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The Physics Teacher

The Original Bell Theorem without Calculus

Jed Brody

Published Mar 31, 2021• 1 citation
Publisher

Abstract

Bell’s theorem is a topic of perennial fascination. Publishers and the general public have a steady appetite for approachable books about its implications. The scholarly literature includes many analogies to Bell’s theorem and simple derivations of Bell inequalities, and some of these simplified discussions are the basis of interactive web pages. Less well known is that the original Bell theorem is virtually unaffected when the calculus is taken away. Indeed, the only use of integrals in Bell’s derivation is to compute averages. We can simply replace integrals with the word “average.” The resulting proof of Bell’s theorem is just as easy to understand, and just as shocking, as the analogies designed to ease comprehension.

DOI
10.1119/10.0004150

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Bell’s theorem is a topic of perennial fascination. Publishers and the general public have a steady appetite for approachable books about its implications. The scholarly literature includes many analogies to Bell’s theorem and simple derivations of Bell inequalities, and some of these simplified discussions are the basis of interactive web pages. Less well known is that the original Bell theorem is virtually unaffected when the calculus is taken away. Indeed, the only use of integrals in Bell’s derivation is to compute averages. We can simply replace integrals with the word “average.” The resulting proof of Bell’s theorem is just as easy to understand, and just as shocking, as the analogies designed to ease comprehension.
The purpose of this lecture is to give a self-contained demonstration of a version of Bell's theorem and a discussion of the significance of the theorem and the experiments which it inspired. The lecture should be comprehensible to people who have had no previous acquaintance with the literature on Bell's theorem, but I hope that explicitness about premisses and consequences will make it useful even to those who are familiar with the literature. All versions of Bell's theorem are variations, and usually generalizations, of the pioneering paper of J. S. Bell of 1964, entitled “On the Einstein–Podolsky–Rosen Paradox.” All of them consider an ensemble of pairs of particles prepared in a uniform manner, so that statistical correlations may be expected between outcomes of tests performed on the particles of each pair. If each pair in the ensemble is characterized by the same quantum state φ, then the quantum mechanical predictions for correlations of the outcomes can in principle be calculated when the tests are specified. On the other hand, if it is assumed that the statistical behavior of the pairs is governed by a theory which satisfies certain independence conditions (always similar to the Parameter and Outcome Independence conditions stated below, though the exact details vary from version to version of Bell's theorem), then it is possible to derive a restriction upon the statistical correlations of the outcomes of tests upon the two particles.
Quantum Physics is a cornerstone of modern science and technology, yet a comprehensive approach to integrating it into school curricula and communicating its foundations to policymakers, industrial stakeholders, and the general public has yet to be established. In this paper, we discuss the rationale for introducing entanglement and Bell’s Inequalities (BI) to a non-expert audience, and how these topics have been presented in the exhibition “Dire l’indicibile” (“Speaking the unspeakable”), as a part of the Italian Quantum Weeks project. Our approach meets the challenge of simplifying quantum concepts without sacrificing their core meaning, specifically avoiding the risks of oversimplification and inaccuracy. Through interactive activities, including a card game demonstration and the staging of CHSH experiments, participants explore the fundamental differences between classical and quantum probabilistic predictions. They gain insights into the significance of BI verification experiments and the implications of the 2022 Nobel Prize in Physics. Preliminary results from both informal and formal assessment sessions are encouraging, suggesting the effectiveness of this approach.
Contrary to what Lambare [arXiv:2008.00369] assumes, in non-Newtonian calculus (a calculus based on non-Diophantine arithmetic) an integral is typically given by a nonlinear map. This is the technical reason why all the standard proofs of Bell-type inequalities fail if non-Newtonian hidden variables are taken into account. From the non-Newtonian perspective, Bell's inequality is a property of a limited and unphysical class of hidden-variable models. An explicit counterexample to Bell's theorem can be easily constructed.
A fundamental Bell-type inequality is derived by referring solely to laws of logic, tertium non datur and the transitivity axiom, that are used to express the local realism assumption. The general form of the achieved inequality makes it a root of other more specific Bell-type theorems involving additional mathematical formalism and experimental conditions. Its universality is demonstrated explicitly by showing that the most famous theorems provided by Bell, Clauser, Horne, Shimony and Holt (CHSH), Clauser and Horne (CH), Eberhard, d'Espagnat and Maccone are direct corollaries of the general inequality. The method to retrieve Bell's inequalities of different kinds, as being based on a single scheme of reasoning, unifies diversity of proofs presented in the literature. Additionally, some of the known inequalities can be easily achieved in a more general form compared to their original versions and still suitable, or possibly even more convenient, for experimental investigations. By extracting local realism in the form of an explicit logical propositions and making it a cornerstone of a systematic reasoning leading to the universal Bell's inequality, the paper offers a synthetic perspective on Bell-type theorems and their logical basis.