomesetofparticlelinestoformaclosedcircle.Hereisanelectronandapositronexchangingtwophotons:

Each line represents a particlewith a certain amount of energy. This energy is conserved

whenlinescometogether:ifoneparticlecomesinandsplitsintotwo,forexample,thesumoftheenergiesofthosetwoparticlesmustequalthatoftheinitialparticle.Buthowthatenergygetssplitupiscompletelyarbitrary,aslongasthesumtotalisfixed.Infact,duetothewackylogicofvirtualparticles, theenergyofoneparticlecanevenbeanegativenumber,suchthattheotheronehasmoreenergythantheinitialparticledid.

Thismeans thatwhenwe calculate the process described by a Feynman diagramwith aninternalclosedloop,anarbitrarilylargeamountofenergycanbetravelingdownanyparticularlinewithintheloop.Sadly,whenwedothecalculationforwhatsuchdiagramscontributetothefinal answer, the result can turn out to be infinitely large. That’s the origin of the infamousinfinitiesplaguingquantumfieldtheory.Obviouslytheprobabilityofacertaininteractioncanbeatmost1,soaninfiniteanswermeanswe’vetakenawrongturnsomehow.

Feynmanandothersmanagedtoworkoutaprocedurefordealingwiththeseinfinities,nowknown as renormalization.When you have a bunch of quantum fields that interact with oneanother,youcan’tsimplyfirsttreatthemseparately,andthenaddintheinteractionsattheend.Thefieldsareconstantly,inevitablyaffectingoneanother.Evenwhenwehaveasmallvibrationin the electron field, which we might be tempted to identify as a single electron, there areinevitably accompanying vibrations in the electromagnetic field, and indeed in all the otherfieldsthattheelectron interactswith. It’s likeplayingapianonote inashowroomwithmanypianos present; the other instruments will begin to gently hum along with the original one,causing a faint echo of whatever notes you are playing. In Feynman-diagram language, thismeans thatevenan isolatedparticlepropagating throughspace isactuallyaccompaniedbyasurroundingcloudofvirtualparticles.

Asaresult, it’shelpfultodistinguishbetweenthe“bare”fieldsastheywouldbehaveinanimaginaryworldwhereallinteractionsweresimplyturnedoff,andthe“physical”fieldsthatareaccompaniedbyotherfieldstheyinteractwith.TheinfinitiesthatyougetbynaïvelyturningacrankintheFeynmandiagramsaresimplyaresultoftryingtoworkwithbarefields,whereaswhatwereallyobservearephysicalones.Theadjustmentrequiredtogofromonetoanotherissometimes informally described as “subtracting off infinity to get a finite answer,” but that’smisleading.Nophysicalquantitiesareinfinite,norweretheyever;theinfinitiesthatquantum-field-theorypioneersmanagedto“hide”weresimplyartifactsoftheverybigdifferencebetweenfields that interact and fields that don’t. (We face exactly this kind of issue when trying toestimatethevacuumenergyinquantumfieldtheory.)

Nevertheless, renormalization comes with important physical insights. When we want tomeasuresomepropertyofaparticle,suchasitsmassorcharge,weprobeitbyseeinghowitinteractswithotherparticles.Quantumfieldtheoryteachesusthattheparticlesweseearen’tsimple point-like objects; each particle is surroundedby a cloud of other virtual particles, or(moreaccurately)bytheotherquantumfieldsitinteractswith.Andinteractingwithacloudisdifferentfrominteractingwithapoint.Twoparticlesthatsmashintoeachotherathighvelocitywill penetrate deep into each other’s clouds, seeing relatively compact vibrations, while twoparticlesthatpassbyslowlywillseeeachotheras(relatively)bigpuffyballs.Consequently,theapparentmassorchargeofaparticlewilldependontheenergyoftheprobeswithwhichwelookatit.Thisisn’tjustasonganddance:it’sanexperimentalprediction,whichhasbeenseenunmistakablyinparticle-physicsdata.

Thebestwaytothinkaboutrenormalizationwasn’treallyappreciateduntiltheworkofNobellaureateKennethWilsonintheearly1970s.WilsonrealizedthatalloftheinfinitiesinFeynman-diagram calculations came from virtual particles with very large energies, corresponding toprocesses at extremely short distances. But high energies and short distances are preciselywhereweshouldhavetheleastconfidencethatweknowwhat’sgoingon.Processeswithveryhigh energies could involve completely new fields, ones that have such highmasses thatwehaven’tyetproducedtheminexperiments.Forthatmatter,spacetimeitselfmightbreakdownatshortdistances,perhapsatthePlancklength.

So,Wilsonreasoned,whatifwe’rejustalittlebitmorehonest,andadmitthatwedon’tknowwhat’sgoingonatarbitrarilyhighenergies?InsteadoftakingloopsinFeynmandiagramsandallowingtheenergiesofthevirtualparticlestogouptoinfinity,let’sincludeanexplicitcutoffinthetheory:anenergyabovewhichwedon’tpretendtoknowwhat’shappening.Thecutoffisinsomesensearbitrary,but itmakessensetoput itatthedividinglinebetweenenergiesaboutwhichwehavegoodexperimentalknowledge,andabovewhichwehaven’tbeenabletopeek.

There can even be a physically good reason to choose a certain cutoff, if we expect newparticlesorotherphenomenatokickinatthatscale,butdon’tknowexactlywhattheywillbe.

Ofcourse, therecouldbe interesting thingsgoingonathigherenergies,soby includingacutoffwe’readmittingthatwe’renotgettingexactlytherightanswer.ButWilsonshowedthatwhatwedoget isgenerallymore thangoodenough.Wecanprecisely characterizehow,androughly by howmuch, any new high-energy phenomena could possibly affect the low-energyworld we actually see. By admitting our ignorance in this way, what we’re left with is aneffectivefieldtheory—onethatdoesn’tpresumetobeanexactdescriptionofanything,butonethatcansuccessfully fit thedataweactuallyhave.Modernquantum field theorists recognizethatalloftheirbestmodelsareactuallyeffectivefieldtheories.

