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HOWTOTEACHQUANTUMMECHANICS1

DavidAlbertColumbiaUniversity

Abstract

Idistinguishbetweentwoconceptuallydifferentkindsofphysicalspace:aspaceofordinarymaterialbodies,whichisthespaceofpointsatwhichIcouldimaginablyplace(say)thetipofmyfinger,orthecenterofabilliard-ball,andaspaceofelementaryphysicaldeterminables,whichisthesmallestspaceofpointssuchthatstipulatingwhatishappeningateachoneofthosepoints,ateverytime,amountstoanexhaustivephysicalhistoryoftheuniverse.Inallclassicalphysicaltheories,thesetwospaceshappentocoincide–andwhatwemeanbycallingatheory“classical”,andallwemeanbycallingatheory“classical”,is(Iwillargue)preciselythatthesetwospacescoincide.Butoncethedistinctionbetweenthesetwospacesinonthetable,itbecomesclearthatthereisnologicalorconceptualreasonwhytheymustcoincide–anditturnsout(andthisisthemaintopicofthepresentpaper)thataverysimplewayofpullingthemapartfromoneanothergivesusquantummechanics.

WhatIwanttoshowhere,bymeansofafewverysimplemechanical

examples,is(inanutshell)thateverythingthathasalwaysstruckeverybodyas

strangeaboutquantummechanicscanbeexplainedbysupposingthattheconcrete

fundamentalphysicalstuffoftheworldisfloatingaroundinsomethingother,and

larger,anddifferent,thanthefamiliar3-dimensionalspaceofoureveryday

experience.1I’mthankfultoJillNorth(andofcourse,lessdirectly,toJohnBell)forsuggestingthistitle.2Thiscouldobviouslydowithsomequalification.Aristotle(forexample)famously

2

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Letmebeginbyintroducingausefulmathematicaldevicewithwhichsome

readersmaybeunfamiliar.

Everyclassicalphysicalsystemcanbeuniquelyassociatedwithaformula

calleditsHamiltonian,whichexpressesthetotalenergyofthesysteminquestion–

whichexpresses(thatis)thesumofthekineticandthepotentialpartsoftheenergy

ofthesysteminquestion-asafunctionofthevaluesofitsphysicaldegreesof

freedom,andofthevaluesofvariousofthederivativesifitsphysicaldegreesof

freedom.

Anditturnsout–andthisiswhytheHamiltonianissuchausefuldevice-

thattheHamiltonianofaclassicalsystemconciselyencodeseverythingthereisto

sayaboutthedynamicallawsofmotionthatthatsystemobeys.Itturnsout(thatis)

thatthewaythatthetotalenergyofsuchasystemdependsonitsdegreesof

freedom–andonvariousofthederivativesofthosedegreesoffreedom-uniquely

determinestheequationsoftheevolutionsofthevaluesofthosedegreesoffreedom

intime.Itturnsout(thatis)thatthereisadirectandstraightforwardandfully

algorithmicprocedureforderivingthoseequations–foranyclassicalsystem-from

itsHamiltonian.

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TosaythattheHamiltoniantellsuseverythingaboutthedynamicallawsofa

classicalsystem(however)isnotquitetosaythatittellsusexactlywhatkindofa

classicalsystemitisthatwearedealingwith.Consider,forexample,averysimple

Hamiltonian–onethatconsistsexclusivelyofkineticenergyterms-like:

H=½m(d2x1(t)/dt2)+½m(d2x2(t)/dt2) (1)

ThisHamiltonianfixesthedynamicallawsofasystemwith2degreesoffreedom–

thetwoxi(t).Buttherearetwoquitedifferentsortsofphysicalsystemsthata

Hamiltonianlikethisonecouldverynaturallybereadasdescribing.Wecouldread

it(thatis)asdescribingapairofparticles,bothofmassm,movingaround,inthe

absenceofanyforces,andwithoutinteractinginanywaywithoneanother,inaone-

dimensionalspace.Orwecouldreaditasdescribingasingleparticle,ofmassm,

movingaround,intheabsenceofanyforces,inanotherwiseemptytwo-

dimensionalspace.AllthattheHamiltoniandoesistodeterminethedifferential

equationsthateachofthetwoxi(t)needtosatisfy.AllthattheHamiltoniandoes–

inthisparticularcase–istodeterminethat

xi(t)=ai+vit (2)

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whereaiandviarecanbeanyrealnumbers.Andthat’spreciselythesortof

behaviorthatwewouldintuitivelyexpectofeitheroneofthetwodifferentphysical

systemsdescribedabove.

*

Good.Let’smakethingsalittlemorecomplicated.

Consider(again)aclassicaluniversewithtwophysicaldegreesoffreedom–x1

andx2–butnowsupposethatthevaluesofx1andx2evolveintimeinaccordwith

theHamiltonian

H=(1/2m1)(dx1(t)/dt)2+(1/2m2)(dx2(t)/dt)2+δ(x1-x2).(3)

ThisHamiltoniandiffersfromtheoneinequation(1)intwoimportantways:it

includesaverysimplepotential-energyterm-δ(x1-x2)–anditallowsforthe

possibilitythatthevaluesofminthetwokinetic-energytermsmaybedifferent.

Herearetwodifferentwaysofdescribingauniversewhosedynamicallawsare

givenbyaHamiltonianliketheoneinequation(3):

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A)Theuniverseconsistsoftwopoint-likephysicalitems,movingaroundina

one-dimensionalspace,andinteractingwithoneanother,oncontact,bymeansof

elasticcollisions.

B)Theuniverseconsistsofasinglepoint-likephysicalitem,movingaround

inatwo-dimensionalspace,withaninfinitepotentialbarrieralongit’sx1=x2

diagonal–asinFigure1.

Figure1

Thesetwodescriptions–likethetwodescriptionsweconsideredinconnection

withtheHamiltonianinequation(1)–arefullymathematicallyisomorphictoone

another.Butinthiscase,unlikeinthepreviousone,thetwodescriptionsarenotapt

tostrikeusasequallynatural.Takealmostanybody,withalmostanykindofan

educationinphysics,andwakethemupinthemiddleofthenight,andaskthemto

describethesortofworldthatmighthaveaHamiltonianliketheoneinequation(3)

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asitsfundamentallawofmotion–andyouarelikelytogetsomethingthat’smuch

closertothelanguageofdescription(A)thanitistothelanguageofdescription(B).

Andthereasonsforthatwillbeworthpausingover,andthinkingabout.

Tobeginwith,themassassociatedwiththekineticenergyofx-motionandthe

massassociatedwiththekineticenergyofy-motion–intheexampleweare

consideringhere-aredifferent.Andweareusedtoassociatingasinglemasswitha

singlematerialobject.Youmightevensaythatitispartandparcelofourveryidea

ofwhatitistobean‘ordinarymaterialobject’thateverysuchobjectisinvariably

associatedwithsomesingle,determinate,valueofit’smass.Andoureveryday

conceptionoftheworldseemstohavesomethingtodowithit’sbeingthehabitation

ofobjectslikethat.Andoureverydayconceptionofspaceseemstohavesomething

todowiththesetofpointsatwhichanordinarymaterialobjectmightinprinciple

belocated,orwiththestageonwhichsuchobjectsseemtomaketheirwayabout.

Good.Butwhatifthemasseshappentobethesame?Won’titbejustasnatural

(inthatcase)tothinkofthisuniverseasconsistingofasinglematerialparticle,

movingaroundina2-dimensionalspace,withaninfinitepotentialbarrieralongthe

diagonallinex1=x2?

Well,no.Thereareotherissueshereaswell.Itseemstobeanimportantpartof

oureverydayconceptionofthespaceinwhichmaterialparticlesmaketheirway

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about(forexample)thatitisbothhomogeneousandisotropic.2Itseemstobean

importantpartofoureverydayconceptionofthespaceinwhichmaterialparticles

maketheirwayabout(thatis)thatitshouldbejustaseasy,insofarasthe

fundamentallawsofphysicsareconcerned,foramaterialparticletobeinone

locationasitisforittobeinanother,andthatitshouldbejustaseasy,insofaras

thefundamentallawsofphysicsareconcerned,foramaterialparticletobemoving

inonedirectionasitisforittobemovinginanother.Andthetwo-dimensional

pictureofthesortofworldweareconsideringhereobviouslyfeaturesa

fundamentallawthatdistinguishesbetweenpointsonthediagonalandpointsoffof

it.Butifyoulookatthatsamelawinthecontextoftheone-dimensionalpicture–if

youlook(thatis)atthepotentialtermintheHamiltonianinthecontextoftheone-

dimensionalpicture-allitsaysisthatthetwoparticlescan’tpassthroughone

another.Andthatwayofputtingitobviouslymakesnodistinctionswhatever

betweenanytwopointsintheone-dimensionalspace,oramongeitherofitstwo

directions.

Whycouldn’twethinkofthepresencepotentialbarrierinthe2-dimensional

picture(then)notasamatteroffundamentallaw,but(rather)asarisingfromthe

merelydefactoconfigurationofafield?Well,thatwouldamounttodenyingthatthe

Hamiltonianinequation(3)isinfactthefundamentalHamiltonianoftheuniverse.

2Thiscouldobviouslydowithsomequalification.Aristotle(forexample)famouslythoughtotherwise.Butthereisanintuitiveandwell-knownandlong-standingclassical-mechanicalconceptionofspacethatIamgesturingathere,whichItakeitisrecognizabletoeveryone,andwhich(withalittlework)canbeformulatedinsuchawayastoapplytospecialandgeneralrelativityaswell.

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Inthatcase(toputitinaslightlydifferentway)thefundamentalHamiltonianofthe

worldisgoingtobesomethingmoreelaboratethantheoneinequation(3),

somethingwhichoffersadynamicalaccountnotonlytheevolutionsoftheco-

ordinatesx1andx2,buttheconfigurationsofthefieldsaswell(something,thatis,

thatanswersquestionsabouthowthefieldsgotthere,andhowtheyevolve,how

theyareaffectedbychangesinthex1andx2degreesoffreedom,andsoon).And

thatnewfundamentaltheoryisgoingtobringwithitallsortsofnewphysical

possibilities,andnewcounterfactualrelations,thatwerenotpresentintheoriginal

2-dimensionalHamiltonianthatweweredealingwithabove.

So,whatfeelsmorefamiliaraboutthefirstofthesedescriptionsisthatit

featuresaspacewhichishomogenousandisotropic,andwhichconsistsofthesorts

ofpointsatwhichordinarymaterialparticles–particles(thatis)whichare

associatedwithuniqueanddeterminatequantitiesofmass–mightinprinciplebe

located.Let’srefertospaceslikethat(then)asspacesofordinarymaterialbodies.

Andnote(sinceitwillbeimportanttowhatfollows)thatwhatisandisn’tgoingto

count,forthisorthatparticularphysicaluniverse,asaspaceofordinarymaterial

bodies,isnotamatterofitsfundamentalmetaphysicalstructure,but(rather)ofits

dynamicallaws.

