Einstein’s boxes

Max Jammer, The Philosophy of Quantum Mechanics (Wiley, New York, 1974), pp. 109–114.

David

Bohm

, “,” , – ().

J. S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge U.P., New York, 1987).

Erwin

Schrödinger

, “, in , – (), reprinted in Ref. 5.

John A. Wheeler and Wojciech H. Zurek, Quantum Theory and Measurement (Princeton U.P., New York, 1983).

For a general survey of Einstein’s opposition to quantum theory, including some discussion of the boxes thought experiment, see

R.

Deltete

and

R.

Guy

, “,” , – ().

Niels Bohr, “Discussions with Einstein on epistemological problems in atomic physics,” in Albert Einstein: Philosopher-Scientist, edited by P. A. Schilpp (The Library of Living Philosophers, Evanston, 1949), pp. 200–241.

A.

Einstein

,

B.

Podolsky

, and

N.

Rosen

, “,” , – ().

Credit for this name should probably go to Arthur Fine, who used the phrase in passing in The Shaky Game (The University of Chicago Press, Chicago, 1996), p. 37.

Lucien

 

Hardy

presented an equally clear account of the thought experiment in a recent paper: “,”  (), – ().

For an interesting theoretical discussion of some potential practical difficulties with this experiment, see

Julio

 

Gea-Banacloche

, “,”  (), – ();

T.

Norsen

, “,” (7), ().

L. de Broglie, The Current Interpretation of Wave Mechanics: A Critical Study (Elsevier, Amsterdam, 1964). Note also that this passage comes just after a discussion of the thought experiment given by Einstein at the 1927 Solvay conference and that de Broglie introduced the discussion with this comment: “In a recent article… I have revived the above objections [Einstein’s] in a new form.”

The article is

L. de

Broglie

, “,” , ().

N.

Bohr

, “,” , – ().

This is one very plausible and influential interpretation of Bohr’s reply; see, for example, Ref. 1, p. 196. But other commentators have understood Bohr’s rather obscure prose differently or not at all.

See, for example, the comments of J. S. Bell, Ref. 3, pp. 155–156.

Actually, as Tim Maudlin, Quantum Non-Locality and Relativity (Blackwell, Malden, MA, 1994), pp. 140–141 has eloquently pointed out, neither does the original EPR thought experiment. The fact that one can, in the EPR situation, predict with certainty just the position of particle 2 (without disturbing it in any way) is sufficient, according to the reasoning set up by EPR, to conclude that, if local, quantum theory is incomplete. For that alone means that the position of particle 2 is an element of reality, and one that has no counterpart in the original, pre-measurement entangled wave function. Maudlin refers to the attempt to (in addition) beat the uncertainty principle by also establishing the real existence of particle 2’s momentum as “an unnecessary bit of grandstanding” which plunged “the previously simple EPR argument into the muddy waters of [modal logic and counterfactual definiteness].” As we have seen, it is precisely this question of counterfactuals that Bohr seems to have jumped on in his reply. In any case, Einstein’s Boxes (which completely avoids the issue of counterfactual definiteness) should help clarify this point.

Reference 9, p. 35.

See, for example, Ref. 1, pp. 115–117; Ref. 11, pp. 24–26.

L. E.

Ballentine

, “,” , – ().

L. E. Ballentine (Ref. 18) translates the French from “Electrons et Photons,” Institut International de Physique Solvay, Rapports et Discussions du Cinqui’eme Conseil de Physique (Gauthier-Villars, Paris, 1928), pp. 253–256.

Reference 1, pp. 24–33.

Reference 1, pp. 44, 160–162.

Reference 1, p. 116.

Reference 1, p. 116. See also Ref. 18, p. 1765.

Reference 9, Chap. 3.

Reference 9, p. 35.

Reference 9, p. 69.

Reference 9, p. 69.

For a complete list of references, see

Don

Howard

, “,” , – ().

Reference 28, pp. 178–179.

Reference 9, p. 71.

Reference 9, Chap. 5.

Einstein wrote that the question of establishing elements of reality for both position and momentum of the distant particle “ist mir wurst”—which Fine, Ref. 9, p. 38, translates as “I couldn’t care less.”

Reference 28, p. 180.

A. Einstein, from the “Reply to criticisms” section of Albert Einstein: Philosopher Scientist, edited by P. A. Schilpp (Harper and Row, 1959), p. 681.

Reference 9, pp. 37–38.

Reference 1, p. 187.

J. S. Bell, “La nouvelle cuisine,” in Between Science and Technology, edited by A. Sarlemijn and P. Kroes [Elsevier Science (North-Holland), North Holland, 1990]. Bell gives an equivalent classical analog in “Bertlemann’s socks and the nature of reality,” reprinted in Ref. 3.

David Bohm, Quantum Theory (Prentice Hall, Englewood Cliffs, NJ, 1951), pp. 611–623.

Gregor

Weihs

,

Thomas

Jennewein

,

Christoph

Simon

,

Harald

Weinfurter

, and

Anton

Zeilinger

, “,” , – ().

