After the corpuscleinterpretationhasbeenestablished asanadmissibleinter- pretation, wecan ask the questionofthe consequencesof thisinterpretation.
Itis,in particular,the questionof causalitywhichinterestsus.We sawthatin the classical-statistical conception there are causal laws which connect the states g,p with future states, and with other entities u. Are there-similar laws introduced with ourcorpuscle interpretation?
Ashortanalysisshowsthatthisisnot thecase. Let usassumea sequenceof values corresponding to (3), 25, in the form
q* mq qi pk mp pk (1)
and letu be an entity which isneither commutative with q nor with p. We
then have no way of measuring u between the measurements mq and mp
without disturbing the situation. We can only introduce a probability
Accordingto(9), 25, thisprobabilityisequalto
P(mq.qi.pklum) = P(m
q.qi,um) = P(m
q.qi.mu,um) (2)
Thismeans thatudoes notdependonthe valueofpexisting simultaneously with qitandthat,ontheother hand, thedependenceofuonqissoconstructed that to one valueq there iscoordinated a spectrumof possiblevalues u. We
thereforecannotintroduce acausalfunction
(3) such that the statistical relations of quantum mechanics are reduced to the
classical-statistical case.
Thelattercase isconstructed in such away that itpossesses causalchains which establish the connection between d(q) and d(p) on the one hand and
d(u) onthe other hand. Our considerations therefore show that such causal chains cannot be constructed in the interpretation of definitions 1 and 2.
Generalizing ourinvestigation inthedirectionindicatedattheendof 24,we
shall ask whether in the interpretation given by these definitions at least probabilitychainscan beconstructed. The meaningof thistermmust nowbe definedprecisely.
For this purpose let us consider a structure in which a relative probability function
d(q,p;u) (4)
takesthe place ofthe causal function (3) and establishesthe connection be- tween thedistributions d(q) andd(p) onthe onehand, andd(u) onthe other hand. Relative probability functions correspond to relative probabilities;
123 theyare integrated only over thevariable followingthe semicolonanddeter- minethe probability ofthisvariable relative tovaluesofthevariablesbefore thesemicolon. Thesefunctionsareageneralization of causallawsintoproba- bility laws.1 We shall say that the interpretation so introduced is given in termsof probability chainsif the function (4) isindependentof the functions d(q) andd(p). The termchain structure therefore characterizesa structure in whichtheprobability ofa valueuisdeterminedifthe valuesqand pare given.
We can interpret this feature as meaning that the connection between d(q) andd(p) on the one side, and d(u) on the other side, goes through the values qandp.
Our results show that we cannot even construct probability chains in the sense defined.Becauseof (2),the function (4) degeneratesintoa function
d(q;u) (5)
of q alone. Whereas, in this case, the value ofp has no influence on u, it is differentafterameasurementofphasbeenmade; wethenhaveaprobability
function ,, N //%x
) (6)
which shows u to be independent of q. Thismeans thatthe probability ofa valueu, inthe giveninterpretation, depends sometimes onqalone, and some- timeson p alone.In thefirstcase, therefore,thevalueofp, whichwe assumed
asexisting,thoughnot being measured, hasnoinfluenceonu;whereasinthe secondcasethe valueof qhasnoinfluenceonu.Nowthetwocasesdifferwith respect to the probability distributions d(q) and d(p), since in the first case d(q) isaconcentrateddistribution,andd(p) isnot,whereasinthesecondcase the converse holds. We thus find that the dependence of u on q or p varies with thedistributions d(q) andd(p);we therefore do nothave a function (4) which is independent of d(q) and d(p). In consideration of the requirement statedabove, theexistingrelations,therefore,cannot beinterpreted asa chain structure.
Thisresult is made clearerifwe startfrom a general situations. We then have, accordingto definition2,
d(?,p;i) = d.(u) (7)
where ds(u) isthedistribution ofurelativetothesituations. Thismeansthat theprobability ofa valueudoes notdepend ontheparticularcombinationof values q,p, nor on either of these values. Instead, it depends directly on s,
and with this on d(q) and d(p). The fact that these latter distributions, although they determine thesituations toa great extent, do not completely determine s (cf. 20),hasno influenceon our result;it proves onlythat, be- sides d(q) andd(p),there are further factors determining s. We therefore do nothave a chainstructure in thesense defined.
1Cf.the author's Wahrscheinlichkeitslehre (Leiden,1935), 44.
124
The questionariseswhetherthisnegativeresultisconfinedtotheinterpre- tation of definitions 1and2.Weshallshowthatthisisnot thecase, andthat alsowithotherdefinitions ofthe valuesofunobservedentitieswecannotintro- duce a chain structure. In order to show this we shall proceed by steps, choosingmore and moregeneralformsofdefinitions.
