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gives no information about this. “We might note here how the usual interpre- tation leads to a paradox in the case of exper- iments with a negative result. Suppose that the particle is charged, and that in the box B2 in Tokyo a device has been installed which en- ables the whole of the charged particle located in the box to be drained off and in so doing to establish an observable localization. Now, if nothing is observed, this negative result will signify that the particle is not in box B2 and it is thus in box B1 in Paris. But this can reasonably signify only one thing: the parti- cle was already in Paris in box B1 prior to FIG. 1: A single particle is confined to the box B, into which the drainage experiment made in Tokyo in a barrier is inserted thus splitting the box and the particle’s box B2. Every other interpretation is absurd. wave function in two. The two half boxes (B1 and B2) are How can we imagine that the simple fact of then separated, at which point the boxes may be opened and having observed nothing in Tokyo has been their contents examined. able to promote the localization of the parti- cle at a distance of many thousands of miles away?”11 Ψ1, is located in B1 and the other, Ψ2, in B2. The wave function is thus of the form Although de Broglie did not make his reasoning ex- Ψ= c Ψ + c Ψ , where c 2 + c 2 = 1. 1 1 2 2 | 1| | 2| plicit, he gives some hints that allow us to plausibly re- “The probability laws of wave mechanics now construct the intended logical structure of this argument tell us that if an experiment is carried out in for incompleteness. The rhetorical question at the end box B1 in Paris, which will enable the pres- implies that de Broglie thinks it is impossible (“absurd”) ence of the particle to be revealed in this box, for the negative result of a measurement in Tokyo to have the probability of this experiment giving a a decisive causal effect on the contents of the box in Paris. 2 positive result is c1 , whilst the probability Likewise for a positive result: “If we show that the par- | | 2 of it giving a negative result is c2 . Accord- ticle is in box B1, it implies simply that it was already ing to the usual interpretation,| | this would there prior to localization,” and therefore, by implication, have the following significance: because the that it was already not in box B2 before the observa- particle is present in the assembly of the two tion. Generalizing, the measurement at B1 cannot have boxes prior to the observable localization, it affected the contents of the box B2, and therefore the would be immediately localized in box B1 in contents of B2 (which are known after inspection of B1) the case of a positive result in Paris. This must have been already definite before that inspection. does not seem to me to be acceptable. The According to at least one commonly held version of only reasonable interpretation appears to me the orthodox interpretation, however, it is the very act to be that prior to the observable localization of observation that disturbs the physical system in ques- in B1, we know that the particle was in one tion and brings about a definite result. Why should de of the two boxes B1 and B2, but we do not Broglie think that the act of opening B1 and examining know in which one, and the probabilities con- its contents can’t affect the contents of B2? Because B2 sidered in the usual wave mechanics are the is spatially distant. Its contents could not be affected by consequence of this partial ignorance. If we any reasonable physical mechanism by something done show that the particle is in box B1, it implies to the well-separated B1. That is evidently the point of simply that it was already there prior to lo- taking one box to Paris and the other to Tokyo before calization. Thus, we now return to the clear opening them. classical concept of probability, which springs There is thus a locality or separability assumption at from our partial ignorance of the true situa- work here. Given that B2 is physically separated from tion. But, if this point of view is accepted, B1 by a great distance, nothing we do to B1 can af- the description of the particle given by the fect the contents of B2. (Indeed, we could consider the customary wave function Ψ, though leading contents of B2 at space-time coordinates that lie outside to a perfectly exact description of probabili- the light cone of the observation event at B1. Then any ties, does not give us a complete description of “disturbance” by the measurement at B1 on B2’s con- the physical reality, because the particle must tents would evidently violate relativity’s prohibition on have been localized prior to the observation superluminal causation.) So when we open B1 and de- which revealed it, and the wave function Ψ termine its contents, we learn immediately whether or 3 not B2 contains the particle. And because B2’s contents can be properly attached, we see that the argumentation cannot have been in any way affected by the opening of of the mentioned authors does not justify their conclu- B1, B2 must have either contained or not contained the sion that quantum-mechanical description is essentially particle all along. And, finally, because “the wave func- incomplete.”12 tion Ψ gives no information about” the particle’s actual Bohr seems to have meant13 that although we do not pre-measurement location (in particular, Ψ attributes no “mechanically” disturb the second particle by measuring definite particle content to B2), that description must the position or momentum of particle 1, we do enact a have been incomplete. decisive choice when we decide which quantity to mea- De Broglie seems to have had this reasoning in mind. sure. For whichever quantity we decide to actually mea- Notably, it is structurally identical to the argument of the sure, we forgo the possibility of subsequently learning more famous EPR paper.8 EPR begins with a 2-particle the value of the complementary variable in the distant wave function particle. Bohr discusses this decisive choice in terms of the need for a “discrimination between different exper- ik(x1−x2−a) ψ(x1, x2)= δ(x1 x2 a)= dk e , (1) imental procedures which allow of the unambiguous use − − Z of complementary classical concepts.”12 Bohr is evidently which is a simultaneous eigenstate of the relative posi- referring to the mutually exclusive experimental arrange- tion operator (ˆx xˆ ) (with eigenvalue a) and of the ments needed to measure the position and momentum of 1 − 2 particle 1. total momentum operator (ˆp1 +p ˆ2) (with eigenvalue 0). The authors also assume the two particles to be suffi- In short, although we can choose to learn about the ciently well-separated that any measurement performed position of particle 2 or the momentum of particle 2, we on particle 1 can have no physical effect on particle 2. cannot with any single experimental arrangement learn They then argue as follows: if the experimenter chooses about both the position and the momentum. And, says to measure the position of particle 1, he will immedi- Bohr, this fact demonstrates the invalidity of EPR’s si- ately be able to infer the position of particle 2; should he multaneous use of the two concepts in regard to par- choose instead to measure the momentum of particle 1, ticle 2: the experimental arrangement which warrants he may immediately infer the momentum of particle 2. use of one concept makes the other simply inapplicable. Therefore, because nothing the experimenter does in the Bohr believed that this “complementarity” perspective region near particle 1 can affect particle 2 in any way, allows one to consistently maintain that the “quantum- both the position and momentum of particle 2 must al- mechanical description of physical phenomena ... fulfill[s] 12 ready have had definite values prior to any measurement. ... all rational demands of completeness.” That is, it Both quantities are “elements of reality” according to the eludes the EPR argument from locality to incomplete- famous EPR criterion of reality: “If, without in any way ness. disturbing a system, we can predict with certainty (that Although widely accepted in the physics community, is, with probability equal to unity) the value of a physical the adequacy of Bohr’s reply to EPR has been questioned quantity, then there exists an element of physical reality many times by those who have carefully scrutinized the corresponding to this physical quantity.”8 issue.14 We will not fully engage this long-standing dis- Having also defined a necessary condition for the com- pute, but we can shed some fresh light on the debate pleteness of a theory (“every element of the physical re- over the validity of EPR’s conclusion by returning to Ein- ality must have a counterpart in the physical theory”8), stein’s Boxes and seeing how this simpler thought exper- EPR concluded that the quantum mechanical description iment fares in escaping what appears to have been the is incomplete because it doesn’t permit simultaneous at- main thrust of Bohr’s response to EPR. tribution of definite position and momentum values to It seems reasonable to infer from de Broglie’s for- particle 2. mulation of the Boxes argument (quoted above) that Niels Bohr is widely believed to have refuted the EPR de Broglie meant to assert the existence of a pre- argument by pointing out an “essential ambiguity” in measurement “element of reality” corresponding to the 12 EPR’s criterion of reality. The alleged ambiguity was contents of (at least) the distant box, B2, and that he contained in EPR’s phrase “without in any way disturb- would argue for its existence on the basis of the EPR cri- ing a system.” In his response, Bohr appears to concede terion of reality: “without in any way disturbing” box B2, EPR’s locality/separability principle when he writes that we can determine whether it contains a complete particle “there is in a case like that just considered no question or nothing. The presence or non-presence of the particle of a mechanical disturbance of the system under inves- in B2 is therefore an element of reality even before the tigation during the last critical stage of the measuring box B1 is opened and its contents examined. Finally, procedure.” However: “even at this stage there is essen- because the initial quantum mechanical wave function Ψ tially the question of an influence on the very conditions attributed neither (definite) state to box B2, there is at which define the possible types of predictions regarding least one element of reality which fails to “have a coun- the future behavior of the system. Because these con- terpart in the physical theory.” And therefore quantum ditions constitute an inherent element of the description theory is incomplete by precisely the EPR completeness of any phenomenon to which the term ‘physical reality’ criterion. 4

The logic of this elaboration of de Broglie’s argument exactly matches the logic of the EPR paper. But be- cause the Boxes thought experiment in no way relies on a choice between two complementary quantities to measure,15 it seems immune from Bohr’s criticism that there is a kind of “semantic disturbance”16 effected by this choice. There is no question with the Boxes experi- ment about which quantity will be measured. We are go- ing to open B1 and look for the particle there, period. So there appears to be no room for “an influence on the very conditions which define the possible types of predictions regarding the future behavior of the system.”12 Given the assumption of locality/separability (with which Bohr ap- pears to agree given his denial of a “mechanical distur- bance”), the Einstein’s Boxes thought experiment estab- lishes the conclusion of incompleteness (that is, it estab- lishes the dilemma between locality and completeness) in a straightforward way that seems immune to Bohr’s criticism of EPR.

