THE CURRENT SITUATION IN QUANTUM MECHANICSnonloco-physics.0catch.com/cat.pdf · THE CURRENT...

THE CURRENT SITUATION IN QUANTUM MECHANICS

ERWIN SCHRODINGER(1887-1961)

A translationof:Die gegenwartige Situation in der Quantenmechanik

Die Naturwissenschaften,48, 807-812,823-828,844-849(1935).

CONTENTS

1. Thephysicsof models 12. Thestatisticsof variablesin quantummodels 33. Probabilisticforecasts:examples 44. Is anensembleinterpretationtenable? 55. Are variablesreally indistinct? 66. Epistemologicalreform 77. Wave functionsasexpectationcatalogues 88. Measurementtheory-I 99. -functionsasstatedescriptions 1010. Measurementtheory-II 1111. Entanglementresolution,aninterventionof consciousness 1412. An example-I 1513. Example-II:all possiblemeasurementsareentangled 1614. Timevariationof entanglement;statusof time 1815. A law of natureor just analgorithm 20

1. THE PHYSICS OF MODELS

In thesecondhalf of thenineteenthcenturyan Idealof theexactdescriptionof naturewasachievedthat is thefruit of hundredsof yearsof researchandanaspirationfor eons.Built on the kinetic theoryof gasesandthe mechanicaltheoryof heat,it is describedasclassical, andhasthefollowing characteristics.

For thosenaturalobjectsfor which onewould like to comprehendtheir observedbe-havior, oneconstructsanimage,basedonexperimentbut withoutexcludingintuition, thatis accuratein all details,actually, in view of practicallimits, moreexact thanobtainableby any possibleempiricallymeans.Theprecisionof this imageshouldcomparewith thatof a mathematicalconstructionor a geometricalfigure,which, from a limited numberofspecifications,allows thecalculationof all otherdetails. For example,knowing onesideandtwo anglesof a triangle is sufficient to calculatethe remainingsidesandangle,theinscribedcirclesandtheir radii, etc. Suchan imagewould differ from geometricfiguresonly in thatit existsin timeandmustbeconfiguredin four dimensionalspace-timeaspre-ciselyasageometricalfigurein three.Thatis, thedifferenceis (asis obvious)thatsuchanimageevolvesin time andcantake on variousstates;and,if a statein all detailsis given

1

2 ERWIN SCHRODINGER(1887-1961)

at a particlartime, thennot only areall otherdetailsfor this momentto becalculable(asfor a triangle),but alsoall detailsfor latertimesaswell. An integralpartof suchanimageis its intrinsic ability to evolve in a particularway; that is, if left alone,it will evolve intimecontinuouslythroughaparticularprogressionof states.This is its essentialnature,itshypotheticalbasistheexistenceof which,asnotedabove,onepresupposeson intuition.

Naturally, no oneis so simplemindedasto think that a totally faithful imagecanbeachieved for anything in the real world. This implicit reservation is revealedalreadybythefactthatsucha fabricatedimageis calledapictureor model. With theuncompromisedclarity notobtainablewithoutcertainarbitrarysimplifications,oneseeksto show only thata particularhypothesiscanbe verified in detail without introducingeven morearbitraryinput into the tediouscalculationsexploring its variousconsequences.However, becauseof practicalconstraints,oneactuallyachievesonly whata clever fellow could interpolatedirectly from empiricaldata. In any case,it shouldbeclearwhat input in thehypothesesof the model is arbitraryandis, therefore,to be revisedif calculationdoesnot conformto observationaneventualitythatonemustbe forever preparedto accept.Whenmanyexperimentsconfirm the model, it is consideredto faithfully representthe reality of theobjectin all importantaspects.On theotherhand,if disagreementis found,it is not takenasa setback,ratherasthefortuitousidentificationof an inadequacy just wherethemodelis to beimprovedtherebyfosteringconvergenceto anevermoreaccuratemodel.

This classicalmethodof achieving precisionof a modelaimsat isolatingtheunavoid-ablekernelof arbitrarinessin thehypothesesset,likenucleiin cells,readyfor modificationin thecourseof thedevelopmentsspurredon by new empiricaldata. Onemight saythatthis methodis basedon the belief thatsomehowthe initial conditionsactuallydetermineuniquelytimeevolution,or thatacompletemodelwouldagreeexactlywith reality therebypermittingcalculationof outcomesfor all experiments.Perhapseven, its the otherwayaround,this belief is basedon themodel. Nevertheless,it is mostlikely that this processof modeldevelopmentconsistsof an endlessseries,andthat, the notion of a completemodel is oxymoronic,similar to thephraselargestwholenumber.

Thefoundationfor all that follows is a clearconceptionof whatis meantby thespeci-ficationsof a statein themodel. It is especiallyimportantto distinguishthedifferencebe-tweena particularmodelanda particularstatewithin thatmodel.For example,RUTHER-FORDs modelof thehydrogenatomconsistsof two point masses.As specificationsonecouldselectthe six coordinatesandsix momentumcomponentsof the two massesthatmakestwelve itemsaltogether. On the otherhand,onecouldalternatelytake the coordi-natesof thelocationandthevelocityof thecenterof massandalsothevectorseparationofthetwo massesandits angularorientationwith time derivatives;againthis makestwelveitems. It is not partof RUTHERFORDs modelthat thesetwelve itemshave particularval-ues,which do, however, specify a particular state. A clear overview of the totality ofpossiblestateswithoutrelationshipsto oneanotherconstituentsthe model or themodelin anarbitrarystate. However, amodelis notcompletegivenjustaparticularstate,but mustalsoencompasswhatever is neededto specify the time evolution from statetostatewhenever thereareno external influences.(For half of the specifications,the otherhalf providessomeinformation,but thathalf mustbedeclared.)Thisknowledgeis latentin thestatement:Thetwo pointparticleshavemassm, andM, andcharges

eand e, and

are,therefore,attractedtogetherby theforcee2 r2 if r is their separation.Theseparameters,i.e., particularvaluesfor m M and e (but naturally not r) belong

to the specificationof the model(not a particularstate),andthey cannotbe calledstatespecifications. On theotherhand,thevalueof r is exactlysuchastatespecification.In the

THE CURRENTSITUATION IN QUANTUM MECHANICS 3

secondsetof specificationsgivenabove,however, r doesin factappearseventhon thelist;but still in thefirst setr is notanindependent13thspecification,asit canbededucedfromtheothersusing

r x1 x2 2 y1 y2 2 z1 z2 2 The numberof specificationitems (usually calledvariables in distinction to model pa-rameters, e.g.,m M e) is unlimited. Twelve appropriatevaluesaresufficient to specifyall othersat any givenmoment,that is, the completestateat thatmoment.No particulartwelve values,however, areprivilegedto be thespecification. Examplesof other, particu-larly importantvariablesinclude:energy, thethreecomponentsof momentumwith respectto thecenterof mass,andthekineticenergyof thecenterof mass.Thislatterchoicehastheparticularcharacteristicthat they arevariablesthatcanhave differentvaluesfor differentstates,but retaintheirvaluesfor thatsequenceof statesactuallyseenundertimeevolution,andareknown, therefore,asconstantsof themotion.

2. THE STATISTICS OF VARIABLES IN QUANTUM MODELS

At the crux of the currentformulationof QuantumMechanics(QM) thereis certaindogma,which maystill suffer somemodification,but which, I sense,will remaindogma.It consistsin the belief that modelswith specificationscapableof preciselydeterminingreality, aswerepresupposedin classicalphysics,do not exist.

Onemight think, that for thoseacceptingthis dogma,classicalmodelshave exhaustedtheir potential. But this is not so. Rather, they areusedstill, not only to criticize nega-tive aspectsof thenew dogma,but alsoto describethereducedmutualdeterminationthatremainsin the new theoryamongthe samevariablesof just thoseclassicalmodelsusedearlier. This transpiresasfollows:

A. The classicalconceptof a stateis abandoned,insofar asa well chosenhalf of theformer set of variablesallows full statedetermination. For the RUTHERFORD model,for example,the six positionvariables,or themomentumcomponents(otherchoicesarepossible)suffice. The otherhalf thenremainsundetermined,i.e., thesenewly redundantspecificationscanexhibit variousdegreesof uncertainty. In generalin a full set(for theRUTHERFORD model,twelve in number)all canbeknown only with uncertainty. Thede-greeof uncertaintycanbebestdetermined,with guidancefrom HAMILTONianmechanics,whencareis taken in the choiceof variablesso that they areorderedpairwiseascanon-ical conjugates,for which the mostelementaryexampleis onein which thecoordinatexis matchedwith the momentumin the samedirection px (i.e., masstime velocity). Suchpairsmutually limit eachothers precisionwith which they canbedeterminedin that theproductof their tolerancesor variations(symbolizedwith ) cannotbe lessthana givenconstant1ie.:

(1) xpx h (TheHEISENBERG uncertaintyrelationship.)

