The above analysis shows that neither the corpuscle interpretation nor the waveinterpretation can be carriedthrough without causal anomalies. Using theparticle interpretationwecanexplainsomeexperimentsinsuchawaythat
the laws of phenomena and interphenomena are the same; but then we en- counter anomaliesinothercases.Usingthewaveinterpretationwecanexplain these other cases in such a way that the laws of phenomena and interphe-
nomenaarethesame; but then anomaliesappearinthe explanationof experi- mentsofthefirstkind. Finally,a combinationofthetwointoaninterpretation ofpilotwaves showsother anomalies.
The question arises whether there is another interpretation, perhaps un- knowntous,whichisfreefromcausal anomalies.Theprecedinginvestigations cannot be considered as a proof that thereis no such interpretation. Such a proofcannotbegivenbytryingoutoneinterpretation afteranother;wethen are never sure whether a better interpretation which escaped our attention remains.Theproofmustbebasedona general theoryoftherelationsbetween quantummechanicalentities. Weshallgivethisproofin 26;the conception of causalityassumed for it, whichtakes account of possiblemodifications of this notion,willbeexplainedattheendof 24.Ourresultscan be formulated as follows: It isimpossible to give a definition of interphenomena in such a
9Thisfollowsfromconsiderations similar tothoseindicatedinfn. 3,p. 28.
10Theanomaliesofthis interpretation areveryclearlypresentedinL.deBroglie'sbook, Introductiond V EtudedelaMecaniqueondulatoire(Paris, 1930),chap.9.
33 way that the postulates of causality are satisfied. Theclassof descriptions of interphenomenacontainsno normalsystem. This canbe proved tobe a conse- quenceofthebasic principles ofquantummechanics.We shall call thisresult theprinciple ofanomaly.
In view of this negative result two different conceptions can be carried through. The first calls for a duality of interpretations. Among the class of
equivalent descriptions we have two, the corpuscle interpretation and the waveinterpretation,whicharemoreexpedientthantheothers; sincewehave no normalsystem, wecanuse,instead, either ofthesetwointerpretations asa
minimumsystem,i.e., asystemforwhichthe deviationsfromanormal system constituteaminimum.Inthisconceptioncausalanomaliescannot beavoided;
buttheycanatleastbe reducedtoaminimum.
Thesecond conceptionrepresents a moreradical remedy. Sinceno normal description ofinterphenomenaexists,ithasbeensuggestedweshouldrenounce anydescription ofinterphenomena;weshouldrestrictquantummechanicsto
statementsabout phenomena thennodifficultiesof causalitywill arise. The
impossibility ofanormal systemis construed, in thisconception, as areason forabandoningalldescriptions ofinterphenomena.Weshall callconceptionsof this kind restrictive interpretations of quantummechanics, since they restrict the assertions of quantum mechanics to statements about phenomena. The
ruleexpressing thisrestrictioncan assumevarious forms, and we shallthere- forehave severalrestrictiveinterpretations. Interpretationswhich donot use restrictions, like the corpuscle and the wave interpretation, will be called exhaustive interpretations, since they include a complete description of inter- phenomena.
Theadherentsofrestrictiveinterpretationshave maintainedthatadescrip- tion of interphenomena is unnecessary; for the purpose of observational predictions,theysay,it issufficienttohaveaninterpretationwhichrefersonly to phenomena. The latter statement is true; but it cannot be considered as proof that exhaustive descriptions should be abandoned. We should clearly keep in mind that neither of the two conceptions can be proved to betrue.
Theseconceptions representvolitionaldecisionsconcerning theformofphysics;
either ofthemisasjustifiableastheother.