*

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Good.Let’smakethingsalittlemorecomplicated.Focusonthesecondofthe

twodescriptions–thelessfamiliarone,the2-dimensionalone-ofthesimple

universethatweweretalkingaboutabove.Andnowconsideradifferentuniverse,a

slightlymorecomplicatedone,whichweobtainbyintroducingasecondpoint-like

physicalitemintothetwo-dimensionalspace–anitemwhichwestipulatetobe

intrinsicallyidenticaltothefirst,andwhichfloatsaroundunderthegovernanceof

exactlythesamesortofHamiltonianastheoneinequation(3)(seefigure2).The

completeHamiltonianofauniverselikethat(then)isgoingtobe:

H=(1/2m1)(dx1(t)/dt)2+(1/2m2)(dx2(t)/dt)2+(1/2m3)(dx3(t)/dt)2+

(1/2m4)(dx4(t)/dt)2+δ(x1-x2)+δ(x3-x4)(4)

wherex1andx2aretheλandμco-ordinatesofitem#1,andx3andx4aretheλandμ

co-ordinatesofitem#2,respectively.

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Figure2

Auniverselikethisonemightbedescribedasconsistingoffourordinary

materialparticles–callthemparticle1andparticle2andparticle3andparticle4-

movingaround(asbefore)inahomogenousandisotropicone-dimensionalspace.

x1willthenrepresenttheone-dimensionalpositionofparticle1,andx2will

representtheone-dimensionalpositionofparticle2,andsoon.Butthewaythose

particlesmovearoundisnowgoingtobekindoffunny.Suppose(justtokeep

thingssimpleforthemoment)thatm1=m3andm2=m4.Thenparticles1and3are

goingtobequalitativelyidenticaltooneanother,andparticles2and4aregoingto

bequalitativelyidenticaltooneanother,andyetthewaythatparticle1interacts

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withparticle2isgoingtobedifferentfromthewaythatparticle3interactswith

particle2,andthewaythatparticle2interactswithparticle1isgoingtobe

differentfromthewaythatparticle4interactswithparticle1(particle1,for

example,isgoingtobounceoffofparticle2,butitwillpassrightthroughparticle4–

andparticle4isgoingtobounceoffofparticle3,butitwillpassrightthrough

particle1).Andso,unlikeinthetwo-particlecaseweconsideredbefore,a

qualitativedescriptionofthephysicalsituationofthisworld,atsomeparticular

time,intheone-dimensionalspace(thatis:acompletespecificationofwhichfour

pointsinthisone-dimensionalspaceareoccupiedbyparticles,togetherwitha

specificationofthevelocitiesoftheparticlesateachofthosepoints,togetherwitha

specificationoftheintrinsicpropertiesoftheparticlesateachofthosepoints)isnot

goingtogiveusenoughinformationtopredict,eveninprinciple,thequalitative

situationofthisworldatothertimes.3

3Thefirstthingthat’slikelytopopintoone’shead,onbeingconfrontedwiththis,isthatparticle1mustnot(infact)bequalitativelyidenticaltoparticle3,and(similarly)thatparticle2mustnotbequalitativelyidenticaltoparticle4.Let’sthinkabouthowthatmightwork.Supposethatparticle1werenotqualitativelyidenticalwithparticle3.ThentherewouldhavebesomepairofphysicalpropertiesPandQsuchoneofthemisPandtheotherisQ.Andifparticle2werenotqualitativelyidenticaltoparticle4,thentherewouldhavetobesomepairofpropertiesRandS(whichmightormightnotbedifferentpropertiesfromthepropertiesPandQ)suchoneofthemx’sisRandtheotherisS.Andwiththesenewpropertiesinhand,onecouldofcoursewritedownalaw–whichwouldaccountforthemotionsoftheseparticles-totheeffectthat(say)particlesthatarePonlybounceoffofparticlesthatareR,andthatparticlesthatareQonlybounceoffofparticlesthatareS.Orsomethinglikethat.Butnothinglikethatcanberight.Whatwearesupposedtobeimagininghere(remember)isauniversethatconsistsoftwointrinsicallyidenticalpoint-likephysicalitems,floatingaroundinatwo-dimensionalspace,inaccordwiththeHamiltonianinequation(4).Whatwearesupposedtobeimagining(toputitslightlydifferently)isthataspecificationofthelocationsofthosetwointrinsicallyidenticalpoint-likephysicalitemsinthetwo-dimensionalspace,atanyparticulartime,amountstoacompletequalitativedescriptionoftheworldatthetimein

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Maybethethingtosay(then)isthatthesortofworldthatweareimagininghere

isjustnotthoroughlylawful–maybethethingtosay(thatis)isthatthereissimply

notanyfullygeneralruleabouthowthecompletephysicalconditionsofauniverse

likethisoneatdifferenttimesarerelatedtooneanother.Butthat’sobviouslynot

righteither.Ifwelookbackatthetwo-dimensionalrepresentationofthisparticular

universe(afterall)theneverythingimmediatelysnapsintoplace:acomplete

specificationofthequalitativesituation,atanyparticularinstant,inthetwo-

dimensionalspace(thatis)isgoingtogiveusenoughinformationtopredict,in

question–adescription(thatis)onwhichalloftheotherqualitativefeaturesoftheworld,atthatinstant,supervene.ButifthereareP’sandQ’sandR’sandS’softhesortthatwehavejustnowbeenimagining–then(asthereadercaneasilyconfirmforherself)therearegoingtobetwoqualitativelydifferentsituationsintheone-dimensionalspacecorrespondingtoeverypairoflocationsinthetwo-dimensionalspace–andsoaspecificationofthelocationsofthetwointrinsicallyidenticalpoint-likephysicalitemsinthetwo-dimensionalspacewouldnotamounttoacompletequalitativedescriptionoftheworldafterall.Andso–inthesortofworldthatweareimagininghere,andifwearegoingtobeinthebusinessofattributinganyphysicalpropertiesatalltotheparticlesthataremovingaroundintheone-dimensionalspace–particle1mustbeintrinsicallyidenticaltoparticle3,andparticle2mustbeintrinsicallyidenticaltoparticle4.

Andevenifwesetasidetheideathatthesefourmaterialparticlesare“shadows”ofsomethingelsemovingaroundinahigher-dimensionalspace–even(thatis)ifweimagineauniversethatconsistsofnothingwhateveroverandabovethosefourmaterialparticles,movingaroundinaone-dimensionalspace,asiftheywereshadowsoftwoidenticalpoint-likephysicalitemsmovingaround,inatwo-dimensionalspaceinaccordwiththeHamiltonianinequation(4)–itwouldstillmakenosensetoimaginethatparticle1issomehowintrinsicallydifferentfromparticle3,andthatparticle2issomehowintrinsicallydifferentfromparticle4,becauseitfollowsfromthequalitativeidentityofthetwo(imaginary)point-likephysicalitemsinthe(imaginary)two-dimensionalspacethatparticle1isgoingtorespondtoanyexternallyimposedforce-fieldinexactlythewaythatparticle3does,andthatparticle2isgoingtorespondtoanyexternallyimposedforce-fieldinexactlythewaythatparticle4does.

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principle,howthatsituationisgoingtoevolveintothefuture.Andfromthat(of

course)wearegoingtobeabletoreadoffallofthefuturequalitativesituationsin

theone-dimensionalspaceaswell.

Inthecaseweconsideredbefore,theone-dimensionalrepresentationofthe

universeandthetwo-dimensionalrepresentationoftheuniversewere

straightforwardlyisomorphictooneanother.Inthecaseweconsideredbefore(that

is)therewasexactlyonepossiblestateofthepoint-likeitemfloatingaroundinthe

two-dimensionalspacecorrespondingtoeveryindividualoneofthepossiblestates

ofthetwomaterialparticlesfloatingaroundintheone-dimensionalspace.Buthere

(asImentionedabove)therearetwoqualitativelydifferentstatesofthetwopoint-

likephysicalitemsfloatingaroundinthetwo-dimensionalspacecorrespondingto

everyindividualqualitativestatethefourmaterialparticlesfloatingaroundinthe

one-dimensionalspace.Andsothehistoryoftheuniversewearedealingwithhere

–thehistory(thatis)ofthisparticularpairofpoint-likephysicalitemsfloating

aroundinthisparticulartwo-dimensionalspace-cansimplynotbepresentedinthe

formofahistoryofthemotionsoffamiliarmaterialbodies,andthedynamicallaws

ofauniverseliketheonewearedealingwithherecansimplynotbewrittendownin

theformoflawsofthemotionsofeverydaymaterialbodies.Inthecasewe

consideredbefore(toputitslightlydifferently)thebasicphysicalstuffoftheworld

–thestuffonwhosehistorythehistoryofeverythingelsesupervenes,thestuffto

whichthefundamentaldynamicallawsapply-wasthestuffofthematerialparticles.

Butitseemsliketherightthingtosayaboutauniverselikethisoneisthatthebasic

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physicalstuffisthestuffofthetwopoint-likeitemsinthetwo-dimensionalspace–

andthatthereasonthateverythinglookssooddasviewedfromtheperspectiveof

theone-dimensionalspaceisthattheone-dimensionalspaceisn’twherethingsare

reallygoingon,andthatthematerialparticlesthatmovearoundinthatspaceare

reallyjust“shadows”(asitwere)oftheactual,fundamental,physicalitems.

Sotherearegoingtobetwosortsofspacethatareworthtalkingaboutina

universelikethisone.Thereis,tobeginwith,theone-dimensionalhabitationof

ordinarymaterialbodies.Andthenthere’sthespaceinwhichonecanrepresent

everythingthat’sgoingon,inwhichonecankeeptrackofeverythingthat’sgoingon,

merelybysayingwhatitisthat’sgoingonateveryindividualoneofitspoints-the

space(youmightsay)ofthetotalityofatomicopportunitiesforthings,atany

particulartemporalinstant,tobeonewayoranother.Callthat“thespaceofthe

elementaryphysicaldeterminables”.Andwhatwehavejustseenisthatthespaceof

theelementaryphysicaldeterminables,inaworldliketheonewearedealingwith

here,hastwodimensions.

Thespaceofordinarymaterialbodiesandthespaceofelementaryphysical

determinablesturnouttobeverydifferentkindsofthings.Itispartandparcelof

ourideaofthespaceofordinarymaterialbodies(forexample)thatallofthepoints

initaregoingtobeintrinsicallyidenticaltooneanother–buttheaboveexample

makesitclearthatweshouldhavenosuchexpectations,asageneralmatter,about

thespaceofelementaryphysicaldererminables.Thespaceofordinarymaterial

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thingsisthesetofpointsatwhichyoucouldimagine,inprinciple,placingthetipof

yourfinger.Buttheitemsthatmovearoundinthespaceoftheelementaryphysical

determinables,atleastinthecaseofthesortofuniverseweareconsideringnow,

arenotmaterialbodiesatall.

