Quantum Violation of Bell's Inequality: a misunderstanding based on a mathematical error of neglect

Abstract

The fabled violation of Bell's inequality by the probabilistic specifications of quantum mechanics is shown to derive from a mathematical error. The inequality, designed to assess consequences of Einstein's principle of local realism, pertains to four polarization products on the same pair of photons arising in a gedankenexperiment. The summands of the CHSH quantity $s(λ)$ inhere four symmetric functional relations which have long been neglected in analytic considerations. Its expectation $E[s(λ)]$ is not the sum of four ``marginal'' expectations from a joint distribution, as quantum theory explicitly avoids such a specification. Rather, $E[s(λ)]$ has four distinct representations as the sum of three expectations of polarization products plus the expectation of a fourth which is restricted to equal a function value determined by the other three. Analysis using Bruno de Finetti's fundamental theorem of prevision (FTP) yields only a bound for $E(s)$ within $(1.1213, 2]$, surely not $2\sqrt{2}$ at all. The 4-D polytope of cohering joint $P_{++}$ probabilities at the four stipulated angle settings are displayed as passing through 3-D space. Aspect's ``estimation'' is based on polarization products from different photon pairs that do not have embedded within them the inhering functional relations. When you do actively embed the restrictions into Aspect's estimation procedure, it yields an estimate of 1.7667, although this is not and cannot be definitive.