[Paper Review] Bell's Theorem Begs the Question

Joy Christian|arXiv (Cornell University)|Feb 19, 2023

Philosophy and History of ScienceArts and Humanities3 citations

TL;DR

This paper argues that Bell's theorem relies on circular reasoning by unjustifiably assuming the additivity of expectation values for non-commuting observables in hidden variable theories, which is equivalent to presupposing the $±2$ bound it seeks to prove. When this flawed assumption is corrected, the local-realistic bound on the Bell-CHSH correlation sum is found to be $±2\sqrt{2}$, consistent with quantum mechanics, thereby invalidating Bell's conclusion that local realism is ruled out by experiments.

ABSTRACT

I demonstrate that Bell's theorem is based on circular reasoning and thus a fundamentally flawed argument. It unjustifiably assumes the additivity of expectation values for dispersion-free states of contextual hidden variable theories for non-commuting observables involved in Bell-test experiments, which is tautologous to assuming the bounds of $\pm2$ on the Bell-CHSH sum of expectation values. Its premises thus assume in a different guise the bounds of $\pm2\,$ it sets out to prove. Once this oversight is ameliorated from Bell's argument by identifying the impediment that leads to it and local realism is implemented correctly, the bounds on the Bell-CHSH sum of expectation values work out to be ${\pm2\sqrt{2}}$ instead of ${\pm2}$, thereby mitigating the conclusion of Bell's theorem. Consequently, what is ruled out by any of the Bell-test experiments is not local realism but the linear additivity of expectation values, which does not hold for non-commuting observables in any hidden variable theories to begin with. I also identify similar oversight in the GHZ variant of Bell's theorem, invalidating its claim of having found an inconsistency in the premisses of the argument by EPR for completing quantum mechanics. Conceptually, the oversight in both Bell's theorem and its GHZ variant traces back to the oversight in von Neumann's theorem against hidden variable theories identified by Grete Hermann in the 1930s.

Motivation & Objective

• To identify and expose a circular reasoning flaw in Bell's theorem that undermines its conclusion against local realism.
• To demonstrate that the assumption of linear additivity of expectation values for non-commuting observables in hidden variable theories is unjustified and tautological.
• To re-derive the correct bounds on the Bell-CHSH sum under proper local-realistic modeling, showing they are $±2\sqrt{2}$, not $±2$.
• To clarify that Bell-test experiments do not rule out local realism but rather the invalid assumption of additivity for non-commuting observables.
• To establish that dispersion-free hidden variable theories inherently violate linear additivity for non-commuting observables, making the assumption physically unjustifiable.

Proposed method

• Analyzes the mathematical structure of Bell-CHSH inequalities and identifies the assumption of additivity of expectation values for non-commuting observables as the core flaw.
• Reconstructs the derivation of the Bell-CHSH sum using dispersion-free hidden variable states, where eigenvalues are definite and expectation values are eigenvalues of the operators.
• Applies the correct algebraic treatment of non-commuting observables in hidden variable theories, avoiding the unjustified linear combination of expectation values.
• Derives the corrected upper bound on the Bell-CHSH sum as $±2\sqrt{2}$ by analyzing the vector magnitude $||{\bf n}||$ in the context of spin measurement settings.
• Uses geometric and algebraic methods to bound the magnitude of a vector derived from cross products of measurement settings, showing $0 \leq ||{\bf n}|| \leq 2$, which leads to the corrected bound.
• Demonstrates that the $±2$ bound arises only from the flawed assumption of additivity, not from local realism itself.

Experimental results

Research questions

• RQ1 Does Bell's theorem rest on an unjustified assumption about the additivity of expectation values in hidden variable theories?
• RQ2 Is the $±2$ bound on the Bell-CHSH sum logically derived or tautologically assumed?
• RQ3 What is the correct local-realistic bound on the Bell-CHSH correlation sum when additivity is not assumed?
• RQ4 Why do Bell-test experiments not rule out local realism, according to this analysis?
• RQ5 How does the non-commutativity of observables affect the additivity of expectation values in dispersion-free hidden variable theories?

Key findings

• The assumption of linear additivity of expectation values for non-commuting observables in Bell's theorem is circular, as it presupposes the $±2$ bound it aims to prove.
• When this assumption is removed, the correct local-realistic bound on the Bell-CHSH sum is found to be $±2\sqrt{2}$, matching quantum mechanical predictions.
• The $±2$ bound is not a consequence of local realism but of the unjustified additivity assumption.
• Dispersion-free hidden variable theories do not support linear additivity of expectation values for non-commuting observables, making the assumption physically invalid.
• Bell-test experiments do not rule out local realism but instead rule out the linear additivity of expectation values, which does not hold in such theories anyway.
• The vector magnitude $||{\bf n}||$ derived from measurement settings is bounded between 0 and 2, leading to the corrected bound of $±2\sqrt{2}$ on the Bell-CHSH sum.