Thisleavesuswithagoodnews/badnewssituation.Thegoodnewsisthatweareabletosayan enormous amount about the behavior of particles at low energies, using the magic ofeffectivefieldtheory,evenifwedon’tknoweverything(oranything)aboutwhat’shappeningathigherenergies.Wedon’tneedtoknowallthefinalanswersinordertosaysomethingreliableandtrue.That’sabigpartofwhywecanbeconfidentthatthelawsofphysicsgoverningtheparticlesandforces thatmakeupyouandmeandoureverydayenvironmentsarecompletelyknown:thoselawstaketheformofaneffectivefieldtheory.There’splentyofroomtodiscovernewparticlesand forces,buteither theymustbe toomassive (highenergy) tohaveyetbeenproducedinexperiments,ortheyinteractwithussoincrediblyweaklythattheycan’tpossiblyhaveaneffectontablesandchairsandcatsanddogsandotherpiecesofthearchitectureofourlow-energyworld.

Thebadnewsisthatwewouldverymuchliketolearnmoreaboutwhat’sreallygoingonathighenergiesandshortdistances,butthemagicofeffectivefieldtheorymakesthatextremelyhard.It’sgoodthatwecanaccuratelydescribelow-energyphysicsnomatterwhatisgoingonat higher energies, but it’s also frustrating because this seems to imply that we can’t inferwhat’sgoingonuptherewithoutsomehowprobingitdirectly.Thisiswhyparticlephysicistsareso enamored of building ever larger and higher-energy particle accelerators; that’s the onlyreliablewayweknowoftodiscoverhowtheuniverseworksatverysmalldistances.

ACKNOWLEDGMENTS

Everybook isacollaboration,andthisonemore thanothers.There ismuchtobesaidaboutquantummechanics,andtherewasdefinitelyatemptationtosayitall.Thatmighthavebeenafunbooktowritebutitwouldhavebeenatediouschoretoread.Ioweavarietyofgenerousandinsightfulreadersfortheirhelpinwrestlingthemanuscriptdowntosomethingmanageableand,hopefully,inparts,fun.IshouldspecificallymentionhelpfulcommentsfromNickAceves,Dean Buonomano, Joseph Clark, Don Howard, Jens Jäger, Gia Mora, Jason Pollack, DanielRanard,RobReid,GrantRemmen,AlexRosenberg,LandonRoss,ChipSebens,MattStrassler,and David Wallace. In ways stretching from small—offhandedly mentioning something inconversation that later ended up in the book—to large—reading every chapter and offeringusefulinsights—thesegenerousfolkshelpedrescuemefromwritingabookthatwouldnothavebeennearlyasgood.IwanttogivespecialthankstoScottAaronson,whoisthebesttest-readeraphysicist/author

couldaskfor,givingthetextathoroughreadingandofferinginvariablyusefulfeedbackonbothsubstance and style. I’ll alsomention GiaMora again, because shewas inexplicably omittedfromtheacknowledgmentsofTheBigPicture,andIfeelbadaboutthat.ItgoeswithoutsayingthatI’velearnedanenormousamountaboutquantummechanicsand

spacetime froma largenumberofextremelysmartpeopleover theyears,andtheir influencepervadesthisbookevenifIdidn’ttalkspecificallyaboutthewordswrittenhere.ManythanksgotoDavidAlbert,NingBao,JeffBarrett,CharlesBennett,AdamBecker,KimBoddy,CharlesCao, AidanChatwin-Davies, SidneyColeman,EdwardFarhi, AlanGuth, JamesHartle, JenannIsmael,MatthewLeifer,SethLloyd,FrankMaloney,TimMaudlin,SpirosMichalakis,AlyssaNey,Don Page, Alain Phares, John Preskill, Jess Reidel, Ashmeet Singh, Leonard Susskind, LevVaidman,RobertWald,andNicholasWarner,nottomentionthenumerousothersIamdoubtlessforgetting.Thanks as usual to my students and collaborators for tolerating my occasional absences

while trying to finish thebook.And thanksalso to the students in125C, the thirdquarterofCaltech’s course on quantum mechanics for juniors, who tolerated me teaching them aboutdecoherenceandentanglementratherthanjustthefamiliarroutineofsolvingtheSchrödingerequationoverandover.AmillionthankstomyeditoratDutton,StephenMorrow,whosepatienceandinsightwere

moresorelyneededfor thisbookthantheyhavebeen in thepast.Heeven letme includeanentirechapterindialogueform,althoughit’spossibleIjustworehimdown.Anauthorcouldn’timagine an editor who cared more about the quality of the final product, and much of thequalityhereisduetoStephen.Thanksalsotomyagents,KatinkaMatsonandJohnBrockman,who alwaysmake a process that could potentially be nerve-racking into something tolerable,possiblyevenenjoyable.AndthemostthanksofalltoJenniferOuellette,theperfectpartnerinwritingandinlife.Not

onlydidshesupportmeincountlesswaysalongthejourney,butshetooktimeoutofherownverydemandingwritingscheduletogothrougheverypageherecarefullyandofferinvaluableinsightandtoughlove.Ididn’tdeletenearlyasmuchasshesuggested,andprobablythebookispoorerthereby,buttrustme,it’swaybetterthanitwasbeforeshegottoit.ThanksalsotoJenniferforbringingintoourlivesArielandCaliban,thebestwriting-partner

catsanauthorcouldaskfor.Noactualcatsweresubjectedtothoughtexperimentsduringthecompositionofthisbook.

FURTHERREADING

Therehaveobviouslybeenalargenumberofbookswrittenaboutquantummechanics.Hereareafewthatarerelevanttothethemesofthisbook:

Albert,D.Z.(1994).QuantumMechanicsandExperience.HarvardUniversityPress.Ashortintroductiontoquantummechanicsandthemeasurementproblemfromaphilosophicalperspective.

Becker,A.(2018).WhatIsReal?TheUnfinishedQuestfortheMeaningofQuantumPhysics.BasicBooks.Ahistoricaloverviewofquantumfoundations,includingalternativestoMany-Worldsandtheobstaclesthatmanyphysicistsfacedinthinkingabouttheseissues.

Deutsch,D.(1997).TheFabricofReality.Penguin.AnintroductiontoMany-Worldsbutalsomuchmore,fromcomputationtoevolutiontotimetravel.

Saunders,S.,J.Barrett,A.Kent,andD.Wallace.(2010).ManyWorlds?Everett,QuantumTheory,andReality.AcollectionofessaysforandagainstMany-Worlds.

Susskind,L.,andA.Friedman.(2015).QuantumMechanics:TheTheoreticalMinimum.BasicBooks.Aseriousintroductiontoquantummechanics,taughtatthelevelofanintroductorycourseforphysicsstudentsatagooduniversity.

Wallace, D. (2012). The Emergent Multiverse: Quantum Theory According to the Everett Interpretation.Oxford University Press. Somewhat technical, but this is the now-standard reference book on Many-Worlds.