III. EINSTEIN AND THE BOXES

FIG. 2: Einstein’s original version of the boxes thought ex- Although it is not specifically formulated in terms periment: particles incident from the left penetrate a narrow of a particle being split between a pair of boxes, Ein- aperture, where they diffract, resulting in essentially spheri- stein presented a thought experiment at the 1927 Solvay cal Schr¨odinger waves propagating toward the hemispherical Conference17,18 on which de Broglie’s later formulation detection screen. was based. The experiment involves a particle inci- dent on a diaphragm with a single aperture, behind which lies a large, hemispherical detection screen. (See wave and Born rule probabilities is incomplete because it Fig. 2.) With a sufficiently narrow aperture, the inci- fails to describe the actual path taken by each individual dent electrons will diffract, resulting in essentially spher- particle between the slit and the screen. ical Schr¨odinger waves propagating toward the screen. If Interpretation 2, on the other hand, denies that each a single electron is sent through the aperture, the elec- individual electron has a definite trajectory. Instead, tron will eventually be detected at some distinct point each electron is assumed to be spread out in a way that on the screen. Einstein’s comments focus on the appar- is accurately described by the spherical wave. This inter- ent conflict between the spreading spherical wave and the pretation was probably meant to capture the view held distinct point of eventual detection. by Schr¨odinger at the time,20 according to which Ψ 2 | | Einstein described two possible interpretations of this represented an actual charge density for an individual scenario. According to Interpretation 1, “The de Broglie- electron. It also is a fair description of the views of Jor- Schr¨odinger waves do not correspond to a single electron, dan and (the later) Heisenberg who held that, prior to the but to a cloud of electrons extended in space. The the- moment of detection, there existed a spatially dispersed ory does not give any information about the individual potentiality for particle localization, these potentialities processes, but only about the ensemble of an infinity of being triggered into realities by the eventual interaction elementary processes.” In Interpretation 2, “The theory with a measurement device.21 has the pretension to be a complete theory of individual As Einstein went on to point out, however, the par- processes.”19 ticle is in fact always found at a definite point on the Implicit in Interpretation 1 is the idea that each par- screen. Thus, at the moment it is detected there, the ticle follows some definite trajectory from the slit to the Schr¨odinger wave (representing charge density or some screen; the spreading of the wave merely indicates igno- kind of irreducible potentiality for particle localization) rance about the exact identity of this trajectory for any must suddenly collapse to zero at all other points on the particular particle. The wave represents a kind of en- screen. And, as long as we hold (with Interpretation semble average over a large number of such individual 2) that the initially spherical wave accurately and com- trajectories. The standard Born rule can then be un- pletely captures the physical reality of the propagating derstood as follows: Ψ 2 gives the probability that any particle, this collapse evidently involves a kind of action- particular electron in| this| (real or imagined) ensemble in at-a-distance. fact has (just prior to detection) a given location. Thus, Max Jammer summarizes Einstein’s objection to Inter- according to Interpretation 1, the description in terms of pretation 2 as follows: “If ψ 2 is interpreted [this way], | | 5 then, as long as no localization has been effected, the had been “smothered by the formalism.”25 Let us see how particle must be considered as potentially present with Einstein reworked the EPR argument in this letter, and almost constant probability over the whole area of the also examine the role played therein by the boxes. screen; however, as soon as it is localized, a peculiar Einstein began the letter by clarifying what he meant action-at-a-distance must be assumed to take place which by locality and completeness. He asked Schr¨odinger to prevents the continuously distributed wave in space from consider a classical particle (a ball) confined to a box. A producing an effect at two places on the screen.”22 This partition is then inserted, and the two half-boxes are car- possibility will be further explored in Sec. IV. ried to distant locations where they are separately opened Einstein continued23: “It seems to me that this dif- and their contents examined. As in his discussion at the ficulty cannot be overcome unless the description of the Solvay Conference, Einstein surveyed two possible ways process in terms of the Schr¨odinger wave is supplemented of analyzing this experiment: by some detailed specification of the localization of the particle during its propagation. I think M. de Broglie “Now I describe a state of affairs as follows: [who, recall, was toying with the pilot-wave theory at The probability is 1/2 that the ball is in the this time] is right in searching in this direction. If one first box. Is this a complete description? works only with Schr¨odinger waves, the Interpretation 2 NO: A complete description is: the ball is (or of ψ 2, I think, contradicts the postulate of relativity.” is not) in the first box. That is how the char- | | The relevant aspect of the “postulate of relativity” is acterization of the state of affairs must appear of course precisely the locality/separability assumption in a complete description. mentioned previously: there can be no causal dependence YES: Before I open them, the ball is by no between two space-like separated events. Relativity thus means in one of the two boxes. Being in a prohibits the action-at-a-distance entailed by Interpreta- definite box only comes about when I lift the tion 2. covers. This is what brings about the statisti- cal character of the world of experience, or its Summarizing, Einstein thought that any possible in- empirical lawfulness. Before lifting the covers terpretation in which the wave function was regarded as the state [of the distant box] is completely a complete description of physical reality would have to characterized by the number 1/2, whose sig- entertain unacceptable (relativity-violating) action-at-a- nificance as statistical findings, to be sure, is distance associated with the collapse of the wave func- only attested to when carrying out observa- tion on measurement. Thus he rejected Interpretation 2 tions. Statistics only arise because observa- in favor of Interpretation 1, according to which the wave tion involves insufficiently known factors for- function was an incomplete description of individual pro- eign to the system being described.”26 cesses, to be (presumably) completed with something like de Broglie’s pilot-wave model. Note that the NO and YES alternatives map exactly Just like the (later) EPR argument, this simple onto Interpretation 1 and Interpretation 2 from the 1927 thought experiment was intended to pose a dilemma be- discussion. According to the NO view (and Interpreta- tween locality and completeness: if the wave function tion 1), the description of the state of the system in terms represents a complete description of physical reality, then of probabilities is incomplete, there being, in reality, an physical reality contains relativity-violating action-at-a- actual fact of the matter about the location (trajectory) distance. Or equivalently, if we insist on respecting rel- of the particle. According to the YES view (and Inter- ativity’s prohibition, we must regard the quantum me- pretation 2), the description in terms of probabilities is chanical description of reality as incomplete. complete because the actual fact of the matter regarding As is hopefully now clear, de Broglie’s re-formulation of the location of the ball (particle) only comes into exis- the thought experiment in terms of literal boxes simply tence with the act of measurement. exaggerates the spatial separation of two parts of the For the case of a classical particle like the ball, the total wave function, and thus brings out more clearly the YES view is not very plausible, and Einstein asserts implications of the locality assumption. But the essential that “the man on the street would only take the first structure of the argument de Broglie presents is identical . . . interpretation seriously.”27 But for a single electron, to the original Einstein argument of 1927. the YES view is essentially the completeness claim of the Some years after the 1927 Solvay Conference, Einstein Copenhagen interpretation of quantum mechanics. Ein- did discuss the specific scenario involving a single particle stein didn’t accept this view and wanted to argue that the divided between two half-boxes, although he used the first view (NO) was the correct one not only for the classi- boxes as merely a classical analog to clarify his definitions cal particle but for the electron as well. But he recognized of locality and completeness. The discussion occurred in that merely arguing from the classical analogy wasn’t ad- a 1935 letter to Schr¨odinger, written shortly after the equate, and went on in the letter to Schr¨odinger (and in publication of the EPR paper. As Arthur Fine24 pointed several other places28 over the subsequent decades) to out, in this letter Einstein disclosed that Podolsky, not construct a complete argument. Einstein, wrote the EPR paper and that he (Einstein) The necessary additional premise — one that was was disappointed with how it turned out: the main point taken for granted for the case of the ball but needed 6 to be made explicit to make the argument rigorous for has not changed. That it happens to change to a state the electron — was the locality/separability assumption. function that can (according to the standard eigenstate- Howard29 translates Einstein, again from the same 1935 eigenvalue link) be interpreted as attributing a definite letter: “My way of thinking is now this: properly consid- value for some property (an “element of reality”) is sim- ered, one cannot [refute the completeness doctrine, Inter- ply irrelevant.32 pretation 2] if one does not make use of a supplementary According to Einstein “Now what is essential is exclu- principle: the ‘separation principle.’ That is to say: ‘the sively that [the wave functions relating to the distant par- second box, along with everything having to do with its ticle] are in general different from one another. I assert contents is independent of what happens with regard to that this difference is incompatible with the hypothesis the first box (separated partial systems).’ If one adheres that the ψ description is correlated one-to-one with the to the separation principle, then one thereby excludes physical reality (the real state). After the collision, the the second . . . point of view, and only the [first] point real state of [the two particle system] consists precisely of view remains, according to which the above state de- of the real state of [particle 1] and the real state of [par- scription is an incomplete description of reality, or of the ticle 2], which two states have nothing to do with one real states.” another. The real state of [particle 2] thus cannot depend Einstein continues: “The preceding analogy [that is, upon the kind of measurement I carry out on [particle 1]. the ball-in-boxes scenario] corresponds only very imper- (‘Separation hypothesis’ from above.) But then for the fectly to the quantum mechanical example in the [EPR] same state of [particle 2] there are two (in general arbi- paper. It is, however, designed to make clear the point of trarily many) equally justified ψ[2], which contradicts the view that is essential to me. In quantum mechanics one hypothesis of a one-to-one or complete description of the 33 describes a real state of affairs of a system by means of real states.” a normed function ψ of the coordinates (of configuration In summary, by the separation principle, the real, space). The temporal evolution is uniquely determined physical state of particle 2 is unaffected by the mea- by the Schr¨odinger equation. One would now very much surement at 1. Yet according to the collapse postulate, like to say the following: ψ stands in a one-to-one cor- the wave function associated with particle 2 changes. At respondence with the real state of the real system. The most one of the two wave functions can constitute a com- statistical character of measurement outcomes is exclu- plete description. Thus the quantum mechanical descrip- sively due to the measuring apparatus, or the process of tion is shown to be, in the general case, incomplete. measurement. If this works, I talk about a complete de- Although Einstein dropped the ball-in-boxes thought scription of reality by the theory. However, if such an experiment (and returned to the two-particle entangled interpretation doesn’t work out, then I call the theoreti- state of EPR) when presenting this reformulated argu- cal description ‘incomplete’.”30 ment from locality to incompleteness, he need not have Note that Einstein defines “completeness” here as a done so. Despite his statement that the ball-in-boxes one-to-one correspondence between quantum wave func- analogy “corresponds only very imperfectly to the quan- tions and real states of real systems. This definition is tum mechanical example in the [EPR] paper,” the EPR somewhat different from the one (cited in Sec. II) used dilemma can be established equally well with a quantum in the EPR paper; Arthur Fine refers to it as “bijective mechanical version of the boxes. completeness” in contrast to “EPR completeness.”31 Consider the quantum state of the single particle whose wave function has been split between two spatial loca- Einstein then dropped the ball-in-boxes example and tions (boxes): returned to the original EPR scenario involving two en- tangled particles. He reconstructed the EPR logic as a 1 ψ = (ψ1 + ψ2), (2) simple argument from the separation principle to a failure √2 of bijective completeness, modeled after the discussion of the boxes. The argument is the following: consider the where ψ1,2 are completely localized in two well-separated entangled state of the two particles discussed in EPR. Be- regions of space (B1,2). According to the standard recipe, fore the measurement on particle 1, the entangled wave if and when a position measurement is made on the par- function attributes to particle 2 no definite position and ticle, its wave function immediately collapses to zero at no definite momentum. After a measurement on 1 (what- every point except the place where the particle was found. ever is measured), the wave function collapses and parti- Applying this recipe here, imagine that we open box B1 cle 2 is in a state of some definite position or some definite and find the particle there. At this moment, the wave momentum or perhaps some other quantity altogether – function in B2 immediately goes to zero (indicating that which state of course depends on what kind of measure- there is no longer any probability of finding the parti- ment is made on particle 1 and what its outcome is. But cle there). But according to the locality premise, the that aspect (which to Bohr seemed so central to the EPR actual physical state of B2 (whatever that might have argument) is now completely irrelevant. All that mat- been) cannot have changed. Hence, we have two dif- ters is that the wave function associated with the second ferent quantum mechanical descriptions (in particular, particle has changed in a situation where (by the sepa- wave functions with different values in B2) of the same ration principle) the actual physical state of particle 2 one unchanged physical contents of B2 – a clear failure 7 of Einstein’s (bijective) completeness criterion. ever, this view blasts the framework of the This reformulation of Einstein’s argument for incom- ‘orthodox quantum theory’.”34 pleteness using the ball-in-boxes scenario is astonishingly simple compared to the EPR argument for the same con- Or as Arthur Fine eloquently summarizes this point: “It is important to notice that the conclusion Einstein draws clusion. Our reformulation highlights that, unlike our re- construction of de Broglie’s boxes argument and unlike from EPR is not a categorical claim for the incomplete- ness of quantum theory. It is rather that the theory poses the argument presented in EPR, the argument from lo- a dilemma between completeness and separation; both cality to incompleteness need not rely on actually deter- cannot be true.”35 mining the identity of any elements of reality (by means Perhaps this elaboration of the logic of Einstein’s argu- of the EPR criterion of reality). In the EPR scenario in- ment against the completeness doctrine — especially the volving the two entangled particles, there is an almost ir- simplified version involving the boxes — helps explain resistible tendency to attempt to establish definite prop- erties for the distant particle; after all, there is a particle why Einstein never believed that Bohr (or anyone else) had adequately addressed the dilemma between locality there that one is tempted to think has those properties. 36 With the (one particle) Einstein’s Boxes argument, how- and completeness posed in EPR. ever, the contents of the distant box are less clear, and the temptation to attribute definite properties is conse- IV. BELL AND THE BOXES quently reduced – which is a good thing, from the point of view of emphasizing that it does not matter to the argu- ment what the contents of the box are. All that matters is In the last paper of his tragically short life, J. S. Bell that according to the locality principle, whatever is there prefaced a discussion and derivation of his celebrated in- doesn’t change as a result of the measurement performed equality with a classical analogy reminiscent of Einstein’s boxes. Bell wrote: in B1. That alone is sufficient to establish the conclusion that Einstein’s bijective completeness criterion fails for “Most physicists were (and are) unimpressed (an assumed local) quantum theory. by [the correlations exhibited in the EPR sce- It also bears repeating that Einstein’s actual argument nario]. That is because most physicists do not in terms of the pair of previously interacting particles really accept, deep down, that the wavefunc- and the reformulation here in terms of the ball-in-boxes tion is the whole story. They tend to think scenario both completely avoid any mention of the non- that the analogy of the glove left at home is commuting observables that seemed (to Bohr at least) a good one. If I find that I have brought only to play such a large role in the actual EPR paper. It is one glove, and that it is right-handed, then I intriguing to speculate about how Bohr would have re- predict confidently that the one still at home sponded to this simplified version of the argument for in- will be seen to be left handed. But suppose completeness. We won’t, however, undertake such specu- we had been told, on good authority, that lation here. Suffice it to credit Einstein’s Boxes for help- gloves are neither right- or left-handed when ing to un-smother Einstein’s true arguments against the not looked at. Then that, by looking at one, quantum completeness doctrine. we could predetermine the result of looking Although Einstein strongly believed in the local- at the other, at some remote place, would be ity/separability premise of this argument and therefore remarkable. Finding that this is so in prac- regarded it as an argument for incompleteness, he real- tice, we would very soon invent the idea that ized that, strictly speaking, what the argument estab- gloves are already one thing or the other even lishes is only that, for quantum mechanics, locality en- when not looked at. And we would begin to tails incompleteness. That is, quantum mechanics must doubt the authorities that had assured us oth- give up either the completeness claim or the locality prin- erwise. That common-sense position was that ciple: taken by Einstein, Podolsky and Rosen, in re- spect of correlations in quantum mechanics. “By this way of looking at the matter it be- They decided that the wavefunction, making comes evident that the paradox forces us to no distinction whatever between one possibil- relinquish one of the following two assertions: ity and another, could not be the whole story. 1. the description by means of the ψ-function is And they conjectured that a more complete complete. story would be locally causal. 2. the real states of spatially separated objects “However it has turned out that quantum me- are independent of each other. chanics can not be ‘completed’ into a locally causal theory, at least as long as one allows, On the other hand, it is possible to adhere as Einstein, Podolsky and Rosen did, freely to (2) if one regards the ψ-function as the operating experimenters. The analogy of the description of a (statistical) ensemble of sys- gloves is not a good one. Common sense does tems (and therefore relinquishes (1)). How- not work here.”37 8

