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Bell's Inequality is Violated in Classical Systems as Well as Quantum Systems

Shiro Ishikawa

Published Oct 1, 2017• 0 citations
OpenAlex

Abstract

Bell's inequality is usually considered to belong to mathematics and not quantum mechanics. We think that this makes it difficult to understand Bell's theory. Thus in this paper, contrary to Bell's spirit (which inherits Einstein's spirit), we try to discuss Bell's inequality in the framework of quantum theory with the linguistic Copenhagen interpretation. And we clarify that whether or not Bell's inequality holds does not depend on whether classical systems or quantum systems, but depend on whether a kind of simultaneous measurements exist or not. And further we assert that our argument ( based on the linguistic Copenhagen interpretation) should be regarded as a scientific representation of Bell's philosophical argument (based on Einstein's spirit).

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Bell's inequality is usually considered to belong to mathematics and not quantum mechanics. We think that this makes it difficult to understand Bell's theory. Thus in this paper, contrary to Bell's spirit (which inherits Einstein's spirit), we try to discuss Bell's inequality in the framework of quantum theory with the linguistic Copenhagen interpretation. And we clarify that whether or not Bell's inequality holds does not depend on whether classical systems or quantum systems, but depend on whether a kind of simultaneous measurements exist or not. And further we assert that our argument ( based on the linguistic Copenhagen interpretation) should be regarded as a scientific representation of Bell's philosophical argument (based on Einstein's spirit).
Bell's inequality itself is usually considered to belong to mathematics and not quantum mechanics.We think that this is making our understanding of Bell' theory be confused.Thus in this paper, contrary to Bell's spirit (which inherits Einstein's spirit), we try to discuss Bell's inequality in the framework of quantum theory with the linguistic Copenhagen interpretation.And we clarify that the violation of Bell's inequality (i.e., whether or not Bell's inequality holds) does not depend on whether classical systems or quantum systems, but depend on whether a combined measurement exists or not.And further we conclude that our argument (based on the linguistic Copenhagen interpretation) should be regarded as a scientific representation of Bell's philosophical argument (based on Einstein's spirit).
About twenty years ago, we proposed the mathematical formulation of Heisenberg's uncertainty principle, and further, we concluded that Heisenberg's uncertainty principle and EPR-paradox are not contradictory. This is true, however we now think that we should have argued about it under a certain firm interpretation of quantum mechanics. Recently we proposed the linguistic quantum interpretation (called quantum and classical measurement theory), which was characterized as a kind of metaphysical and linguistic turn of the Copenhagen interpretation. This turn from physics to language does not only extend quantum theory to classical systems but also yield the quantum mechanical world view (i.e., the philosophy of quantum mechanics, in other words, quantum philosophy). In fact, we can consider that traditional philosophies have progressed toward quantum philosophy. In this paper, we first review the linguistic quantum interpretation, and further, clarify the relation between EPR-paradox and Heisenberg's uncertainty principle. That is, the linguistic interpretation says that EPR-paradox is closely related to the fact that syllogism does not generally hold in quantum physics. This fact should be compared to the non-locality of Bell's inequality.
The Copenhagen interpretation is the most authorized interpretation of quantum mechanics, but there are a number of ideas that are associated with the Copenhagen interpretation. It is ceratin that this fact is not necessarily desirable. Thus, we propose a new interpretation of measurement theory, which is the linguistic aspect (or, the mathematical generalization) of quantum mechanics. Although this interpretation is superficially similar to a part of so-called Copenhagen interpretation, we show that it has a merit to be applicable to both quantum and classical systems. For example, we say that Bell’s inequality is broken even in classical systems.
An example of a classical system violating Bell’s inequalities is discussed. Existence of a classical system violating Bell’s inequalities takes away the “mysterious ” property usually called “non-locality ” which according to some characterizes quantum-mechanical systems. In recent years there has been an explosion of research on consequences from what initially had been an attempt by Einstein, Podolsky and Rosen (EPR)[1] to put quantum mechanics (QM) in question. Turning the tide is mainly due to the theoretical paper of Bell [2] discussing a two -spin particle system proposed by Bohm [3] as a special case of the system discussed in the initial critique by EPR [1]. Despite the initial questions posed by EPR the problems regarding QM are usually considered nowadays as settled and questioning it is considered highly unusual. Furthermore, claimed
The assumptions added by Bohr and concerning the Hilbert space (formed by all solutions of Schroedinger equation) changed fundamentally the original physical interpretation of these solutions proposed earlier by Schroedinger. This new alternative was refused by Einstein on the basis of the EPR Gedankenexperiment, but accepted fully for microscopic reality by scientific community. Both the quantum alternatives were discussed, however, again later. Bell tried to find a possibility how to decide between them; he generalized Einstein's Gedankenexperiment assuming that also spins of two detected particles would be measured. He derived then some inequality for a special combination of four coincidence probabilities, and it was commonly assumed that his inequality held for the original Schroedinger interpretation but not in Bohr's Copenhagen quantum mechanics; without any actual proof having been given. Corresponding experiments were proposed and finished in 1982. The violation of Bell's inequality was then interpreted as decisive victory of Bohr's theory. However, it will be shown that Bell's inequality has been interpreted mistakenly. It has been based always on some assumption that does not hold in any probabilistic theory (i.e., in the given spin experiment) but only in deterministic classical theory. There is not any argument for preferring the Copenhagen quantum mechanics and against Einstein's critical standpoint. Some other consequences will be mentioned, too.
In recent years the fulfillment of the inequalities established by Bell has been tested in different laboratories in the world. These inequalities make it possible to establish which of the interpretations of quantum mechanics is correct—the interpretation due to Einstein, according to which the properties of a quantum system exist as elements of physical reality irrespective of their observation, or the Copenhagen interpretation of Bohr and Fock, according to which the properties of a microscopic system described by noncommuting operators do not exist independently of the means of measurement. The experiments can be divided into three classes: experiments with optical photons, experiments with γ rays, and experiments with nucleons. The experiments that have been performed convincingly indicate violation of Bell's inequalities. Thus, the Copenhagen interpretation of quantum mechanics and its associated principle of relativity with respect to the means of measuring the properties of a microscopic system give at the present the only description of quantum phenomena that does not contradict experiment.