At the present time, most physicists continue to hold a skeptical attitude toward the proposition of a `hidden variables' interpretation of quantum theory, in spite of David Bohm's successful construction of such a theory and John S. Bell's strong arguments in favor of the idea. Many are convinced either that it is impossible to interpret quantum theory in this way, or that such an interpretation would actually be irrelevant. There are essentially two reasons behind such doubts. The first concerns certain mathematical theorems (von Neumann's, Gleason's, Kochen and Specker's, and Bell's) which can be applied to the hidden variables issue. These theorems are often credited with proving that hidden variables are indeed `impossible', in the sense that they cannot replicate the predictions of quantum mechanics. Many who do not draw such a strong conclusion nevertheless accept that hidden variables have been shown to exhibit prohibitively complicated features. The second reason hidden variables are disregarded is that the most sophisticated example of a hidden variables theory---that of David Bohm---exhibits nonlocality, i.e., it can happen in this theory that the consequences of events at one place propagate to other places instantaneously. However, as we shall show in the present work, neither the mathematical theorems in question nor the attribute of nonlocality serve to detract from the importance of a hidden variables interpretation of quantum theory. The theorems imply neither that hidden variables are impossible nor that they must be overly complex. As regards nonlocality, this feature is present in quantum mechanics itself, and is a required characteristic of any theory that agrees with the quantum mechanical predictions.
In the present work, the hidden variables issue is addressed in the following ways. We first discuss the earliest analysis of hidden variables---that of von Neumann's theorem---and review John S. Bell's refutation of von Neumann's `impossibility proof'. We recall and elaborate on Bell's arguments regarding the theorems of Gleason, and Kochen and Specker. According to Bell, these latter theorems do not imply that hidden variables interpretations are untenable, but instead that such theories must exhibit contextuality, i.e., they must allow for the dependence of measurement results on the characteristics of both measured system and measuring apparatus. We demonstrate a new way to understand the implications of both Gleason's theorem and Kochen and Specker's theorem by noting that they prove a result we call ``spectral incompatibility''. We develop further insight into the concepts involved in these two theorems by investigating a special quantum mechanical experiment which was first described by David Albert. We review the Einstein--Podolsky--Rosen paradox, Bell's theorem, and Bell's later argument that these imply that quantum mechanics is irreducibly nonlocal. We present this discussion in a somewhat more gradual fashion than does Bell so that the logic of the argument may be more transparent.
The paradox of Einstein, Podolsky, and Rosen was generalized by Erwin Schrödinger in the same paper where his famous `cat paradox' appeared. We develop several new results regarding this generalization. We show that Schrödinger's conclusions can be derived using a simpler argument---one which makes clear the relationship between the quantum state and the `perfect correlations' exhibited by the system. We use Schrödinger's EPR analysis to derive a wide variety of new quantum nonlocality proofs. These proofs share two important features with that of Greenberger, Horne, and Zeilinger. First, they are of a deterministic character, i.e., they are `nonlocality without inequalities' proofs. Second, as we shall show, the quantum nonlocality results we develop may be experimentally verified in such a way that one need only observe the `perfect correlations' between the appropriate observables; no further tests are required. This latter feature serves to contrast these proofs with EPR/Bell nonlocality, the laboratory confirmation of which demands not only the observation of perfect correlations, but also an additional set of observations, namely those required to test whether `Bell's inequality' is violated. The `Schrödinger nonlocality' proofs we give differ from the GHZ proof in that they apply to two-component composite systems, while the latter involves a composite system of at least three-components. In addition, some of the Schrödinger proofs involve classes of observables larger than that addressed in the GHZ proof.
Douglas L. Hemmick, Ph.D.
© 1996
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