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Abstract
Bell’s theorem founded on Bell’s inequalities asserts that no local realistic theories can reproduce all quantum mechanical predictions for spin correlation of particle pairs. It is pointed out that the assumption of setting-independent probability makes Bell’s inequalities not impose constraint on all local realistic models and thus constitutes a theoretical loophole of Bell’s theorem. A counterexample is presented by showing that a local realistic model with appropriate probability density reproduces all quantum mechanical predictions. It becomes clear that experiments violate Bell’s inequalities because the real correlation functions are not constrained by these inequalities. It is also exposed that, rigorous logical reasoning of counter factual deduction does not permit to exclude any premises of Bell’s inequalities by a certain amount of experimental violations of these inequalities.
Highlights
Bell’s theorem has influenced some physics researches over a half of century
A counterexample is presented by showing that a local realistic model with appropriate probability density reproduces all quantum mechanical predictions
As all proofs without inequality commence with the quantum mechanical concept of entangled state, they involve inherent inconsistency in logic for experimental test to discriminate between quantum mechanical concepts and classical concepts
Summary
Bell’s theorem has influenced some physics researches over a half of century. As the foundation of the theorem, Bell’s original inequality [1] has inspired two variants CHSH (Clauser-Horne-Shimony-Holt) inequality and CH (Clauser-Horne) inequality for convenience of experimental implementation [2] [3]. Bell had sensed that setting-independent probability might be one of the four possibilities invalidating his theorem [12] He proposed a measure for compensation while the inequality was kept in use [13]. To author’s knowledge, Barut et al advanced an instance with a simple model of a classical break-up process [14], and Gisin et al with a local hidden variable model exploiting detection loophole [15] As these demonstrations are not directly related to the correlation function in Bell’s original proof, and do not reveal the flaw of Bell’s inequalities, they do not disprove Bell’s theorem. A counterexample to Bell’s theorem is presented by demonstrating that a spin correlation function adopted from an extension of Bell’s origin reproduces all quantum mechanical predictions under appropriate probability densities. The counterexample disproves Bell’s theorem firmly, and the cause for which is that Bell’s inequalities do not impose constraint on real correlation functions
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