REFERENCES

PrologueDon’tBeAfraid

“IthinkIcansafelysay”:SeeR.P.Feynman(1965),TheCharacterofPhysicalLaw,MITPress,123.

Chapter2TheCourageousFormulationAustereQuantumMechanics

“Shutupandcalculate”:SeeN.D.Mermin(2004),“CouldFeynmanHaveSaidThis?”PhysicsToday57,5,10.

Chapter3WhyWouldAnybodyThinkThis?

HowQuantumMechanicsCametoBe

“siximpossiblethings”:L.Carroll(1872),ThroughtheLookingGlassandWhatAliceFoundThere,Dover,47.

“Sweetisbyconvention”:QuotedinH.C.VonBaeyer(2003),Information:TheNewLanguageofScience,Weidenfeld&Nicolson,12.

“very revolutionary”: Quoted in R. P. Crease and A. S. Goldhaber (2014), The Quantum Moment: HowPlanck,Bohr,Einstein,andHeisenbergTaughtUstoLoveUncertainty,W.W.Norton&Company,38.

“Thereappearstomeonegravedifficulty”:Quoted inH.Kragh(2012),“Rutherford,Radioactivity,andtheAtomicNucleus,”https://arxiv.org/abs/1202.0954.

“hadwritten a crazy paper”: Quoted in A. Pais (1991),Niels Bohr’s Times, in Physics, Philosophy, andPolity,ClarendonPress,278.

“Averitablesorcerer’scalculation”:QuotedinJ.Bernstein(2011),“AQuantumStory,”TheInstituteLetter,InstituteforAdvancedStudy,Princeton.

“Idon’tlikeit”:QuotedinJ.Gribbin(1984),InSearchofSchrödinger’sCat:QuantumPhysicsandReality,BantamBooks,v.

Chapter4WhatCannotBeKnown,BecauseItDoesNotExist

UncertaintyandComplementarity

For more on the double-slit experiment, see A. Ananthaswamy (2018), Through Two Doors at Once: TheElegantExperimentThatCapturestheEnigmaofOurQuantumReality,Dutton.

Chapter5EntangledUpinBlue

WaveFunctionsofManyParts

A. Einstein, B. Podolsky, and N. Rosen (1935), “Can Quantum-Mechanical Description of Reality BeConsideredComplete?”PhysicalReview47,777.

Forgeneral insight intoBell’s theoremand itsrelationshiptoEPRandBohmianmechanics,seeT.Maudlin(2014),“WhatBellDid,”JournalofPhysicsA47,424010.

“secularpress”:QuotedinW.Isaacson(2007),Einstein:HisLifeandUniverse,Simon&Schuster,450.D.Rauchetal.(2018),“CosmicBellTestUsingRandomMeasurementSettingsfromHigh-RedshiftQuasars,”PhysicalReviewLetters121,080403.

Chapter6

SplittingtheUniverseDecoherenceandParallelWorlds

A good biography of Hugh Everett is P. Byrne (2010), The Many Worlds of Hugh Everett III: MultipleUniverses,MutualAssuredDestruction,andtheMeltdownofaNuclearFamily,OxfordUniversityPress.QuotesinthischapterarelargelyfromthisbookandA.Becker(2018),WhatIsReal?,BasicBooks.

Everett’soriginalpaper(longandshortversions)andvariouscommentariescanbefoundinB.S.DeWittandN.Graham(1973),TheManyWorldsInterpretationofQuantumMechanics,PrincetonUniversityPress.

“Nothinghasdonemoretoconvinceme”:QuotedinA.Becker(2018),WhatIsReal?,BasicBooks,127.H.D.Zeh(1970),“OntheInterpretationofMeasurementsinQuantumTheory,”FoundationsofPhysics1,69.“TheCopenhagenInterpretationishopelesslyincomplete”:QuotedinP.Byrne(2010),141.“Split?”:QuotedinP.Byrne(2010),139.“Lestthediscussionofmypaperdie”:QuotedinP.Byrne(2010),171.“doomedfromthebeginning”:QuotedinA.Becker(2018),136.“Ican’tresistasking”:QuotedinP.Byrne(2010),176.“I realize that there isacertainvalue”:M.O.Everett (2007),Things theGrandchildrenShouldKnow,

Little,Brown,235.

Chapter7OrderandRandomness

WhereProbabilityComesFrom

“Why do people say”: Quoted in G.E.M. Anscombe (1959), An Introduction to Wittgenstein’s Tractatus,HutchinsonUniversityLibrary,151.

“fatnessmeasure”:D.Z.Albert(2015),AfterPhysics,HarvardUniversityPress,169.W.H.Zurek(2005),“ProbabilitiesfromEntanglement,Born’sRulefromEnvariance,”PhysicalReviewA71,

052105.C.T.SebensandS.M.Carroll(2016),“Self-LocatingUncertaintyandtheOriginofProbabilityinEverettian

QuantumMechanics,”BritishJournalforthePhilosophyofScience69,25.D. Deutsch (1999), “Quantum Theory of Probability and Decisions,” Proceedings of the Royal Society ofLondonA455,3129.

Foracomprehensivereviewofthedecision-theoreticapproachtotheBornrule,seeD.Wallace(2012),TheEmergentMultiverse.

Chapter8DoesThisOntologicalCommitmentMakeMeLookFat?

ASocraticDialogueonQuantumPuzzles

“mistaken and even a vicious”: K. Popper (1967), “Quantum Mechanics Without the Observer,” in M.Bunge (ed.),Quantum Theory and Reality. Studies in the Foundations Methodology and Philosophy ofScience,vol.2,Springer,12.

“a completely objective discussion”: K. Popper (1982), Quantum Theory and the Schism in Physics,Routledge,89.

Formoreonentropyandthearrowoftime,seeS.M.Carroll(2010),FromEternitytoHere:TheQuestfortheUltimateTheoryofTime,Dutton.

“Askinghowmanyworlds”:D.Wallace(2012),TheEmergentMultiverse,102.“Despitetheunrivaledempiricalsuccess”:D.Deutsch(1996),“CommentonLock-wood,”BritishJournalforthePhilosophyofScience47,222.

