4 EPR, Bohr, Bell, and two meanings of ’completeness’

Present research on foundations of QT is strongly influenced by a paper published in 1935 by Einstein, Podolsky, and Rosen (EPR) [9]. There is an enormous secondary literature, see e.g. Fine [11], Ballentine [1], Redhead [28], and the author’s website [16]. In this work, EPR claim that the quantum-mechanical description of reality is incomplete. The CI is attacked ’from inside’ because the basic assumptions used in this paper do not reflect the positions of the authors but are part of the CI. The most significant example is EPR’s assumption that ”The state of the particle is completely characterized by a wave function ψ  ”, a statement in sharp opposition to Einstein’s well-documented opinion that ψ  describes an ensemble. EPR’s conclusion was, of course, attacked by CI’s advocates who considered QT as a complete theory. However, a discussion of the specific EPR problem was generally avoided and EPR’s claim of incompleteness of QT was attacked on different routes - circumventing the specific problem. Bohr, in his reply, took a very philosophical, elusive route, which was not really convincing for many people.

According to the prevailing opinion this question was decided in favor of Bohr by the work of John Bell [54], about thirty years later. Bell circumvented the specific EPR problem by relating it to the problem of hidden variables. A (local) hidden variable theory is compatible with all predictions of QT providing, however, at the same time, a more detailed (deterministic) description of reality. Physical intuition tells us that such a thing cannot exist but Bell proved that it cannot exist - at least within the framework of his postulates; all no-go proofs are of course only valid within a certain ’universe of discourse’ (repeated remarks on this important limitation will be omitted from now on for brevity). He formulated general conditions for local hidden variable theories and derived therefrom an inequality which differs from the corresponding prediction of QT. Thus, he showed that hidden variable theories cannot exist if QT is correct. This shows that QT is a ’complete’ theory (with the meaning of ’complete’ given in context). This reasoning seems correct but the question is what can be concluded from it. The simplest conclusion is that EPR’s proof of incompleteness of QT cannot be true because Bell showed that QT is complete. I claim that this simple reasoning is not justified because a subtle semantic trap, concerning the meaning(s) of the word ’complete’, has been overlooked.

The word complete has two different meanings. If used to characterize the predictive power of a physical theory it means: ”All facts that can be observed can be predicted (with certainty)”. This kind of completeness could be called ’predictive completeness’, or ’p-completeness’ for brevity. In order to find out if a physical theory is p-complete one needs solutions (predictions) of this theory and experiments testing these predictions. This first kind of completeness may equivalently be characterized by saying that an ’individuality interpretation’ for this (p-complete) theory exists. The standard example for a p-complete theory is classical mechanics. Classical massless field theories are of a similar nature but do not directly describe individual particles.

The second meaning of the word complete can be described as follows: ”No better theory, in the sense of producing more ’definite’ (deterministic) predictions, exists”. This is a very strong assertion. It entails not only the concrete physical theory under discussion but also an infinite number of other theories (all unknown), which are all not allowed to exist according to the assertion. Such an assertion can of course only be verified within a certain ’universe of discourse’, which may possibly be generalized in later steps. But it can be approached nevertheless. Let us call this second kind of completeness ’metaphysical completeness’, or ’m-completeness’. As an example, we mention classical probabilistic theories where the uncertainty is only in the initial conditions while the movement in phase space is deterministic [19]. These theories are m-incomplete, with classical mechanics playing the role of the ’better theory’.

How are these two kinds of completeness related to each other ? A p-complete theory is also m-complete. It would not make sense to search for a better theory than classical mechanics (in its range of validity) because classical mechanics makes already predictions with probability equal to one. This means, the implication
p-completeness  ⇒  m -completeness,

holds true. This means that m-incompleteness implies p-incompleteness and that p-incompleteness is a necessary condition for m-incompleteness. On the other hand p-incompleteness is not a sufficient condition for m-incompleteness. A p-incomplete theory may be either m-complete or m-incomplete.

Let us now reconsider the EPR-Bohr-Bell question using this refined vocabulary. What Bell proved is obviously m-completeness of QT. In EPR’s paper both kinds of completeness occur. In the last paragraph EPR express their believe that QT is m-incomplete:

”We believe however that such a [more complete] theory is possible”

The communication of EPR’s ’believe’ (which had been known for a long time) is of course not the central message of EPR. The central message is given by the logical deductions reported in the body of the paper, i.e. in the whole paper except the last paragraph. So, which kind of completeness is referred to in the body of EPR’s paper ? The subject of the paper is the problem of predictions of the values of certain observables, thus what EPR mean by completeness is obviously p-completeness. An assertion of m-incompleteness, i.e. a statement that a better theory then QT must exist, can nowhere be found in the relevant part (the body) of EPR’s paper.

The proof of p-incompleteness of QT was of course a necessary prerequisite for EPR’s ’believe’ in a deterministic replacement of QT. If an analysis had led to the conclusion that QT is p-complete, this had also implied that QT is m-complete. On the other hand, EPR were aware of the fact that p-incompleteness is only a necessary and not a sufficient condition for m-incompleteness. Thus, they were aware of the fact that their proof of p-incompleteness of QT did not imply m-incompleteness of QT. They express this in the first sentence of the last paragraph of their paper in a very clear way:

”While we have thus shown that the wave function does not provide a complete description of physical reality, we left open the question of whether or not such a description exists”.

It is a real mystery why this clear statement, separating cleanly the two different kinds of completeness (made even more explicit in the present essay only by means of different names) from each other, has been overlooked by the scientific community.

It follows that Bell’s proof of m-completeness cannot be used, even if we accept the assumptions underlying his proof [12], as an argument against EPR’s proof of p-incompleteness of QT. Bell was in error, when claiming that his results contradict EPR ! If we accept both Bell’s and EPR’s findings we arrive at the final conclusion that QT is p-incomplete and m-complete. This conclusion is compatible with a recent derivation of non-relativistic quantum theory from statistical postulates [1923] and agrees roughly with the common sense assessment of QT as a correct (complete) statistical (incomplete) theory. It implies that an individuality interpretation of QT is not justified.

The present conclusion can only be avoided if one takes a deterministic point of view of the world, namely that everything that can be observed must in principle be predictable. This then means that p-incompleteness implies m-incompleteness (or the existence of a hidden variable theory). But note that this implication, which eliminates our final conclusion, is not a logical requirement but the consequence of a new (in fact, very old) philosophical dogma. From the point of view of physics such a deterministic dogma is not required. Interestingly, this deterministic point of view was shared by Einstein and Bohr (see [16] for a more detailed explanation), despite their otherwise very different opinions. It was denied by several other authors, in particular by Popper [25]. Unfortunately, today’s discussions are still centered around the two alternatives represented by Einstein and Bohr.