But(notwithstandingallthat)thespaceoftheelementaryphysical

determinablesisclearlythemorefundamentalofthetwo.Thesituationinthespace

ofordinarymaterialbodies(onceagain)supervenes,bydefinition,onthesituationin

thespaceoftheelementaryphysicaldeterminables–butthereverseisofcourse

nottrue–ornot(atanyrate)inthesortofworldwearethinkingofhere.Sothe

sortsofdistinctionsthatonecanmakeinthelanguageofthespaceofthe

elementaryphysicaldeterminablesaremorefine-grainedthanthesortsof

distinctionsonecanmakeinthelanguageofthespaceofordinarymaterialbodies.

Moreover,thespaceoftheelementaryphysicaldeterminablesiswhatfixesthe

elementarykinematicalpossibilitiesoftheworld–andsoitis(inthatsense)

somethinglogicallypriortothelawsofdynamics,it’ssomethinglikethearena

withinwhichthoselawsact.Butthespaceofordinarymaterialthings(asI

mentionedbefore)issomethingwhosetopologyandwhosegeometryandwhose

veryexistenceallexplicitlydependonwhatthefundamentaldynamicallawsactually

happentobe–it’ssomethingthatthedynamicscanbethoughtofasproducing,

somethingwhichis(inthatsense)emergent.

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Thereareotherdifferencestoo–thereisaninterestingquestion(forexample)

aboutwhetherthespaceoftheelementaryphysicaldeterminablesneedstobe

thoughtofashavinganygeometryatall–butadiscussionofthosewouldtakeus

toofarafieldatthemoment.4

Theimageof‘space’thatallofusgrewupwith(then)turnsouttobeacrude

andundifferentiatedamalgamofbothaspaceofordinarymaterialbodiesanda

spaceofelementaryphysicaldeterminables.Thatphysicsshouldneverheretofore

havetakennoteofthedistinctionbetweenthesetwosortsofspacesisentirely

unsurprising–becausetheyhappentobeidenticalwithoneanother(justasthey

wereinthetwo-particle,one-dimensionalexampleweconsideredabove)in

NewtonianMechanics,andinMaxwellianElectrodynamics,andinthephysicsof

everydaymacroscopicpracticallife.Themanifestimageoftheworld(youmight

say)includesbothaspaceofordinarymaterialthingsandaspaceoftheelementary

physicaldeterminables-togetherwiththestipulationthattheyare,infact,exactly

thesamething.Andclassicalphysicsnevergaveusanyreasontoimagine

otherwise.But(notwithstandingallthat)thesetwoideaswouldseemtobeworth

carefullypryingapart.Theyhavenothinglogicallytodowithoneanother,anditis

theeasiestthingintheworld(aswehavejustseen)toimagineuniverses,andto

writedownHamiltonians,inwhich(forexample)theyhavedifferentnumbersof

dimensions.

4Theseissuesoftheoriginandsignificanceofdistancearethefocusofarecentunpublishedmanuscriptofminecalled“OntheEmergenceofSpaceandTime”.

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Ok.Let’sgetback,withallthisinmind,totheparticularsystemwewere

thinkingaboutbefore–theonedescribedbytheHamiltonianinequation(4).One

oftheeffectsofintroducingasecondpoint-likephysicalitemintothetwo-

dimensionalspaceis(aswehaveseen)topryapartthespaceofordinarymaterial

bodiesandthespaceofelementaryphysicaldetirminables–tomakethem(in

particular)intotwodistinctandtopologicallydifferentspaces.Andoneofthe

effectsofthiscoming-apartisthatthegoings-oninthespaceofordinarymaterial

bodies–or(rather)thatthegoings-oninthephysicaluniverse,asviewedfromthe

perspectiveofthespaceofordinarymaterialbodies–beginstolookodd.

Particles1and2bounceoffoneanother,andparticles3and4bounceoffone

another,but(eventhoughparticle1isintrinsicallyidenticaltoparticle3and

particle2isintrinsicallyidenticaltoparticle4)theparticles1and2movearoundas

ifparticles3and4simplydidnotexist,andparticles3and4movearoundasif

particles1and2simplydidnotexist.Andsowhatwearepresentedwith,inthe

spaceoftheordinarymaterialbodiesofauniverselikethisone,islesslikea

collectionoffourparticlesfloatingaroundinaone-dimensionalspace,thanitis

(say)likeapairofcausallyunconnectedparallelworlds,ineachofwhichthereisa

pairofparticlesisfloatingaroundinaone-dimensionalspace,orlikeapairof

differentpossibilities,orlikeapairofdifferentscenarios,abouthowoneandthe

samepairparticlesmightbefloatingaroundinaone-dimensionalspace,or

somethinglikethat.Whateveristrueinbothofthescenariosisapparentlytrue

simpliciter–sothat(forexample)ifx1andx3bothhappentobeequalto5,thenthe

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firstparticle(theonewhosemassism1=m3)isunambiguouslylocatedatpoint5in

theone-dimensionalspaceofordinarymaterialbodies–butit’shardertoknow

exactlywhattosayaboutfactsonwhichtheydiffer.

Ifwewereadamantaboutrepresentingauniverselikethistoourselvesinits

one-dimensionalspaceofordinarymaterialbodies,wemightdosowiththehelpof

anadditionalpieceofnotation–apairofbrackets(say),oneofwhichlinksparticle

1withparticle2,andtheotherofwhichlinksparticle3withparticle4-asinfigure

3–toindicatewhichparticlessharethese‘scenarios’withoneanotherandwhich

don’t.

Figure3

Fromthepointofviewofthetwo-dimensionalspaceofelementaryphysical

determinables,thebracketsarejustawayofkeepingtrackoftheconnections

betweenthefourordinarymaterialparticlesintheone-dimensionalspaceandthe

twopoint-likephysicalitemsinthetwo-dimensionalspace.Butifweareresolutein

banishinganythoughtofthatlatterspacefromourminds,thenweareapparently

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goingtoneedtothinkofthebracketsassignifyingsomerealandradically

unfamiliarandnot-further-analyzablephysicalconnectionbetweenpairsof

materialparticlesthemselves–somethingthatcannotbereducedto,somethingthat

doesnotsuperveneon,thespatialdistributionoflocalphysicalproperties.

*

Ok.Let’scomplicatethingsstillmore.Supposethatweweretoaddatermofthe

formδ(x1-x2)δ(x3-x4)totheHamiltonianinequation(4),sothatitlookslikethis:

H=(1/2m1)(dx1(t)/dt)2+(1/2m2)(dx2(t)/dt)2+(1/2m3)(dx3(t)/dt)2+

(1/2m4)(x4(t)/dt)2+δ(x1-x2)+δ(x3-x4)+δ(x1-x3)δ(x2-x4)(5)

Thatwouldamounttoaddinganewandfunnykindofaninteraction-an

interactionnotbetweentwooftheparticlesfloatingaroundinthematerialspace,

but(rather)betweenthetwopoint-likeitemsfloatingaroundinthedeterminable

space–aninteraction(thatis)betweenwhatmightpreviouslyhavelookedtous,

fromtheperspectiveofthematerialspace,liketwodifferentpossibilities,ortwo

differentscenarios,ortwodistinctandparallelworlds.

Note(tobeginwith)thatthisnewtermisstillgoingtopreservetheinvarianceof

theHamiltonianundertranslationsintheone-dimensionalspace–andsothe

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materialspaceofthisnewworld,thespaceinwhichallpointsareintrinsically

identical,thespaceinwhichparticleshaveuniquedeterminatemasses,isstillgoing

tobeone-dimensional.Butthebehaviorsoftheseparticles,asviewedfromtheone-

dimensionalspaceinwhichtheylive,aregettingodderandodder.

Theeffectofaddingthisnewinteractionisgoingtobequantitativelysmall–

becausecollisionsbetweenthetwopoint-likeitemsinthedeterminablespaceare

goingtobemuchmuchrarerthancollisionsbetweeneitheroneofthemandthe

fixeddiagonalpotentialbarrier-butitisnonethelessgoingtobeconceptually

profound.Fromtheperspectiveofthematerialspacethingsarestillgoingtolook

moreorlessasiftherearetwopairsofparticlesfloatingaroundintwoparallel

possiblesituations–linkedtogetherbytheirmysteriousbrackets.Butamore

detailedexaminationisnowgoingtorevealthatthispictureofparallelpossible

situationsdoesnotquiteholdup–becausetheevolutionsofthesetwopossibilities

cansometimes,infact,interferewithoneanother.

Moreover,theeffectsofthisnewinteraction,asviewedfromtheperspectiveof

thespaceofordinarymaterialbodies,aregoingtobebizarrelynon-local.Particles

x1andx2aregoingtocollidewithoneanother(thatis:particles1and3aregoingto

interactwithoneanother,particles1and3aresuddenlygoingtobecomevisibleto

oneanother,particles1and3aresuddenlygoingtobeunabletopassthroughone

another)onlyintheeventthatparticles2and4happentobecollidingwithone

another,somewhereinthematerialspace,anywhereinthematerialspace,atexactly

21

thesametemporalinstant.Andviseversa.Andthemechanismwherebythosetwo

collisionsmakeoneanotherpossibledoesnotdependonanywaywhateveronthe

one-dimensionalphysicaldistancebetweenthem–itonlydependsontheir

primitiveandunanalyzableandnowevenmoremysteriousbracket-connections.

Andallofthisshouldbynowhavebeguntoremindyouofquantummechanics.

Butthebusinessofsayingexactlyhowitshouldremindyouofquantummechanics–

thebusinessofsayingexactlywhatshouldremindyouofwhat-requiresclose

attention.Thethingthatneedstobekeptinmind,thethingthatpeopleseemto

havetroubletakinginwhenallofthisisfirstpresentedtothem,isthatintroducing

anotherpoint-likeconcretefundamentalphysicalitemtothe2-dimensionalspaceis

notatalllikeintroducinganotherparticletothe1-dimensionalspace,orlike

introducinganotherpairofparticlestothe1-dimensionalspace,orlikeintroducing

asecondBohmianMarvelousPointintothecontentsofthesortofuniversethatwe

havebeenthinkingabouthere-but(rather)likeaddinganotherterm,likeadding

anotherbranch,tothequantum-mechanicalwave-functionofthesingle,original,

two-particlesysteminthe1-dimensionalspace.Consider,forexample,apairof

quantum-mechanicalparticles,oneofwhich(particle1)hasonlyposition-space

degreesoffreedom,andtheotherofwhich(particle2)hasbothspinandposition-

spacedegreesoffreedom.Andsupposethatthequantumstateofthatpairof

particles,atacertaintime,is:

(1/√2)[x=α>1[x=γ,éz>2+(1/√2)[x=β>1[x=γ,êz>2(6)

22

Andnotethatifα≠β(whichistosay:ifthetwoversionsofparticle1donothappen

tobelocatedatthesamepointinspace)thenthereduceddensitymatrixofparticle

2isgoingtobeanincoherentmixtureofspin-states–butifα=β(whichistosay:if

thetwoversionsofparticle1dohappentobelocatedatthesamepointinspace)

thenthetwospin-statesofparticle2aregoingtointerferewithoneanother,insuch

awayastoproduceaneigenstateofx-spin.Andnote(aswell)thatnoneofthis

dependsinanywayonhowfarapartinspaceα(orβ)andγmayhappentobe,or

whatmayhappentobegoingoninthespacebetweenthem.5

*

Let’sgoonestepfurther.Thisone(however)willtakeabitofsettingup.