Speakinginterms ofthe classofequivalent descriptions, thesituation can be characterized as follows. The system of phenomena is the samefor each description ofthis class;it isthereforetheinvariant ofthis class.Nowtheclass
is dependent on its invariant; so far, any restrictive interpretation deter- minesthewholeclassofexhaustivedescriptions.Thelatterdescriptions,how-
ever, reveal afeaturewhichwewouldnotknowifwe knewonlyarestrictive description:thisisthefact thatnointerpretation free fromcausalanomalies can begiven.Sincethisisapropertyoftheclassofexhaustivedescriptions, it representsaninherent propertyofeveryrestrictiveinterpretation.This prop- ertyisexpressedintherestrictiveinterpretations throughthe factthat they
34
ruleoutcertainstatements;butthe reasonfor this rulecanonlybe formulated intermsofastatementabouttheproperties oftheclassofexhaustivedescrip- tions.
Wethereforeshall turnnowtoafurther analysis ofthe classof exhaustive interpretations,whilewepostponethediscussion ofrestrictiveinterpretations.
Withinthe firstclass, we said, thetwointerpretations interms of corpuscles
and waveshold aspecial position so far astheyrepresentminimum systems.
Tothiswe now must addasecondstatement whichsecuresauniqueposition to thesetwointerpretations,andwhichatthesametime attenuatesthe*conse- quencesresultingfromthe absenceofanormalsystem.
Althoughwehavenoexhaustivedescription freefromanomalies holdingfor
all interphenomena, we can construct such a description for every interphe-
nomenon byusing either the wave or the corpuscle interpretation. It is this fact whichwe expressin speakingofthe duality ofwave and corpuscle inter- pretation. We mean bythisthatforagivenexperimentatleastoneofthetwo
willbe a normaldescriptionandwillthusdefineinterphenomenainsuch away
that theyfollow the same laws as the phenomena; itis onlyin other experi- mentsthat the interpretation so chosenwill lead to causal anomalies. Let us call thisstatement theprinciple of eliminabilityof causal anomalies. Thediffer- encebetween alland every, whichweused toformulate this principle,iswell
known tosymbolic logic. Using thisgrammatical distinctionin another form we may also say: It is false to say that all interphenomena follow the laws holding for phenomena; but it is correct to say that every interphenomenon does so. We do not have one normal systemfor all inter-phenomena, but we do haveanormalsystemfor everyinterphenomenon.
Asbefore,an analogy fromdifferentialgeometrymayillustratetheseformu-
lations.When weuse asystemoforthogonal coordinatesonthesphere,suchas that given in the circles oflongitude and latitude (such asystem is possible becauseitdoes notconsistthroughoutof straightestlines,thecirclesof latitude notbeinggreatestcircles), thissystemhassingularities atthe NorthPoleand the South Pole; i.e., these points do not have a definite longitude. These
singularities, however, are due only to the system of coordinates; the poles themselves are not distinguished geometrically from any other point ofthe sphere. Thesingularities can therefore be "transformed away" by theintro- duction ofanother system of coordinates; thus, the sailor will use, near the poles, a notation of points which determines positions relative to a chosen initial point and two chosen directions rectangular to each other. These co- ordinates couldeven be producedandusedtocover thewholesphere, atleast ifoneset oflinesisnotassumedto consist of straightestlines;but thensingu-
larities will appear in other points of the system. We may call a system of orthogonal coordinateswithout singularities a normalsystem. Then we may
say that we can introduce a normal system for every extended area on the sphere, but wecannot introduceone normal systemfor allareas, i.e., for the
35 whole sphere. We thus express a statement about the sphere in terms of a statement concerning the classof possible systemsof coordinates.
Thedifferencebetweenthiscase andthe caseconsidered above, whichcon- cerns orthogonal straight-line coordinates, is as follows. Asystem of coordi- nates beingbothorthogonalandstraight-linedispossibleonlyfor infinitesimal areas; for extended areas of some size it cannot even be carried through approximatively. Ifwerenouncetherequirementof straightlineswecancon- struct a system whichcovers great areas ofthe sphere, and which is strictly orthogonal; but such a system will lead to singularities in two points. The advantage of this caseover thefirst isthat,withthis definitionofthenormal system, we obtain a normal system for extended areas, not for infinitesimal areasonly.
