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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
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1
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
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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)
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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.
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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.
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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.
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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.
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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-
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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.
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*
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.
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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
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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
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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
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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
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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.
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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
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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.
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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
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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.
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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.
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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.
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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.
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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.
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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
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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
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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.
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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),
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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.
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‘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
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(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
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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
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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.
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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
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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.
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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
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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
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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.
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Butifweimaginethatthefundamentalconcretephysicalstuffofaworldlike
thisoneisactuallyfloatingaroundinthetwo-dimensionalspace,thenthestrange
andcomplicatedone-dimensionalappearancescanbeunderstood,inthemannerof
allofthebestanddeepestandmostsatisfyingscientificexplanationswehave,in
termsofasimpleandliteralandmechanicalpicture–thesortofpicture(thatis)
thatonecandrawonapieceofpaper-ofwhat’sgoingonunderneaththesurfaceof
thoseappearances.Andexactlythesamesortsofconsiderationscanbeappliedto
thefullmathematicalformalismofquantummechanics,andtoanyofthevarious
solutionsthathavebeenproposedtothemeasurementproblem.