Tobeginwith,replacetheverysharplypeakedpotentialbarrieralongthe

diagonalinthetwo-dimensionalspacewithamoresmoothlyvaryingpotentialwell

alongthediagonalinthetwo-dimensionalspace.Thatis:replacetheδ(x1-x2)+δ(x3-

x4)in(4)withV(|x1-x2|)+V(|x3-x4|),whereV(|r|)issomesmoothlyvaryingand

monotonicallyincreasingfunctionof|r|which,andwhichisnegativeforallfinite5Thereadermightliketoconsiderwhyitisthatthekindofquantum-mechanicalnon-localityondisplayinthisexamplecannotbeexploited,asamatteroffundamentalprinciple,forthesuperluminaltransmissionofinformation.Thereis,ofcourse,afamousargument,duetoVonNeumann,totheeffectthatthenon-localitiesassociatedwiththecollapseofthewave-function,orwiththeeffectivecollapseofthewave-function,cannotbeexploitedforthesuperluminaltransmissionofinformation–buttherearenocollapseshere,andthesortofargumentoneneedsinacaselikethisturnsouttobesomewhatdifferent.

23

valuesof|r|,andwhichasymptoticallyapproacheszeroas|r|approachesinfinity.

Thisamountstoreplacingthesharpcontactrepulsiveforcebetweenparticles1and

2andparticles3and4intheone-dimensionalspacewithaforcethatattracts1

towards2and3towards4,andwhichactsacrossfinitedistances(likeNewtonian

gravitation,say)intheone-dimensionalspace.Andlet’sstipulate,aswell,thatthis

attractiveforcecanbeswitchedonandoffaswewish.

TheHamiltonianwe’redealingwithnow(then)is

H=(1/2m1)(dx1(t)/dt)2+(1/2m2)(dx2(t)/dt)2+(1/2m3)(dx3(t)/dt)2+

(1/2m4)(x4(t)/dt)2+g(t)(V(|x1-x2|)+V(|x3-x4|))+δ(x1-x3)δ(x2-x4)(7)

wheretheg(t)istheresimplytoremindusthatweareallowedtoswitchthatpart

ofthepotentialenergyonandoffatourdiscretion.

Andnote(yetagain)thatnoneofthesechangesaregoingtoalterthefactthat

thematerialspaceofasystemlikethisistheone-dimensionalspace,andthatit’s

determinablespaceisthetwo-dimensionalone.

Good.Nowsuppose(forreasonsthatwillpresentlybecomeclear)thatparticles

1and3aremuchmoremassivethanparticles2and4.Andsetthingsupasfollows:

Theattractiveforceisoff,andparticles2and4areatrestattheorigin,andparticle

1isatthepoint+1andparticle3isatthepoint-1,asdepictedinfigure4.

24

Figure4

Thecorrespondingarrangementinthetwo-dimensionalspace,whichis

depictedinfigure5,hasoneofthepoint-likephysicalitems-itemnumber1–atthe

point(λ=+1,μ=0)andtheother–itemnumber2–at(λ=-1,μ=0).Thesmallarrows

infigures4and5indicatethedirectionsinwhichthetwoparticles(inthecaseof

figure4)andthetwopoint-likeitems(inthecaseoffigure5)willbegintomove

oncetheattractivepotentialisswitchedon.

25

Figure5

Sinceparticles2and4aretouchingoneanotherhere,particles1and3aregoing

bounceoffoneanotheriftheyshouldhappentomeet.Soparticles1and3,inthis

situation,donotrelatetooneanothermuchlikecomponentsoftwodistinct

possibilities,oroftwoparallelworlds.Indeed–andonthecontrary-whatweare

dealingwithhere(solongasthey-particlesremainatrest,andtouchingone

another,andsolongastheattractiveforcesareswitchedoff)isjustthefamiliarcase

oftwoparticles(thex-particles)movingaroundinaone-dimensionalspace,and

interactingwithoneanotherbymeansofarepulsivecontactinteraction–precisely

thecase(thatis)thatwestartedoffwith.

26

Andnowsupposethatweswitchtheattractiveforceson.Atthispointwewill

havemovedthingsintoaregimeinwhichbothofthedimensionsofthe

determinablespaceassociatedwiththeHamiltonianin(7)comedecisivelyinto

play.Andonewaytothinkaboutwhat’sgoingonhereisthatwehaveswitchedona

pairofmeasuring-devicesforthepositionsofparticles1and3–deviceswhose

pointersareparticles2and4.Whentheattractiveforcesareswitchedon,eachof

they-particlesstartstomoveinthedirectionofit’scorrespondingx-particle–each

ofthey-particlesindicates(youmightsay)thedirectioninwhichit’scorresponding

x-particleislocated–anditwaspreciselyinordertobuildtheappropriatesortof

asymmetryintothisindicator-indicatedrelationshipthatwestipulated,afew

paragraphsback,thatthemassesofparticles1and3bemuchlargerthanthe

massesofparticles2and4.

Andnote(andthisisthepunchline)thatassoonasthesemeasurementstake

place,andforaslongastheirdifferentoutcomesarepreservedindifferences

betweenthepositionsofparticles2and4,thewholemetaphysicalcharacterofthe

situation–atleastasviewedfromthe1-dimensionalmaterialspace-appearsto

radicallyshift.Anypossibilityofinteractionbetweenparticles1and3isnow

abolished,andthesystembehaves,again,foralltheworld,asifitwereapairof

mutuallyexclusivescenarios,orofparalleluniverses,inoneofwhichalightparticle

detectsaheavyparticleatposition+1,andintheotherofwhichthesamelight

particledetectsthesameheavyparticle,instead,atposition-1.Andanyonefamiliar

with(say)themany-worldsinterpretationofquantummechanics,orwiththede-

27

coherenthistoriesinterpretationofquantummechanics,isgoingtorecognizethat

whatwehavestumbledacrosshereispreciselytheannihilationoftheoff-diagonal

interferenceterms,inthereduceddensitymatrixofameasuredsystem,bythe

interactionwithameasuring-device-andanyonefamiliarwithBohmianMechanics

isgoingtorecognizethatwhatwehavestumbledacrosshereispreciselythe

phenomenonthatisresponsiblefortheso-calledeffectivecollapseofthewave-

function.

Thereis,ofcourse,asyet,nouniquedeterminatefactofthematteraboutthe

outcomeofthesortofmeasurementdescribedabove.Whatweareleftwith–once

theattractivepotentialisswitchedoff-is(again)somethinglikeapairofparallel

universes,inoneofwhichalightparticlehasdetectedaheavyparticleatposition

+1,andintheotherofwhichthesamelightparticlehasdetectedthesameheavy

particle,instead,atposition-1.Andthebusinessofarrangingforoneortheotherof

thoseuniversestosomehowamounttotheactualoneisjustthefamiliarbusinessof

solvingthequantum-mechanicalproblemofmeasurement:Youeitherfindawayof

makingoneortheotherofthoseuniversesdisappear(that’sthesortofthingthat

happensintheoriesofthe“collapseofthewave-function”)oryoufindawayof

endowingoneortheotherofthoseuniverseswithsomesortofspecialstatus(as

onedoes,forexample,inBohmainMechanics).Butwearegettingaheadof

ourselves.Putthemeasurementproblemtoonesideforthemoment–we’llcome

backtoitlater.

28

*

Wecanedgestillclosertothefamiliarquantum-mechanicalformalismby

replacingthepoint-likephysicalitemsinthespaceofelementaryphysical

determinablesbysomethingmorelikefields.Intheexamplesweconsideredabove,

inwhichdifferentpossiblesituationsinthespaceofordinarymaterialbodies

correspondedtodifferentconcretepoint-likephysicalitemsfloatingaroundinthe

spaceofelementaryphysicaldetirminables,thebusinessofarrangingforthe

possibilityofinterferencebetweendifferentsuchsituationshadtodowiththe

introductionofnewtermsintothefundamentalHamiltonianoftheworld–terms

(forexample)liketheδ(x1-x3)δ(x2-x4)inequations(5)and(7)-wherebythe

differentpoint-likephysicalitemscanliterallypushorpullononeanother.But

fieldscandosomethingelsetooneanother,somethingthathasnothingtodowith

pushingorpulling,somethingthatdoesn’tdependontheintroductionofany

additionaltermsintotheHamiltonian:theycanaddtoorsubtractfromoneanother-

theycaninterfere(thatis)inthewaythatwaterwavesdo.Soifwhatwethinkofas

inhabitingthespaceofelementaryphysicaldetirminablesaresomethinglikefields,

thensomethinglikeaprincipleofsuperposition–then(thatis)thepossibilityof

variousdifferentpossiblesituationsinthespaceofordinarymaterialbodies

actuallyphysicallyinterferingwithoneanother-isgoingtobebuiltrightintothe

fundamentalkinematicsoftheworld,justasitisinquantummechanics,andit’snot

goingtorequireanyspecializedadditionstothelawsofdynamics.

29

Here’saverysimpleexample.

Gobacktocaseofasinglepoint-likephysicalitem,floatingaroundinatwo-

dimensionalspace,withadiagonalpotentialbarrier.Andimaginethatwereplace

thatitemwithascalarfield–afieldwhichalwayshasthevalue+1atexactlyoneof

thepointsinthe2-dimensionalspace,andwhichalwayshasthevaluezero

everywhereelse.Andsupposethatthepointatwhichthefieldhasthevalue+1

movesaroundinthe2-dimensionalspace–justasthepoint-likephysicalsystemdid

intheearlierexample–inaccordwiththeHamiltonianinequation(3).

Itwillbenatural–justasitwasinthecaseofthesinglepoint-likeitemfloating

aroundina2-dimensionalspace,andforexactlythesamereasonsasitwasinthe

caseofthesinglepoint-likeitemfloatingaroundinthe2-simensionalspace–to

describeaworldlikethisasconsistingofapairofordinarymaterialparticles,of

differentmasses,floatingaroundinahomogenousandisotropic1-dimensional

space.