Chapter9OtherWays

AlternativestoMany-Worlds

“clearlybeingassigned”:QuotedinA.Becker(2018),WhatIsReal?,BasicBooks,213.“IfwecannotdisproveBohm”:QuotedinA.Becker(2018),90.“thepaperiscompletelysenseless”:QuotedinA.Becker(2018),199.“Everett phone”: J. Polchinski (1991), “Weinberg’s Nonlinear Quantum Mechanics and the Einstein-

Podolsky-RosenParadox,”PhysicalReviewLetters66,397.Formore on hidden-variable and dynamical-collapsemodels, seeT.Maudlin (2019),Philosophy of Physics:QuantumTheory,Princeton.

R. Penrose (1989), The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics,Oxford.

“the Einstein-Podolsky-Rosen paradox is resolved”: J. S. Bell (1966), “On the Problem of Hidden-

VariablesinQuantumMechanics,”ReviewsofModernPhysics38,447.“asuperfluousideologicalsuperstructure,”and“artificialmetaphysics”:QuotedinW.Myrvold(2003),

“OnSomeEarlyObjectionstoBohm’sTheory,”InternationalStudiesinthePhilosophyofScience17,7.H.C.VonBaeyer(2016),QBism:TheFutureofQuantumPhysics,Harvard.“There is indeed” and “QBism regards”: N. D. Mermin (2018), “Making Better Sense of Quantum

Mechanics,”ReportsonProgressinPhysics82,012002.C.A.Fuchs(2017),“OnParticipatoryRealism,”inI.DurhamandD.Rickles,eds.,InformationandInteraction,

Springer.“TheEverettinterpretation(insofarasit isphilosophicallyacceptable)”:D.Wallace (2018),“On the

Plurality ofQuantumTheories:QuantumTheory as aFramework, and Its Implications for theQuantumMeasurementProblem,”inS.FrenchandJ.Saatsi,eds.,ScientificRealismandtheQuantum,Oxford.

Chapter10TheHumanSide

LivingandThinkinginaQuantumUniverse

M.Tegmark (1998), “The InterpretationofQuantumMechanics:ManyWorldsorManyWords?”FortschriftPhysik46,855.

R.Nozick(1974),Anarchy,State,andUtopia,BasicBooks,41.“All that quantummechanics purports to provide”: E. P.Wigner (1961), “Remarks on theMind-Body

Problem,”inI.J.Good,TheScientistSpeculates,Heinemann.

Chapter11WhyIsThereSpace?

EmergenceandLocality

Italkmoreaboutemergence(andtheCoreTheory)inS.M.Carroll(2016),TheBigPicture:OntheOriginsofLife,Meaning,andtheUniverseItself,Dutton.

“Ithinkmyfather”:JamesHartle(2016),personalcommunication.

Chapter12AWorldofVibrationsQuantumFieldTheory

“It is inconceivable that inanimate brute matter should”: I. Newton (2004), Newton: PhilosophicalWritings,ed.A.Janiak,Cambridge,136.

P.C.W.Davies (1984), “ParticlesDoNotExist,” inB. S.DeWitt, ed.,QuantumTheory ofGravity: Essays inHonorofthe60thBirthdayofBryceDeWitt,AdamHilger.

Chapter13BreathinginEmptySpace

FindingGravitywithinQuantumMechanics

Formoreontheimplicationsandlimitationsof locality,seeG.Musser(2015),SpookyActionataDistance:ThePhenomenonThatReimaginesSpaceandTime—andWhatItMeansforBlackHoles,theBigBang,andTheoriesofEverything,Farrar,StrausandGiroux.

“Iusemorebraingrease”:A.Einstein,quotedbyOttoStern(1962),interviewwithT.S.Kuhn,NielsBohrLibrary & Archives, American Institute of Physics, https://www.aip.org/history-programs/niels-bohr-library/oral-histories/4904.

“PerhapsthesuccessoftheHeisenbergmethod”:A.Einstein(1936),“PhysicsandReality,”reprintedinA.Einstein(1956),OutofMyLaterYears,CitadelPress.

T. Jacobson (1995), “Thermodynamics of Space-Time: The Einstein Equation of State,” Physical ReviewLetters75,1260.

T.Padmanabhan(2010),“ThermodynamicalAspectsofGravity:NewInsights,”ReportsonProgressinPhysics73,046901.

E.P.Verlinde (2011), “On theOrigin ofGravity and theLawsofNewton,” Journal ofHighEnergyPhysics1104,29.

J. S. Cotler, G. R. Penington, and D. H. Ranard (2019), “Locality from the Spectrum,”Communications inMathematicalPhysics,https://doi.org/10.1007/s00220-019-03376-w.

J.MaldacenaandL.Susskind(2013),“CoolHorizonsforEntangledBlackHoles,”FortschrittederPhysik61,781.

C.Cao,S.M.Carroll,andS.Michalakis(2017),“SpacefromHilbertSpace:RecoveringGeometryfromBulkEntanglement,”PhysicalReviewD95,024031.

C.CaoandS.M.Carroll(2018),“BulkEntanglementGravityWithoutaBoundary:TowardsFindingEinstein’s

EquationinHilbertSpace,”PhysicalReviewD97,086003.T.BanksandW.Fischler(2001),“AnHolographicCosmology,”https://arxiv.org/abs/hep-th/0111142.S.B.Giddings(2018),“Quantum-FirstGravity,”FoundationsofPhysics49,177.D. N. Page and W. K. Wootters (1983). “Evolution Without Evolution: Dynamics Described by Stationary

Observables,”PhysicalReviewD27,2885.

Chapter14BeyondSpaceandTime

Holography,BlackHoles,andtheLimitsofLocality

Holography,complementarity,andblackholeinformationarediscussedinL.Susskind(2008),TheBlackHoleWar:MyBattlewithStephenHawkingtoMaketheWorldSafeforQuantumMechanics,BackBayBooks.

A.Almheiri,D.Marolf,J.Polchinski,andJ.Sully(2013),“BlackHoles:ComplementarityorFirewalls?”JournalofHighEnergyPhysics1302,062.

J.Maldacena(1997),“TheLarge-NLimitofSuperconformalFieldTheoriesandSupergravity,”InternationalJournalofTheoreticalPhysics38,1113.

S.RyuandT.Takayanagi(2006),“HolographicDerivationofEntanglementEntropyfromAdS/CFT,”PhysicalReviewLetters96,181602.

B.Swingle(2009),“EntanglementRenormalizationandHolography,”PhysicalReviewD86,065007.M. Van Raamsdonk (2010), “Building Up Spacetime with Quantum Entanglement,”General Relativity andGravitation42,2323.

EpilogueEverythingIsQuantum

“Awonderofthiskind”:A.Einstein(1949),AutobiographicalNotes,OpenCourtPublishing,9.