Bell goes on in the paper to derive his famous the- quantum mechanics” suffer inevitable non-locality. And orem, in the context of Bohm’s version of the EPR Eugene Wigner wrote: “In my opinion, the most con- argument.38 In this version, it is different spin compo- vincing argument against the theory of hidden variables nents of the two particles, rather than their positions was presented by J. S. Bell.”43 and momenta, which play the role of quantum mechani- In summary, it is almost universally believed that ex- cally non-commuting variables which an EPR-like argu- perimental violations of Bell’s inequalities imply fatal ment suggests may nevertheless possess simultaneously problems with attempts to “complete” quantum theory definite values. by adding hidden variables, but pose no particular prob- Assuming a non- (or super-) quantum-mechanical the- lem or puzzle for quantum mechanics itself. This atti- ory in which each particle in the EPR pair carries with tude evidently accounts for the fact that very few physi- it a set of hidden variables (an “instruction set”) telling cists are presently interested in the hidden variables pro- it how to respond when encountering variously oriented gram. It also explains why equally few worry that there is polarizers (that is, a theory with hidden elements of re- any deep contradiction between standard quantum the- ality of precisely the kind suggested by the EPR-Bohm ory and relativity. thought experiment), Bell imposes a locality constraint Bell himself, however, did not see it this way. Like Ein- according to which the outcome of the measurement on stein, Bell believed that quantum mechanics itself faced a each side must depend only on the variables in the par- problematic dilemma between completeness and locality. ticles’ instruction set and the orientation of the local ap- We will discuss the logical implications of this shortly, paratus. The dependence of each outcome on the setting in particular, the implications of combining the EPR of the distant apparatus or the distant outcome is ex- (or Einstein’s Boxes) locality-completeness dilemma with cluded. The mathematical consequence is an inequality Bell’s theorem. restricting the average correlations between the outcomes First, let us give one final version of the argument for of measurements on the two particles. this dilemma by applying Bell’s reasoning to the boxes This kind of inequality is violated by the quantum me- scenario. Instead of applying this reasoning to an EPR- chanical predictions and, it seems, by real particles in motivated hidden variable theory (like Bell did), however, real experiments.39 This violation means that one (or we will assume orthodox quantum mechanics, complete- more) of the assumptions made in deriving the inequal- ness doctrine and all, as our working theory. What we ity must be false. The two assumptions typically called will see is that by using reasoning similar to Bell’s, we into question are the assumption of hidden variables and can arrive, not at a conclusion equivalent to Bell’s fa- the assumed lack of (relativity-violating) causal depen- mous theorem, but rather at a new argument for the dence of the outcomes on distant settings and distant EPR dilemma: quantum mechanics is either incomplete outcomes. And choosing one of these assumptions to or non-local. blame has generally been considered no dilemma at all; Let us begin by assuming that the initial wave function, relativity’s prohibition on super-luminal causation is a central pillar of modern physics, so “obviously” the cor- 1 λ = (ψ1 + ψ2), (3) rect choice is to reject the assumption of hidden variables. √2 Thus, the empirical violations of Bell’s inequalities seem to be regarded by most physicists as the best argument is a complete description of the physical reality of the against the hidden variables program and for the com- pre-measurement situation. (As before, ψ1 and ψ2 repre- pleteness doctrine. sent states in which the particle is localized respectively 40 For example, N. David Mermin wrote that violations in B1 and B2. Note too that we have used the variable of Bell’s inequalities “fatally undermine the position of λ rather than ψ to denote the pre-measurement wave EPR” meaning, presumably, EPR’s claim that a (hidden function. This choice is meant to highlight the similarity variable) completion of quantum theory might be possi- to Bell’s derivation, in which λ is traditionally used to ble. Daniel Styer, answering an alleged “misconception denote a complete specification of the pre-measurement regarding quantum mechanics”41, stated that “The ap- state of the two particles including any necessary hidden pealing view that a state vector describes an ensemble variables. Here, however, we are assuming with Bohr of classical systems [and therefore an incomplete descrip- that the wave function alone provides a complete speci- tion of individual systems] was rendered untenable by fication of the pre-measurement contents of the boxes.) tests of Bell’s theorem which show that no determinis- Let N = 0, 1 represent the number of particles 1 { } tic model, no matter how complicated, can give rise to found in box B1 when it is opened; similarly for N2. all the results of quantum mechanics.” (Bohm’s theory The Born rule implies the following expressions for the of course provides a straightforward counterexample to probabilities of finding N1 = 1 (when/if B1 is opened and this claim, unless Styer intends the tacit premise that examined) and of finding N2 = 1 (when/if B2 is opened the deterministic model must respect locality.) Asher and examined): Peres and Daniel R. Terno42 assert that “Bell’s theo- rem does not imply the existence of any non-locality in 1 P (N =1 λ)= , (4) quantum mechanics itself;” only “classical imitations of 1 | 2 9 and box do not change as a result of the measurement on the first box, and if these physical contents are completely de- 1 P (N2 =1 λ)= . (5) scribed by the wave function λ, then the associated (Born | 2 rule) probability cannot change as a result of the outcome Finally, let us calculate the probability for a double de- of the distant experiment. That is precisely Bell’s local- tection, that is, the probability that the particle is found ity assumption. To demand the use of a new, collapsed in both of the two half-boxes when they are opened. Call wave function for the second box (one that takes into ac- this probability P (N1 =1& N2 =1). If we follow exactly count the information gathered when the first box was Bell’s reasoning in deriving an inequality for hidden vari- opened) is to concede that the description provided by able theories, we can (by Bayes’ formula44) decompose the initial wave function was incomplete. the joint probability into a product as: If the reader finds this response unconvincing, imagine that the same objection was leveled against the parallel P (N1 = 1& N2 =1 λ) assumption in Bell’s theorem by an advocate of a sup- | = P (N1 =1 λ) P (N2 =1 N1 =1, λ). (6) posedly local hidden variable theory. Such an advocate | · | might object to Bell on the grounds that, after the first Now, quoting Bell: “Invoking local causality, and the measurement is made (on particle A say), the state speci- assumed completeness of . . . λ . . . , we declare redundant fication of particle B should be allowed to re-adjust to be certain of the conditional variables in the last expression, in accord with the outcome of (that is, the information because they are at space-like separation from the result obtained during) the experiment on A. “The experimen- in question.”44 In our situation, this same local causality tal violation of Bell’s inequalities,” such a person might requirement implies that say, “in no way proves that my local hidden variable the- ory is untenable. For Bell unjustifiably assumed that the P (N =1 N =1, λ)= P (N =1 λ), (7) 2 | 1 2 | probabilities for various outcomes at B could depend only on the physical state attributed to B before the measure- because, as Bell says, the outcome event at B (N = 1) 1 1 ment at A. But in my local hidden variable theory, the is (let us assume) at a space-like separation from the state attributed to B, and therefore the probability for event in question (N = 1) at B . Thus, the probability 2 2 a particular outcome at B, depends on the result of the of this event at B should not depend on the outcome 2 measurement at A. Thus Bell’s constraint does not apply N = 1; it should instead depend only on those variables 1 and my theory is saved.” (λ) that described (by assumption, completely) the pre- measurement contents of the boxes. The correct response to such an argument would be Thus, the probability of double-detection simplifies to to simply deny that this sort of theory is local. A state attribution for B that changes as a result of the measure-