Supposenowthattherearetwopointsinthe2-dimensionalspaceatwhichthe

fieldisnon-zero,andsuppose(justtomakethingsinteresting)thatthefieldhasthe

value+1atoneofthosepointsandthatithasthevalue-1attheother,andsuppose

thatthetwopointsinquestionmovearoundthe2-dimensionalspaceinaccordwith

theHamiltonianinequation(4).Auniverselikethisonecanbedescribed(as

before)asconsistingoffourordinarymaterialparticles,movingaroundina

30

homogenousandisotropicone-dimensionalspace.Butthewaythoseparticles

moveis(again)goingtobekindoffunny.Ifm1=m3,andifm2=m4,andifwe

assumethatsignsofthefieldsinthe2-dimensionalspacecanmakenodifferenceto

theintrinsicpropertiesoftheirone-dimensionalshadows,thenparticle1and

particle3aregoingtobeintrinsicallyidenticaltooneanother-andyetparticle1is

goingtoelasticallycollidewithparticle2,whereasparticle3isgoingtopassright

troughparticle2,andsoon.Andso(again)aqualitativedescriptionofthephysical

situationatsomeparticulartime,intheone-dimensionalspace,isnotgoingtogive

usenoughinformationtopredict,eveninprinciple,thequalitativesituationofthis

worldatothertimes.Andsothe2-dimensionalspaceofelementaryphysical

detirminablesturnsout(again)tobemorefundamental–inallofthewaysthatwe

havealreadydiscussed–thantheone-dimensionalspaceofordinarymaterial

bodies.Andso(again)thisturnsouttobelesslikeacollectionoffourparticles

floatingaroundinaone-dimensionalphysicalspacethanitisliketwodifferent

possibilitiesabouthowtwosuchparticlesmightbefloatingaroundinaone-

dimensionalspace,orliketwocausallyunconnectedparallelworlds,ineachof

whichapairofparticlesisfloatingaroundinaone-dimensionalspace,orsomething

likethat.

Exceptthathere–evenintheabsenceofanyfurthermodificationofthe

fundamentallawsofmotionwhichisdesignedtoallowthesetwopossibilitiesto

dynamicallyinteractwithoneanother,evenintheabsence(thatis)ofanadditional

termintheHamiltonianliketheδ(x1-x3)δ(x2-x4)ofequations(5)and(7)–theycan

31

nevertheless,nowandthen,andinanaltogetherdifferentandpurelykinematical

way,interferewithoneanother.Note(forexample)thatifthetwofield-pointsin

thetwo-dimensionalspaceofelementaryphysicaldetirminablesshouldever

happentocrosspaths,theirtwofieldswillcanceloneanother.Andwhatthat

meansisthatif(forexample)thepositionsofparticles2and4intheone-

dimensionalspaceofordinarymaterialbodiesshouldeverhappentocoincide,andif

(whilethepositionsof2and4coincide)particles1and3shouldeverhappento

cometogether,then,nomatterhowfarapartparticles1and3mayhappentobe

fromparticles2and4,allfouroftheparticlesaregoingtodisappear!Thisisnotthe

kindofthing(ofcourse)thatcanhappentonon-relativisticquantum-mechanical

particles.Itwouldamount(foronething)toaviolationofunitarity.Butitis–for

allthat,andinallsortsinterestingrespects-notveryfarfromcasesofthoroughly

quantum-mechanicalinterferenceliketheonedescribedinequation(6).Andafew

obviousfurthermodificationswillgetus-asliterallyasyouplease-alltheway

there.

*

Let’sstartbyallowingthefieldtobenon-zeroatanynumber(thatis:anyfinite

number,oranycountablyinfinitenumber,oranyuncountablyinfinitenumber,or

eventheentirecollection)ofpointsinthespaceofelementaryphysical

detirminables.Thedifferentpossiblephysicalstatesoftheworld(then)willconsist

ofthedifferentpossibleconfigurationsofthefield–thedifferentpossiblestatesof

32

theworld(thatis)willconsistofdifferentpossibleassignmentsoffield-valuesto

everyoneofthecontinuousinfinityofpointsinthespaceoftheelementaryphysical

detirminables.Andlet’ssupposethatthefieldF(λ,μ),atanypoint(λ,μ)inthespace

ofelementaryphysicaldetirminables,cantakeoncomplexvalues,andlet’ssuppose

thatthereisalaw(orperhapsaninitialcondition)totheeffectthattheintegralof

|F(λ,μ)|2,overtheentiretyofthetwo-dimensionalspaceofdetirminables,atany

particulartemporalinstant,isequalto1.

Now,everyfunctionofλandμthatobeystheabovestipulationscan–asa

matterofpuremathematics-berepresentedasauniquevector,oflength1,inan

infinite-dimensionalHilbertspace.Andwecandefine–inthefamiliarway-an

innerproductonthatspace.Andwiththatmathematicalapparatusinhand,wecan

stipulatethattheevolutionofthevectorthatrepresentsthefield-configurationof

theworldintimeisgivenbysomedeterministicandlinearandunitarytime-

translationoperatorwhoseinfinitesimalgeneratorisaHermetianoperatoronthat

spacecalled(byanalogywithit’sclassicalcounterpart)aHamiltonian.

Andnow,atlast,whatwehaveinfrontofus,inallitsglory,iswhatisusually

referredtoastheQuantumTheoryofapairofnon-relativisticstructurelessspin-

zeroparticles,floatingaround,andinteractingwithoneanother,inaone-

dimensional“physicalspace”.Butwhatthattheoryactuallyappearstobeabout,if

youcomeatitbywayofthesimpleandmechanicalandflat-footedroutethatwe

havebeenfollowinghere,isafield.Andthespaceofpointsonwhichthatfieldis

33

defined,thespaceofpointsatwhichthatfieldtakesonvalues,hastwodimensions

ratherthanone.Andallofthefamiliartalkofparticlesfloatingaroundinaone-

dimensionalspacehastodowiththewaythingslookfromthepointofviewofthe

spaceofordinarymaterialbodies–which(again)issomethingother,andsmaller,

lessfundamentalthanthestageonwhichthefullhistoryoftheworldplaysitselfout,

andwhichemergesasaby-productoftheactionoftheHamiltonian.

*

Thispictureofthewave-functionasconcretephysicalstuffmayseemhardtofit

together,atfirst,withwhatonethinksoneknowsaboutquantummechanics.

Consider(forexample)thequestionofobservables.Wearetold–instandard

presentationsofquantummechanics-thatnomeasurementcandistinguish,with

certainty,betweenasysteminthestateF(q1…qN)asysteminthestateF’(q1…qN)

unlessthevectorsrepresentingF(q1…qN)andF’(q1…qN)happentobeorthogonalto

oneanother.Butwhyintheworld–onthisnewwayofthinking-shouldanything

likethatbethecase?IftheseF(q1…qN)’sarereallyconcretephysicalstuff–as

opposedtoabstractmathematicaldescriptionsofthestatesofsomethingelse-why

isitthatanythingshouldstandinthewayofourmeasuringtheamplitudeofthat

stuff,toanyaccuracywelike,atanypointweplease,justasweroutinelydowith

(say)electromagneticfields?

Let’ssee.

34

Thecrucialpoint(itturnsout)isthattheveryideaofmeasurementis

inextricablyboundupwiththespaceofordinarymaterialthings.Thepoint(more

precisely)isthatinorderforthisorthatphysicalquantitytocountassomething

measurable,theremustbepossiblephysicalprocesseswherebythevalueofthat

quantitycanreliablybebroughtintocorrelationwiththepositionsofordinary

materialobjects(thepositionsofpointers,thedistributionsofink-moleculeson

piecesofpaper,etc.)inthespaceofordinarymaterialthings.

Let’sstartwithasimpleparadigmcase-fromwhich(thereafter)thereadercan

easilygeneralize,asmuchasshepleases,forherself.

Suppose(then)thatthesymmetriesoftheHamiltonianoftheworldentailthat

thespaceofordinarymaterialbodieshappenstobe1-dimensional,asinthe

examplesthatwewereconsideringabove.Anddividetheqiintothreedisjointsets:

the‘pointer’set{q1},andthe‘object’set{q2….qO},andthe‘rest-of-the-world’set

{qO+1…..qN}.Andpositasingleverysimpleconnection–asingleverysimpleruleof

correspondence-betweenthefield-configurationinthespaceofelementaryphysical

detirminablesandoureverydayempiricalexperienceoftheworld,towit:the

‘pointer’particleisatoraroundpositionxinthespaceofordinarymaterialthingsif

35

andonlyifF(q1…qN)vanishes,oralmostvanishes,outsideoftheregionboundedby

(q1=x-ε)and(q1=x+ε),whereεissmall.6That(itturnsout)willbeallweneed.

Now,twodifferentfield-configurationsofthe‘object’,f(q2….qO)andf’(q2….qO),

canbedistinguishedfromoneanotherbyameasurementifandonlyifthereisat

leastonepossiblefield-configurationofthe‘pointer+rest-of-the-world’–callitg(q1,

qO+1…..qN)-suchthatifthefield-configurationoftheworldatt0isg(q1,qO+1…..qN)

f(q2….qO),thenthepointerparticleendsup,atacertainlatertimet1,atthepointx,

andifthefield-configurationoftheworldatt0isg(q1,qO+1…..qN)f’(q2….qO),thenthe

pointerparticleendsup,atthatsamelatertimet1,atthepointy,wherethedistance

betweenxandyismuchgreaterthanε.7

AnditfollowsfromtheabovecorrespondencerulethatanyF(q1…qN)inwhich

thepointerparticleislocatedatxisorthogonal(ornearlyso)toanyF(q1…qN)in

6ThethoughthereharksbacktoourdiscussionofthesystemdescribedbytheclassicalHamiltonianinequation(4).Thethought(thatis)isthatthe‘pointer’particleisatoraroundxifandonlyifallofwhatwewerepreviouslycallingthe‘scenarios’combinedinF(q1…qN),oralmostallofthem,oralmostallofanappropriatelyweightedcombinationofthem,agreethatitis.Thisparticularruleofcorrespondence(bytheway)willbeappropriatetoversionsofquantummechanics–liketheGRWtheory-whosefundamentalontologiesconsistexclusivelyofthefieldsF(q1…qN).FortheorieslikeBohmianMechanics–whichhavericherfundamentalontologies–adifferentrulewillbeappropriate.ButthereaderwhoisfamiliarwithBohmianMechanicswillhavenotroubleinconfirmingforherselfthatanargumentverymuchanalogoustotheonethatfollows,andwhicharrives,intheend,atexactlythesameconclusion,canbeconstructedinthattheoryaswell.7The“if”hereisjustamatterofreflectingonwhatitmeanstodistinguishbetweentwosituationsbymeansofameasurement–butthe“onlyif”requiresasomewhatmoreelaborateargument.Theinterestedreadercanfindsuchanargumentonpages89-91ofarecentbookofminecalledAfterPhysics.

36

whichthepointerparticleislocatedaty.Andsoitwillfollowfromtheabove

analysisofwhatitistobeabletodistinguishbetweenf(q2….qO)andf’(q2….qO)by

meansofameasurement,andfromthestipulationtheuniversaloperatoroftime-

translationisunitary,thatf(q2….qO)andf’(q2….qO)canonlybedistinguishedfrom

oneanotherbymeansofameasurementiftheyareorthogonaltooneanother.And

itwillfollowfromthatthatiff(q2….qO)andf’(q2….qO)canbedistinguishedfromone

anotherbymeansofameasurement,thentheremustbesomeHermetianoperator

ofwhichbothf(q2….qO)andf’(q2….qO)areeigenfunctions,withdifferent

eigenvalues.

Andfromthere,withouttoomuchfurthertrouble,onecanrecovertheentirety

ofthealgebraofthequantum-mechanicalobservables.