N.

David Mermin

, “,” quant-ph/9712003.

D.

Styer

, “,” , – ().

Asher

Peres

and

Daniel R.

Terno

, “,” , – ().

E. Wigner, “The interpretation of quantum mechanics,” reprinted in Ref. 5.

Reference 37, p. 109.

Arthur Fine (private communication) has criticized this derivation on the grounds that it equates factorizability (of the joint, double-detection probability) with locality. (See also “The shaky game,” pp. 59–63 and “Do correlations need to be explained?,” in Philosophical Consequences of Quantum Theory, edited by J. Cushing and E. McMullin (University of Notre Dame Press, Notre Dame, 1989), pp. 175–194.) He argues that there is no reason (save a kind of primitive metaphysical bias) to insist that distant events can only be statistically correlated if they are causally implicated by one another, that is, either one causes the other or both share a prior common cause. I am not particularly swayed by this objection, because I believe it is a fundamental goal of science to unearth causal explanations of (persistent) observed correlations, and not something that we should apologize for or dismiss as based on “metaphysical bias.” But even if one were to grant this objection, one must grant it equally as applied to both the derivation here of double-detection probabilities for the boxes and to Bell’s original derivation of the famous inequality. That is, standard quantum theory and the hidden variable theories are on equal footing. If Bell’s factorizability assumption (applied to orthodox quantum theory) is invalid, then it is invalid applied to hidden variables theories as well—in which case Bell’s theorem cannot be used the way it typically is used—namely, as an argument against the hidden variables program. For then the empirical violations of Bell’s inequalities simply would not count as an empirical refutation of local hidden variable theories.

To avoid possible confusion, let me just state explicitly that this Bell-inspired derivation is not meant to show that Einstein’s boxes is somehow equivalent to Bell’s theorem. It isn’t. The former establishes that quantum mechanics itself cannot be both complete and local. The latter establishes that no local hidden variable theory can agree with quantum theory’s predictions. Our point here is to show that one can generate a new version of the Einstein’s Boxes argument by borrowing some of the logical steps also used by Bell in his derivation of his inequalities. It will be crucial to the subsequent discussion that these two items—the locality-completeness dilemma and Bell’s theorem—are distinct.

Given that there are, at this point, several distinct definitions of “completeness” and “locality” on the table, the reader may begin to suspect that the dilemma we keep running into is merely due to an improper definition of one of these terms, and that the trouble could be avoided with some clever new definitions. One such attempt will be considered in Sec. V. For now let me simply caution the reader against this attitude. The various definitions we have seen are not truly distinct; they all focus on different aspects of a single, shared concept of “completeness” (the idea that a complete theory shouldn’t leave anything out of its description of reality) and a single, shared concept of “locality” (that what exists/occurs “over here” shouldn’t instantaneously affect what exists/occurs “over there”). In my opinion, the lesson of the various formulations we are considering is not that new definitions might allow us to elude the dilemma, but rather that any reasonable definitions give rise to the dilemma. (As we will see in the next section, a sufficiently radical redefinition in which the basic meaning is changed can “succeed” in removing the dilemma. But this isn’t true success; it’s cheating.)

It is worth clarifying the logic here. The arguments we have been discussing for the locality-completeness dilemma show that, to maintain locality, one must regard the quantum mechanical description of reality as incomplete. Bell’s theorem then demonstrates that this project of saving locality cannot succeed: no local hidden variable completion of quantum mechanics can reproduce quantum theory’s predictions. Thus, Bell’s result, combined with the quantum locality-completeness dilemma, shows that any theory which reproduces the predictions of quantum mechanics—quantum mechanics itself most certainly included—must necessarily involve non-local actions-at-a-distance. Moreover, because Bell’s inequalities are empirically violated—because the predictions of quantum mechanics are correct—these non-local actions-at-a-distance are known to exist in nature. Of course, this leaves the completeness question unanswered. It doesn’t show that a non-local hidden variable theory like Bohm’s is true—but merely that whatever theory one favors (straight QM or a hidden variable theory), the theory will have to be non-local in order to agree with experiment. The motivation for considering non-local hidden variable theories like Bohm’s, then, (as opposed to simply conceding that quantum mechanics is non-local but still complete) comes from the other jobs (such as solving the measurement problem) the hidden variables can do for us. See Bell’s article “Bertlemann’s socks and the nature of reality,” (Ref. 37) and Sheldon Goldstein, “Bohmian mechanics,” Stanford Encyclopedia of Philosophy, 〈http://plato.stanford.edu/entries/qm-bohm/〉 (Sec. 2 in particular) for an especially clear discussion of this point.

Werner Heisenberg, The Physical Principles of the Quantum Theory (Dover, New York, 1949), p. 39.

See, for example, Abner Shimony, “Search for a worldview which can accommodate our knowledge of microphysics,” in Philosophical Consequences of Quantum Theory: Reflections on Bell’s Theorem, edited by James T. Cushing and Ernan McMullin (University of Notre Dame Press, 1989).