Returningtoquantummechanics,we mustsay that thesituationthere cor- respondsto the second case. The causal anomalies can be transformed away
strictly for "extended areas", i.e., for a whole experiment, by a suitable
description. Theywillreappear onlyforother experiments or forquestionsin which experiments of different kinds are compared; for the answer to such questions we then can introduce a new description such that once more the anomaliesdisappear.Thereason thatthisisalwayspossibleisgivenin there- lation of indeterminacy. If we could observe a particle passing through the
slitBiintheexperimentof figure5,wecould not introduce awavedescription,
andthereforewouldhave no normaldescription ofthe experiment, i.e., node- scriptionfreefromanomalies.Onthe other hand,ifwecouldprovethatawave arrived simultaneouslyat differentpointsCofthe screenintheexperimentof figure 4, wecould not introduce a corpuscle description, andtherefore would have no normal description of this experiment. We see that the principle of elimindbility ofanomaliesismadepossiblethroughtheprinciple of indeterminacy, since thelatterprinciplemakesitimpossible evertoconstructacrucial experi-
mentbetweenwaveinterpretationandcorpuscleinterpretation.1
Theultimate rootofthe dualityofwaveinterpretationandcorpuscleinter- pretationisthereforegivenintheprinciple ofindeterminacy;butthisprinciple alsopoints thewayoutofthe dilemmaofcausalanomalies, aresultstatedby us inthe principle of eliminability. We spoke above ofthe skilldisplayedby
physicists in applying sometimesthe waveinterpretation and sometimes the corpuscle interpretation; we now see that we can give a justification of this change of interpretations which proves that the switching over to a normal interpretationisalegitimatemeansofphysicalanalysis.Whenthephysicist, in face ofaparticularexperiment, introduces asuitable descriptionwhich elimi- nates causal anomalies within theframeofhisquestion, hemaybecompared
1Itmaybe questionedwhetherit isactually possible in allcases to eliminate causal anomaliesbyasuitable description.Whatcan beshownisthat theprinciple of elimina- bilityholds, at leastforsingleparticles orforswarmsofparticleswhich donotinteract with each other suchas electronswarmsor light rays.Difficultiesariseforcomplicated structurescomposedofseveralparticles. Cf. 27.
36
to thesailorwho,attheNorth Pole, discontinuesdetermininghis position in termsof longitude,andprefers touseanothersystemofcoordinatesfreefrom
singularities. Such a procedure is permissible because nature has not deter-
minedonenormal systemforallinterphenomena, but only a separatenormal systemforeach interphenomenon.
Letus considersomeexamples.Usingthewaveinterpretation,wearriveat the questionwhythewholewavedisappearsafteraflashhasbeenobservedon one point of the screen. We eliminate the causal anomaly presented in this description oftheinterphenomenonbyintroducing theparticleinterpretation.
The wavethen is transformed intoa probability, and, instead of the disap- pearance of a wave, we have the simple statement that although the proba- bilityP(A,C2)of findingaflashonthe screeninapointC2has acertain positive value,theprobabilityP(A.Ci,Cz) of findingaflashinC2afteraflashhasbeen observed inCi, iszero. Insteadof the contractionof a waveinto a point, we haveherethetrivial logicalfactthatprobabilities arerelative. Itwas bycon-
siderations ofthiskind thatBorn wasled tothe introductionofthe statistical interpretation ofthewaves whichoriginallyhadbeenconceivedbySchrodinger aswavesofelectricaldensity.
Another example wherethenormaldescriptionisgivenbytheparticle inter- pretationisrepresentedbythefollowing consideration.Theprobability that a particleleaving the sourceA willpassthrougheither of theslits isthe sumof the probabilities that a particle will pass through one of the slits; we may
indicatethisbythe symbolic expression:
P(A,BiV&) = P(A,BO + P(A,B8 ) (1)
(The logistic sign "
V" means "or".) This relation follows from the rules of probabilitybecausetheparticlescan go throughonlyoneoftheholes atatime.