We begin with the following set of definitions. We leave definition 1 un- changed,butretain definition2 onlyfor qandp, whereasotherentitiesuare considered as being dependent on p and qeven when u is notobserved. We
ask whether this dependence of u on p and q can be so constructed-that a function (4) exists. If this interpretationistorepresenta chainstructure, we must demand, as explained above, that the same function (4) holds for all possible physical situations s. Now it can be shownthat it is impossible to defineafunction (4) ofthe required property. Thisisprovedbythe following considerations.
Letd(p)beadistributioncompatible with a rather exactmeasurementofg;
thus, d(p) willrepresentaflatcurve. Wedividetheaxisqintosmallintervals
Ag correspondingtothegreatestexactnesswithwhichqcan bemeasuredwhen phas thedistributiond(p);let
<frbethemeanvalueofanintervalA<?i.Further-
more,let di(q) be adistributionwhichispractically entirelywithin theinterval
Ag; when wehave suchadistributionwecan saywithpracticalcertainty that the valueof qis#. A^-functionwhichunitesthe twodistributionsd(p) and
di(q)maybewritten^(q).Since ourdefinitionleavesdi(q)openwithin thelimits oftheinterval Agt-(andbesides forthe reasons explainedin 20),thereis,not one,butaclassoffunctions\l/i(q) satisfyingthiscondition. Letusassumethat arulewhichdeterminesone^-functionofthis classforeveryqihasbeenintro- duced;thisfunctionmaybewritten ^(q^q).Thefunctiondl(q) =
\^(q^q)\
2then
isapproximatelygivenbya Diracfunction. Theclassof situationscharacter- izedbythe functions$(qi,q),foranyintervalA<ft,maybecalledtheclassA.
If, in asituation ofthe classA, we measure an entity u notcommutative withq,the probabilityofobtainingavalueuisdeterminedbythecoefficients
<roftheexpansion
(8)
Since ^(q^q) representsameasurementofqwiththeresult qiywecanwritethe probabilityunderconsiderationintheform
P(mq.qi.mu ,u) = |er(#,w)|2
(9)
Applyingdefinition1 intheform (1), 25,wecan droptheterm muontheleft
handside,andthusarriveat
(10)
125 Usingthe d-symbol,we can writethelefthandside inthe formdi(qi*,u)9 and
thereforehaveforallsystemsoftheclassA:
*(;*)= kCfetOI1 (11)
Nowletus considera classB ofsystems which allpossessthe distribution d(p) previouslyused,but which haveanyofthedistributions d(q) compatible withd (p).Thisclass will includethesystemsofclassA, butwillfurthermore includesystems withratherflat distributionsd(q).For each systemofclassB
we haVe, according to the rule of elimination holding for probabilities, the
relation: ^
dB(q;u)
=JdB(q;p) d(q,p;u)dp (12)
Here we have given no subscript B to the second term under the integral because this function, introduced in (4), is supposed to be the same for all situations.Nowapplyingdefinition2, 25,toqandp,andusing thedefinition ofclassBywehave
dB(q;p) = dB(p) = do(p)
(13)
Wethereforeinfer
\u) = /d(p) - d(q,p;u)dp (14)
X
Thismeansthat thedistributiondB(q]u)isthesameforallsystemsoftheclass Bjsincetherighthandside of (14)has thesamevalueforallsystemsofB. We
can nowdetermine the function dB(q]u) as follows. Since the systemsofthe
classAbelongtotheclass5,wehave, usinganyoftheintervalsA<ft:
dB(qi]u) =
di(qi\u) =
\<r(qi,u)\* (15)
Since this relation holds for all i, we can omit the subscript i from gt, and
thusarriveat , . >. , f >.,2 ,1fiN
dB(q\u) = |cr(g,w)|2
(16)
NowthemeaningofdB(q\u)isgivenby
P(*B.q,u) = dB
(q;u) (17)
whereSBrepresentsanysituation oftheclassB. Using(9)wederivenow from
(16)and(17)therelation
P(sB.q,u) = P(m
q.q.mu,u) (18)
Applying(1), 25,tothelefthandsideand (22), 22, totherighthandside, wecanwritethis intheform:
P(sB.q,u) = P(sB.q.mv
,u) =P(sB.mq.q.mu,u) = P(sB.m
q.q,u) (19)
when weapply(1), 25,oncemoreinthelaststep.Usingconsiderationssuch
126
as presentedin 22, wecanshowthatthisresultleads tocontradictionswhen weuse therule of elimination. When weapplythelatterruleformallyto the term onthelefthandside ineachofthe followingrelations,wehave
P(sB.q,u)dq (20a)
P(sB-mq,u) = IP(sB.mq,q) P(8B.m
q.q,u)dq
^
(206)
Now the first terms under the integrals are equal because of (1), 25; the equalityofthesecondtermsisstatedin (19). Wetherefore derive, using (1),
25,forthesecondline:
P(8B,u) = P(8B.m
q,u) (21a)
P(sB-mu,ii) = P(sB.m
q.mu,u) (216)
The latter equation contradicts the inequality (18), 22, when we replace,
in thelatterrelation,the expressionmu.UibySB,thetermw byu,andtheterm
vbyq. Moreprecisely: Forall those entitiesu for which the inequality (18), 22,holds, (216) willbefalse. Itfollowsthat there areentitiesusuchthat it isimpossibletouse thesamefunctiond(q,p;u) forallsituations ofthe classB.