Shimony has since acceded to the position taken by Bell in the quote in the next paragraph. See Abner Shimony, “Bell’s theorem,” Stanford Encyclopedia of Philosophy, 〈http://plato.stanford.edu/entries/bell-theorem/〉.

Reference 37, p. 111.

See Ref. 15 for a clear and systematic discussion of the possible positions one might take on this issue of the consistency of quantum non-locality and relativity.

A.

 

Ádám

,

L.

 

Jánossy

, and

P.

 

Varga

,  , ().

See also

Eric

 

Brannen

and

H. I. S.

 

Ferguson

,  , – ();

John F.

 

Clauser

, “,”  , – ();

P.

 

Grangier

,

G.

 

Roger

, and

A.

 

Aspect

, “,”  , – ();

G. Greenstein and A. G. Zajonc, The Quantum Challenge (Jones and Bartlett, Sudbury, MA, 1997), Chap.2, and

J. J.

Thorn

,

M. S.

Neel

,

V. W.

Donato

,

G. S.

Bergreen

,

R. E.

Davies

, and

M.

Beck

, “,” , – ().

John F. Clauser, “Early history of Bell’s theorem” in Quantum [Un]speakables: From Bell to Quantum Information, edited by R. A. Bertlmann and A. Zeilinger (Springer, New York, 2002).

Here, by “photons,” we mean, strictly speaking, the particles that are actually detected. There is nothing here to rule out the possibility that, in addition, there exists (say) a wave that splits in half at the mirror. This is the story told by Bohm’s theory: the wave packet splits in half at the mirror and the photon particle ends up in one or the other of the two packets.

This is not surprising from the point of view outlined in Ref. 47. Because nature is non-local, it stands to reason that all local theories would be inconsistent with experiment.

After reading an early draft of this paper, Roderich Tumulka (private communication) suggested another version of the argument in which there seems to be no explicit assumption of locality: consider two different Lorentz frames $(F$ and $F′)$⁠, relative to which the observation of the contents of box 1 is simultaneous with two different space–time points $(P$ and $P′,$ say, with $P$ earlier than $P′)$ on the worldline of box 2. According to the collapse postulate of orthodox quantum theory, an observer in $F$ should attribute a definite particle content to box 2 between $P$ and $P′,$ the measurement on box 1 having collapsed the wave function in box 2 at $P.$ However, according to an observer in $F′,$ box 2 remains in a superposition of containing and not containing a particle until the event $P′.$ It seems this disagreement can be put together with the principle of relativity (according to which all Lorentz frames are equally valid platforms from which to describe reality) to yield a violation of Einstein’s bijective completeness condition: because the accounts of both observers are equally valid, there are evidently two simultaneously valid but distinct quantum mechanical descriptions of the same physical contents of box 2. (However, this argument involves a switch to an occupation number representation in which we talk of the contents of box 2 rather than the state of the particle per se; it probably therefore deserves further thought and scrutiny.) Whereas in the earlier formulations we had arguments from locality to incompleteness, here we have an argument from the principle of relativity to incompleteness. This argument, perhaps even more clearly than those presented in the main text, brings out the fact that, if quantum mechanics is regarded as complete, the collapse postulate is in deep conflict with relativity as suchand not merely with some narrow aspect thereof, for example, “locality,” about whose definition Heisenberg and others are wont to quibble. This point supports the claim made earlier that the dilemma cannot be escaped simply by redefining one or more of the terms involved.

A similar point is made in Chap. 5 of Ref. 15.

See, for example,

Harvey R.

Brown

and

Michael L. G.

Redhead

, “,” , – ().

Christopher A.

Fuchs

and

Asher

Peres

, for example, assert in a discussion of wave function collapse that “a wavefunction is only a mathematical expression for evaluating probabilities and depends on the knowledge of whoever is doing the computing” and that “, (), – ().

Reference 28, p. 178.

See

N. David

Mermin

, “,” (), – ().

Some, however, definitely do take it seriously: see, for example, Ref. 61, and

Ole

Ulfbeck

and

Aage

Bohr

, “,” (), – ().

H. P.

 

Stapp

makes this point in “,”  , – (), p. 1108.

Thanks to Sheldon Goldstein (private communication) for pointing out that Tim Maudlin also stresses this point in “Space-time in the quantum world,” in Bohmian Mechanics and Quantum Theory: An Appraisal, edited by James T. Cushing, Arthur Fine, and Sheldon Goldstein (Kluwer Academic, Dordrecht, 1996), p. 305. Maudlin writes that “Physicists have been tremendously resistant to any claims of non-locality, mostly on the assumption (which is not a theorem) that non-locality is inconsistent with Relativity. The calculus seems to be that one ought to be willing to pay any price—even the renunciation of pretensions to accurately describe the world—to preserve the theory of Relativity. But the only possible view that would make sense of this obsessive attachment to Relativity is a thoroughly realistic one! These physicists seem to be so certain that Relativity is the last word in space-time structure that they are willing even to forego any coherent account of the entities that inhabit space-time.”