AppendixTheStoryofVirtualParticles

FormoreonFeynmandiagrams, seeR.P.Feynman (1985),QED:TheStrangeTheoryofLightandMatter,PrincetonUniversityPress.

INDEX

acceleration,14–15,48,71,257,300actionatadistance.SeespookyactionatadistanceAdS/CFTcorrespondence,303–304Aesop’sfables,4–5Aharonov,Yakir,177Albert,David,141,177amoebadividinganalogy,123–124amplitudesdescriptionof,19–20probabilitiesand,86–87,130–131,142–146probabilitydistributionand,187unequal,147–148wavefunctionsand,33

angularmomentum,55Anscombe,Elizabeth,129anti—deSitterspace,303,304area,279,284,285Aristotle,13,15arrowoftime,158–159atomsausterequantummechanicsand,34blackbodyradiationand,49–50branchingand,138–139compatibilismand,218Daltonon,45descriptionof,18electronsand,45–46asemptyspace,34,73entropyand,158,160,276,297–298GRWtheoryand,184historyof,45matterand,48obeyingquantummechanicsrules,36radioactivedecayof,120Rutherford’smodel,45–46,52–55statisticalmechanicstheory,29

austerequantummechanics(AQM),32–36,104,245.SeealsoEverettformulationofquantummechanics;Many-Worldstheory

Banks,Tom,285Bauer,Edmond,222Bayes,Thomas,136Bayesianinference,198Bayesianism,136Bekenstein,Jacob,297,300Bekenstein-Hawkingentropy,297–299Bell,John,30–31,178,190Bellstates,102–103Bell’stheoremonentanglement,102–106,190,233BigBang,93–94,113,287blackholeinformationpuzzle,294blackholescomplementarity,304–306degreesoffreedominside,298–299descriptionof,291–292dynamicalnonlocalityand,296emittingradiation,293–296entropyand,297–299evaporation,295eventhorizonand,293ingeneralrelativity,293,301–302generalrelativityand,270,297,305holographyfor,302–303

maximum-entropynatureof,300–301membraneparadigm,305no-cloningtheoremand,296particlesand,291,293–294asregionofspacetime,293,299representinghighest-entropystates,299–300temperatureof,293

blackbodyradiation,49–50blackbodyspectrum,50Bohm,David,30,103,178,189–190Bohmianmechanicsasalternativeformulationofquantummechanics,192–193Einsteinon,194haphazardconstructionof,202Heisenbergon,194nonlocalityand,191Oppenheimeron,194particleobservationand,41,190–194particlesmomenta,195Paulion,194problemsin,194–195uncertaintyprinciplein,195–196wavefunctionsin,193

Bohr,Niels,28,31,35,54–58,66–67,74–75,109Boltzmann,Ludwig,158–163,276Born,Max,20,33,58–59,65–66,67Bornruledescriptionof,19–20inMany-Worldstheory,146–148particlelocationsand,191–193probabilitiesand,130–131,145,167–168asPythagoras’stheorem,87,142self-locatinguncertainty,171

Bousso,Raphael,300brains,ascoherentphysicalsystems,220branchcounting,142–144branchingatomsand,138–139causeof,213–214decisionmakingand,213–216decoherenceand,119–120,122–123,137–138,183,186descriptionof,157–161asemergentworlds,239withfourconsecutivespinmeasurements,134inMany-Worldstheory,169–172Many-Worldstheoryand,138–140asnonlocalprocess,171–172quantumsystemsand,216Schrödinger’sequationand,116

Bunn,Ted,117

Cao,ChunJun(Charles),285categoricalimperative(Kant),210Caves,Carlton,198CFT(conformalfieldtheory),303choice-making.Seefreewillclassicalelectromagnetism,250,269classicalmechanicsatoms(Seeatoms)fields(Seefields)Hamiltonian,64,196,239–240,281–282momentumin,16,70–71,239Newtonand,14–15particles(Seeparticles)positionin,16,70–71,239rulesof,21–22

closeduniverse,287–288cloudofprobability,19,37Coleman,Sidney,129collapsetheory,212compatibilism,218complementarity,74–79,304–306compositeparticles,46–47conformalfieldtheory(CFT),303consciousness,122–123,219–224consequentialism,210Copenhageninterpretationofquantummechanics,23,57,110,116,221CoreTheory,230,249,252cosmologicalconstant,256–257

cosmologicalconstantproblem,258–259credences,136–137,141–142,211curvedspacetime.Seegeneralrelativity

Dalton,John,45Davies,Paul,256deBroglie,Louis,28,30,60,62–65,188deBroglie—Bohmtheory,41.SeealsoBohmianmechanicsdecisionmaking,asclassicalevents,213–216decisiontheory,148–149,212decoherencebranchingand,119–120,122–123,137–138,183,186descriptionof,117–120linkingausterequantummechanicstotheworld,245multipleworldsand,233–234Penroseand,186asrapidprocess,140reversalof,160Schrödinger’scatthoughtexperiment,241–243worldsinterferencewithoneanother,157Zehand,178–179

degreesoffreedom,71,262–263,283–284,298–299Democritus,44demonthoughtexperiment(Laplace),16,57,63,162–163,235Dennett,Daniel,238–239deontology,210determinism,216–218Deutsch,David,126,148,174,194DeWitt,Bryce,126,288Dirac,Paul,65disappearingworldstheory,117distributionofprobabilities.Seeprobabilitydistributionsdouble-slitexperiment,75–79,120–123,191dualism,223dynamicallocality,233,281–282dynamicalnonlocality,296dynamical-collapsemodels,181–186

effectivefieldtheory,320–321Einstein,AlbertBohmand,189–190onBohmianmechanics,194Bohrdebatewith,28–29,31,109compassstory,310oncosmologicalconstant,256–257deathof,309generalrelativitywork,110,112,185,230–231,279–280.SeealsogeneralrelativityonHeisenberg’sapproachtoquantumtheory,271atInstituteforAdvancedStudy,110labelingquantummechanicsasspooky,11lightquantumproposal,51–52,60,66onmatrixmechanics,59physicaltheory,102Podolskyand,101quantumentanglementand,31onquantummechanics,96,102,268asrelativitypioneer,31onspacetime,269–270specialrelativitytheory,99,233onuncertaintyprinciple,91,109

