Wigner and Bell Inequalities relationships and Kolmogorov's Axioms

TL;DR

This paper explores how violations of Bell's and Wigner's inequalities are linked to the violation of Kolmogorov's axioms, especially the non-negativity of probabilities, and proposes a new interpretation to avoid negative probabilities.

Contribution

It demonstrates that Bell's and Wigner's inequality violations stem from the failure to respect Kolmogorov's probability axioms and suggests a new probability interpretation to prevent negative probabilities.

Findings

Bell's inequality violation relates to non-compliance with Kolmogorov's axioms.

Negative joint probabilities occur when Bell's inequality is violated.

Proposes a new probability interpretation to avoid negative probabilities and inequality violations.

Abstract

In this work, we show that Bell's inequality violation of arise from the fact that the condition imposed upon the development of inequality is not respected when it is applied in the idealized experiment. Such a condition is that the quantities taken by probability must be non negative, and such a condition is represented by $|\cal{P(w)}|=\cal{P(w)}$. We will also show that, when trying to define the values of the joint probabilities of $ (Z_1, Z_2, Z_3) $, through the values obtained from the $ (Z_j, Z_k) $ pairs, we find that these values are negative, so not Kolmogorov's axiom is respected: $\cal{P(w)}\geq0$ in cases where Bell's inequality is violated, and we also show that only such violation is possible if Wigner's inequality, in a certain arrangement, is violated, and that both violations are related to the violation of one of Kolmogorov's axioms. At the end of the paper, we suggest a new interpretation of the probabilities involved, in order to avoid the situation of negative probabilities and the violation of Bell's inequality and, consequently, Wigner's inequality.