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arXiv:0707.1333

Disproof of Bell's Theorem: Further Consolidations

Joy Christian

Published Jul 10, 2007• 6 citations
arXivHTMLPDF

Abstract

The failure of Bell's theorem for Clifford algebra valued local variables is further consolidated by proving that the conditions of remote parameter independence and remote outcome independence are duly respected within the recently constructed exact, local realistic model for the EPR-Bohm correlations. Since the conjunction of these two conditions is equivalent to the locality condition of Bell, this provides an independent geometric proof of the local causality of the model, at the level of microstates. In addition to local causality, the model respects at least seven other conceptual and operational requirements, arising either from the predictions of quantum mechanics or the premises of Bell's theorem, including the Malus's law for sequential spin measurements. Since the agreement between the predictions of the model and those of quantum mechanics is quantitatively precise in all respects, the ensemble interpretation of the entangled singlet state becomes amenable.

DOI
10.48550/arxiv.0707.1333

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•Jan 1, 2011•0 citations
The failure of Bell's theorem for Clifford algebra valued local variables is further consolidated by proving that the conditions of remote parameter independence and remote outcome independence are duly respected within the recently constructed exact, local realistic model for the EPR-Bohm correlations. Since the conjunction of these two conditions is equivalent to the locality condition of Bell, this provides an independent geometric proof of the local causality of the model, at the level of microstates. In addition to local causality, the model respects at least seven other conceptual and operational requirements, arising either from the predictions of quantum mechanics or the premises of Bell's theorem, including the Malus's law for sequential spin measurements. Since the agreement between the predictions of the model and those of quantum mechanics is quantitatively precise in all respects, the ensemble interpretation of the entangled singlet state becomes amenable.
It is shown that Bell's theorem fails for the Clifford algebra valued local realistic variables. This is made evident by exactly reproducing quantum mechanical expectation value for the EPR-Bohm type spin correlations observable by means of a local, deterministic, Clifford algebra valued variable, without necessitating either remote contextuality or backward causation. Since Clifford product of multivector variables is non-commutative in general, the spin correlations derived within our locally causal model violate the CHSH inequality just as strongly as their quantum mechanical counterparts.
A parameterized operational representation of an EPRB-like experiment is presented that facilitates direct comparisons among mathematical models that characterize the concepts of local hidden variables, the factorization condition, remote context independence, remote outcome independence, local causality, determinism, predictability, and quantum mechanics. The different models and conditions are derived from a common template, differentiated by parameter adjustment only. This perspective enables new and easier proofs of some existing results. It also clarifies the relationships among the various notions of locality and their relationship to the Bell-CHSH inequalities. Included are tables that make proofs short and intuitive, and figures that dramatize the differences between sets of correlations of quantum mechanics (QM) and those of realistic, local, deterministic, or predictable models. In addition, a novel proof of the no-communication theorem for QM provides a new perspective on why it is true.
Bell's theorem is 50 years old. Still there is a controversy about its implications. Much of it has its roots in confusion regarding the premises from which the theorem can be derived. Some claim that a derivation of Bell's inequalities requires just locality assumption, and nothing more. Violations of the inequalities are then interpreted as ``nonlocality'' or ``quantum nonlocality''. We show that such claims are unfounded and that every derivation of Bell's inequalities requires a premise -- in addition to locality and freedom of choice -- which is either assumed tacitly, or unconsciously, or is embedded in a single compound condition (like Bell's ``local causality''). The premise is equivalent to the assumption of existence of additional variables which do not appear in the quantum formalism (in form of determinism, or joint probability for outcomes of all conceivable measurements, or ``additional causes`'', or ``hidden variables'', ``complete description of the state'' or counterfactual definiteness, etc.). A certain irony is that perhaps the main message of violation of Bell's inequalities is that our notion of locality should be based on an operationally well-defined no-signalling condition, rather than on local causality.
In the debate of the foundations of quantum theory (QM), Bell’s theorem [1] is an important milestone. Based on Einstein’s criticism of completeness [2], Bell formulated an expression for the correlation between distant spin measurements such as described by Bohm [3]. In Bell’s expression, hidden variables to restore locality and causality to the theory (LHV’s) are introduced through a probability (mass) density function and through their influence upon the elementary measurement functions in the two separate wings (denoted by the Aand the B-wing) of the experiment. Many experiments and theoretical developments arose from Bell’s original paper. The most important experiment was performed by Aspect [4]. Aspect’s results were interpreted as a confirmation of the completeness of quantum mechanics. From that point, QM was considered a non-local theory. In a previous paper [5] the present author argued that there was insufficient ground for this conclusion. The author would like to point that merely an appeal to the simplicity and apparent logical truth of Bell’s theorem is advanced to support the exclusive non-locality interpretation. Beyond that, there is no proof that experimental results must exclusively be interpreted in this manner. According to the author the inequalities of Bell were given the status of theorem without sufficient supportive evidence save simplicity. In this paper it will be demonstrated that such a conclusive supportive argument of Bell’s theorem does not exist because an approximate classical model is possible. The principles of the proposed model can also be expressed in numerical terms. This enables numerical simulation of the experiments on a computer. Let us shortly describe a typical idealized Bell experiment. In such an experiment, from a single source, two particles with opposite spin are sent into opposite directions. For instance, we could think of a positron and an electron arising from para-Positronium that are drawn apart by dipole radiation. Subsequently, in the respective wings of the experimental set-up, the spin of the individual particle is measured with a SternGerlach magnet. The measurements are found to be correlated with the, unitary, parameter vectors of the magnets, a = (a1, a2, a3),b = (b1, b2, b3). It is well-known that the QM correlation for singlet state electron and positron is equal to,