B. If all the variablesat a given momentare not determined,then naturally at latertimes they cannotbe determinedfrom former values. Onecould call this the Principleof Causality, but in view of A, it is nothing really new. If at no time a classicalstatecanbe specified,this situationcannotbe changedby force. Whatdoeschange, andthenby compulsion,arethestatisticsor probabilities. Individual variablescanbeprecise,but

1h 1 041 10 27erg-sec.andin theliteratureis denotedusuallyby thesymbol , whereh itself is then2timesthis value.


othersthenmadeimprecise. All in all it is so, however, that the overall precisionof adescriptiondoesnot changethroughtime, which is a consequenceof the fact that thelimits explicatedin A areat all timesthesame.

But now, what exactly do the terms: imprecise,statistics,andprobability actuallymean? QM respondsas follows. It commandeersany conceivablemeasurablevariableor specificationfrom classicalmodelsasdirectly measurable,evenacceptingits unlimitedprecision,with theproviso thatonly oneis selected.If useof a well selectedandlimitednumberof measurementsmaximally determinedan objects state,asper A is possible,thenthe new theoryhasthe mathematicalmachineryto specifya statisticaldistributionat the samemomentor for any other time for any variable. That is, it is able to givethe percentageof casesfalling within specificrangesof all variables(i.e., probabilities).The generalopinion is that theseprobabilitiesin fact are thosefor the relevant variablegiving the likely valueto beobtainedby a measurement.With a singlemeasurementtheaccuracy of suchaprobabilityforecastcannotbeverified,exceptpartiallyif thedistributionis sharplypeakedin a small interval. In orderto fully verify thesedistributions,thewholeexperiment,including orientationandpreparationmeasurements,mustbe fully repeatedvery often, and thenconsideronly thosecaseswherethe orientationmeasurementsareidentical.In thosecases,for aspecificvariablegiventhesameorientationresults,statisticalpredictionswill beverifiedby measurementsuchis thepopularopinion.

Onemusttakespecialcarein criticizing thisopinion,becauseit is verydifficult to parse;a situationwhich is a consequenceof our language.Thereis, however, anothercriticismthatarisesnaturally. No physicistin theclassicalepoch,having conceiveda model,wouldhave beensorashasto believe thatspecificationsfor objectsin naturearemeasurabledi-rectly in fact. Usually only derived consequencesfrom suchmodelsactually turnedoutto be amenableto experiment.Moreover, it would have beenanticipatedon the basisofmuchexperience,that long beforeadequateexperimentaltechniquesaredeveloped,themodelwould have beenalreadysubstantiallymodifiedto accommodatenewly foundem-pirical facts. While thenew theorydeclaredtheclassicalmodelincompetentto regulatethe interrelationshipsamongspecifications(for which its originatorshasintendedit), itdoesanointitself competentto certify which measurementsof the objectarein principledoable;this, its foundersshouldhave consideredaninsolentusurpationof possiblefuturedevelopments.Now, is it not presumptuousfantasy, to think that researchersfrom earliertimes,who onehearsnowadaysdid not evenknow whatmeasurementactuallyis, never-thelesshave bequeathedto usunintentionallythe instrumentto judgewhat is measurableonhydrogentoday?

I hopeto show below that this reigningopinionwasbornof necessity. But first I con-tinuewith its description.

3. PROBABIL ISTIC FORECASTS: EXAMPLES

Nowadaysas before,all predictionsrelateto the specificationsof a classicalmodel,i.e., to the locationsandvelocitiesof masspoints,or energy andmomentumandthe like.Thenonclassicalaspectis thatonly probabilitiescanbepredicted.Ideally with QM, it isconsideredthat the taskalways involvesprojectingthe maximally accurateprobabilitiesthatnatureallowsfor resultsof measurementsto bemademomentarilyor atanotherfuturetime basedon currentmeasurements.But now, what is theactualsituation?In importantandin typicalcases,it is asfollows:

If onemeasurestheenergy of a PLANCK oscillator, theprobabilityof findinganenergyvaluebetweenE andE canbedifferentfrom zeroif betweenE andE thereis oneof the


values3h 5h 7h 9h

For all intervalsnot includingoneof thesevalues,theprobabilityis zero;i.e.,othervaluesareexcluded. The acceptablevaluesareoddmultiplesof the modelparameterh (h PLANCK s constant, oscillatorfrequency). Two thingsstandouthere.One,thereis norelationto previousmeasurementsthey areunnecessary. Two, thevaluedoesnot sufferuncertainty, in factit is moreprecisethanany possiblemeasurement.

Anothertypicalexampleis measurementof angularmomentum.In Fig. 1 let M bebeamobilemasspoint,wherethearrow depictsboththemeasureanddirectionof its momen-tum (mass velocity), O is anarbitrarypivot point in space,the origin of the coordinatesystem,say, that is not a point of any physicalsignificance,just a geometricallandmark.Theangularmomentumwith respectto O in classicalmechanicsis givenby theproductofthemomentumvectorandthelengthof theperpendicularto its line-of-flight OF.

In QM, analogousto the energy of an oscillator, angularmomentumis quantized;theprobabilityof its valuesfalling in intervalsnot includingthevalues

(2) 0 h 2 h 2 3 h 3 4 h 4 5 is zeroagain;in otherwords,only oneof thesevaluescanresultfrom measurement.Onceagain, it is not dependanton previous measurements.One can well imagine just howsubstantialthis precisepredictionis, much more so than knowledgeof which value orthe probability for eachpermissablevalue in any particularcase. It is alsonotablethatthereis no mentionof theorientationpoint O; wherever it is put, theseriesof admissiblevalues,(2), is the same. Thus, in a model, this point is irrelevant, sincethe lever armOF changescontinuouslyasthepointO is displacedwhile themomentumvectorremainsconstant.This exampleillustrateshow QM usesa modelto readoff which variablescanbemeasured,while it still mustdesignatewhich interrelationshipsamongthesevariablesthatarevalid simultaneously.

Doesonenot get the feeling in both casesthat the essentialcontentof whatis to besaid,canbeshoe-hornedonly with greateffort into apredictionof theprobabilityto encounterthisor thatmeasurementresultof a variableof the classicalmodel? Doesonenothavetheimpressionthattheissuehereis thefundamentalcharacteristicof new groupsof indicationshaving no morethana namein commonwith classicalcounterparts?This uneaseconcernsnot just exceptionalcases,eventruly importantpredictionshave this character. While thereareactuallysometasksapproachingthesort for which this meansof expressionis tailor made,they do nothavenearlythesameimportance.All themorethosethatonenaively fabricatesasdidacticexamples,thathave absolutelyno importancereally. Giventhelocationof anelectroninhydrogenat time t 0; oneconstrueswhattheprobabilityof its locationis ata latertime.Whocares?

Literally taken,suchpredictionsdo concernthemodel. But theusefulconclusionsarenot imaginable;and,whatis imaginable,is virtually useless.

4. IS AN ENSEMBLE INTERPRETATION TENABLE?

In QM a classicalmodelplaysa proteanrole. Eachof its specificationscanbe, de-pendingon circumstances,an item of interestandacquirea certainreality. But never alltogethernow it is these,thenit is those,andalwaysonly half of afull complimentneededfor a cleardeterminationof themomentarystate.Whathappenedto therestof them?Do


they not have any reality? Maybe(seebelow) in just a vague,fuzzy way? Or, do all ofthemhaveuniquemeanings,but suchthatknowledgeof only half is possible?

Thissecondsuppositionis commodiousfor statisticaltheoryof thesortdevelopedin thelatterhalf of thepreviouscentury;especiallysowhile thenew theorywasborn from it inanalysisby PLANCK from December1899on thetheoryof heatradiation.Theessenceofthismodeof thoughtis foundin thefactthatonepracticallyneverknowsall specificationsfor a system,but only a few. For thedescriptionof a realisticbodyat a givenmomentonedoesnot call on a stateof themodel,rathera so-calledGIBBSian ensemble,which is anideal,merelyimaginedcollectionof statescorrespondingto theactualknowledgein hand.The body is thenconsideredto executejust oneof the statescontainedin the ensemble.This viewpoint hasenjoyedgreatsuccess.Its greatesttriumphswerethere,whereeachoftheoptionscannotbefoundpertainingto asinglememberof theensemble.Thebodythenreally doesbehave erratically, exactly correspondingto probabilisticexpectations(think:thermodynamicfluctuations).Thus,oneis temptedto attributequantumuncertaintyto theplurality of suchpossibilitiesbut suchthatjustwhich oneis realizedremainsunknown.