We may try to avoid the contradiction by abandoning definition 2, 25, which we omitted already for w, also for the entities q and p. In this case qand parenolongerindependentofeachother,and wehavea functionrf(g;p) suchthat
(22) Since p isnowdependent on#, therequirement of a chain structure must beappliedtothefunctiond(q\p).ThismeansthatforallsystemsoftheclassB
the functionds(q',p) isthesame.But then (12) leads, as before, to theresult thatds(q]u)isthesameforallsituations oftheclassB; andwiththisthesame contradictionsasbeforeare derivable.
Itmay bequestionedwhetherwe are entitled toextend the postulate ofa chainstructure tothefunctiond(q;p).Althoughthepiinthiscase aredepend- ent on the qt, one might argue that this is not to be regarded as a kind of causaldependence;i.e., thisdependenceshould notbeconstruedasa physical law, but as being due to accidental conditions determining the probability situationandvaryingfromcasetocase. Theg andp-should, rather, becon- ceived astheindependent parametersofphysical occurrencesinthe sense that they can bearbitrarily chosen, andcausaldependence inthe senseofa chain structureshouldbeassumedto hold only forother entitiesu withrespectto the g andp^ Ifthisobjectionismaintained,weargueas follows.Ifthe proba- bility situation variesfromcase to case, all kindsof situationsshould occur;
mathematically speaking, this means that the three probabilities d(q), d(p),
d(q;p), represent the fundamentalprobabilitieswhich can be assumed arbi- trarily.Wethereforecanmakethefollowingassumption.
AssumptionF: Amongthe systemsof class Bthere isa subclassB' of sys- tems havingthesamedistributiond (q',p)jandforevery ^-functioninBthere
isanumberofsystemshavingthis^-functionandbelongingtoB',i.e.,having atthesametime thedistributionsd (p)andd(q;p).
Thismeansthatwecancoordinateadistributiond(q\p) toasituationhav- ingtherdistributions d(q) and d (p), and that there must be systems having these three distributions. It is true that we have no means of determining observationally the lattersystems, i.e., thesystems ofthe classB'. Assump-
tionThasthereforethe characterofa convention.Butshoulditbeimpossible to carrythrough this convention, i.e., should thepossibility of definingsuch systems beexcluded, thiswouldmeanthat theprobability d(q\p) dependson the distribution d(q), since thedistribution d (p) is the same for allsystems ofB. In other words,thiswouldmeanthat the relative probability ofhaving avaluep whena value qisgivendepends onthe distribution ofthevaluesq.
Theprobabilitythataparticle attheplace qhas themomentumpthenwould depend on wherethe otherparticles are situated. Thiswouldnot only contra- dictourassumptionaccordingtowhichtheqandprepresent theindependent parameters, butwouldintroducea kindofdependenceoftheponthe qwhich contradicts the requirements of a chain structure. Using assumption F, we nowwrite(12)fortheclass 5',intheform:
7;^) =
dB'(q',u) = Ja
JdB'(q;p) d(q)p'Ju)dp (23)
Sinceds'(q]p) isnowthe sameforallsystemsofB1',namely, = d(q]p), itfol- lows thatalsodB'(q\u} isthe same for all systemsof J3'. With thisresult the samecontradictions as beforeare derivable, since, accordingtoassumptionF, B' includes a correspondingsubclass A' ofA, namely, a class A! of systems with functions$(qi,q).
Wehavecarriedthroughthisconsiderationforcontinuousvariables qand p because position and momentum are, in general, represented by variables of this kind. Since a sharp measurement of continuous variables is impossible, our result must be stated, more precisely, in the form: However small the intervals Agare chosen, it is alwayspossible to find anentityu forwhichthe numericalcontradictions arising from (21b) are sufficiently large. We do not wanttosaybyour proof that the classesA,A', B, B1 mustremainthesame
forsmaller A; this isobviously not thecase. Astatement, therefore, of what happensincase A = isnotderivable;but suchstatementsare inanyevent beyond the limits of assertability in quantummechanics. If we were touse discrete variables <frandpk,suchstatements could bemade; butin this case our proofcan begiveninasimplerway.Ifqisexactlymeasuredasbeing =q^
wehave afunctiond(qi'9pk);theassumptionthatina generalsituations,p is
independentofg,i.e., thatda(qi;pk) = da(pk),thencontradictsourdefinitionof achainstructure, sincethend8(qi',pk) dependsond(q)andd(p),namely,isdif- ferentaccording assrepresents,ordoes notrepresent,ameasurementofq.If,
onthe otherhand,weput d9(qi*,pk) = dfe
;p*)>wecanconstruct,using(12)with the subscriptsinsteadofB, thesamecontradictionsasabove.Forcontinuous variablesqand pthelogicalsituationisdifferent,because here sharp valuesdo notexist,andforeverymeasurementofq within the exactnessA wecan assume thatd(qi',pk) =
d(pk),withoutarriving atcontradictionsifonlyd(pk) iswithin thelimitsgivenbytheHeisenbergrelation. Forthiscasewethereforeneed a special proof showing that contradictions arise for other entities u, as given above.