Einstein-Podolsky-Rosen(EPR)thoughtexperiment,96–102,109,191,233,285electriccharge,48nelectricfields,47–48electricity,46electromagneticfieldFeynmandiagramsand,315gravitonsand,274leadingtoparticle-likephotons,255Maxwellon,47–48

electromagneticradiation,49–50,66.Seealsolightelectromagneticwaves,60,249–250electromagnetism,classical,250,269electronsatomsand,45–46Bohr’squantizedorbits,55–58,66–67cloudofprobability,19definitionof,18discoveryof,45double-slitexperiment,75–79

entanglementand,92,120–122Feynmandiagramsand,315interferencebands,121interferencepatternof,77–78innaturalhabitat,18orbiting,46particlesvs.waves,49,75spinoutcomes,80–83,97–99insuperposition,34

elementaryparticles,46–47elementsofreality,100emergence,234–239empiricaltheories,155–156emptyspaceatomsas,34,73energyof,256–257entropyand,278noparticlesin,260–261quantumvacuumin,256–257,259–261quantumversionof,302asstationary,260

energy,173,184,253,256–257,281–282,291.Seealsovacuumenergyentangledsuperposition,114–116entanglement.SeealsoBell’stheoremonentanglementinaction,94–95todefinedistances,276–277degreesoffreedomin,263,283–284descriptionof,37–38,91indifferentregionsofspace,261–265Einsteinand,31electronsand,92,120–122entropyand,160withenvironment,118–119EPRpaperon,96–102,109examplesof,91–92inGRWtheory,182–183momentumand,92nonlocalnatureof,178no-signalingtheorem,97–99inquantumfieldtheory,249quantumstatesand,118,261–262quantumsystemsand,159–160Schrödinger’sequationand,38two-qubitsystem,95–96vacuumenergyand,262–265

entanglemententropy,160,277,283entropicarrowoftime,158–159entropyareaand,285arrowoftime,158–159atomsand,158,160,276,297–298Bekenstein-Hawking,297–299ofblackholes,297–299Boltzmannformulafor,158–163inclosedsystems,158cutoffs,278ofemptyspace,278entanglement,160eventhorizonassimilarto,297limitson,300low,159fromobjectivetosubjectivefeature,161–162quantummechanicsand,276–278ofquantumsubsystems,277withinthermodynamics,279invacuumstate,279

epistemicprobabilities,135–140epistemology,30,197–201equalprobability,144–146equal-amplitudes-imply-equal-probabilitiesrule,144–146ER=EPRconjecture,285nERP.SeeEinstein-Podolsky-Rosen(EPR)thoughtexperimenteventhorizon,293,296–297,301–305Everett,Hughdeathof,127DeWittand,126leavingacademia,178lifestylechoices,127

Many-Worldsformulationand,39Petersenand,125onquantumgravity,111–114,122–125,272onquantumimmortality,207onquantummeasurements,164–165atWeaponsSystemsEvaluationGroup,125–126

Everett,Mark,127EverettformulationofquantummechanicsasassaultonBohr’spicture,123–124implicationsof,41ingredientsfor,40–41measurementsand,104–105,114–117,123–125overviewof,38–40assimpleandelegant,202

Everettphone,180

fatnessmeasure,141Feynman,Richard,2,27n,111,314Feynmandiagramsdescriptionof,314–316electromagneticfieldand,315explicitcutoff,320infinitiesin,319internalclosedloop,317–318particlephysicists’useof,315–317particlesand,314–316renormalization,318–319virtualparticlesin,316

fieldmetric,273fieldsdefiningfeatureof,47definitionof,44inGRWtheory,185inquantumfieldtheory,250–252

FifthInternationalSolvayConference,27–28firewallproposal,296–297nFischler,Willy,285forces,examplesof,16formalismofquantummechanics,30,152–153foundationsofquantummechanicsAlbertand,177Bell’stheoremonentanglement,102,105Bohr-Einsteindebateson,109consensuson,178Everett’sproposal,111,123measurementproblemsand,17Nobelprizeawardedfor,59physicistsresponseto,196–197,272,311Popperand,157spacetimeand,6

Franklin,Benjamin,48nfreewill,216–218frequency,measuring,50–51frequentism,133–135.SeealsoBayesianismFuchs,Christopher,198,200

Geigercountersasquantumsystems,221–222generalrelativity.Seealsoquantumgravity;specialrelativitybehaviorofspacetimein,280BigBangand,287blackholesand,270,293,297,301–302,305Einstein’sworkon,110,112,185,230–231,269–270expansionoftheuniverseand,270asfieldtheory,230,250gravitonsand,273–274loopquantumgravityand,275metricfield,273Penrose’sworkon,185–186predictionsin,156quantizing,230–231,270–271replacingLaplace’stheory,248universe’szeroenergyand,287–288Wheeler’sworkon,110

genericquantumstate,282geometry,270–271,306Gerlach,Walter,80Ghirardi,Giancarlo,181“ghostworld”scenario,120Giddings,Steve,285

gluons,46GospelofMatthew,27nGoudsmit,Samuel,178gravitationalfields,48,248,273–274gravitationalwaves,53–54gravitons,273–274gravity,6–7,165,185–186,230,267,270–273.Seealsogeneralrelativity;quantumgravity;relativitytheoryGreen,Michael,274GRWtheory,181–186,196–197,202–203

Habicht,Conrad,52Hamilton,WilliamRowan,64Hamiltonianformulationofclassicalmechanics,64,196,239–240,281–282Hammeroff,Stuart,219Hawking,Stephen,113,291–292,293h-barversion,56Heisenberg,Werner,28,35,57–59,63,67,194Heisenbergcut,35Heisenbergmethod,271hiddenvariables,187–190hidden-variabletheories,188–189,190,260Hilbertspace,85,154,164–166,263,306Hobbes,omas,218holographicprinciple,302–304Hooft,Gerard’t,302horizontalspin,81–83HouseUn-AmericanActivitiesCommittee,189humanchoice-making.Seefreewillhumanconsciousness,219–224Hume,David,177

idealism,223–225imposingacutoff,258indeterminism,216–218infinities,inFeynmandiagramcalculations,317–319InstituteforAdvancedStudy,Princeton,NewJersey,110interference,76–78,120–122interpretationofquantummechanics.SeeCopenhageninterpretationofquantummechanics