P (N1 = 1& N2 =1 λ) (8) ment outcome at A is precisely the kind of non-locality | Bell’s locality assumption is meant to exclude. But this = P (N =1 λ) P (N =1 λ). 1 | · 2 | response applies equally well to quantum theory itself, if If we substitute the two 50% probabilities on the right- we are assuming that quantum theory provides, with the hand side, we evidently find that a double-detection wave function alone, a complete description of physical should occur with probability 25%. That is, one quar- reality. Recalling Bell’s reasoning, it is precisely the as- ter of the time, we should find the particle in both boxes. sumption of “the assumed completeness of . . . λ” that One expects, of course, that this will never happen. warrants the removal of the A-side outcome from the list (Such an occurrence would violate several fundamental of variables on which the B-side result is conditioned. conservation principles.) Therefore one (or more) of the Despite the conventional wisdom to the contrary, Bell’s assumptions leading to this prediction must be wrong. reasoning is not uniquely applicable to hidden variable In particular, it seems that we must reject either the theories or any other particular kind of theory. Rather assumption of locality or the assumption of completeness it is a way of teasing out the implications of any theory – the same dilemma established earlier using the boxes that is assumed to be both complete and local. If we use in a different way.45 the Einstein’s Boxes scenario, this reasoning (applied to The reader might object that we neglected to collapse quantum theory without any hidden variables) provides the wave function and thus erred in treating the second yet another way to establish Einstein’s dilemma: quan- measurement as based on the same quantum state (λ) as tum theory either violates relativity’s demand for local- the first. As Einstein pointed out in 1927,22, it is pre- ity/separability or is incomplete.46 cisely the collapse postulate which prevents a single par- Note that this Bell-inspired argument for the quan- ticle, whose wave function before measurement is spread tum completeness-locality dilemma relies on definitions out over some finite region of space, from being detected of completeness and locality that are different from those at two different places at once. encountered in Secs. II and III. Locality is defined here Our claim, however, is that this collapse violates the formally in terms of the stochastic irrelevance (or redun- locality assumption if the wave function description is re- dancy) of facts that are outside the past light cone of garded as complete. If the physical contents of the second a given event: P (N = 1 N = 1, λ) = P (N = 1 λ). 2 | 1 2 | 10

And completeness here amounts to the standard asser- two parts, a reflected and a transmitted packet. There is tion that the probability of detecting a particle is given then a definite probability for finding the photon either in by the absolute square of the wave function, and that this one part or in the other part of the divided wave packet. probability cannot be accounted for by ignorance regard- After a sufficient time the two parts will be separated ing any additional hidden variables – i.e., that there is by any distance desired; now if an experiment yields the nothing other than the value of λ in B2 that determines result that the photon is, say, in the reflected part of the the probability for detecting a particle there.47 packet, then the probability of finding the photon in the We return then to the earlier point about the use of other part of the packet immediately becomes zero. The Bell’s theorem as an argument against the hidden vari- experiment at the position of the reflected packet thus ables program (that is, its use as an argument for the exerts a kind of action (reduction of the wave packet) orthodox completeness doctrine). It seems that we need at the distant point occupied by the transmitted packet, to reconsider the implications of Bell’s theorem when it and one sees that this action is propagated with a velocity is combined with the completeness-locality dilemma. If greater than that of light.”49 Bohr can be considered to have adequately refuted the There are several interesting points to note. First, argument for this dilemma, that is, if quantum mechan- Heisenberg converted the Boxes thought experiment into ics can be validly interpreted as both complete and local, a more realistic (and experimentally realizable; see the then Bell’s theorem would indeed argue strongly against following) situation where the carrying apart of two half- any attempt to supplement the theory with additional boxes is replaced by the natural motion of a particle after variables. But what if (perhaps motivated by Einstein’s being either transmitted or reflected by a half-silvered Boxes) we regard Bohr’s response as inadequate? That mirror. Second, it is interesting that Heisenberg seemed is, what if we agree with Einstein that a dilemma between to accept the dilemma between locality and complete- locality and completeness exists for quantum theory? ness that Einstein’s argument is intended to establish There seem to be two options. We might choose to and to favor the “complete but non-local” response to regard the quantum theory as complete but non-local. that dilemma. But then we are no longer in a position to use the required Yet as he claimed in the sentence just following the non-locality of hidden variable theories (required, that previous quote, the admitted “action at a distance” that is, by Bell’s theorem) as a reason to dismiss them. If is exerted by the nearby experimental apparatus doesn’t non-locality is unacceptable to physical theories, not just lead to a contradiction with relativity: “However, it is Bohm’s theory (for example), but quantum theory itself also obvious that this kind of action can never be utilized would have to be dismissed as in conflict with relativity for the transmission of signals so that it is not in conflict theory. with the postulates of the theory of relativity.”49 Perhaps we are therefore tempted to back up and take So evidently Heisenberg’s position was to concede to the other horn of the dilemma – to regard quantum me- Einstein that quantum mechanics was non-local, but to chanics as local but incomplete. But then it seems clear deny that its particular type of non-locality was at odds that we ought to attempt to complete it, presumably by with relativity’s prohibitions on superluminal causation. adding hidden variables (which, it turns out, will have to (This view is widely held and was eloquently advocated be non-local). recently by Abner Shimony50 who described the situa- Either way, Bell’s theorem ceases to be a valid argu- tion as a “peaceful coexistence” between quantum non- ment against the hidden variables program – a conclusion locality and relativity.51) The tacit premise here is that opposite the standard interpretation of Bell’s result. the only kind of superluminal causation that relativity If the argument for the locality-completeness dilemma theory forbids is the kind that can be used “for the trans- is sound – and I think the various formulations in terms of mission of signals.” Einstein’s Boxes make this argument rather compelling This particular attempt to wriggle free of the dilemma compared to the original EPR argument – then Bell’s re- was assessed by Bell: sult turns into a powerful argument in favor of taking se- riously non-local hidden variable theories like Bohm’s.48 “Do we then have to fall back on ‘no signaling faster than light’ as the expression of the fun- damental causal structure of contemporary V. HEISENBERG, LOCALITY, AND theoretical physics? That is hard for me to EXPERIMENTAL METAPHYSICS accept. For one thing we have lost the idea that correlations can be explained, or at least this idea awaits reformulation. More impor- Heisenberg also discussed something like Einstein’s tantly, the ‘no signaling’ notion rests on con- Boxes: “. . . one other idealized experiment (due to Ein- cepts which are desperately vague, or vaguely stein) may be considered. We imagine a photon which applicable. The assertion that ‘we cannot sig- is represented by a wave packet built up out of Maxwell nal faster than light’ immediately provokes waves. It will thus have a certain spatial extension and the question: also a certain range of frequency. By reflection at a semi- transparent mirror, it is possible to decompose it into Who do we think we are? 11

We who can make ‘measurements’, we who not surprising to anyone today, it is hoped that the pre- can manipulate ‘external fields’, we who can ceding discussion underlines the apparent implications of ‘signal’ at all, even if not faster than light? this outcome: if we accept the locality principle, we must Do we include chemists, or only physicists, evidently conclude that photons56 do not split at the mir- plants, or only animals, pocket calculators, ror, but instead choose one or the other path exclusively. or only mainframe computers?”52 That is, we must conclude that the quantum mechanical description of the photon before detection, In addition, Heisenberg’s move here undermines one of the principal arguments against non-local hidden vari- 1 ψ = (ψ1 + ψ2), (9) able theories like Bohm’s. For although that theory does √2 include non-local, action-at-a-distance effects, these ef- fects (just like the similar effects Heisenberg concedes is incomplete because it fails to attribute a definite posi- are present in orthodox quantum mechanics) cannot be tion to the particle. used to transmit signals. In Bohm’s theory this inability Alternatively, we could uphold the completeness of the arises from the inevitable uncertainty about the initial wave function description by denying the locality premise locations of particles within their guiding waves. So if, and regarding the collapse of the wave function on mea- as Heisenberg argues, it is only the possibility of send- surement as describing a real physical change in the state ing superluminal signals that conflicts with relativity, we of the distant packet’s referent. must evidently conclude that Bohm’s theory and ortho- There is thus an interesting parallel between the AJV dox quantum mechanics enjoy an equally peaceful coex- experiment and