2

Here’swhat’shappenedsofar:

Westartedoffbylookingattwowaysofrepresentingaclassicalsystemwith

twodynamicaldegreesoffreedom,whoseHamiltonianconsistsofthestandard

kineticenergytermsandasimplecontactinteraction.Oneoftheserepresentsthe

37

systemasapairofparticlesfloatingaroundinaone-dimensionalspaceofordinary

materialbodies,andtheotherrepresentsthesystembymeansofasinglepoint-like

physicaliteminthetwo-dimensionalspace–thespaceofthepossibleone-

dimensionalconfigurationsofthepairofparticlesfloatingaroundinthespaceof

ordinarymaterialbodies.Becausethesetwowaysofrepresentingthesystemare

bothcomplete,andbecausetheyarefullymathematicallyequivalenttooneanother,

andbecausethetwo-dimensionalrepresentationlooks(inallsortsofways)less

natural,andlessfamiliar,andlesslikethemanifestimageoftheworldthantheone-

dimensionalrepresentationdoes,thereseemedtobenocompellingreasontotake

thetwo-dimensionalspacephilosophicallyseriously.

Butassoonasweimagineanadditionalpoint-likephysicalitemfloatingaround

inthetwo-dimensionalspace,allofthisabruptlychanges.Oncethetwo-

dimensionalspaceisinhabitedbymorethanasinglesuchitem,theone-

dimensionalrepresentationandthetwo-dimensionalrepresentationarenolonger

mathematicallyequivalenttooneanother–andeachofthemseemstohavea

distinctandphilosophicallyinterestingroletoplay.Theone-dimensionalspaceis

stillthespaceofordinarymaterialbodies–buttherepresentationofthesystemin

thatspaceisnolongermathematicallycomplete.Andthesmallestspaceinwhich

thesystemcanberepresentedinacompleteandseparableway8–thespace(that

is)oftheelementaryphysicaldetirminables-isnowtwo-dimensional.

8Whatitmeansforarepresentationtobeseparable,bytheway,isforthatrepresentationtotaketheformofaspatialdistributionoflocalphysicalproperties.

38

Moreover,thegeneraldirectionofthesechangesisunmistakablyquantum-

mechanical.Itturnsoutthataddinganotherconcretepoint-likefundamental

physicalitemtothehigher-dimensionalspaceisnotsomuchlikeaddingmore

concretephysicalmaterialtothelower-dimensionalspaceasitislikeadding

anotherlow-dimensionalworld,oranotheractualizedlow-dimensionalpossibility,

oranotherterminaquantum-mechanicalsuperposition.Andthesedifferent

possibilitiescanbemadetointeractwithoneanother,inwaysthatareverymuch

reminiscentofquantum-mechanicalinterference,bymeansoftheadditionof

anotherverysimpletermtotheHamiltonian.Andtheadditionofsuchatermalso

generatesdistinctlyquantum-mechanicalsortsofnon-locality,anddistinctly

quantum-mechanicalimagesofmeasurement,andsoon.

And(ontopofthat)ithappenstobeacharacteristicofclassicalphysicaltheories

thatthespaceofordinarymaterialthingsandthespaceofelementaryphysical

detirminablesareexactlyandinvariablyoneandthesame.Anditseemsnaturalto

wonderwhetherallofthispointstosomekindofadiagnosis,orsomekindofan

explanation,oftheactualun-classicalweirdnessoftheworld.Itseemsnaturalto

wonder(thatis)whetheritispreciselythiscoming-apartofthespaceofordinary

materialthingsandthespaceofelementaryphysicaldeterminablesthatturnsoutto

beatthebottomofeverythingthat’sexceedinglyandparadigmaticallystrange

aboutquantummechanics.Whatitmeans(thatis)forarepresentationtobeseparable,inthelanguageIintroducedafewpagesback,isforitnottoinvolveanyentanglement.

39

Buthowcanthatpossiblybetrue?Forquantum-mechanicalsystemsconsisting

ofjustasinglestructurelessspin-zeroparticle(afterall)thespaceofordinary

materialbodiesandthespaceofelementaryphysicaldetirminablesareprecisely

oneandthesame,justastheyareforclassicalsystems.Andyetahellofalotof

whateverybodyagreesisexceedinglyandparadigmaticallystrangeaboutquantum

mechanicscanalreadybeencounteredinsystemslikethat.9Andthiswillbeworth

thinkingthroughinsomedetail.Andthebusinessofthinkingitthroughwillbethe

workofthissection.

Consider(then)asinglestructurelessparticle,ina3-dimensionalspaceof

ordinarymaterialthings,whosequantum-mechanicalwave-functionhappenstobe

non-zero,atacertainparticulartime,intwoseparateandcompactanddisjoint

regionsofthatspacecalledAandB.

What’sstrangeaboutsituationslikethatisthatbothofthefollowingclaims

abouttheparticleinquestionareapparently,simultaneously,true:

1) Thereisaperfectlyconcreteandobservablesenseinwhichthe

particle,orsomethingverycloselyassociatedwiththeparticle,isin

9Feynmanfamouslysays(forexample)thattheonlymysteryinquantummechanicsistheonethatonethatcomesupinconnectionwiththedouble-slitexperiment–andthedouble-slitexperimentseems(onthefaceofit)toinvolvenothingoverandaboveasinglestructurelessparticlemovingaroundinthepresenceofacomplicated(double-slitted)externalpotential.

40

bothregions.(WhatIhaveinmindhere,whenIspeakofa

‘concreteandobservable’senseinwhichtheparticleisinboth

regions,isofcoursethepossibilityofmeasuringtheeffectsof

interferencebetweenthebranchofthewave-functionthat’slocated

inAandthebranchofthewave-functionthat’slocatedinB–as,for

example,inthedouble-slitexperiment)

2) Thereisaperfectlyconcreteandobservablesenseinwhichthe

particle,andeverythingsufficientlycloselyassociatedwiththe

particle,isinonlyoneofthoseregions.(AndwhatIhaveinmind

here,whenIspeakofa‘concreteandobservable’senseinwhich

theparticleisinonlyoneregion,isthefactthatifwemeasurethe

particle’sspatiallocation,wewilleitherfindaparticleinAand

nothingwhateverinB,orwewillfindaparticleinBandnothing

whateverinA)

Note(tobeginwith)thatthereisnothingparticularlyunintelligible,inandof

itself,aboutclaim(1).(1)iswhatBohrandhiscircleusedtocallthe‘wave’aspect

ofquantum-mechanicalparticles–andonecouldthinkofthat,intheabsenceof(2),

assuggestinganovelbutbynomeansunfathomablepictureofthesubatomic

structureofmatter,accordingtowhichparticlesaretobeunderstood,atthe

microscopiclevel,assomethingakintocloudsorfluidsorfieldsthatcan(incertain

circumstances)spreadthemselvesoutoverfiniteandevendisjointregionsofthe

spaceofordinarymaterialthings.

41

Whathasalwayscompletelyfreakedeverybodyout(ontheotherhand)isthe

combinationof(1)and(2).Anditturnsoutthatallofthewaysthatwehaveof

imaginingthat(1)and(2)could(somehow)bothbetruearegoingtoinvolvetelling

storiesaboutsystemsthatconsistofmorethanasingleparticle,systems(thatis)

whosequantum-mechanicalwave-functionstakeonvaluesatpointsinspacesof

morethan3dimensions,systems(thatis)forwhichthespaceofelementary

physicaldetirminablesdivergesfromthespaceofordinarymaterialthings.

Letmetrytosay,alittlemoreconcretely,whatIhaveinmind.

Note(tobeginwith)thatthebusinessoffiguringouthow(1)and(2)couldboth

betrueisnothingotherthanthebusinessofsolvingthequantum-mechanical

measurementproblem.Andsothevariousattemptsatcomingtotermswith(1)

and(2)togetherthatweoughttohaveinthebackofourmindsherearethingslike

theGRWtheory,andBohmianMechanics,andtheMany-Worldsinterpretation.And

itturnsoutthatallofthoseattempts,andallofthestrategiesthatanybodyhasever

somuchashintedatforsolvingthequantum-mechanicalmeasurementproblem,

dependonthephenomenonofentanglement.Andthephenomenonof

entanglementis,aswehavenotedbefore,andasastraightforwardmatterof

definition,thephenomenonofthedivergenceofthespaceofelementaryphysical

detirminablesfromthespaceorordinarymaterialthings.

42

Consider(forexample)thecaseofBohmianMechanics.Thephenomenathat

pertainto(1)havetodo–inthecontextofBohmianMechanics–withthefactthat

thewave-functionofthesortofparticleweweredescribingaboveisnon-zeroboth

inregionAandinregionB.Anditmightlook,atfirstglance,asifthebusinessof

accountingforthephenomenathatpertainto(2),inthecontextofBohmian

Mechanics,amountstonothingmorethanthesimpleobservationthat–

notwithstandingthatthewave-functionofthisparticleisspreadoverbothregionA

andregionB–theBohmiancorpuscleitselfislocatedeitherinregionAorinregion

B.Andallofthatcanofcoursebepresentedintheformofastoryaboutwhatthings

arephysicallylike,atvariousdifferenttimes,atvariousdifferentpointsinthe

familiar3-dimensionalEuclidianspaceofordinarymaterialthings,andofour

everydayempiricalexperienceoftheworld.

Butthis(onalittlereflection)isallwrong.What(2)isaboutisnotmerelythat

theparticleiseitherinregionAorinregionB,but(inaddition)thatwhenwelook

fortheparticleweseeiteitherinregionAorinregionB,and(moreover)thatwhat

weseeisinfactreliablycorrelatedwithwheretheparticleactuallyis,andthatwhat

weseematchesupintheappropriatewaywithwhatwewouldseeifwewereto

lookagain,andwithwhatsomebodyelsewouldseeiftheyweretolookfor

themselves,andwithhowtheparticleitselfwillbehaveinthefuture,andsoon.Andif

notforallthat,therewould(indeed)benothingheretopuzzleover.Andthe

variousbusinessesofaccountingforallthat,inthecontextofBohmianMechanics,

alldepend(again)onthefactthattheprocessofmeasurementinvariablyand

43

ineluctablygeneratesquantum-mechanicalentanglementsbetweenthemeasuring-

devicesandthemeasuredparticle.10

10Maybeitwillbeworthtakingaminutetorubthisin. Supposethattheinitialwave-functionofthecompositesystemconsistingofaparticle(p)andameasuring-device(d),whichisdesignedtorecordthepositionofthatparticle,is: [ready>d(α[A>p+β[B>p), (i)where[ready>disthephysicalstateofthesystemdinwhichdispluggedinandproperlycalibratedandfacingintherightdirectionandinallotherrespectsreadytocarryoutthemeasurementofthepositionofp,and[A>pisthestateofpinwhichpislocalizedinthespatialregionA,and[B>pisthestateofpinwhichpislocalizedinthespatialregionB.

Andnote,tobeginwith,thatanysatisfactoryscientificaccountofwhyitisthatifwemeasurethepositionofaparticlelikethis‘wewilleitherfindaparticleinAandnothingwhateverinB,oraparticleinBandnothingwhateverinA’hasgottobeanaccountnotonlyofthebehaviorofpundercircumstanceslike(i),butalsoofthebehaviorofdundercircumstanceslike(i).