Jacobson,Ted,279Jordan,Pascual,58,67Jumpers(play),129

Kant,Immanuel,210,223

Laplace,Pierre-Simon,16,48,235,248Lewis,David,175light,49–50lightquanta(Einstein),51–52,60,66LIGOgravitational-waveobservatory,53localityprinciple,99,171–172,232–233,240,292London,Fritz,222loopquantumgravity,275low-probabilityworlds,168

magneticfields,47“makingadecision,”213–216Maldacena,Juan,285n,303ManhattanProject,189Many-Worldstheory.Seealsoausterequantummechanics(AQM);Everettformulationofquantummechanicsattachingprobabilitiesto,132Bornrulein,146–148branchingand,138–140,169–170Everettand,39formulasimplicity,179frequentismand,133–135life-spanofapersonand,139–140aslocaltheory,171–172low-probabilityworlds,168measurementand,122,169,179asmorallyrelevant,212–213asnonlocalprocess,233overviewof,38–40quantum-firstperspectiveof,231–232seedsof,113assimpleandelegant,203–204wavefunctionin,234

matrixmechanics,57–59,63,65,67matterwaves,60,62,65

MatthewEffect,27nMaxwell,JamesClerk,47–48,269measurementlocality,233measurementproblemofquantummechanicsalteringSchrödingerequationfor,180–181austerequantummechanicsand,36collapsingwavesystems,22–24,112consciousnessand,224consensuson,17definitevs.indefiniteoutcomes,104–105Everett’stheoryon,104–105,114–117,123–125GRWtheoryand,184Humeon,177Many-Worldstheoryofbranchingand,179Oppenheimeron,178textbookapproachto,242

membraneparadigm,305Mermin,N.David,27,198,200,201Merton,Robert,27nmetric,field,273Michalakis,Spyridon,285microtubules,219–220MilkyWaygalaxy,298–299Misner,Charles,113modeofthefield,253–254modesofthestring,60–61momentuminclassicalmechanics,16,70–71,239entanglementsand,92positionand,69inwavefunctions,71–72

momentumspace,240morality,210–213multipleworlds,6,39,119,180,184,233–234

naturequantumfieldtheoryand,229–230quantummechanicsdescribing,174–175

neuroscience,224neutrons,46newquantumtheory,57.SeealsoquantummechanicsNewton,Isaac,14–15,24,48,239,247–248Newtoniangravity,247–248Newtonianmechanics,14–15Newton’slawsofmotion,21,195,238no-cloningtheorem,296nonlocalprocess,171–172,178,191,233no-nonsenseutilitarianism,211no-signalingtheorem,97–99nuclearfission,110

observableuniverse,94,164–166,182,299,301observer,122–123Occam’srazor,152oldquantumtheory,52–56ontologicalcommitments,152Oppenheimer,Robert,178,189–190,194

Padmanabhan,Thanu,280Page,Don,288Parfit,Derek,139participatoryrealism,200particlesbehavinglikegravitons,274blackholesand,291,293–294Bohmianmechanicsand,41,190–194composite,46–47definingfeatureof,47definitionof,44double-slitexperiment,75–79ofEarth,236elementary,46–47Feynmandiagramsand,314–316inquantumfieldtheory,250virtual,316wavesvs.,49,75

Pauli,Wolfgang,28,57,63,188,194Penrose,Roger,185–186,219personalidentitythroughtime,137–140

Petersen,Aage,113,124,125physicalreality,100–102physicalism,223,224physics,13–15pilotwave,187–188pilot-wavetheories,194Planck,Max,50,66Planck’sconstant,50–51,56Podolsky,Boris,96,101pointerstates,244–245,255Polchinski,Joe,180Popper,Karl,154–157positioninclassicalphysics,16,70–71,239momentumand,69inquantummechanics,70–71

positrons,Feynmandiagramsand,314–315,317post-decoherencewavefunction,234preferred-basisproblem,241–245PrincipiaMathematica(Newton),14probabilitiesamplitudesand,86–87,130–131,142–146Bornruleand,130–131,145,167–168credencesand,136–137decision-theoreticalapproachto,212epistemic,135–140equalprobability,144–146fatnessmeasure,141frequentismand,133–135givenbyamplitudessquared,131,145,147self-locatinguncertaintyand,140–142,149

probabilitydistributions,29–30,187,198probabilityrule,59protons,46Pythagoras’stheorem,86,87,142,146–147,273

quantumarrowoftime,158–159QuantumBayesianism(QBism),41,198–201quantumentanglement.Seeentanglementquantumfieldtheoryentanglementin,249fieldsin,250–252lowest-energystateof,253modeofthefield,253–254natureand,229–230particlesin,250pointerstatesof,255transitionsbetweenstates,255vacuumstatein,254,256–257wavefunctionsin,250–252,254

quantumfluctuations,259–260quantumgravity.Seealsogravityconceptualissues,272–273constructingtheoryof,292Everetton,111–114,122–125,272locationinspace,196loopquantumgravity,275numberofquantumstatesand,165problemoftimein,288spacetimeand,271–272stringtheoryand,274–275technicalchallengesof,113

quantumimmortality,207–209quantumlogic,74quantummeasurementprocess,consciousnessand,164–165,222–224quantummechanicsalternativeformulationof(SeeBohmianmechanics;GRWtheory)atomsand,36discardingclassicalphysics’framework,16–17Einsteinon,96,102,268Einstein-Bohrdebate,28–29,31electromagneticwaves,249–250entropyand,276–278lackofunderstandingof,24–25asonespecificphysicalsystem,229positionin,70–71presentationsof,13rulesof,22–23spacetimeand,271–272

specialrelativityand,269spookinessof,11–12understandingof,1–4violatinglogic,73–74

quantumrandomnumbergenerator,205–206quantumstates.SeealsoBellstatesdisappearingworldstheoryand,117entanglementand,118,261–262evolvingunderSchrödingerequation,232,243,287–288asfundamental,241Hilbertspaceand,85,165uncertaintyprincipleand,73–74,89

quantumsubsystems,277quantumsuicide,208quantumsystemsbranchingand,216classicaldividewith,18,35–36entangled,159–160Geigercountersas,221–222GRWtheoryand,182mathematicaldescriptionof,3measuring,117–118wavefunctionsdescribing,21

quantumutilitymaximizingdevice(QUMaD),211quantumvacuum,254,256–257,259–261quantum/classicaldivide,35–36quarks,46qubits,83–86,87,164,277

radiation,blackholesemitting,293–296radioactivedecay,120radioactiveemissions,242randomness,294–295random-numbergenerator,134Reeh-Schliedertheorem,264regionofspace,65,156,261–265,280–284,299–300relativitytheory,30–31,97–99,112,268–269.Seealsogeneralrelativity;spacetime;specialrelativityrenormalization,318–319Rimini,Alberto,181Rosen,Nathan,96Rutherford,Ernest,45,57Rutherfordatom,45–46,52–55Ryu,Shinsei,303–304

