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Journal of Changzhou Institute of Technology

A Study on Bell's Theorem

Tan Tian-rong

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

This paper proves that Bell's inequality is originated from classical probability theory,and thereby is related neither to locality nor to hidden variables.But it is also possible to derive spin correlation formula in quantum mechanics according to the classical probabilistic theory.This fact indicates that on cer- tain conditions,classical probabilistic theory would be inconsistent with quantum mechanics,but under other conditions,classical probabilistic theory and quantum mechanics would,reach the same goal by different means.Still further,this paper confirms that all of the probability operation rules in classical probability theo- ry,containing joint probability operation rules,are suitable for micro processes,but the event operation rules in classical probability theory are unsuitable.The reason obtaining Bell's inequality lies in the steps applying Boolean algebra rules on the non-Boolean micro event space.

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This paper proves that Bell's inequality is originated from classical probability theory,and thereby is related neither to locality nor to hidden variables.But it is also possible to derive spin correlation formula in quantum mechanics according to the classical probabilistic theory.This fact indicates that on cer- tain conditions,classical probabilistic theory would be inconsistent with quantum mechanics,but under other conditions,classical probabilistic theory and quantum mechanics would,reach the same goal by different means.Still further,this paper confirms that all of the probability operation rules in classical probability theo- ry,containing joint probability operation rules,are suitable for micro processes,but the event operation rules in classical probability theory are unsuitable.The reason obtaining Bell's inequality lies in the steps applying Boolean algebra rules on the non-Boolean micro event space.
Bell’s inequality can be traced back to a classical probabilistic formula that is invalid clearly, while spin correlation formula in quantum mechanics to a formula that has confirmed both by facts and quantum mechanics. Bell’s two misunderstandings about Bell’s theorem are pointed out: Firstly, the hidden variable theory Bell’s used, which is a special form of such theories, is regarded as a general one. Secondly, there are two theses obtainable from Bell’s hypotheses; one is compatible with quantum mechanics, the other leads to Bell's inequality. Unfortunately, the former is regarded as a property of local hidden variable theory, while the latter as self-evident. G.Lochak has revealed the first thesis, and thereby he pointed out that Bell’s inequality has nothing to do with locality, but he did not find the second one, so that he still analyzed this problem starting from hidden variable theory, which is actually without regard to Bell’s inequality.
It is proved that Bell's inequality can be traced back to a classical probabilistic formula that is invalid clearly, while the spin correlation formula in quantum mechanics to a formula that has confirmed both by facts and quantum mechanics. Two misunderstandings about Bell's theorem are pointed out: Firstly, the hidden variable theory Bell's used is a special one, but it is regarded as a general form of such theories. Secondly, there are two outcomes obtainable from Bell's hypotheses: one is compatible with quantum mechanics and the other leads to Bell's inequality. Unfortunately, the former is regarded as a property of local hidden variable theory, while the latter as self-evident. G. Lochak has revealed the first misunderstanding, and thereby he pointed out that Bell's inequality has nothing to do with locality, but he did not find the second one, so that he still analyzed this problem starting from hidden variable theory, which is actually irrelative to Bell's inequality. (The Journal of American Science. 2005;1(2):42-50).
It is pointed that Bell had two misunderstandings for his work: Firstly,the hidden variable theory that he applied for deriving Bell's inequality keeps wholly intact the operating rules of classical probabilistic theory,but he regarded it as the general local hidden variable theory.Secondly,in Bell's work,the promise actually deriving Bell's inequality is not the one that Bell regarded as the character of local hidden variable theory.Also,other versions can prove Bell's inequality without local hidden variable theory and the Lochak's objection against Bell's theory are examined.
It is not generally known that the inequality that Bell derived using three random variables must be identically satisfied by any three corresponding data sets of ±1’s that are writable on paper. This surprising fact is not immediately obvious from Bell’s inequality derivation based on causal random variables, but follows immediately if the same mathematical operations are applied to finite data sets. For laboratory data, the inequality is identically satisfied as a fact of pure algebra, and its satisfaction is independent of whether the processes generating the data are local, non-local, deterministic, random, or nonsensical. It follows that if predicted correlations violate the inequality, they represent no three cross-correlated data sets that can exist, or can be generated from valid probability models. Reported data that violate the inequality consist of probabilistically independent data-pairs and are thus inconsistent with inequality derivation. In the case of random variables as Bell assumed, the correlations in the inequality may be expressed in terms of the probabilities that give rise to them. A new inequality is then produced: The Wigner inequality, that must be satisfied by quantum mechanical probabilities in the case of Bell experiments. If that were not the case, predicted quantum probabilities and correlations would be inconsistent with basic algebra.
Using CH model of Bell’s theorem and only by applying the conditional probability definition in ”classical ” probability theory,without imposing the locality condition,we are able to show that the stochastic realistic (CH) model cannot reproduce all of the predictions of quantum mechanics. PACS number: 03.65.Bz It is generally claimed that Bell inequalities are inconsistent with experimental data and with the predictions of quantum mechanics.A condition often used in the derivation of Bell inequalities is the so-called locality condition.The temptation is to attribute Bell inequalities violation to the violation of locality. In this paper we are going
The Bell theorem and inequality were derived as consequences of seemingly reasonable physical and statistical hypotheses. Bell’s assumptions were used to deduce cross-correlations of three spin measurements on two entangled particles neglecting non-commutation. The assumed correlation functions, later confirmed for certain quantum measurements, violate the Bell inequality. The present paper reviews a more general derivation of the Bell inequality showing that it is identically satisfied by finite data sets whether deterministic or random, after assuming merely that they exist. It is thereafter concerned with the consequences of this result for interpretations of the inequality that result in its violation. A primary finding is that correlation functions have differing forms due to quantum commutation, non-commutation, and conditions of measurement, and result in satisfaction of the Bell inequality used consistently with its derivation. A stochastic process having the same correlation function for all variable pairs is shown to be inconsistent with experimentally reported data. The logic of the three and four variable inequalities is shown to be similar. Finally the inequalities in probabilities are shown to follow from those in correlations with quantum mechanical results satisfying either when properly implemented.