Unfortunately, the examplegivenabove involving angularmomentumshows that thissurmiseis not tenable.Consideran ensemblein which eachmemberis relatedto a dif-ferentpoint O andwith differentlinear momenta.Thenonecanchosethe lengthof themomentumvectorwith respectto any pivot point suchthatany of the admissiblevalues,(2), is obtained. But then for any otherpivot point, O , the valuesarisingmight not beamongtheadmissibleones.Transplantingthe issueto anensembledoesnot resolve thisdiscrepancy. Anotherexampleis thatof theenergy of anoscillator. Considerthecaseinwhich its energy hasa sharpvalue,e.g.,the lowestone,3h, say. Theseparationof thetwo masspoints(thatconstitutetheoscillator)is thenvery imprecise. In orderto beableto transferthesefactsto an ensemble,the statisticsof the separationsshouldbe limitedfrom abovesothat thepotentialenergy never exceeds3h. But this is not whathappensin fact,arbitrarily largeseparationsarealsoincluded,albeitwith diminishingprobability.Moreover, this defectis not anancillarycalculationalquirk to becorrectedwithout affect-ing theessentialkernelof thetheory. Amongmany othereffects,this featureis thebasisof GAMOV s quantumexplanationfor radioactive decay. Suchexamplesareto be foundwithout limit. Notethatin all of this thereis no mentionof time. Nor would it helpatallto acceptmodelsincorporatingnonclassicalfeatures,sayinstantaneousjumping. Evenatasingleinstant,thatwouldnotwork; theredoesnotexist acollectionof classicalmodelmomentarystateswhich would adequatelycover thetotality of quantumpredictions.Thiscanalsobeexpressedasfollows: if I for any instantattributea particular(but unknown tome)stateto amodelsystem,or equivalentlyfix all values(again,justnotknown to me)forthemodels specifications,therestill is no conceivablepredictionwhich is not in conflictwith a portionof thequantumconclusions.

This is not entirelywhatoneexpectswhenonehearsthattheresultsof thenew theory,in comparisonto classicaltheory, areneversharp.

5. ARE VARIABLES REALLY INDISTINCT?

Thealternativewouldbeto ascribesreality to whatevervariablesaredistinctandsharp;or. moregenerallyput: to attributeto variablesexactly thatdegreeof realitycorrespondingto thesharpnesswhichquantumtheoryallows them.

That it is not impossibleto expressin just onecompletelyclear construct,the degreeandsortof indistinctnessof all variables,is manifestedby the fact thatQM hasandusesjustsuchaninstrument,namelytheso-calledwaveor -function,or sometimescalledthe


systemvector. We shall have muchmore to say aboutit below. Suchfunctions,beingabstractmathematicalentitieswithout visualizablecontent,as is oft notedwith respectto the new modeof thought in general,is devoid of consequence.In any caseit is aconceptualentity encompassingthe indistinctnessof all variablesat any given instantasclearlyandexactly asa classicalmodeldoestheir exactitudewith numbers.In addition,its law of time evolution is in no way lessclearandprecise,for isolatedsystems,thanthat for a classicalmodel.Consequently, -functionsareappropriatefor situationswhereindistinctnessis limited to the scaleof unverifiableatomicdimensions.In fact, onehasextractedfrom suchfunctionsquiteimaginableandcommodiousnotions,for example,thenegative charge cloud surroundingan atomic nucleusetc. Seriousreservationsarise,however, whenever this indistinctnessshouldpertainto macroscopicobjectsfor which theis simply misplaced.Thestateof a radioactivenucleushaspresumablyjust sucha degreeandtypeof indistinctnessin thatneitherthetime nor directionof a decayof an-particleare fixed. Confinedto the interior of a nucleus,this doesnot disturb us. The ejectedparticlecanbedescribed,if oneseeksanimage,asasphericalwaveexpandingaway fromthe nucleussuchthat it would impacta sphericaldetectorcenteredon the nucleusoveris whole surface. Suchdetectors,however, do not respondthis way, but flashesat singlespots,or, to fully respecttruth,flashesoccurarbitrarily hereandthereasit is not possibleto carryout this exercisewith a singleradioactive atom. If insteadof a sphericaldetectora spacefilling ionizablegasis used,the tracksseendepict linear trajectories2, departingfrom the-emittingnucleus(i.e., WILSON cloudchambertracksof condensationon ionsengenderedby theexpelledparticles).

Onemightevenconsideraburlesqueillustration.Supposeacatis confinedin abox to-getherwith, but isolatedfrom, adevilish gadgetconsistingof a GEIGER-counterto triggerahammerto smashavile of lethalprussicacid.Furthersupposethecounteris setbeforeaminisculeamountof radioactivematerialwhich within anhourhasanequallikelihoodof,or of not, one-decay. Now, a -functionfor thetotal system,catandgadget,accordingQM principles,afteronehouris amix of liveanddeadcat!

Thesignificantpointhereis thatwhatoriginally waslimited to themicroscopicatomicdomain,is transferedto a macroscopicarenaavailableto directobservation.This illustra-tion stronglythrottlesa naive understandingof any indistinct model asa representationof reality. In andof itself, thereis nothingunclearor contradictoryhere. It simply illus-tratesthedifferencebetweenanout-of-focusphotographandoneof a cloudor fog bank.

6. EPISTEMOLOGICAL REFORM

In 4 we saw that it is not possiblesimply to adaptthe classicalmodel andascribeprecisevaluesto indistinct or unknown variables. In 5 we saw that indeterminacy isnot just indistinctness,astherearecaseswherean easilymadeobservationcompensatesany indeterminacy. So, what remains?To escapethis dilemma,epistemologyis calledto the rescue.Oftenwe areassuredthat thereis no realdifferencebetweenthestateof anaturalobjectandwhatoneknows aboutthis object,or ratherwhatonecouldknow withsufficienteffort. Reality it is positedisactuallyjust that,whatoneperceives,observesor measures.Thus, if I have at any momentthe bestpossibleinformationon an objectallowedby natures laws, thenI amentitledto rejectall furtherquestionsaboutthatobjectasvacuous, at leastinsofarasI amconvincedthatnoadditionalobservationcouldincrease

2For exampleson thepagesof this journal,seeFigs. 5 & 6, p. 375of theyear1927andp. 734of lastyear,1934,wherehowever, thetracksweremadeby hydrogennuclei.


my knowledgewithout,however, destroying alreadyheldinformation,e.g.,by changingits state.

This throws somelight on the genesisof the remarkmadeat the endof 2, that I de-scribedas rather far reaching: namely that all of a models quantitiesin principle aremeasurable.This, asan article of faith with universaldominion,simply cannotbe dis-regardedif onefinds oneselfcompelledto call upona rescuingomnipotencefor aid inphysicsmethodology.

Reality resistsmentalimitation via models.Soonegrantsnaive realismfreereignandfalls backdirectly on the indubitablethesisthat in theendreality (for a physicist)is onlyobservationor measurement.In whichcaseall ruminationsonphysicsproblemsarebasedon andpertainto theresultsof feasiblemeasurements,asthoughall othersortsof realityandmodelsthereofareimpotent.All numericalquantitiesthatarisein physicscalculationsthenmustbe designatedmeasurementresults.But, aswe werenot born yesterdayso asto be startingafreshtoday to do science,andalreadypossesa well developedquantumcalculus,which now, aftersubstantialsuccesses,we do not wish to abandon,we seemtofeel obliged to dictate from the desk just which measurementsarepossiblein princi-ple, i..e, which mustexist so as to verify our calculus. Sincethis allows a sharpvaluefor every variableof the modelseparately(if for only a half set), eachseparatelymustbe preciselymeasurable.We mustallow ourselvesno less,having lost our naive realistinnocence.We have nothingbetterthanour calculusto show uswhereNaturehasdrawnthe ignoramus-border, that is, what constitutesthe bestpossibleknowledgeof an object.Shouldour calculusbe inadequate,then our measurement-realitywould dependon theconscientiousnessof experimentersto garnerinformation. We mustinstructthemon justhow far they maygo, if clever enough.Otherwise,it is to befearedthat just there,wherefurtherquestioningis forbidden,somethingusefulmight turnup.