Itcan betestedbyobservations, as follows:Toascertainthe valueoftheleft handsidewecountallflashesoccurringonthe screenwhenbothslitsareopen;
todetermine thetwovaluesofthe righthandsidewe countallflashes onthe screen occurring, respectively,when oneoftheslits isclosed.Sincestatistics so compiledshowthat (1)holds, thisrelationmustholdalso for thewaveinter- pretation. Here, however, the explanation leadstocausal anomalies.We then must assume that whenever a wave leaves A in the direction of Bi there is
anotherwaveleavingsimultaneouslyinthedirection ofB2, butthat thewave goingtowardtheopenslitBIwill sometimesdisappeardueto aninfluenceof theclosedslitB2(namely, inallthosecaseswhen, inthe corpuscleinterpreta- tion, aparticleisemitted onlyinthedirection ofJ32),andis thuscontrolledby aninfluencewhichrepresentsanactionatadistance.Thephysicistwhowishes to explain the well-confirmed relation (1) will therefore prefer the corpuscle interpretation by the use of which he can derive the relation without the assumptionofanomalies.
On the other hand,there are questionswhich onlybythe use ofthe wave
KINDS 37 nterpretationcan be answered withoutreferenceto anomalies. We sawthat he corpuscle interpretation of the experiment indicated in figure 5, 7,
nvolvesanactionata distance 'between thetwoslitsBIandJB2. Inthewave nterpretationthis actionata distanceiseliminated andreplacedbya state-
nentabout a phaserelationbetweenthewavesarriving inBIand 2,whichis lueto theircommonoriginfromthe sourceA. In answeringquestionsof this dndthephysicistwilltherefore preferthewaveinterpretation.
Our examples show that it is even preferable to speak, not of thenormal
jysteniforeveryinterphenomenon,but, ofthenormal systemforevery question joncerning interphenomena. It is the question which determines the normal
;ystem, and, relative to the same experimental arrangement, different ques- ionscan be asked whichrequiredifferentnormalsystems. Thus the question joncerning the probabilityrelation(1)isasked withrespect toanexperimental irrangement which, for other questions, necessitates a wave interpretation.
SVhen we say "for every question", we mean, of course, that the question is sufficientlylimited,and not constructedasan"and"-combination of different questions. With this qualificationwe can formulate theprinciple of elimina- aility as stating: We have no normal system for all questions concerning interphenomena,butwe do have a normal system for everysuch question.
Theswitching overfrom oneinterpretation toanotherisjustifiable, wesaid,
asameansofeliminating causal anomalies.Thisis,however, itsonlyjustifica- tion, and itwould beincorrect to adducereasons ofanother kind. Wesome- timesread that questionslike"what becomesofthewave aftera flashonthe screenhas been observed" or "why does a particle going throughthe slitBI movedifferentlyaccording as the slitBzisclosedoropen" mustbeforbidden becausetheyare not adequate to therespective interpretation. But thisword
"adequate" means only that the answer to such questions leads to causal anomalies. Theoccurrenceof such anomalies does notmakethe question, or the answer, unreasonable. If we have decided to use one of the exhaustive interpretations, such questions are not meaningless. We then must become accustomed to the fact that for a given interpretation there are always questionswhich canonlybeansweredbythe assumptionofcausal anomalies.
Ifwe prefer tousein theanswer tosuch questions aninterpretation which is freefromanomalies,wehave goodreasonstodoso;butweshouldnotbelieve that theanswer so constructed isthe onlymeaningful answer, or the only ad- missibleanswer, ortheonlytrueanswer.Allthe meritsofsuchaninterpretation consist inthefactthatit isfreefromcausalanomaliesfortheinterphenomenon considered;butneitherdoesthis factmakeitmoretruethanothers, norisita necessary condition of meaning within an exhaustive interpretation. Judg- mentsof thiskindarebasedonaconfusionofexhaustiveandrestrictiveinter- pretations. Onlyfor thelatter will the mentioned questions be meaningless;
but for such an interpretation the complete description of the experiment which is free from anomalies is meaningless as well. Within an exhaustive