Review of Nelson's analysis of Bell's theorem

TL;DR

This review analyzes Nelson's critique of Bell's theorem, discussing how local hidden variable theories can violate Bell's inequalities if outcomes are not predetermined, and explores the assumptions underlying Bell's original results.

Contribution

It clarifies the distinction between Bell's two theorems, relates Bell's assumptions to Nelson's analysis, and connects these to the EPR reality criterion.

Findings

Bell's inequalities can be violated without nonlocality if outcomes are not predetermined.

Bell's first theorem restricts models where outcomes depend locally on detector settings.

Bell's second theorem allows violations via non-deterministic or causality-violating theories.

Abstract

This article contains a review of Nelson's analysis of Bell's theorem. It shows that Bell's inequalities can be violated with a theory of local random variables if one accepts that the outcomes of these variables are not predetermined prior to measurement. The article describes the relation between Bell's theorem and the Strong Free Will theorem of Conway and Kochen. Then, the original articles of Bell are analyzed in detail. Following an article of Faris, it is explained that Bell's work on the hidden variable question in fact describes two separate theorems. Bell's first theorem says that there can be no model for the singlet state where an outcome does not depend locally on the settings of the detector where the outcome was measured. Bell's second theorem shows that Bell's inequalities can be violated by a theory that is either not deterministic, or violates causality in the sense of relativity or the free will assumption of the experimenters. It is shown in detail where Bell implicitly makes the various locality assumptions that Nelson has shown to be necessary for deriving Bell's inequality. The article closes by relating the various assumptions needed to derive Bell's theorem with the reality criterion of EPR