7. WAVE FUNCTIONS AS EXPECTATION CATALOGUES

In thecontinuationof the explicationof orthodoxy, let us turn againto the or wavefunction.It is theinstrumentof predictionfor probabilitiesof measurementresults.Withinit is to befoundthetotality of valid futureexpectations,a catalogueasit were.It providesthebridgeof relationshipsandconditionsbetweenmeasurementsandsubsequentmeasure-ments,similar to themodelandits currentstatein classicaltheory, which is somethingithasin commonwith wavefunctions.It canbe,in principle,uniquelydeterminedby asuit-ableselectionof measurementson the object,half asmany aswould benecessaryin theclassicalcase.In thisway, thecatalogueof expectationsis compiled.Thereafterit changeswith time, just like thestatein a modelin classicaltheory, inevitably andunambiguously(causally)theevolution of a -function is governedby a partialdifferentialequation(first order in time andsolved from t). This correspondsto an unperturbedmotionin classicaltheory. But this proceedsonly so long asno measurementis made. At eachmeasurement,oneis requiredto attributeto a-functionanabruptmodificationdependingonthenumericalresultsof themeasurementthatitself is notpredictable. All of whichsaysthat this secondarysortof changehasabsolutelynothingto do with theorderlyevolutionof a -function betweenmeasurements.The abruptchangeoccasionedby measurementrelatescloselyto themattersconsideredin 5,whicharethemostinterestingaspectsof thetheory. It constitutesexactly thepoint thatrequiresa breakwith naive realism,andwhichprecludesequatinga -functionwith a statein classicaltheory, not becauseunpredictableabruptalterationscannotbeattributedto ontologicalobjectsor encompassedin a model,


but becausean observationof a naturalprocessis asmucha part of natureasthe objectitself.

8. MEASUREMENT THEORY-I

Rejectingrealismhaslogical consequences.A variablehas in generalno particularvaluepreceedingits measurement;thatis, measuringit doesnot just revealthevaluethatithadapriori. Whatdoesit mean,really?Theremustexist acriterionfor whetherameasure-mentis right or wrong,amethodgoodor bad,accurateor inaccuratewhetherit deservesthetitle measurementor not. Everymachinationwith ameterin thevicinity of anobject,from which oneat somepoint takesa reading,cannotreally constitutea measurementofthis object. Thus, it is ratherclear, if reality doesnot determinemeasuredvalues,thenmeasuredvaluesat leastdeterminereality, whichmustbeavailableafter ameasurementinthesensecertifiedby ourepistemology, i.e.,asmeasuredvalues.In otherwords,thesoughtcriterioncanbeonly: reiterationof a measurementmustyield the samemeasuredvalue.With suchserialmeasurementrepetitionthen,theaccuracy of themeasuringprocesscanbecheckedandtherebydemonstratedthat it is not just emptytheatrics.It is comforting,thatthis instructioncoincideswith theexperimentalistsprocedurefor whomatrue valueis unknown in advance.We capturetheessenceof this pointasfollows:

Theintentional interaction of two systems(measured objectand measurementinstru-ment)is calleda measurementof thefirstsystemif somediagnosticindicatorof thesecondsystem(measurementinstrumentreading)for an immediaterepetitionof this interaction(where no interveninginfluenceon themeasuredsystemis allowed)alwaysyields,withingiventolerances,identicalresults.

This definition surelyneedsrefinement,it is not without defects.The ontic is alwaysmorecomplex thanits mathematicalrepresentationanddifficult to formulatein smoothprose.

Before thefirst measurementany quantumtheoreticalforecastcanbevalid. After sucha measurementin all cases: further measurementresultsmust fall within relevant toler-ances.That is, thecatalogueof predictions(the-function) is alteredwith respectto themeasuredvariableby measurement.Whenthemeasurementprocedureis known to be re-liable, thenthefirst measurementreducesthetheoreticalpalletof expectationsto bewithinthe tolerances,regardlessof its previous extent. This constitutesthe abruptalterationof-functionsmentionedabove. But the fact is thatnot only the measuredvariables cata-logueof expectationsis unpredictablyaltered,but alsothatfor othervariables,in particularfor conjugatevariables. If a rathersharppredictionfor a particles momentumprevailsbeforeits locationis measuredbeyond that allowed by Eq. (1), thenthe forecastfor themomentummustalsohave beenaltered. The quantumcalculusautomaticallytakesthiseffect into account;thereexists no -function from which with appropriatemethodsanexpectationnot in accordwith Eq. (1) canbeobtained.

In that the expectationcatalogueis alteredby measurement,the objectitself thenbe-comesno longerusefulfor checkingthecompletecatalogueof probabilitiesof theincom-ing state,in particularfor the measuredvariablewhich retainsits first revealedvalue. Inorderto checkthewholepalette,in otherwordsto checktheexperimentthat is theratio-nalbehindtheprescriptionin 2,namelyto checktheprobabilitycataloguesencompassedin a wave function, the measurementproceduremustbe faithfully repeatedab ovo. Onemustprepareidenticalobjectswith -functionsidenticalto theonevalid for thefirst mea-surement.(Note, this repetition,beingof the experimentnot just of a measurement,is


fundamentallydifferentfrom thatmentionedabove!) Thismustbedonenot just twice,butveryoften.Thusly, forecaststatisticscanbeverifiedpercurrentopinion.

Thedifferencebetweenerrorlimits andstatisticalscatterof measurementsononehand,andtheoreticalexpectationstatisticsontheothermustbekeptin mind. They areunrelated,they engagetwo typesof repetition, aswasjust emphasized.

At this point notionsregardingconstraintson measurementcanbe refined. Therearemeasurementapparatuswhich hold their readingsafteruse;or, whats worse,thepointermight getstuckaccidentally. Rerunsthenyield thesameresultrepeatedly;and,theexper-imentermight conclude,following theabove principle,thathis measurementis somehowspecialandparticularlyaccurate.But this is a resultpertainingnot to theobject,but to theinstrument. In fact, therefore,the above principle is incompletein an importantrespect,onethatearliercouldnot beconsideredeasily, namelyby neglectingtheessentialdiffer-encebetweentheobjectandmeasuringinstrument(that only the latterdeliversreadings,is a superficiality).For example,oftenaninstrumentmustbereset,aswe have just noted,to its initial conditionbeforereuseasis well known to experimenters.Theoreticallythismatteris dealtwith bestby proscribingthatall instrumentsberecalibratedsothatfor eachrepetitionthesame-functionpertainswhenre-exposedto theinstrument.In addition,anydeliberateperturbinginteractionwith the objectitself mustbe preventedduring controlmeasurementsi.e., repetitionof thefirst sort (leadingto measurementerrorstatistics).That is a characteristicdifferencebetweenobjectandinstrument.For a repetitionof thesecondsort (to checkquantumforecasts)theobject-instrumentdistinctionvanishes;i.e.,it is reallyquiteinsignificant.

From this we learnthat for a secondmeasurementan identically constructedandpre-paredinstrumentmaybeused,it neednot bethe identical instrument;in fact,sometimesinstrumentexchangesaremadejust to checktheoriginal one.Eventotally differenttypesof instrumentscanbeusedfor sequentialmeasurements(repetitionof thefirst sort)if theyyield compatibleresults,i.e., they measureessentiallythesamevariableandaremutuallycalibrated.

9. -FUNCTIONS AS STATE DESCRIPTIONS

Rejectingrealismengenderscertainresponsibilities.From a point of view basedonclassicalmodels,-functionsare,in termsof theircontent,incomplete,they include,seem-ingly, only 50% of a total description.Fromthe QM standpoint, however, they mustbecompletefor reasonsthatwerementionedat theendof 6. It mustbeimpossibleto attachstill morevalid forecaststo themor elseonelosestheright to disregardall furtherdemandsfor morespecificity, aspointless.

Therefromit follows that two differentcataloguesfor thesamesystemunderdifferentconditionsor at differenttimes,canoverlappartially, but never so thatoneis completelycontainedin the other. Wereit otherwise,it would be possibleto enlargeonewith addi-tional information,namelythe differenceof the overlap. The theorytakesthis automati-cally into account;thereexistsno -function yielding all the sameexpectationsassomeother, but alsoadditionalones.

Therefore,whenever a -function is altered,be it from within by itself, or from with-out by measurement,therevampedfunctionalwaysyieldssomedifferentexpectationsnotextractablefrom theunalteredversion.That is, thecataloguewill have not only new pre-dictions,but will alsohave lost somepreviousones. Now, informationcanbe obtained,but not thereafterlost. Thelossof expectationscanonly meanthenthatwhatwascorrect


to begin with, becomesfalsethereafter. A correctpronouncementcanbecomefalse,if itsessencechanges.I find thefollowing conclusionin this regardfaultless:

THEOREM I: If there are different-functionsfor a givensystem,thenthesystemis indifferentstates.

If the matterconcernssystemsfor which thereexists only one -function, then theinverseobtains:

THEOREM I I : Givenidentical-functions,thesystemis in thesamestate.This inversedoesnot follow directly from thefirst without alsoassumingcompletion.

Wereoneto hold thata differencefor the samecataloguesaspossible,this would admitthatit still doesnot giveanswersto all questions.Thelanguageusedby nearlyall authorsreflectsthispoint, in thatthey areessentiallyfabricatinganew sortof reality, andI believefully legitimately. Suchlanguageis, moreover, not tautological,not simply a definitionfor the word state. Without the assumptionof cataloguecompleteness,alterationsto-functionswouldbeeffectedsimplyby acquisitionof new information.