Good.Supposethatpanddareallowedtointeractwithoneanother,inthefamiliarway,whenastatelike(i)obtains.Thenitwillfollow,inthefamiliarway,fromthelinearityofthequantum-mechanicalequationsofmotion,andfromthestipulationthatdisaproperly-functioningdeviceforthemeasurementandrecordingofthepositionofp,thatthestateofthiscompositesystemoncethisinteractioniscompletewillbe:

α[‘A’>d[A>p+β[‘B’>d[B>p, (ii)

where[‘A’>disthestateofdinwhichthepositionofd’spointerindicatesthattheoutcomeofthemeasurementofthepositionofpis‘A’,and[‘B’>disthestateofdinwhichthepositionofd’spointerindicatesthattheoutcomeofthemeasurementofthepositionofpis‘B’.

AndconsiderhowitisthatBohmianMechanicsmanagestoguaranteethat,incircumstanceslike(ii),thepositionsoftheBohmaincorpusclesthatmakeupthepointerofdareproperlyandreliablycorrelatedwiththepositionoftheBohmiancorpusclep–theposition(thatis)oftheBohmaincorpusclewhosepositionhasjustnowbeenmeasured.Note(inparticular)thatthatcorrelationdependscruciallyonthefactthatthewave-functionin(ii)vanishesinthoseregionsoftheconfiguration-spaceofthecompositesystemconsistingofpanddinwhichpreciselythosecorrelationsdonotobtain.Andnote(moreover)thatitmustvanishinthoseregionswithoutvanishingthroughoutthoseregionsofthatconfiguration-spaceinwhichpislocatedinA,andwithoutvanishingthroughoutthoseregionsofthatconfiguration-spaceinwhichpislocatedinB,andwithoutvanishingthroughout

44

Let’shavealookatexactlyhowthatworks.Startwithasingle,structureless,

particle–callitp-andtwoboxes.OneoftheboxesiscalledA,andislocatedatthe

point(x=+1,y=0,z=0),andtheotheriscalledB,andislocatedatthepoint(x=-1,

y=0,z=0).11Andlet[A>pbethestateofpinwhichpislocatedinA.Andlet[B>p

bethestateofpinwhichpislocatedinB.Andsupposethatatt=0,thestateofpis

(1/√2)[A>p+(1/√2)[B>p. (8)

Now,statesliketheonein(8)famouslyresistanyinterpretationassituationsin

whichpiseitherinboxAorinboxB.Andwhatfamouslystandsinthewayofsuch

aninterpretationisthefactthatifweopenbothboxes,whenastatelike(8)obtains,

thenthesubsequentobservablebehaviorsoftheparticle–theprobabilities(for

example)offindingtheparticleatthisorthatpointinspace–areingeneralgoingto

beverydifferentfromthebehaviorofaparticlereleasedfromboxA,andvery

different(aswell)fromthebehaviorofaparticlereleasedfromboxB,andvery

thoseregionsofthatconfiguration-spaceinwhichd’spointerislocatedin‘A’,andwithoutvanishingthroughoutthoseregionsofthatconfiguration-spaceinwhichd’spointerislocatedin‘B’.Andnote(andthis,finally,istheheartofthematter)thattheprevioustwosentencescanonlysimultaneouslybetrueofawave-function(liketheonein(ii))inwhichpanddarequantum-mechanicallyentangledwithoneanother–note(thatis)thattheprevioustwosentencescanonlysimultaneouslybetrueofawave-functionwhich(liketheonein(ii))cannotberepresentedasafunctionoverthepointsofanythree-dimensionalarena.11Inordertokeepthingsassimpleaspossible,wewilltreattheseboxesnotasphysicalsystems,but(rather)asexternallyimposedpotentials–andwewilltreattheopeningsandclosingsofthoseboxesnotasdynamicalprocesses,but(rather)asvariationsinthoseexternallyimposedpotentialswithtime.

45

different(aswell)fromanythingalongthelinesofaprobabilisticsumoraverageof

thosetwobehaviors.

Andallofthis,asImentionedabove,canbeexplained,inthecontextofBohmain

Mechanics,bymeansofastoryaboutwhatthingsarephysicallylike,atvarious

differenttimes,atvariousdifferentpointsinthefamiliar,material,3-dimensional

spaceofoureverydayempiricalexperienceoftheworld.Theparticleitselfstarts

outineitherboxAorboxB–butit’swave-function,it’sso-calledpilot-wave-is

non-zero(whenastatelike(8)obtains)inbothboxes.Andso,whentheboxesare

opened,andthetwobranchesofthewave-functionflowoutwards,andfillupthe

three-dimensionalspacearoundthem,andoverlapwithoneanother,theyinterfere

–andthatinterferenceobservablyaffectsthemotionoftheparticlethatthosetwo

branches,together,areguiding.

Andthepuzzle(again)isthatmeasuringthepositionofaparticlelikethat

somehowmakesoneortheotherofthosebranchesgoaway.Andthequestionis

how.Thequestion(toputitasnaivelyandasliterallyandasflat-footedlyasone

can)iswhere,exactly,thatotherbranchgoes.

Consider(then)aradicallysimplifiedstand-inforameasuring-device–callitM-

whichconsists(justasinthecaseweconsideredbefore)ofasinglestructureless

particle,andwhichisconstrained(wewillsuppose)tomovealongtheX-axis.And

let[ready>MbethestateofMinwhichMislocatedatthepoint(x=0,y=0,z=0),

46

andlet[‘A’>MbethestateofMinwhichMislocatedat(x=+1/2,y=0,z=0),andlet

[‘B’>MbethestateofMinwhichMislocatedat(x=-1/2,y=0,z=0).Andsuppose

thatthekinetictermintheHamiltonianofMhappenstobeidenticallyzero.And

supposethatthereisaninteractionbetweenMandp,whichwecanswitch‘on’and

‘off’asweplease,andwhich(whenit’sswitched‘on’)produces(overthecourseof,

say,theensuingsecond)evolutionslikethis:12

[ready>M[A>p---->[‘A’>M[A>pand[ready>M[B>p---->[‘B’>M[B>p(9)

Whentheinteractionisswitched‘on’(then)Mwillfunction,atleastunderthe

sortsofcircumstancesenvisionedabove,asameasuring-instrumentfortheposition

ofp.

Notethatwhereasthespaceoftheelementaryphysicaldeterminablesofa

singlestructurelessquantum-mechanicalparticlepisthree-dimensional,thespace

oftheelementaryphysicaldeterminablesofthecompositequantum-mechanical

systemconsistingofpandM–callthatFpM–isgoingtobefour-dimensional.We

canassignuniqueaddressestopointsinthatarenausingthethreeco-ordinates

(callthemxp,yp,andzp)thatcorrespondtotheparochialthree-dimensional

12Hereagain,justtokeepthingssimple,wearegoingtotreatthebusinessofturningthisinteraction‘on’and‘off’notasavariationinanydynamicaldegreeoffreedom,but(rather)asavariationinanexternallyimposedeffectiveHamiltonian.Noneofthesesimplifications–asthereadercaneasilyconfirmforherself–involvesanylossingenerality.

47

‘positionofp’andonemore(callitxM)thatcorrespondstotheparochialone-

dimensional‘positionofM’.

Supposethatweinitiallythatpreparecompositesysteminthestate

[ready>M((1/√2)[A>p+(1/√2)[B>p), (10)

withtheinteractionswitched‘off’,andthenopentheboxes.Inthiscase,theM

remainscompletelyunentangledwithp,andoncetheboxesareopened,onebranch

ofthewave-functionofthecompositesystemwillspreadoutwardfromthepoint

(xp=+1,yp=0,zp=0,xM=0),andtheotherbranchwillspreadoutwardfromthe

point(xp=-1,yp=0,zp=0,xM=0),andeachofthemwillfillupthethree-

dimensionalhypersurfacexM=0ofthedeterminablespaceofthecompositesystem,

andtheywilloverlapwithoneanother,andinterferewithoneanother,andbothof

themwillcontributetodeterminingtheBohmaintrajectoryoftheworld-particle.

(Andnotethatallthis–exceptforthepresenceoftheworld-particleitself–is

exactlyanalogoustowhatwasgoingoninthesystemdescribedbytheHamiltonian

inequation(7)whenparticles2and4arebothattheorigin)

If(ontheotherhand)weinitiallypreparethecompositesysteminthestate

in(8)withtheinteractionswitched‘on’,thenitwillfollowfrom(9),togetherwith

thelinearityofthequantum-mechanicalequationsofmotion,thatthestateofp+M

willbecome

48

(1/√2)[‘A’>M[A>p+(1/√2)[‘B’>M[B>p (11)

NowMandparemaximallyentangledwithoneanother,andiftheboxesare

openedatthispoint,thenonebranchofthewave-functionofthecompositesystem

willspreadoutwardfromthepoint(xp=+1,yp=0,zp=0,xM=+1/2),andfillupthe

three-dimensionalhypersurfacexM=+1/2,andtheotherbranchwillspread

outwardfromthepoint(xp=-1,yp=0,zp=0,xM=-1/2),andfillupthethree-

dimensionalhypersurfacexM=-1/2,andthetwowillnotoverlapwithoneanother,

andwillnotinterferewithoneanother,andonlyoneofthem–theonethat’snon-

zeroonthehypersurfacewheretheworld-particlehappenstobelocated-will

contributetodeterminingthetrajectory.Andthereadershouldnotethatitis

absolutelycriticaltothewayallthisworks–itisabsolutelycritical(inparticular)to

theveryideaofanentanglingofthemeasuring-devicewiththemeasuredparticle–

thatthedimensionofthedeterminablespacealongwhichthewave-function

spreadsoutwhenMisinmotionisorthogonaltoallofthedimensionsofthatspace

inwhichthewave-functionspreadsoutwhenpisinmotion.(Andnotethatthis–

except(again)forthepresenceoftheworld-particleitself–isexactlyanalogousto

whatwasgoingoninthesystemdescribedbytheHamiltonianinequation(7)when

theattractivepotentialisswitchedon)

Andsotheanswertothequestionofwheretheotherbranchgoes,whenwe

measurethepositionofp,isliterally,andflat-footedly,thatitgetspushedoffinto

49

anotherdimension.Andthis(inmicrocosm)isthesortofthingthathappens

wheneverwedomeasurementsonquantum-mechanicalsystems.What’sstrange

aboutquantummechanics,whatmakesitlooklikemagic,eveninacaseassimpleas

thatofasinglestructurelessparticle,isthatthethree-dimensionalspaceofordinary

materialbodiesistoosmalltocontainthecompletemicroscopichistoryofthe

world.