the recent experimental tests of Bell’s istence with it.53 inequalities.39 Because Bell’s inequality represents a con- Let us finally return to the previous comment about straint on local hidden variable theories, empirical viola- the apparent experimental realizability of the Heisenberg tions of the inequalities imply that local hidden variable version of Einstein’s Boxes. According to John Clauser,55 theories aren’t empirically viable. Likewise, Eq. (8) rep- Schr¨odinger was intrigued by the clarity with which resents a constraint on a local version of quantum theory Einstein’s Boxes brought out the locality-completeness (assumed complete). So the AJV finding that this con- dilemma for quantum theory, and persuaded Ad´am,´ straint is violated has implications paralleling those of the J´anossy, and Varga (AJV) to perform Heisenberg’s ver- Bell tests: the local version of quantum theory (assumed sion of the experiment.54 Let us quote from Clauser’s complete) is not empirically viable. Either locality or the description of the AJV experiment: orthodox completeness assumption must be abandoned. This conclusion is not new, but the same dilemma we “In their experiment, two independent photo- have seen now several times. detectors are placed respectively in the trans- The extent to which orthodox quantum theory and mitted and reflected beams of a half-silvered the hidden variable alternatives are on equal footing is mirror. If photons have a particle-like charac- striking: local versions of each are evidently refuted by ter, that is, if their detectable components are experiment.57 Thus it seems the program of “experimen- always spatially bounded and well localized, tal metaphysics” (to use one of Shimony’s apt phrases50) then photons impinging on the half-silvered began more than 30 years before the recent tests of Bell’s mirror will not be split in two at this mirror. inequality. On the other hand, if they are purely wavelike in nature, (as with classical waves) then they can and will be split into two independent VI. DISCUSSION classical wave packets at the mirror. This fact then implies that if they are purely wave- We have presented several different formulations and like (in this classical sense), then the two de- revisions of the argument that Einstein first presented at tectors will show coincidences when a single the 1927 Solvay conference. The goal of this argument in temporarily localized photon is directed at all of its versions is to establish an inconsistency between said mirror. One of these independent wave the following two claims: packets will be transmitted to illuminate the first detector, and the other will be reflected The quantum mechanical description of reality is to illuminate the second detector, and both • complete. detectors will then have a finite probability of detecting the same photon (classical wave Quantum mechanics describes a world with only • 58 packet) . . . [AJV] found no such anomalous local interactions. coincidences, and thereby concluded that . . . Although the various formulations of the argument differ photons do not split at the mirror and thus slightly in terms of logical structure and definitions of key do exhibit a particle-like character.”55 terms, they point to the same conclusion and ultimately Note that this experiment is precisely an empirical test pivot around the same fulcrum: the collapse of the wave of Eq. (8) . Although the outcome of the experiment is function. 12

The collapse rule itself is absolutely essential to the Of course, the existence of a clean distinction between quantum formalism, as evidenced, for example, by the these two views of wave function collapse presupposes derivation of Eq. (8) and the subsequent discussion in a realist philosophy that Bohr and many of his follow- Sec. IV. However, the exact meaning of the rule is some- ers deny (or even claim to have refuted). Bohr, for ex- what opaque in the standard interpretation. The essen- ample, is said to have declared that “there is no quan- tial lesson of Einstein’s Boxes (and this is a point Einstein tum world.”63 Although this anti-realist attitude contin- seems to have understood quite clearly as early as 1927) is ues to receive lip service, it seems that most physicists that we face a very simple choice when it comes to inter- don’t take the literal denial of a quantum world very preting the collapse. Either it represents a real, physical seriously.64 Most of us believe in the real existence of change of the physical state of the affected system, or it atoms, electrons, and other sub-atomic particles. What- doesn’t.59 If it does represent such a physical disturbance, ever strange quantum properties they may have, surely this entails a non-local action-at-a-distance that seems to there is a “they” that have them. conflict with relativity’s prohibition on superluminal cau- We will by no means resolve the debate about realism sation. (Or, if the physical disturbance produced by the here. But it is important to recognize that anti-realism collapse were to propagate from the measurement event is not a sensible response to the alleged incompatibility at a sub-luminal speed, the quantum predictions would of the completeness and locality claims. It is, rather, a be empirically wrong in cases of spatially separated corre- wholesale denial of the very issues of completeness and lations like those illustrated in the AJV experiment.) On locality: there is no sense, for example, to a claim that the other hand, if the collapse of the wave function does there is no quantum world, but quantum mechanics pro- not represent a physical change in the state of the real vides a complete description of it.65 system, then the quantum mechanical description must One cannot respond to Einstein’s dilemma by claim- be incomplete (in at least Einstein’s bijective sense). For ing that an anti-realist attitude warrants a simultaneous quantum theory would then attribute two distinct state belief in the completeness and locality of quantum me- descriptions to one and the same physical state. chanics. For according to the anti-realist attitude, there The orthodox claim, initiated by Bohr in his reply is simply no such issue as completeness (there being no to EPR, is that quantum mechanics is both complete quantum world for quantum mechanics to completely or and local. On further scrutiny, however, it seems this incompletely describe) and likewise no such issue as lo- claim can be maintained only with a kind of double-speak cality. This wholesale denial of the issues is not what about the collapse postulate. To protect the complete- Bohr seems to have expressed explicitly, and it does not ness claim, the collapse must be interpreted as a real seem consistent with the views of most contemporary de- physical state change, usually backed up by some ver- fenders of the Copenhagen interpretation who think, for sion of the disturbance theory of measurement.60 On the example, that locality is a meaningful and important re- other hand, the locality claim can be defended only by in- quirement for physical theories. terpreting the wave function collapse as merely a change It is only from the point of view of the quantum realist 61 in knowledge. that these issues are meaningful; and from this point of This double-speak was perhaps first identified by Ein- view, they are crucial and must be addressed before we stein. In the 1935 letter to Schr¨odinger that was dis- can claim to truly understand the meaning and status cussed earlier, Einstein wrote: “The talmudic philoso- of quantum theory. That is a program Einstein helped pher doesn’t give a hoot for ‘reality’, which he re- initiate in the 1920s and, sadly, it is one that has been gards as a hobgoblin of the naive, and he declares that largely abandoned by the physics community – no doubt the two points of view differ only as to their mode of because many came to believe that the standard interpre- expression.”62 The “Talmudist” here is Bohr, and the tation was adequate on both counts. But, from the point “two points of view” are the two interpretations of the of view of Einstein, this adequacy was largely overesti- boxes discussed previously. But these interpretations mated: the standard interpretation of quantum theory map precisely onto the two just-mentioned interpreta- does suffer from a troubling dilemma, one with major tions of wave function collapse. If, on the one hand, the implications not only for the theory itself but also for wave function description is incomplete (Interpretation 1 possible alternatives to or extensions of it. Indeed, when from Sec. III), then the collapse postulate can be safely combined with the empirical violations of Bell’s inequal- interpreted as merely updating our knowledge. But if, ities, Einstein’s dilemma has major implications for our on the other hand, the wave function description is re- understanding of nature. garded as complete (Interpretation 2), then the collapse It is fascinating that, after more than 75 years of must represent a real physical change in the state of the heated discussion, Einstein’s criticisms of the quantum system. theory still seem effective and fresh. There is still much Evidently Einstein believed that the claim that wave work to be done before we achieve total clarity about the function collapse represents a physically real change of foundations of quantum theory, the kind of clarity that state, and the claim that it represents a mere change Einstein continuously sought. As we approach the cen- in knowledge, do not “differ only as to their mode of tennial celebration of Einstein’s “miraculous year”, let expression.” us acknowledge that Einstein still has much to teach us 13 about how this clarity can be achieved. It is hoped that ful discussions and critical comments on drafts of the the Einstein’s Boxes argument in particular will continue paper. A special thanks is also due to Arthur Fine and to stimulate further discussions on these fascinating and Don Howard, on whose Einstein scholarship my own un- important topics. derstanding (and much of the present work) is largely based.