Thereis still oneotherobjectionto THEOREM I thatmustbeassessed.It couldbesaid,that eachindividual statementof conviction involved here,the actualobjectof concern,is really no morethana probability statementwhich in the cataloguerightly or wronglydoesnot concernan individual item but only the collective, that arisesin that one haspreparedthe samesystema thousandtimes (so as to carry out identicalmeasurements;see8). That is true,but still all membersof this collective mustbeverifiedasabsolutelyidentical,becausefor eachthe identical-function obtains,i.e., they have the identicalcatalogueof expectationsanddifferencesnotalreadyin thecatalogueareinadmissible(seethereasoningfor THEOREM I I ). Thiscollectiveconsists,then,of identicalsingleitems.Ifanexpectationfor it is false,theneachsingleitem musthave beenaltered,otherwisethecollectivewouldbefalsetoo.

10. MEASUREMENT THEORY-I I

Discussion(7)andclarification(8)aboveledto thenotionthatmeasurementsuspendsthe law of continuous-function time evolution, interruptingit with a lawlessalterationdictatedby themeasurementresult. However, duringmeasurementsomeotherunnaturallaw cannotreign;measurementis asnaturalaprocessasany other, andthereforeunabletoviolatenaturallaw. As -functionsarein factinterrupted,they (aswasnotedin 7)cannotbe consideredcandidatedepictionsof objective eventsaswith classicalmodels.Exactlythis is theideacrystallizedin theprevioussection.

Lets try, in outline, to draw contrasts:1. Thecollapseof theexpectationcatalogueisunavoidable; if measurementis to makesensethena goodmeasurementresultmustbenolessthanits numericaloutcome.2. The abruptchangecannotbe regulatedby the timeevolution law, asit dependson the numericalresult,which arisesfirst at the momentofmeasurement.3. This changeincludes(becauseof completeness)lossof knowledge,andasknowledgeassuchis indestructible,this meansthat the item itself haschangedalsoabruptlyandunpredictably, in contrastto normalevolution.

Doesthismakesense?Thesituationis notatall simple;andconcernsthemostdifficultandinterestingaspectof the theory. To begin, we mustattemptto capturetheessenceofthe interactionof objectandinstrument.To do so,first someabstracttechnicalitiesneedbeexplicated.

Thecrux is this. If onehastwo fully separatedbodies,or bettersaid,for eachaseparateexpectationcatalogue(i.e., maximal knowledge), then naturally one also hascomplete


knowledgefor thepair together, that is, consideringthemasa totality, not separateunits,whosecombinedfutureis of interest3.

But now, theinverseis not true. Maximalknowledge of thepair doesnot includemax-imal knowledge of the parts,evenwhenthey at the momentare widelyseparatedanddonot interact. It canbe,however, thata portionof whatoneknows pertainsto the interre-lationshipor conditionalitybetweenthepair (we limit ourselveshereto two items)thusly:if a particularmeasurementgivesa particular resultfor one,thentherewill beparticularexpectationstatisticsfor the other;but if the first resultweresomethingelse,thenother,differentexpectationstatisticsobtain for the second. Or further, if a third result is ob-tainedon the first, thenstill otherexpectationstatisticsobtainfor the other, andso forth.Thisstructureresemblesacompletedisjunctionof all numericalresultsthattheconsideredmeasurementof the first variablecandeliver. In this way the measurementprocess(orwhat is the same,a variableof the secondsystem)is coupledto an asyet undeterminedvalueof thefirst variable,andof course,visaversa.Whenthatis thecase,i.e.,whensuchconditionalprobabilitiesarein the total systemcatalogue,thenit cannotbe maximalforthecomponentsystems.Thus,whenthecontentof two maximalsinglecataloguesshouldsufficeby itself asa total catalogue,theseconditionalprobabilitiesareinadmissible.

Suchqualifiedpredictionsare,moreover, notsomethingnew poppinguphere.They ex-ist in everyexpectationcatalogue.If oneknows the-functionandmakesa measurementgiving aparticularresult,thenonestill knowsthe-function,voila tout. But in thecaseathand,asthewholesystemconsistsof two fully separatedparts,thesituationis quitediffer-ent.Hereit makessenseto differentiatebetweenameasurementononepartorontheother.This in turn justifiesthedesirefor a completecataloguefor eachindividually; otherwise,it couldbepossiblethata portionof that informationon themutualconditionalaspectsiswasted,so-to-speak,andleavesthe wish for separatecataloguesunfulfilleddespitethefactthatthecataloguefor thewholeis complete,thatis, its -functionis known.

Let usdwell herefor amoment.Thejustmaderemarkin all its abstractionactuallysayseverything. Thebestpossibleknowledgefor a total systemdoesnot includednecessarilythe samefor its parts. Translatedinto the terminologyof 9: The total systemis in aparticularstate,but thepartsby themselvesnot so.

-How so?Any systemmustbein somestate.=No! Stateequals-function, and is maximalknowledgethisdoesnot imply that

actuallyI have gainedpossessionof suchknowledge.MaybeIm lazy, in which casethesystemwouldbein no stateatall!

-Fine, thenthe agnosticquestion-prohibitionis not yet ratifiedandI canconsiderthatfor eachpartthereis astate(i.e.,-function), and,just dont know which.

=Not so! SayingI just dont know is unacceptable.After all, maximalinformationfor thetotal systemis indeedat hand.

Theinsufficienceof a -functionasan ersatzfor a model,resultsexclusivelyfromthefact, thatonedoesnot alwayshaveit. Onceonedoeshave it, it surelyis agoodandtrustydescriptionof thestate.But, actuallyonedoesnt really have it in somecaseswhereoneexpectsto have it. In which caseonemaynot assertthat actually thereis one,but I justdont know it; thechosenunderlyingepistemologyforbidsthat. It is namelya sumofknowledge,andknowledgeknown to nobodyis no knowledgeatall.

3Naturally, informationontherelationshipof thetwo to eachothercannotbemissing.If it were,thenit couldenterinto oneor theotherof the-functions;andthat is precludedby definition.


We continue.That a portion of the knowledgein the form of disjoint conditionalitiesbetweenthe partshangsin the blue, cannotbe true for the casewherethe partsareas-sembledfrom oppositeendsof theuniverseandhave hadno interaction,somuchso thatthepartswould betotally unaware of eachother. Measuringonecannotrevealanythingabouttheother, its future,its fateor whatever. Whenever in factthereis anentanglementof predictions, obviously it canariseonly if thepartsatsometimein thepastconstitutedasinglesystem,thatis, interactedsoasto leave tracesthereof.If two separatebodieswhichseparatelyaremaximallyknown cometo a situationin which they affect eachother, andthenmoveapartagain,a conditionalpredictablyarrisesthatI havecalledentanglementofour knowledgeof the parts. The commonexpectationcataloguelogically consistsorigi-nally outof thesumof thecataloguesof theparts;but duringinteraction,cataloguesevolveor developaccordingto well known laws (of course,heremeasurementis not involved).Knowledgeremainsmaximal,but in theend,asthepartsmove apartthecatalogueis notrestoredto thesumof thosefor theparts.Whatremainstherefrom, is,maybeevenseverely,lessthanmaximal.Noticethecontrasthereto classicalmodeltheory, wherenaturallytheinitial stateandtheknown outcomesor final statesareall individually known.

Themeasurementprocessdescribedin 8 falls directly underthis scheme,if we applyit to the total object complex and instruments. In constructingan objective picture ofthis process,aswould bedonefor any otherprocess,we might hopeto clarify theweirdjumping associatedwith -functions,maybeeven banishit altogether. Moreover, onebodyis now theobject,while theotheris theinstrument.In orderto precludeall externalinfluences,the setupshouldbe so imaginedthat the measuringinstrumenthasbuilt intoit an automaticcontrol mechanismso it cansneakup on the object,measure,and latersneakaway. Theactualreadingof the metershallbe postponedbecausefirst we wish todeterminejust what objectively transpired;but we have arrangedthat themeasurementresultitself berecordedandreadout later, asis oftendonein factnowadays.

So,whatdoesthis automaticmeasurementamountto? Again,we startwith a maximalexpectationcatalogue;but, the read-outis, of course,not in it. With respectto the in-strument,thecatalogueis quite incomplete,anddoesnot eventell uswheretherecordingpendid its writing. (Recall the poisonedcat.) So onegetsthe ideathat our knowledgehassublimatedcertainconditionalities: if the penreachesfiler mark 1, then for the ob-ject probability is this or that, if filer mark2, thenanotherprobability, if 3, thena third,etc. Hasthe-functionof theobjectmadea jump? Or hasit, on theotherhand,evolvedaccordingto the appropriatelaw (the differentialequation)?Neither, actually; it ceasedto exist. It has,accordingto inviolable law for the total -function,gottentangledwiththatof theinstrument.Theexpectationcataloguefor theobjecthassplit into a conditionaldisjunctionof expectationcatalogues,likeanartful folding map.In eachfold onefindstheprobability of its own occurrencetransferedfrom the original cataloguefor the object.But the issuenow is: which fold of themapis relevant? Theansweris givenby thefilermark.