We now mustaskwhether ournegative resultcanbeevadedbytheaban- donmentof definition 1, 25. In ordertoanswerthisquestionwe mustenter intosomegeneral considerationsconcerning adefinitionofthe valuesofunob- servedentities.
Ifoursystemincludednodefinitionofunobservedentities, itwouldbeeasy to carry through a chain structure. We then would assume that there were unknown values qand pwhich were connected with theunknownvalue uin suchawaythatafunction (4)existed;weevencouldassumethatthisfunction was degenerated into a causal function (3), 24. Furthermore, we could give arbitrary values to the unobservedentities for each experiment in such awaythat the functions(4), 24,or(3), 24,aresatisfied.Thechosenvalues could neverbeshown tobeinvalidifonly weassumethat everyobservation disturbs the values in an unknown way. Such an interpretation, however, cannot be called a permissible supplementation of observations because the unobservedvalues are not connected with the observed values by means of generalrules. It thereforemustbeconsidered asarequirementofan interpo- lated chain structure that there be general rules which determine the un- observedvalues as functions oftheobserved values, and that these rules be the same for every situation s. The latter qualificationis necessary because otherwise the disturbancebythe observationwould dependonthe probability distributions ofthemeasuredentities;in this case, the disturbance, which in itselfrepresentsacausaloccurrence,wouldnot conformtoourdefinition of a chainstructure.
Now we knowthatinasituationswecanmakeonlyoneobservation,either ofq,orofp, orof u;afurtherobservationthenwillstartfromanewsituation createdbythe preceding observation. Therequiredrulesthereforemustbeso given that they determine the unobservedvalue of any entity asa function oftheobservedvaluealone. Letu betheobservedvalueofanentity;thenwe can introduce two functions fb(u) and fa(u) which determine the values beforeand afterthemeasurement, so thatwehave
Ub= fb(Uo) Ua =
/a(tto) (24)
27- WAVE 1*9 Nowthisassumptionleads tothefollowingdifficulties.When wehave asitua- tionwhich is definitein u, such as exists after a measurement mu with the result u , and thenmake a measurement of u, the value afterthis measure- ment mustbethesameasbeforethismeasurement. Otherwise the valueofu would change between the two measurements of u\ this would be a change without cause, contradicting the principle of strict causality as well as the principle of a probabilitychainstructure. For such a situationthe twofunc- tions/6and famustthereforebe the same. Butthenthesetwofunctionscannot be different in any other situation. Since suchadifferent situation isdistin- guishedfromthe previouscaseonlybyadifference intheprobability distribu- tions, anassumption stating that the twofunctions/& and fadepend on the situation s would mean that the disturbance by the measurement depends, not onlyonthe valuesoftheentities,butalsoontheprobabilitieswithwhich these values occur. This is the case we excluded above as contradicting the requirementsofa chainstructure.
Itfollowsthatforallsituationswe musthave
/6=/a (25)
Butthen the generalization introducedbythe function/istrivial, anddoes not alter our previous results. Instead of considering the observed value uc
asthe value beforeandafterthemeasurement, wethenwouldregard a value f(uo) as the value beforeand after the measurement; andthe preceding con- siderationsleadingtocontradictionscouldthenbecarriedthroughinthesame wayas before.
The considerations given representa general proof that itisimpossible to interpret the statistical relations of quantum mechanics in terms of a chain
structure. This includes a proof that these relationscannotbeinterpreted in termsofcausal chains, sincethelatterconstituteaspecial case ofprobability chains. Withthese results the proofof the principle ofanomaly, towhichwe
referred in 8,isgiven.
The demonstrations of this section are illustrated by the analysis of the interference experiment which we discussed in 7. We saw there that we cannotintroduce a probabilityP(A.Bi,C) whichisindependentof the occur- rences at the place B%. This is a special case of our general theorem stating that we cannot introduce an invariant function (4); the function (4) will
alwaysdependonthewholedistributions d(q) andd(p)yi.e.,onthesituations.