Quantum Violation of Bell’s Inequality: A Misunderstanding Based on a Mathematical Error of Neglect  

作　　者：Frank Lad Frank Lad(Department of Mathematics and Statistics, University of Canterbury, Otautahi/Christchurch, New Zealand)

机构地区：[1]Department of Mathematics and Statistics, University of Canterbury, Otautahi/Christchurch, New Zealand

出　　处：《Journal of Modern Physics》2021年第8期1109-1144,共36页现代物理（英文）

摘　　要：This paper intends to show how the fabled violation of Bell’s inequality by the probabilistic specifications of quantum mechanics derives from a mathematical error, an error of neglect. I have no objection to the probabilities specified by quantum theory, nor to the inequality itself as characterized in the formulation of Clauser, Horne, Shimony, and Holt. Designed to assess consequences of Einstein’s principle of local realism, the inequality pertains to a linear combination of four polarization products <em>on the same pair of photons</em> arising in a gedankenexperiment. My assessment displays that in this context, the summands of the relevant CHSH quantity<em> s</em>(<span style="white-space:nowrap;"><em>λ</em></span>) inhere four symmetric functional relations which have long been neglected in analytic considerations. Its expectation E[<em style="white-space:normal;">s</em><span style="white-space:normal;">(</span><em>λ</em><span style="white-space:normal;">)</span>] is not the sum of four “marginal” expectations from a joint distribution, as quantum theory explicitly avoids such a specification. Rather, I show that <span style="white-space:normal;">E[</span><em style="white-space:normal;">s</em><span style="white-space:normal;">(</span><em style="white-space:normal;">λ</em><span style="white-space:normal;">)</span><span style="white-space:normal;">]</span> has four distinct representations as the sum of <em>three</em> 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 <em>E</em>(<em>s</em>) within <span style="white-space:nowrap;">(1.1213,2]</span> , surely not <img src="Edit_91a32f90-4b68-4415-98bc-3819733feca8.png" alt="" />at all as is commonly understood. I exhibit slices of the 4-dimensional polytope of joint<em> P</em>₊₊ probabilities actually motivated by quantum theory at the four stiThis paper intends to show how the fabled violation of Bell’s inequality by the probabilistic specifications of quantum mechanics derives from a mathematical error, an error of neglect. I have no objection to the probabilities specified by quantum theory, nor to the inequality itself as characterized in the formulation of Clauser, Horne, Shimony, and Holt. Designed to assess consequences of Einstein’s principle of local realism, the inequality pertains to a linear combination of four polarization products <em>on the same pair of photons</em> arising in a gedankenexperiment. My assessment displays that in this context, the summands of the relevant CHSH quantity<em> s</em>(<span style="white-space:nowrap;"><em>λ</em></span>) inhere four symmetric functional relations which have long been neglected in analytic considerations. Its expectation E[<em style="white-space:normal;">s</em><span style="white-space:normal;">(</span><em>λ</em><span style="white-space:normal;">)</span>] is not the sum of four “marginal” expectations from a joint distribution, as quantum theory explicitly avoids such a specification. Rather, I show that <span style="white-space:normal;">E[</span><em style="white-space:normal;">s</em><span style="white-space:normal;">(</span><em style="white-space:normal;">λ</em><span style="white-space:normal;">)</span><span style="white-space:normal;">]</span> has four distinct representations as the sum of <em>three</em> 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 <em>E</em>(<em>s</em>) within <span style="white-space:nowrap;">(1.1213,2]</span> , surely not <img src="Edit_91a32f90-4b68-4415-98bc-3819733feca8.png" alt="" />at all as is commonly understood. I exhibit slices of the 4-dimensional polytope of joint<em> P</em>₊₊ probabilities actually motivated by quantum theory at the four sti

关 键 词：Bell Inequality Defiance CHSH Formulation Fundamental Theorem of Probability Probability Bounds 4-Dimensional Cuts 

分 类 号：O17[理学—数学]