Whathappensif it is not read?Say, it is recordedonphotographicpaperthatis acciden-tally over-exposedbeforedevelopment,or just lost. Thenthis unfortunatemeasurementnotonly teachesusnothingnew, but actuallyhasreducedourknowledge.Knowledgethatonealmosthadearlier, is lost forever. Onehasto now carefully reorganizeeverythingsoasto regainwhatwentastray.

So, what hasthis analysisbroughtus? First, insight into the disjunctive split of ex-pectationcatalogues,which transpiresquitecontinuouslyvia embeddinginto thecommoncataloguefor objectand instrument. From suchmultiplicity the objectcanbe liberated


finally only whena mortal experimenterfinally readsthe instrument.At sometime thatmusthappen,if the enterpriseis really to constitutea measurementsincemortalscon-scientiouslyalwayswork with maximalobjectivity. That is thesecondbit of insight: onlyafter readingthemeter, which resolvesthemultiplicity, doessomethingdiscontinuousorjump-like,occur. Oneis temptedto perceive it asa mentalact,sincetheobjectitself hasin fact alreadymoved on and is no longer physically involved, whatever elsehappens.But still it would not be totally correctto saythat a -function for the object,otherwisealwaysevolving accordingto a differentialequationindependentof anobserver, changesnow jumpwiseasa consequenceof a mentalact. Therefor aninstant,it existedno more.Whatisnt, cant change.Thensuddenlyit is resurrected,resuscitatedoutof theentangledknowledgeheldby anobserverby aprocessthathasnophysicaleffectontheobject.Fromtheformer-functionto thenew one,thereis no continuouspathasit passesthroughan-nihilation. But by fore-and-aftercontrasts,it appearsjump-like. In fact thereis importantdevelopmentsin between:namelytheeffectof thetwo partsononeanother, duringwhichtheobjectpossessedno privatecatalogue,andhadno right to one,becauseit did not existindependently.

11. ENTANGLEMENT RESOLUTION, AN INTERVENTION OF CONSCIOUSNESS

Let usreturnto thegeneralissueof entanglementwithout focusingon measurement.Considertheexpectationcataloguesfor two objects,A andB, entangledby causeof tem-porary interaction. Now let the two be separated.Thenonecanselectone,B say, andhisunder-maximalknowledgeof it andboostit by meansof measurementsup to maximalknowledge.It canbesaid,thatassoonasthat is accomplished,but not sooner, first of alltheentanglementwill have beenresolved,andsecondly, by exploitationof thecondition-alities,thebestmaximalknowledgeof A will havebeengainedalso.

In the first placeknowledgeof the whole systemremainsmaximal always, becausegoodandaccuratemeasurementsneverwould expungeknowledge.Secondly, conditionalstatementsof theform: if atA , thenatB , cannolongbevalid assoonasamaximalcataloguefor B is compiled,which would not be conditionedandso, no moreinforma-tion concerningB canbe investedinto it. Third, conditionalstatementsin the oppositedirection(if at B , thenat A ) canbeconvertedinto statementsonly on A, becauseall probabilitiesfor B arecompletelyknown in advance.Theentanglementis therebyto-tally resolved,andasknowledgeof thewholesystemremains,it mustbeattributedto themaximalcataloguefor B, aswell asanotheronefor A.

It cannothappenthat A indirectly throughmeasurementson B, becomesmaximallyknown, beforeeven B does. If it did, thenall argumentswould work in reverse,that is,for B also. The systemsbecomemaximally known mutually. Incidentally, we notethat,thatwould besoalsoif measurementwerenot focusedon just oneof the two parts.Thecuriousfactorhereis, that focuscan be limited to onepart andstill permit gettinggoodresults.

Which measurementsandin which order they aremadeis up to the discretionof theexperimenter. He neednot selecta particularvariablein orderto exploit theconditionali-ties. He mayconfidentlymake a planbringinghim maximalknowledgeof B, evenwhenheknew nothingof B. No damageis therebydone;even if he,wonderingaftereachstepwhetherheis finished,just checkssoasto spareuselessfurthereffort.

Which A-catalogueof this type indirectly arises,dependsnaturallyon the numericalmeasurementresultsderivedfrom B (beforetheentanglementis completelyresolved,butnotby latermeasurements,in casefurthersuperfluousonesaremade).Supposenow, I had


found this sort of A-cataloguein a particularcase.ThenI cancontemplateandconsiderwhetherI would have found a differentA-cataloguehasI usedanotherplan to measureB. BecauseI have not hadcontactwith thesystemA really, nor would I have hadit in thecontemplatedalternative,sothepredictionsof theothercatalogue,whatever it couldhavebeen,areall alsocorrect.They mustall be in thefirst cataloguealso,asit wasmaximal.Thatwastruetoo for thesecond;sothey mustbeidentical.

Peculiarlythemathematicalformalismdoesnot automaticallysatisfythis requirement.Evenfurther, examplescanbeconstructedwherethis requirementis violatednecessarily.While for eachattemptonly oneschemeof measurements(alwayson B) canbeexecutedin that assoonasit is, the entanglementis resolved andonefinds, via further measure-mentson B, no moreknowledgeoverA. But therearecasesof entanglement,in which formeasurementson B, therearetwo distinctprograms,for which 1.) theentanglementmustberesolved,and2.) thatleadto anA-catalogueto whichtheotherabsolutelycannotleadwhateverthenumericalmeasurementresultin thisor thatexperiment.It is simplythecase,that two rowsof A-catalogues,that for oneor the otherprogramcould be employed,areperfectlydistinctandhaveno elementsin common.

Thereareparticularlycritical cases,in which this situationarisesoften. In generalonemustconsidermattersmorecarefully. If two programsfor themeasurementof B arethenavailable,andalsotwo rowsof A-cataloguesto whichthey couldlead,thenit is notenoughthatthetwo rowshaveoneor morecommonelementsin orderto beableto say:well, oneor theothercanalwaysbeused, andtherebyput asidetheaboverequirementaspresum-ably fulfilled. That is not enoughsinceoneknowstheprobabilityof every measurementon B, seenasa measurementon thetotal system,andby multiple ab-ovo-repetitionseachmust enterwith its attributed frequency. The two rows of A-cataloguesmust thereforeagreeelementfor elementandfurthertheprobabilitiesin eachmustbethesame.And thatnotonly for thetwo programs,ratherfor eachof endlesslymany conceivablevariants.Butthis is by nomeanstheissuehere.ThedemandthattheA-catalogue,whichonegets,mustbethesameregardlessof which measurementsareexpectedto bemadeon B, is never inany wayaccomplished.

Let usconsidernow a peculiar example.

12. AN EXAMPLE-I4

For thesakeof simplicity, let usconsidertwo systemswith only onedegreeof freedom.That is, eachshall be characterizedby only onevariableq andits canonicallyconjugatepartnerp. Theclassicalimageis thatof a masspoint confinedto onedimensionalmotionlike a beadin anabacus.p is theproductof massandvelocity. For thesecondsystemweusetheuselargeQ andP. Althoughwe do not insiston having themmountedon a wire,if they arewe neverthelessdo not insist that the origin for both variablesis the same,sothat q Q doesnot imply that they arecoincident. The two systemscanbe completelyseparated.

In thecitedpaperthetwo possiblyentangledsystems,for whichat a particular instant,all that followsrelates,aresuccinctlydescribedby theequations

q Q and p PThatis: I knowif ameasurementof q anthefirst systemyieldsaparticularvalue,thesamevaluewill result for Q-measurementon the second,andvisa versa;and I know that if a

4A. EINSTEIN, B. PODOLSKY andN. ROSEN, Phys. Rev. 47, 777 (1935). Their paperstimulatedme towrite thisshouldI call it screedor testimony?


p-measurementmadeon thefirst givesa particularvalue,a P-measureyieldstheoppositevalue,andvisa-versa.

A singlemeasurementof q or p or Q or P resolvestheentanglementandrendersbothsystemsknown. A secondmeasurementon the samesystemmodifiesthena forecastonit, but saysnothing aboutthe other. That is, both equationscan not be verified in oneexperiment.But onecanendlessrepeatit ab ovo; duplicatingtheentanglement;andthenasonewishes,testoneor theotherequation,to find it confirmed.We considerthis hereinto havebeendone.