Acknowledgments

Thanks to three anonymous referees and to Roderich Tumulka, Arthur Fine, and Sheldon Goldstein for help-

∗ Electronic address: [email protected] (1935). 1 Max Jammer, The Philosophy of Quantum Mechanics 13 This is one very plausible and influential interpretation (John Wiley and Sons, New York, 1974), pp. 109–114. of Bohr’s reply; see, for example, Ref. 1, p. 196. But 2 David Bohm, “A suggested interpretation of the quantum other commentators have understood Bohr’s rather ob- theory in terms of ‘hidden’ variables, I and II,” Phys. Rev. scure prose differently or not at all. 85, 166–193 (1952). 14 See, for example, the comments of J. S. Bell: Ref. 3, p. 3 J. S. Bell, Speakable and Unspeakable in Quantum Me- 155–6. chanics (Cambridge University Press, New York, 1987). 15 Actually, as Tim Maudlin, Quantum Non-Locality and 4 Erwin Schr¨odinger “The present situation in quantum me- Relativity (Blackwell Publishing, 1994), pp. 140–141 has chanics,” translated by John D. Trimmer, in Proc. Amer. eloquently pointed out, neither does the original EPR Philosophical Soc. 124, 323–38 (1980), reprinted in Ref. 5. thought experiment. The fact that one can, in the EPR sit- 5 John A. Wheeler and Wojciech H. Zurek, Quantum Theory uation, predict with certainty just the position of particle and Measurement (Princeton University Press, 1983). 2 (without disturbing it in any way) is sufficient, according 6 For a general survey of Einstein’s opposition to quantum to the reasoning set up by EPR, to conclude that, if local, theory, including some discussion of the boxes thought ex- quantum theory is incomplete. For that alone means that periment, see R. Deltete and R. Guy, “Einstein’s opposi- the position of particle 2 is an element of reality, and one tion to the quantum theory,” Am. J. Phys. 58, 673–683 that has no counterpart in the original, pre-measurement (1990). entangled wave function. Maudlin refers to the attempt 7 Niels Bohr, “Discussions with Einstein on epistemolog- to (in addition) beat the uncertainty principle by also es- ical problems in atomic physics,” in Albert Einstein: tablishing the real existence of particle 2’s momentum as Philosopher-Scientist, edited by P. A. Schilpp (The Library “an unnecessary bit of grandstanding” which plunged “the of Living Philosophers, Evanston, 1949), pp. 200–241. previously simple EPR argument into the muddy waters 8 A. Einstein, B. Podolsky, and N. Rosen, “Can quantum of [modal logic and counterfactual definiteness].” As we mechanical description of physical reality be considered have seen, it is precisely this question of counterfactuals complete?,” Phys. Rev. 47, 777–780 (1935). that Bohr seems to have jumped on in his reply. In any 9 Credit for this name should probably go to Arthur Fine, case, Einstein’s Boxes (which completely avoids the issue of who used the phrase in passing in The Shaky Game (The counterfactual definiteness) should help clarify this point. University of Chicago Press, 1996), p. 37. 16 Ref. 9, p. 35. 10 Lucien Hardy presented an equally clear account of the 17 See, for example, Ref. 1, pp. 115–117; Ref. 11, pp. 24–26. thought experiment in a recent paper: “Spooky action at 18 L. E. Ballentine, “Einstein’s interpretation of quantum me- a distance in quantum mechanics,” Contemporary Phys. chanics,” Am. J. Phys. 40, 1763–1771 (1972). 39 (6), 419–429 (1998). For an interesting theoretical dis- 19 L. E. Ballentine18 translates the French from “Electrons et cussion of some potential practical difficulties with this ex- Photons,” Institut International de Physique Solvay, Rap- periment, see Julio Gea-Banacloche, “Splitting the wave ports et Discussions du Cinqui’eme Conseil de Physique function of a particle in a box,” Am. J. Phys. 70 (3), 307– (Gauthier-Villars, Paris, 1928), pp. 253–256. 312 (2002) and T. Norsen, “Splitting the wave function,” 20 Ref. 1, pp. 24–33. Am. J. Phys. 70 (7), 664 (2002). 21 Ref. 1, pp. 44, 160–162. 11 L. de Broglie, The Current Interpretation of Wave Me- 22 Ref. 1, p. 116. chanics: A Critical Study (Elsevier Publishing Company, 23 Ref. 1, p. 116. See also Ref. 18, p. 1765. 1964). Note also that this passage comes just after a dis- 24 Ref. 9, Chap. 3. cussion of the thought experiment given by Einstein at 25 Ref. 9, p. 35. the 1927 Solvay conference and that de Broglie introduced 26 Ref. 9, p. 69. the discussion with this comment: “In a recent article . . . 27 Ref. 9, p. 69. I have revived the above objections [Einstein’s] in a new 28 For a complete list of references, see Don Howard, “Ein- form.” The article is L. de Broglie, “L’interpr´etation de la stein on locality and separability,” Stud. Hist. Phil. Sci. m´ecanique ondulatoire,” J. Phys. Rad. 20, 963 (1959). 16, 171-201 (1985). 12 N. Bohr, “Can quantum-mechanical description of physical 29 Ref. 28, pp. 178–179. reality be considered complete?,” Phys. Rev. 48, 696–702 30 Ref. 9, p. 71. 14

31 Ref. 9, Chap. 5. itself cannot be both complete and local. The latter es- 32 Einstein wrote that the question of establishing elements tablishes that no local hidden variable theory can agree of reality for both position and momentum of the distant with quantum theory’s predictions. Our point here is to particle “ist mir wurst” – which Fine, Ref. 9, p. 38, trans- show that one can generate a new version of the Einstein’s lates as “I couldn’t care less.” Boxes argument by borrowing some of the logical steps also 33 Ref. 28, p. 180. used by Bell in his derivation of his inequalities. It will be 34 A. Einstein, from the “Reply to criticisms” section of crucial to the subsequent discussion that these two items Albert Einstein: Philosopher Scientist, edited by P. A. – the locality-completeness dilemma and Bell’s theorem – Schilpp (Harper and Row, 1959), p. 681. are distinct. 35 Ref. 9, pp. 37–38. 47 Given that there are, at this point, several distinct defi- 36 Ref. 1, p. 187. nitions of “completeness” and “locality” on the table, the 37 J. S. Bell, “La nouvelle cuisine” in Between Science and reader may begin to suspect that the dilemma we keep Technology, edited by A. Sarlemijn and P. Kroes (Elsevier running into is merely due to an improper definition of Science (North-Holland), 1990). Bell gives an equivalent one of these terms, and that the trouble could be avoided classical analog in “Bertlemann’s socks and the nature of with some clever new definitions. One such attempt will reality,” reprinted in Ref. 3. be considered in Sec. V. For now let me simply caution 38 David Bohm, Quantum Theory (Prentice Hall, 1951), pp. the reader against this attitude. The various definitions we 611–623. have seen are not truly distinct; they all focus on different 39 Gregor Weihs, Thomas Jennewein, Christoph Simon, Har- aspects of a single, shared concept of “completeness” (the ald Weinfurter, and Anton Zeilinger, “Violation of Bell’s idea that a complete theory shouldn’t leave anything out inequality under strict Einstein locality conditions” Phys. of its description of reality) and a single, shared concept of Rev. Lett. 81, 5039–5043 (1998). “locality” (that what exists/occurs “over here” shouldn’t 40 N. David Mermin, “Nonlocality and Bohr’s reply to EPR”, instantaneously affect what exists/occurs “over there”). In quant-ph/9712003. my opinion, the lesson of the various formulations we are 41 D. Styer, “Common misconceptions regarding quantum considering is not that new definitions might allow us to mechanics,” Am. J. Phys. 64, 31–34 (1996). elude the dilemma, but rather that any reasonable defi- 42 Asher Peres and Daniel R. Terno, “Quantum Information nitions give rise to the dilemma. (As we will see in the and Relativity Theory”, Rev. Mod. Phys. 76 (2004) 93. next section, a sufficiently radical redefinition in which the 43 E. Wigner, “The interpretation of quantum mechanics,” basic meaning is changed can “succeed” in removing the reprinted in Ref. 5. dilemma. But this isn’t true success; it’s cheating.) 44 Ref. 37, p. 109. 