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The history of Bell's inequality



Tests of Bell's inequality are essential to determining if my approach is correct and to motivating the research need to develop my approach. Thus it is useful to briefly review the history that led to Bell's inequality.

In the early 1930's von Neumann  published a paper that claimed to prove that no more complete theory could be consistent with the predictions of quantum mechanics[33,34]. Two years later Einstein, Podolsky and Rosen, in a classical paper frequently referred to today as EPR  , argued that quantum mechanics is an incomplete theory because properties that are conserved absolutely such as momentum must have some objective reality to them and does not model this objective reality[15]. Einstein later argued in a famous series of debates with Bohr 

that the that there must be additional hidden variables that provide a more complete description of a quantum mechanical system. In the early 1950's Bohm  published what he called a ```hidden'' variables theory' that was nonlocal but deterministic and consistent with the predictions of [6]. This suggested there was something wrong with von Neumann's proof. At the time Bohm felt that his theory was different than at some point and thus escaped von Neumann's proof but this apparently is not true and it is the nonlocal nature of Bohm's theory that allows it to reproduce the predictions of in a deterministic model.

In the 1960's Bell 

showed that von Neumann's assumptions were too restrictive[4]. Bell felt it was not possible to prove the general result von Neumann claimed and started looking for additional constraints that might allow one to distinguish between and hidden variable theories. Influenced by Bohm's work, he developed an inequality that the statistics of events with a space-like separation must obey in any local hidden variables theory. (Events are space-like  separated if they are close enough in time and far enough apart in space so that no signal traveling at the speed of light can go from one to the other.)

Bell proved predicts this inequality is violated[3]. In the 1970's Eberhard 

derived Bell's result without reference to local hidden variable theory, it applies to all local theories[12,13]. Eberhard also showed that the nonlocal effects predicts cannot be used for superluminal communication. Eberhard assumed a statistical law known is contrafactual definiteness in his derivation. That is he assumed that he could average over all the possible outcomes of a single experiment. Previously Stapp had argued that Bell's inequality was evidence that this property does not hold for quantum mechanical effects[30]. This assumption is a standard one in statistical analysis. Some argue that the uncertainty principle is a reason for doubting this principle for quantum effects. How can there be a definite outcome for an experiment not performed in ? The reason one can make such arguments is the claim that probabilities are irreducible in . There is no mathematics of irreducible probabilities and reason to doubt that one can develop such mathematics. Thus one can argue that probabilities in are something different than in classical mechanics and completely outside of any known mathematics. In theory one can ascribe any property one wants to probabilities in . Thus, for example, Youssef has argued that probabilities in are complex and not real valued[40].

I doubt all of this. I think the difference between classical and is in the conceptual framework of the theories as discussed in Section gif on page gif

and not in basic principles of mathematics or statistics.

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