If aftermany runsonedecidesto measureq onthefirst systemandP onthesecond,andgets:

q 4 P 7;mayonedoubtthat

q 4 p 7 wouldbea correctpredictionfor thefirst system;or

Q 4 P 7 correctfor the secondsystem?Quantumpredictionsarenever fully testablein a singleexperiment,but correctneverthelessbecausewhoeverhasthemin handis in no dangeroflaterdisappointmentnomatterwhich half hedoesin facttest.

Thisis beyonddoubt:Eachandeverymeasurementonasystemis thefirst measurementon it. Therecanbeno direct influencesfrom othermeasurementson separatedsystems,aphenomenonwhich it existedwould bepuremagic.Neithercanit bejust randomcoinci-denceasthousandsof measurementson virginalsystemsdo coincide.

Theforecastcatalogueq 4 andp 7 would be,naturally, maximal.13. EXAMPLE-I I : ALL POSSIBLE MEASUREMENTS ARE ENTANGLED

Actually a forecastwith this degreeof comprehensiveness,accordingto QM principlesexaminedhereinto their last consequence,is not evenpossible.Many of my friendsaretherebycomfortedandassertthat what a systemwould have told an experimenterif hasnothingto do with realmeasurementsandis not derivedfrom our epistemologicalstandpoint.

Let usagainbecompletelyclear. Focusingon thesystemwith variablesp q (let uscallit the small one),I canposevia directmeasurements,oneof two questions,eitherwhatis q or p. BeforeI do so, if I like I could have measureda fully separatesystem(to beconsideredanancilla) to answereitherof thesequestions,or I might just intendto do solater. My small system,like anexaminedstudent,canknowneitherwhetherI have donesonor whichI havedone;or intendto dolater. Frommany previousrunsI know, that,asitwere,this studentalwaysgetsthefirst questionright. That impliesthathemustknow theanswerto both. Thatthevery processof questingsoupsetsthestudentthatall subsequentquestionshedoesnt getright, changesnothingon this conclusion.No responsibleschoolprinciplewouldconcludeotherwise,nomatterhow muchhewouldwonderwhatthecauseof thisstrangebehavior is. In any case,it wouldnotoccurto him thatbecausetheexaminercheckedhis notebookthat the studentansweredcorrectly, or even that the notebookwasalteredafter-the-factsoasto conformto thegivenanswer.

Thus,thesmallsystemkeepshandyananswerfor whicheverquestioncomesfirst. Thisanswer-in-reserve cannotbe fix ed by measuringQ on theancillarysystem(in theanal-ogy: that theteacherchecksoneof thequestionsin his notebookandtherebythepage onwhich theanswerto theotherquestionis found,is renderillegible). TheQM practitioner


assertsthat after a Q measurementon the ancilla, my small systemgetsa -function inwhich q is fully sharp,but p is completelydiffuse. But still no fix changedmy smallsystemsothatfor it the p questionholdsa preciseanswerready, andthatit is just thatonefoundearlier.

The situationis really worse. The studenthasan answernot only for the q- and p-question,but alsofor a thousandothersalthoughI haveno ideahow hedoesit. p andq arenot theonly variablesthatcanbemeasured;any combination,e.g.,

p2

q2

corresponds,accordingto QM principles,to a particularmeasurementalso. It turnsout5

that for this combinationalsothereexistsa measurementon theancilla to bemadeto getthesmall answer, namelyonP2 Q2, andhereagaintheanswersis identical.Accordingto generalQM rulestheresultmustbeoneof

h 3h 5h 7h 9h The answerfor the p2 q2 -question(when this is the first measurementto which itis subject)that the small systemhasreadymust be from this series. Likewise for themeasurementof

p2

a2q2 wherea is anarbitrarypositiveconstant.In thiscase,perQM, theresultmustbeoneof

ah 3ah 5ah 7ah For eachvalueof a thereis a new question,andfor eachthesmall systemhasananswerfrom theaboveseriesready(with theappropriatevalueof a).

Now, theastoundingfact is: theseanswerscanbeinconsistentwith eachother! As, letq betheanswer, which for theq-question,p theanswer, which for the p-questionis heldready, thenit is not possiblethat

p 2 a2q 2ah

anoddwholenumberholdsfor any valuesq andp andfor anypositivenumbera . Thisis not justamachinationwith dreamedup numbershaving only formal meaning.Two of themeasuredresultsonecanget,e.g.,q andp , theoneby direct,theotherindirectmeasurement.And thenonecanconvinceoneself(seebelow), thattheaboveexpressionwith q andp andthearbitrarya,is not anoddnumber.

The deficit of insight into the interrelationshipsamongthe alreadyheld answers(us-ing thestudents memorytechnique)is quiteextensive; thegapsdo not arisefrom a newQM algebra.Thedeficit is evenmorestrangein thatonecanprove: the entanglementisalreadyuniquelyspecifiedby the requirementsby q Q and p P. If we know thatthecoordinatesareequalandthe momentumopposed,thenthereexistsa particular andunambiguousquantumattributionfor all possiblemeasurements.For anymeasurementonthe small system,onecanget a correspondingoneon the large system,andeachsuchlattermeasurementorientssimultaneouslyon resultsthata particularmeasurementon thesmallyielded.(Naturallyin thesamesenseasabove: only avirginalmeasurementoneachsystemis valid.) As soonaswe have broughtthetwo systemstogether, suchthat (brieflysaid)thentheirpositionandmomentumtakeonacorrespondenceanddoall othervariablesalso.

5E. SCHRODINGER, Proc. Camb. Phil. Soc.(in press).


But just how the valuesamongthemselvesof all thesevariableinterrelate,we knownothing,althoughthesystemfor eachmusthave ananswerready, asit canbedeterminedby directmeasurementson theancilla.

Shouldone think, that becausewe know nothing about the relationshipsamongthevariablesof onesystem,thatnosuchrelationshipsexist,or thatevenarbitrarycombinationscanarise?Thatwould meanthatsucha systemwith onedegreeof freedomusesnot justtwo variables,asdoesclassicalmechanics,for its specification,but many more,perhapsinfinity many. But thenit would be remarkable,that two systemsagreealwaysover allvariables,if they agreeon any two. Onemight think that this is sobecauseof our lack offinesse,thatwe areunableto bring two systemstogetherasa singleonewith agreementregardingtwo variableswithout nolensvolensgettingagreementfor the othervariables,althoughit wouldnot in itself benecessary. Thesetwo assumptionsmustbemadein ordernotto experienceembarrassmentfor thedeficitoninsightinto theinterrelationshipsamongthevariableswithin a system.

14. TIME VARIATION OF ENTANGLEMENT; STATUS OF TIME

It is perhapsnot idle to recall that all in 12 and13 pertainto a single instant. Butentanglementis not timeconstant.It persistsuniquelyasanentanglementof all variables,but its attribution amongthemchanges.That is to say, at a later time t onecansurelylearnby measurementon theancillathemomentaryvalueof q or p, but themeasurementsto do so on the ancilla aredifferent. What they shouldbe is easilyseenin simplecases.It dependsnaturallyon whatever forcesarein play betweenthe systems.Let us assumethat thereareno suchforces.Themasses,for simplicitys sake, aresetequalanddenotedm. Thusin a classicalmodelthemomentap andP remainconstant,asthey arejust masstimesvelocity;and,thecoordinatesat time t (distinguishedwith subscript t) (qt Qt ) canbefoundfrom theinitial onesby

qt q pt mQt Q Pt m

Consideringfirst thesmallsystem,themostnaturalwayto describeclassicallyit attimet isbygiving coordinatesandmomentaat thistimebyqt andp. But, therearealsoalternatives.Insteadof qt one,in analogyto personaldata,cangive q correspondingto age(48 in mycase),or qt correspondingto birth date(1887in my case).Now, from aboveonegets:

q qt pmt;andlikewisefor thelargesystem.Thus,for specificationswemaytake

for smallsystem qt pt m and pfor largesystem Qt Pt m and P

Theadvantageis thatbetweenthese,entanglementremainsinvariant,namely:

qt pt m Qt Pt mp P;

orqt Qt 2tP m

p PWhat changeswith time thenis that the coordinatesfor the small systemcannot be ob-tainedasa coordinatemeasurementon the largesystem,but ratherby a measurementof


theaggregate

Qt 2tmPOneshouldnot imaginethatonemeasuresQt andP separately;thatdoesnotwork. Ratheronemust,as is alwaysthe casein QM, recall that thereis a direct measurementfor theaggregate.Otherwise,all thatwassaidin 12and13still holds,for eachmomentthereis theoneuniqueentanglementof all variableswith theirusualconsequences.

It is preciselythesamewhenforcesbetweenthesystemsarein play, but thenq andpareentangledwith variablesmorecomplicatedthatjust sumsof Qt andP.