AndmuchthesamesortofthingistrueontheGRWtheory.Thismayseem,at

first,likeapuzzlingclaim.Thereadermaywanttoobjectthatwhathappensonthe

GRWtheoryisnotthatoneofthebranchesgets‘pushedoffintoanotherdimension’,

but(rather)thatoneofthebranchessimplydisappears.Butconsiderthe

mechanismofthatdisappearance.Thewave-functionoftheworld,whichisa

functionofpositioninthespaceofelementaryphysicaldeterminables,ismultiplied

byanotherfunction,theso-called‘hitting’function–whichisalsoafunctionof

positioninthespaceofelementaryphysicaldeterminables.Andthismultiplication

ofthewave-functionbythehittingfunctionsomehowmanagestoleaveoneofthe

abovebranchesofthewave-functionintact,andcausestheotheronetovanish.And

thatcanonlyoccurifthesetwobranchesofthewave-function,whichoverlap

everywhereinthe3-dimensionalspaceofordinarymaterialbodies,somehow

managenottooverlapanywhereinthespaceofelementaryphysicaldetirminables.

Andthatcanonlyoccurifthespaceofelementaryphysicaldeterminableshasat

leastonemoredimensionthanthespaceofordinarymaterialbodies–andifthe

50

twobrancheshavesomehowbecomeseparatedfromoneanotheralongthat

additionaldimension.

Andthereadercanconfirmforherselfthatmuchthesamethingwouldbetrue,

aswell,onthemany-worldsinterpretationofquantummechanics–ifthemany-

worldsinterpretationwerenototherwiseincoherent.

Andso,attheendoftheday,theredoesseemtobeanintimateandinvariable

connectionbetweenthecoming-apartofthespaceofordinarymaterialthingsand

thespaceofelementaryphysicaldeterminables(ontheonehand)andeverything

that’sexceedinglyandparadigmaticallystrangeaboutquantummechanics(onthe

other).Quantum-mechanicalsortsofbehaviorseemtorequirethatthespaceofthe

elementaryphysicaldeterminablesisbiggerthanthespaceofordinarymaterial

things–andwheneverthespaceoftheelementaryphysicaldeterminablesspaceis

biggerthanthespaceofordinarymaterialthings,quantum-mechanicalsortsof

behaviorseemtoquicklyensue.Anditbeginstolookasifwhatwehavestumbled

acrosshereis(indeed)adiagnosis,oranexplanation,ofthefactthattheworldis

quantum-mechanical.

3

Let’sseewhereallthisleavesus.

51

Thefactthatthespaceoftheelementaryphysicaldeterminablesoftheworld

andthespaceoftheordinarymaterialbodiesoftheworldareconceptuallydistinct

fromoneanother–thefactthatthereisnoapriorireasonwhateverwhythey

shouldcoincidewithoneanother,orhavethesametopologyasoneanother,orhave

thesamedimensionalityasoneanother–isapurelylogicalpoint,apointwhich

mightinprinciplehavebeennoticed,bymeansofpurelyconceptualanalysis,long

beforetheempiricaldiscoveriesthatgaverisetoquantummechanics.Andwehave

seenhoweasyitis,merelybyplayingaroundwiththesimplestimaginable

HamiltoniansofclassicalNewtonianparticles,tostumbleontophysicalsystemsfor

whichthespaceoftheelementaryphysicaldeterminableshasadifferentnumberof

dimensionsthanthespaceofordinarymaterialbodies.But(asIhavealready

remarked)thereisnothingmysteriousorsurprisingaboutthisdistinction’shaving

infactgoneunnoticedaslongasitdid.Itis(afterall)afundamentalprincipleofthe

ManifestImageoftheWorld–andallthemoreso(indeed)becausewearenoteven

awareofeveractuallyhavingadoptedit–thatthematerialspaceoftheworldand

thedeterminablespaceoftheworldareexactlythesamething.Andthatprinciple

hassincebeenendorsed,andfurtherfortified,inthecourseofscientific

investigation,byNewtonianMechanics,andbyMaxwellianElectrodynamics,andby

theSpecialandGeneraltheoriesofRelativity,andeven(insofarasthesecanbe

consideredinisolationfromquantummechanics)bythehigh-dimensional

geometriesofstringtheory,and(indeed)bytheentireedificeofclassicalphysics.

Youmightevensaythattheprinciplethatthematerialspaceoftheworldandthe

52

determinablespaceoftheworldareexactlythesamethingistheveryessenceofthe

classicalpictureoftheworld,andthesimplestandmostilluminatingwayof

pointingtowhatsetsitapartfromquantummechanics.

Buttherelationshipbetweenthematerialspaceandthedeterminableoneis(for

allthat,andfortheNthtime)anobviouslycontingentmatter.Andoneofthe

lessonsofthesimpleexerciseswehavebeenworkingourwaythroughhereisthat

themomentthatwetakethatin,themomentthatweevenraisethequestionofwhat

theworldmightbelikeifthosetwospacesdifferedfromoneanother,something

paradigmaticallyquantum-mechanicaljustflopsrightout.Anditseemsfairtosay

thatiftheconceptualdistinctionbetweenmaterialspaceoftheworldandthe

determinablespaceoftheworldhadmadeitselfcleartoanybody(say)ahundred

andfiftyyearsago,thenthe20thcenturyphysicsofsub-atomicparticlesmighthave

amountedtolessofashockthan,infact,itdid–itseemsfairtosay(thatis)thatthe

elucidationoftheconceptualdistinctionbetweenthematerialspaceoftheworld

andthedeterminablespaceoftheworldoffersusawayoflookingatquantum

mechanicsassomethingnatural,andbeautiful,andsimple,andunderstandable,and

maybeeventobeexpected.Indeed,inthelightofthesortsofconsiderationsthat

wehavebeenthroughhere–theClassicalcaseistheonethatlooksexceptional,and

conspiratorial,andsurprising.

53

Noneofthis(mindyou)seemstomepointinthedirectionofanydifferentor

deeperormoregeneralormorefundamentaltheoryfromwhichquantum

mechanicsmightimaginablybederived.

IfwhatIhavebeenattemptingheresucceeds,thenwhatitdoesforquantum

mechanicsis(rather)somethingalongthelinesofwhatMinkowskididforSpecial

Relativity:Ittakesafinishedandwell-formulatedfundamentalphysicaltheory–a

theorywhichisinnostrictlylogicalorempiricalneedofanyfurtherelaboration–

andoffersusacrispandelegantandprofoundwaysummingupwhatthetheoryis

tellingusabouttheworld,awayofsayingwhatthetheorymeans,awayofisolating

(youmightsay)itsessence.AndwhatItakemyselftobeproposinghereisan

accountoftheessence–inexactlythesensejustdescribed–ofquantumtheory.

*

Here’sanotherwaytoputit:

WhatItakemyselftobeproposinghereisabetterandmorestraightforward

andmoreintuitivewayofteachingquantummechanics.Theidea(inanutshell)is

thatithelpstopicturetheconcretefundamentalphysicalstuffoftheworldas

floatingaroundinsomethingother,andlarger,andmorefundamental,thanthe

spaceofordinarymaterialbodies–becausepicturingthingsthatwaymakesiteasy

toseewhyeverythinglookssoodd,andwhyitlooksoddinaparadigmatically

54

quantum-mechanicalsortofway,fromthepointofviewofthespaceofordinary

materialbodies.

Ofcourse,theobservationthatithelpstopicturethingsinacertainwaydoesn’t

settleanyquestions,inandofitself,abouthowthingsactuallyare.Butitisn’t

irrelevanttosuchquestionseither.Andwhatitsuggests,Ithink,isthatanyattempt

atinsistingonthecontrary,anyattempt(thatis)atinsistingthatthehabitationof

theconcretefundamentalphysicalstuffoftheworldisthefamiliar3-dimensional

spaceofordinarymaterialbodies,anyattempt(forexample)atthinkingaboutthe

quantum-mechanicalwave-functionassomethingmerelynomic,orassome

incrediblycomplicatedkindofapropertyofordinarymaterialparticles,orasa

multi-field,orwhathaveyou,islikelytocomeatasteepcostintermsofexplanation

andunderstanding.

Whatwesawintheearlysectionsofthispaperwasthatapairofconcretepoint-

likephysicalitems,floatingaroundina2-dimensionalspace,inaccordwitha

simple,classical,localHamiltonianliketheoneinequation(6),cangiveriseto

paradigmaticallyquantum-mechanicalweirdnessinanemergentone-dimensional

spaceofordinarymaterialbodies.Theordinarymaterial“shadows”ofthoseitems

movearoundintheone-dimensionalspaceasiftheywereinteractingwithone

anothernon-locally,andcollidewithoneanother,orfailtocollidewithoneanother,

accordingtorulesthatcannotbewrittendownintermsoftheirintrinsicphysical

55

properties,andseemtobeorganizedintoparallelpossibleworldsorscenariosthat

canneverthelessinterferewithoneanother,andsoon.

Ona“primitiveontological”versionofaworldlikethisone,allthattherereally

actuallyontologicallyisareordinarymaterialparticlesintheone-dimensionalspace

–andweareofferednothingalongthelinesofanexplanationofthebehaviorsof

thoseparticlesatall.Thefactthatthoseparticlesbehaveinthebaroqueand

astonishingwaysthattheydo-thefactthattheybehave(thatis)asiftheywere

shadowsofaconcretepoint-likephysicalitemsfloatingaroundinatwo-

dimensionalspace–isstipulatedtobeamatteroffundamentalphysicallaw.

Period.13

Andona“multi-field”versionofaworldlikethisone,theelementaryand

indivisibleandnot-further-analyzableconcretephysicalitemsoftheworldofthe

world–orsomeofthem,atanyrate14-aresupposedtobelocated,inawaythat

resistsanystraightforwardattemptatvisualization,atpairsofpointsinthe

fundamentalone-dimensionalphysicalspaceoftheworld.

13Thedetailsofa“primitiveontological”versionofaworldlikethisonearegoingtodepend,ofcourse,onexactlyhowweendupsolvingthemeasurementproblem.OnaprimitiveontologicalversionofBohmainMechanics(forexample)therearegoingtobetwoordinarymaterialparticlesfloatingaroundintheone-dimensionalspace,whereasonaprimitiveontologicalversionofaMany-Worldstheorytherewillbefour,andonaprimitiveontologicalversionofatheoryofthecollapseofthewave-function,therewillbefour,twoofwhicheventuallygoaway.14Hereagain–asinfootnote13–thedetailsaregoingtodependonexactlyhowweendupsolvingthemeasurementproblem.

56

Butifweimaginethatthefundamentalconcretephysicalstuffofaworldlike

thisoneisactuallyfloatingaroundinthetwo-dimensionalspace,thenthestrange

andcomplicatedone-dimensionalappearancescanbeunderstood,inthemannerof

allofthebestanddeepestandmostsatisfyingscientificexplanationswehave,in

termsofasimpleandliteralandmechanicalpicture–thesortofpicture(thatis)

thatonecandrawonapieceofpaper-ofwhat’sgoingonunderneaththesurfaceof

thoseappearances.Andexactlythesamesortsofconsiderationscanbeappliedto

thefullmathematicalformalismofquantummechanics,andtoanyofthevarious

solutionsthathavebeenproposedtothemeasurementproblem.