Kwalish Kid wrote:
Bell's Inequalities are a form of the Kochen-Specker examples of possible observables for a system where it is, mathematically, impossible to assign a precise value to every observable in the system. If one sticks to only assigning probabilities to results, then there is no problem. If one wants to account for the probabilities because something is going on "behind the scenes", then one is forced to include within this behind the scenes the flexibility that lets actions at one point in spacetime change the determined outcome at another point that is spacelike separated from the first. This is a violation of locality. However, if we can accept that certain observables are just spread over a certain spacelike region, then there is not necessarily any violation of locality. That is to say, certain systems are just not separate. When two particles are entangled, we must consider an observation on one to be part of the observable of both.

2. The Impossibility of Hidden Variables ... or the Inevitability of Nonlocality?

John von Neumann, one of the greatest mathematicians of the twentieth century, claimed to have mathematically proven that Einstein's dream, of a deterministic completion or reinterpretation of quantum theory, was mathematically impossible. He concluded that (von Neumann 1932, p. 325 of the English translation) It is therefore not, as is often assumed, a question of a re-interpretation of quantum mechanics — the present system of quantum mechanics would have to be objectively false, in order that another description of the elementary processes than the statistical one be possible.This claim of von Neumann was almost universally accepted among physicists and philosophers of science. For example, Max Born, who formulated the statistical interpretation of the wave function, assured us that (Born 1949, p. 109) No concealed parameters can be introduced with the help of which the indeterministic description could be transformed into a deterministic one. Hence if a future theory should be deterministic, it cannot be a modification of the present one but must be essentially different.

Bohmian mechanics is, quite clearly, a counterexample to the claims of von Neumann, so something has to be wrong with von Neumann's argument. In fact, according to John Bell (Mermin 1993, p. 805), von Neumann's assumptions (about the relationships among the values of quantum observables that must be satisfied in a hidden-variables theory) are so unreasonable that the "the proof of von Neumann is not merely false but foolish!" Nonetheless, some physicists continue to rely on von Neumann's proof, although in recent years it is more common to find physicists citing the Kochen-Specker Theorem and, more frequently, Bell's inequality as the basis of this refutation. We still find, a quarter of a century after the rediscovery of Bohmian mechanics in 1952, statements such as these (Wigner 1976):
The proof he [von Neumann] published ..., though it was made much more convincing later on by Kochen and Specker, still uses assumptions which, in my opinion, can quite reasonably be questioned. ... In my opinion, the most convincing argument against the theory of hidden variables was presented by J. S. Bell (1964).


Now there are many more statements of a similar character that could have been cited. This quotation owes its significance to the fact that Wigner was not only one of the leading physicists of his generation, but, unlike most of his contemporaries, he was also profoundly concerned with the conceptual foundations of quantum mechanics and wrote on the subject with great clarity and insight.

There was, however, one physicist who wrote on this subject with even greater clarity and insight than Wigner himself, namely the very J. S. Bell whom Wigner praises for demonstrating the impossibility of a deterministic completion of quantum theory such as Bohmian mechanics. Here's how Bell himself reacted to Bohm's discovery (Bell 1987, p. 160):
But in 1952 I saw the impossible done. It was in papers by David Bohm. Bohm showed explicitly how parameters could indeed be introduced, into nonrelativistic wave mechanics, with the help of which the indeterministic description could be transformed into a deterministic one. More importantly, in my opinion, the subjectivity of the orthodox version, the necessary reference to the ‘observer,’ could be eliminated. ...
But why then had Born not told me of this ‘pilot wave’? If only to point out what was wrong with it? Why did von Neumann not consider it? More extraordinarily, why did people go on producing ‘‘impossibility’’ proofs, after 1952, and as recently as 1978? ... Why is the pilot wave picture ignored in text books? Should it not be taught, not as the only way, but as an antidote to the prevailing complacency? To show us that vagueness, subjectivity, and indeterminism, are not forced on us by experimental facts, but by deliberate theoretical choice?

Wigner to the contrary notwithstanding, Bell did not establish the impossibility of a deterministic reformulation of quantum theory, nor did he ever claim to have done so. On the contrary, over the course of the past several decades, until his untimely death in 1990, Bell was the prime proponent, for a good part of this period almost the sole proponent, of the very theory, Bohmian mechanics, that he is supposed to have demolished.

Bohmian mechanics is of course as much a counterexample to the Kochen-Specker argument for the impossibility of hidden variables as it is to the one of von Neumann. It is obviously a counterexample to any such argument. However reasonable the assumptions of such an argument, some of them must fail for Bohmian mechanics.

Wigner was quite right to suggest that the assumptions of Kochen and Specker are more convincing than those of von Neumann. They appear, in fact, to be quite reasonable indeed . However, they are not. The impression that they are arises from a pervasive error, a naive realism about operators, that will be discussed below in the sections on quantum observables, on spin, and on contextuality.

One of the achievements of John Bell was to replace the "arbitrary axioms" (Bell 1987, page 11) of Kochen-Specker and others by an assumption of locality, of no action-at-a-distance. It would be hard to argue against the reasonableness of such an assumption, even if one were so bold as to doubt its inevitability. Bell showed that any hidden-variables formulation of quantum mechanics must be nonlocal, as, indeed, Bohmian mechanics is. But he showed much much more.

In a celebrated paper published in 1964, Bell showed that quantum theory itself is irreducibly nonlocal. This fact about quantum mechanics, based as it is on a short and mathematically simple analysis, could have been recognized soon after the discovery of quantum theory in the 1920's. That this did not happen is no doubt due in part to the obscurity of orthodox quantum theory and to the ambiguity of its commitments. It was, in fact, his examination of Bohmian mechanics that led Bell to his nonlocality analysis. In the course of his investigation of Bohmian mechanics he observed that (Bell 1987, p. 11):
in this theory an explicit causal mechanism exists whereby the disposition of one piece of apparatus affects the results obtained with a distant piece.
Bohm of course was well aware of these features of his scheme, and has given them much attention. However, it must be stressed that, to the present writer's knowledge, there is no proof that any hidden variable account of quantum mechanics must have this extraordinary character. It would therefore be interesting, perhaps, to pursue some further "impossibility proofs," replacing the arbitrary axioms objected to above by some condition of locality, or of separability of distant systems.

In a footnote, Bell added that "Since the completion of this paper such a proof has been found." This proof was published in his 1964 paper, "On the Einstein-Podolsky-Rosen Paradox," in which he derives Bell's inequality, the basis of his conclusion of quantum nonlocality. (For a discussion of how nonlocality emerges in Bohmian mechanics, see Section 13.)

It is worth stressing that Bell's analysis indeed shows that any account of quantum phenomena must be nonlocal, not just any hidden variables account. Bell showed that nonlocality is implied by the predictions of standard quantum theory itself. Thus if nature is governed by these predictions, then nature is nonlocal. [That nature is so governed, even in the crucial EPR-correlation experiments, has by now been established by a great many experiments, the most conclusive of which is perhaps that of Aspect (Aspect et al., 1982).]