SagittariusA*,299Schack,Rüdiger,198Schrödinger,Erwin,28,59,62Schrödinger’sCatthoughtexperiment,241–245Schrödinger’sequationaltering,180beamsplitter,206branchingand,116descriptionof,21entanglementand,38formulafor,63–64Geigercountersand,221measurementproblemsand,180–181quantumstatesevolvingunder,232,243,287–288spaceandtimetreatmentby,286–287timedefinedin,281wavefunctionsand,21–22,32,64,86,94

Schwartz,John,274scientifictheories,characteristicsof,155–156secondlawofthermodynamics,158,297theself,137–139self-locatinguncertainty,140–142,149,171,211“Shutupandcalculate!,”27sinewave,72,253SolvayConference,27–31,67,96,109,188spaceandtimedegreesoffreedomin,263localityand,240measurementoutcomesof,85–86insuperpositionstate,288–289treatmentof,286–287

spacetimeapplyingquantummechanicsto,271–272blackholesasregionsof,293,299curvatureof(Seegeneralrelativity)

degreesoffreedom,298Einsteinon,269–270foundationsofquantummechanicsand,6geometryof,270–271,306maximumentropyin,300metricin,273quantumgravityand,271–272unified,269warpingof,305wormholesin,285n

specialrelativity,99,170,233,269–270,273.Seealsogeneralrelativityspeedoflightrestrictions,97–98,99spin,79–83spinoutcomes,80–83,87–89,97–99,101spin+apparatussystem,114–116spin-measuringapparatus,118spontaneouscollapseofwavefunctions,181,184–185,192spookyactionatadistance,98,99,105,247–248,309StandardModelofparticlephysics,31,180–181statisticalmechanicstheory,29–30Stern,Otto,80Stern-Gerlachexperiment,133Stoppard,Tom,129stringtheory,274–275superdeterminism,104superpositionsdescriptionof,34–38ofmacroscopicobjects,116Schrödinger’sCatthoughtexperiment,243asseparateworlds,117spaceandtimein,288–289timein,288–289

Susskind,Leonard,285n,302,304–305Swingle,Brian,303–304symmetries,222

TajMahaltheorem,264Takayanagi,Tadashi,303–304Tegmark,Max,207thermodynamics,158,279,297Thomson,J.J.,45Thorn,Charles,302Thorne,Kip,111thoughtexperimentEinstein-Podolsky-Rosen(EPR),96–102,109,191,233,285idealworldof,260–261Schrödinger’sCat,241–245

time.Seespaceandtimetinoxide,44–45two-qubitsystem,95–96ultravioletcatastrophe,50uncertaintyprinciple(Heisenberg)Bohmianmechanicsand,195–196descriptionof,70–73,83Einsteinon,91,109emptyspaceand,260localityand,240quantumstatesand,73–74,89spinoutcomesand,87–89,101

unified“you,”138–139universalwavefunction,113–114,118–119UniverseSplitter,205–206utilitarianism,no-nonsense,211utility,210–212

vacuumenergycosmicaccelerationand,257cosmologicalconstantproblem,258–259ofemptyspace,256–257entanglementand,262–265gravitationalinfluenceof,256–257measuring,50–51,256negative,304positive,304sizeof,257–258

vacuumstateareaand,284emptyspacein,278entropyin,279

entropyproportionaltoboundaryarea,299–300inquantumfieldtheory,254,256–257,260,264

Vaidman,Lev,140VanRaamsdonk,Mark,303–304vectorfieldselectricfieldas,47–48magneticfieldas,47

vectors,84,86–88,131velocityinclassicalphysics,16–17,239measuring,18–19probabilityofmeasuring,71

Verlinde,Erik,280verticalspin,81–83Virgogravitational-waveobservatory,53virtualparticles,316volition,attributing,218vonNeumann,John,74,159–160,188–189,276–278

Wallace,David,129,163,202,239wavefunctionsasabstract,79amplitudesand,33assigninganamplitude,33Bohmianmechanicsand,193Bornon,65–66branchingof,137–138,213–214changingwithtime,62–63collapseof,22–24,33,112,219–220,222consciousnessand,222deBroglie’sview,65descriptionof,19–21distinctpersonsonbranchesof,208–209double-slitexperiment,120–123Hamiltonian,64influencingitself,120inMany-Worldstheory,234measurementoutcomesof,30–31momentumand,71–72foroneparticle,94pilotwaverole,187–188inquantumfieldtheory,250–252,254quantumsystemsand,21ofaqubit,84representingdensityofmassinspace,65Schrödinger’sequationand,21–22,32,64,86,94forsingleparticles,71spacewithin,276spontaneouscollapseof,181,184–185,192fortwoparticles,91–93unifyingparticlesandfieldsinto,44asvectors,86–88

wavemechanics,59–62,65,67wave-function-is-everythingview,33–34wavesdouble-slitexperiment,75–79particlesvs.,49,75

WeaponsSystemsEvaluationGroup,125–126Weber,Tullio,181Weinberg,Steven,180–181Wheeler,JohnArchibald,110–111,123–126,270–272,281,288Wheeler-DeWittequation,288Wilson,Kenneth,319–320TheWire(televisionshow),5Wittgenstein,Ludwig,129Wootters,William,288wormholes,285n

Zeh,HansDieter,117,178–179zeroenergy,256,287–288

ABOUTTHEAUTHOR

SEANCARROLLisatheoreticalphysicistattheCaliforniaInstituteofTechnology.Hisresearchhasfocusedoncosmology,gravitation, fieldtheory,quantummechanics,statisticalmechanics,and foundationsofphysics.Hehasreceivednumerousawards, including theAndrewGemantAwardfromtheAmericanInstituteofPhysics,theRoyalSocietyPrizeforScienceBooks,andaGuggenheim Fellowship. His other books include From Eternity to Here: The Quest for theUltimateTheoryofTime;TheParticleattheEndoftheUniverse:TheHuntfortheHiggsandtheDiscoveryofaNewWorld;andTheBigPicture:OntheOriginsofLife,Meaning,andtheUniverseItself.HealsohoststheweeklyMindscapepodcast.

www.preposterousuniverse.com

AOneworldBook

FirstpublishedinGreatBritain,theRepublicofIrelandandAustraliabyOneworldPublications,2019

Thisebookpublished2019

Copyright©SeanCarroll2019

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ISBN978-1-78607-633-5eISBN978-1-78607-634-2

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