Non-Kolmogorov probability models and modified Bell's inequality

We analyse the proof of Bell's inequality and demonstrate that this inequality is related to one particular model of probability theory, namely Kolmogorov measure-theoretical axiomatics, 1933. We found a (numerical) statistical correction to Bell's inequality. Such an additional term in the right hand side of Bell's inequality can be considered as a probability invariant of a quantum state. This is a measure of nonreproducibility of hidden variables in different runs of experiments. Experiments to verify Bell's inequality can be considered as just experiments to estimate this constant. It seems that Bell's inequality could not be used as a crucial reason to deny local realism. We consider deterministic as well as stochastic hidden variables models.

[1]  P. H. Eberhard,et al.  Bell’s theorem and the different concepts of locality , 1978 .

[2]  A. Kolmogoroff.  Grundbegriffe der Wahrscheinlichkeitsrechnung , 1933 .

[3]  A. Fine.  Hidden Variables, Joint Probability, and the Bell Inequalities , 1982 .

[4]  B. D'espagnat.  Veiled Reality: An Analysis Of Present-day Quantum Mechanical Concepts , 1995 .

[5]  D. Greenberger,et al.  The Wave–Particle Dualism—A Tribute to Louis de Broglie on His Ninetieth Birthday , 1986 .

[6]  J. Bell.  On the Problem of Hidden Variables in Quantum Mechanics , 1966 .

[7]  G. Roger,et al.  Experimental Test of Bell's Inequalities Using Time- Varying Analyzers , 1982 .

[8]  Leslie E Ballentine,et al.  The statistical interpretation of quantum mechanics , 1970 .

[9]  Helly.  Grundbegriffe der Wahrscheinlichkeitsrechnung , 1936 .

[10]  De Broglie’s Initial Conception of De Broglie Waves , 1984 .

[11]  S. Berman,et al.  Nuovo Cimento , 1983 .

[12]  Andrei Khrennikov,et al.  p-Adic Valued Distributions in Mathematical Physics , 1994 .

[13]  Andrei Khrennikov,et al.  Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models , 2011 .

[14]  Harald Bergstriim.  Mathematical Theory of Probability and Statistics , 1966 .

[15]  Andrew Khrennikov,et al.  p-adic probability interpretation of Bell's inequality , 1995 .

[16]  Luigi Accardi,et al.  The Probabilistic Roots of the Quantum Mechanical Paradoxes , 1984 .

[17]  L. Broglie,et al.  La physique quantique restera-t-elle indéterministe ? , 1952 .