This by way of preparationfor thefollowing. Time variationof entanglementis causefor pause.Mustall consideredmeasurementstranspireinstantaneouslyto procuretheir in-evitableconsequences?Cantheexotic conclusionsbeprecludedby calling on thefinitudeof the durationof measurement?Not at all. Oneneedsfor eachsingleexperimentonlyonemeasurement;only thevirginal oneis admissible,othersaresimply besidethepoint.Measurementdurationcanbeignoredbecausenofollow-onmeasurementis intended.Thetwo virginal measurementsmustbesoexecutablethat they give valuesfor thesamepres-electedinstant, known in advancebecausewe mustmeasurea pair known to beentangledat thechoseninstant.

-Is it alwayspossibleto do so?=Maybenot. I suspectnot in fact. Still, at presentQM must require it; QM is so

formulatedthat its forecastsaremadefor a particularinstant. As theseforecastsconcernnumericalresults,they wouldbepointlesswerethemeasurementsnotdoableataparticularinstant,regardlessof how longor shorttheprocedureitself takes.

Justwhentheexperimenterfinally learnsconcretelywhattheresultis, makesnodiffer-ence,however. Theoreticallyit is asimmaterialasthefactthatit takesamonthto integratetheequationspredictingtheweatherfor asingleday. Thefarfetchedexampleof theexam-inationactually, while relevantin spirit, in factis not faultlesslygermane.Theexpressionthe systemknows maynot meanthattheresultderivedfrom aninstantaneousevent,butactuallycamefrom a successionspreadover anextendedinterval. But this neednot con-cernusso long asthe systemsomehow revealedthe resultwithout externalintervention,exceptthat (via experimentalprocedures)it is told which questionis to be answeredandwhentheresultcanbeattributedto a particularinstant,which perQM, for betteror worsemustbeassumedor theforecastrenderedempty.

Herein thisdiscussionwestumbleonanew possibility, namely, if it makessensethataquantumforecastrelatesneveror seldomto a sharpinstant,thenit neednot berequiredofthenumericalresultseither. This,astheentangledvariablesexchangeplacesin thecourseof time,wouldsubstantiallyimpedeemergenceof anantinomy.

That a temporallysharpforecastis a mistake, is probablefor otherreasonsalso. Anynumericaltime readingis alsoanobservationalresult. May oneadmitexceptionalstatusfor aclock reading?Shouldit not,aswith all others,relateto a variable,having in generalnosharpvalueand,in any case,simultaneouslycannothaveonewith everyothervariable?If forecastsof the valuefor someothervariableat a particularinstant,needonenot fearthatneithercanbeknown sharply?Within QM, this issuecannotbeinvestigated,becausetime is considereda priori aspreciselyknown, althoughit mustbekept in mind thatanysortof clock-readingdisturbstheclock itself to someuncontrollabledegree.

I emphasize,we do not now have a formulationof QM from which forecastsdo notpertainto a particularinstant. It seemsto me that this deficit is mademanifestby theantinomynotedabove. By which I do not want to say, however, that this is the onlymanifesteddeficit.


15. A LAW OF NATURE OR JUST AN ALGORITHM

That sharptime is an inconsistency within QM, andindependentlythat the specialstatusof time encumberstheapplicationof theprincipleof relativity to QM, arepointsIhave emphasizedrepeatedlyin recenttimes,regrettablywithout beingableto suggesttheshadow of a remedy.6 Theoverview of thewholesituation,asattemptedherein,leadstoa remarkof anothersortpertainingto thevoraciouslysought,but still not truly achieved,relativization of QM.

The contrivedtheoryof measurement,the apparentjumpsof a -function andfinallythe antinomy of entanglementall originateby causeof the simplicity of the meansbywhich the quantumcalculusallows two separatesystemsto meld into a unity, for whichit seemsartificially tailored. Whentwo systemsenterinto interaction,it is not their -functionsthatenterinto interaction,ratherthesefunctionsimmediatelyceaseto exist anda new onefor thecombinedsystemreified. It consists,briefly put, to begin simply astheproductof thetwo individualfunctions;for whichonedependsonquitedifferentvariationsthandoestheother, andit is a functionof all thesevariationsin a region of muchhigherdimensionalitythanfor theindividual functions.As soonasthesystemsstartto influenceeachother, thetotalfunctionceasesto beaproduct;andlaterwhenthesubsystemsseparateit doesnotsplit upinto thefactorsthatcanbeattributedto theindividualsubsystems.Thus,asprecursor(until the entanglementis resolved by observation)onehasonly a commondescriptionof the two in the higherdimensionalzone. That is the reasonknowledgeoftheindividualsystemsis reducedto aminimum,eventotally destroyed,while thecommonknowledgeremainsmaximal. Thebestpossibleknowledgeof thecombinedsystemdoesnot includethebestpossibleknowledgeof thepartsandthatis thewholemystery.

Whoever reflectson it, must thoughtfully cogitateover the following fact. The con-ceptualmeldingof two or moresystemsinto oneencountersgreatdifficultiesassoonasoneseeksto introduceSpecialRelativity in QM. Thesingleelectronproblemwassolvedsevenyearsagoby P. A. M. DIRAC7 in anastoundinglysimplemanner. A seriesof exper-imentalverificationsornamentedwith thetechnicaljargonelectronspin,pair production,positron, etc., leave no room for doubtin the rectitudeof his solution. But still it stepssmartlyaway from the paradigmaticschemeof QM (which I tried to describeherein)8;moreover, oneencounterssever problemsas soonasan attemptis made,following theexampleof classicaltheory, to developa relativistic many-bodyelectrontheory. (That asolutionto this problemmustbeoutsidetheconventionalscheme,is revealedby the factthat this problemexists alreadyfor the combinationof the simplestsort of subsystems.)In this effort I presumeto make no evaluationof attemptsin this directionthathave beenmade9. That they have hadthe lastword I doubt,however, becauseeventhey themselvesdonot makesucha claim.

Thesituationwith respectto theelectromagneticfield is justasproblematic.Its lawsareindeedtheveryembodimentof relativity; anonrelativistic treatmentis simplyimpossible

6E. SCHRODINGER, Berl. Ber. (16Ap 1931);Ann.delInstitue H. POINCARE, p. 269(1931);CursosdelauniversidadinternationaldeveranoenSantanderI (Signo,Madrid,1935)p. 60.

7P. A. M. DIRAC, Proc. Roy. Soc.Lond.A 117, 610(1928).8P. M. A. DIRAC, Theprinciplesof quantummechanics, (ClarendonPress,Oxford,1930)p. 239;and2nd

Ed. (1935),p. 252.9Someof the more importantcitations: G. BREIT, Phys. Rev. 34, 553 (1929) and 37, 616 (1932); C.

MLLER, Z. Phys. 70, 786 (1931); P. M. A. DIRAC, Proc. Roy. Soc. Lond. A 136, 453 (1932)andProc.Cambridge Phil. Soc., 30, 150(1934);R. PEIERLS, Proc. Roy. Soc.Lond.,146, 420(1934); W. HEISENBERG,Z. Phys.90, 209(1934).


in spiteof thehistoricalfactthatthisfield, in its classicalavatarasheatradiation,providedtheimpetusfor QM, i.e., it wasthefirst systemto bequantized. Thiswasachievedwithsimplemeansbecausephotons,aslight atoms, do not interactwith eachother10, ratheronly with theintermediaryof electriccharge. Thus,we still do not have a really faultlessquantumtheoryof theelectromagneticfield11 Onegetquite far with theconstructionof acompositesystem(DIRACs theoryof light12), but not to thegoal.

Perhapsthe elementarymethodsofferedby the nonrelativistic quantumtheoryarenomorethanconvenientalgorithms,which having deepinfluence,have stronglyprejudicedourwholeapproachto nature.Acknowledgement. I warmly thankthe ImperialChemicalIndustriesLtd. for supportforthiswork.

Translatedby A. F. KRACKLAUER c

2006

OXFORD

10This may be only approximatelytrue. See: M. BORN and I . INFELD, Proc. Roy. Soc. A, 144, 425and147, ibid., 522 (1934); ibid., 150, 141 (1935). This is the most recentattemptat formulating QuantumElectrodynamics.

11Somefurther importantcitations,someof which belongto the lastbut onefootnote. P. JORDAN andW.PAULI, Z. Phys.47, 151(1928);W. HEISENBERG andW. PAULI, Z. Phys.56, 1 (1929),ivid. 59, 168(1930);P. A. M. DIRAC, V. A. FOCK andB. PODOLSKY, Phys. Z. der Sov. Uni. 6, 468 (1932); N. BOHR and L.ROSENFELD, Danske VidenskaberneSelskab. math-phys.Mitt. 12, 87 (1933).

12An excellentcitation: E. FERMI, Rev. Mod. Phys.4, 87 (1932).

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