48 It is worth clarifying the logic here. The arguments we 45 Arthur Fine (private communication) has criticized this have been discussing for the locality-completeness dilemma derivation on the grounds that it equates factorizability (of show that, to maintain locality, one must regard the quan- the joint, double-detection probability) with locality. (See tum mechanical description of reality as incomplete. Bell’s also “The Shaky Game,” pp. 59–63 and “Do Correlations theorem then demonstrates that this project of saving lo- Need to be Explained?” in Philosophical Consequences of cality cannot succeed: no local hidden variable comple- Quantum Theory, edited by J. Cushing and E. McMullin tion of quantum mechanics can reproduce quantum the- (University of Notre Dame Press, Notre Dame, 1989) pp. ory’s predictions. Thus, Bell’s result, combined with the 175–194.) He argues that there is no reason (save a kind of quantum locality-completeness dilemma, shows that any primitive metaphysical bias) to insist that distant events theory which reproduces the predictions of quantum me- can only be statistically correlated if they are causally im- chanics – quantum mechanics itself most certainly included plicated by one another, that is, either one causes the other – must necessarily involve non-local actions-at-a-distance. or both share a prior common cause. I am not particularly Moreover, because Bell’s inequalities are empirically vio- swayed by this objection, because I believe it is a funda- lated – because the predictions of quantum mechanics are mental goal of science to unearth causal explanations of correct – these non-local actions-at-a-distance are known (persistent) observed correlations, and not something that to exist in nature. Of course, this leaves the completeness we should apologize for or dismiss as based on “metaphys- question unanswered. It doesn’t show that a non-local hid- ical bias.” But even if one were to grant this objection, den variable theory like Bohm’s is true – but merely that one must grant it equally as applied to both the deriva- whatever theory one favors (straight QM or a hidden vari- tion here of double-detection probabilities for the boxes able theory), the theory will have to be non-local in or- and to Bell’s original derivation of the famous inequality. der to agree with experiment. The motivation for consid- That is, standard quantum theory and the hidden variable ering non-local hidden variable theories like Bohm’s, then, theories are on equal footing. If Bell’s factorizability as- (as opposed to simply conceding that quantum mechan- sumption (applied to orthodox quantum theory) is invalid, ics is non-local but still complete) comes from the other then it is invalid applied to hidden variables theories as jobs (such as solving the measurement problem) the hid- well – in which case Bell’s theorem can not be used the den variables can do for us. See Bell’s article “Bertlemann’s way it typically is used – namely, as an argument against Socks and the Nature of Reality,”37 and Sheldon Goldstein, the hidden variables program. For then the empirical vio- “Bohmian mechanics,” Stanford Encyclopedia of Philos- lations of Bell’s inequalities simply would not count as an ophy, empirical refutation of local hidden variable theories. (Sec. 2 in particular) for an especially clear discussion of 46 To avoid possible confusion, let me just state explicitly this point. that this Bell-inspired derivation is not meant to show that 49 Werner Heisenberg, The Physical Principles of the Quan- Einstein’s Boxes is somehow equivalent to Bell’s theorem. tum Theory (Dover Publications, 1949), p. 39. It isn’t. The former establishes that quantum mechanics 50 See, for example, Abner Shimony, “Search for a worldview 15

which can accommodate our knowledge of microphysics,” a violation of Einstein’s bijective completeness condition: in Philosophical Consequences of Quantum Theory: Re- because the accounts of both observers are equally valid, flections on Bell’s Theorem, edited by James T. Cushing there are evidently two simultaneously valid but distinct and Ernan McMullin (University of Notre Dame Press, quantum mechanical descriptions of the same physical con- 1989). tents of box 2. (However, this argument involves a switch 51 Shimony has since acceded to the position taken by Bell to an occupation number representation in which we talk in the quote in the next paragraph. See: Abner Shimony, of the contents of box 2 rather than the state of the par- “Bell’s Theorem, ” Stanford Encyclopedia of Philosophy, < ticle per se; it probably therefore deserves further thought http://plato.stanford.edu/entries/bell-theorem/>. and scrutiny.) Whereas in the earlier formulations we had 52 Ref. 37, p. 111. arguments from locality to incompleteness, here we have 53 See Ref. 15 for a clear and systematic discussion of the an argument from the principle of relativity to incomplete- possible positions one might take on this issue of the con- ness. This argument, perhaps even more clearly than those sistency of quantum non-locality and relativity. presented in the main text, brings out the fact that, if 54 A. Ad´am,´ L. J´anossy, and P. Varga, Acta Phys. Hung. 4, quantum mechanics is regarded as complete, the collapse 301 [xx I don’t have either the title or exact page num- postulate is in deep conflict with relativity as such and not bers for this paper; it’s in an obscure journal, and I only merely with some narrow aspect thereof, for example, “lo- know of it via the reference in Clauser’s paper. xx] (1955). cality,” about whose definition Heisenberg and others are See also: Eric Brannen and H. I. S. Ferguson, Nature, Vol. wont to quibble. This point supports the claim made earlier 9 (1956) 481–482; John F. Clauser, “Experimental Dis- that the dilemma cannot be escaped simply by redefining tinction Between the Quantum and classical field-theoretic one or more of the terms involved. predictions for the photoelectric effect”, Phys. Rev. D 9, 59 A similar point is made in Chap. 5 of Ref. 15. 853–860 (1974); P. Grangier, G. Roger and A. Aspect, “Ex- 60 See, for example, Harvey R. Brown and Michael L. G. perimental evidence for a photon anti-correlation effect on Redhead, “A critique of the disturbance theory of inde- a beamsplitter,” Europhys. Lett., Vol. 1 (1986) 173-179; terminacy in quantum mechanics,” Found. Phys. 11, 1–20 G. Greenstein and A. G. Zajonc, “The Quantum Chal- (1981). lenge”, Jones and Bartlett Publishers, 1997, Chapter 2; 61 Christopher A. Fuchs and Asher Peres, for example, assert and J.J. Thorn, M.S. Neel, V.W. Donato, G.S. Bergreen, in a discussion of wave function collapse that “a wavefunc- R.E. Davies and M. Beck, “Observing the quantum behav- tion is only a mathematical expression for evaluating prob- ior of light in an undergraduate laboratory,” Am.J.Phys. abilities and depends on the knowledge of whoever is doing 72 1210-1219 (2004). the computing” and that “Collapse is something that hap- 55 John F. Clauser, “Early history of Bell’s theorem” in pens in our description of the system, not to the system Quantum [Un]speakables: From Bell to Quantum Informa- itself.” A. Fuchs and Asher Peres, Phys. Today, Vol. 53 (3), tion, edited by R. A. Bertlmann and A. Zeilinger (Springer, 70–71 (March 2000). 2002). 62 Ref. 28, p. 178. 56 Here, by “photons,” we mean, strictly speaking, the par- 63 See N. David Mermin, “What’s Wrong With This Quan- ticles that are actually detected. There is nothing here to tum World?”, Phys. Today, Vol. 57 (2), 10–11 (Feb. 2004). rule out the possibility that, in addition, there exists (say) 64 Some, however, definitely do take it seriously: see, for ex- a wave that splits in half at the mirror. This is the story ample, Ref. 61, and Ole Ulfbeck and Aage Bohr, “Genuine told by Bohm’s theory: the wave packet splits in half at fortuitousness. Where did that click come from?,” Found. the mirror and the photon particle ends up in one or the Phys. 31 (5), 757–774 (2001). other of the two packets. 65 H. P. Stapp makes this point in “The Copenhagen inter- 57 This is not surprising from the point of view outlined in pretation,” Am. J. Phys. 40, 1098–1116 (1972), p. 1108. Ref. 47. Because nature is non-local, it stands to reason Thanks to Sheldon Goldstein (private communication) for that all local theories would be inconsistent with experi- pointing out that Tim Maudlin also stresses this point in ment. “Space-time in the quantum world,” in Bohmian Mechan- 58 After reading an early draft of this paper, Roderich Tu- ics and Quantum Theory: An Appraisal, edited by James mulka (private communication) suggested another version T. Cushing, Arthur Fine, and Sheldon Goldstein (Kluwer of the argument in which there seems to be no explicit as- Academic Publishers, 1996), p. 305. Maudlin writes that sumption of locality: consider two different Lorentz frames “Physicists have been tremendously resistant to any claims (F and F ′), relative to which the observation of the con- of non-locality, mostly on the assumption (which is not a tents of box 1 is simultaneous with two different space- theorem) that non-locality is inconsistent with Relativity. ′ ′ time points (P and P , say, with P earlier than P ) on The calculus seems to be that one ought to be willing to the worldline of box 2. According to the collapse postulate pay any price – even the renunciation of pretensions to of orthodox quantum theory, an observer in F should at- accurately describe the world – to preserve the theory of tribute a definite particle content to box 2 between P and Relativity. But the only possible view that would make P ′, the measurement on box 1 having collapsed the wave sense of this obsessive attachment to Relativity is a thor- function in box 2 at P . However, according to an observer oughly realistic one! These physicists seem to be so certain ′ in F , box 2 remains in a superposition of containing and that Relativity is the last word in space-time structure that ′ not containing a particle until the event P . It seems this they are willing even to forego any coherent account of the disagreement can be put together with the principle of rel- entities that inhabit space-time.” ativity (according to which all Lorentz frames are equally valid platforms from which to describe reality) to yield