Bell's Theorem Begs the Question

TL;DR

Bell's theorem is challenged as begging the question by relying on the additivity of expectation values for non-commuting observables; the author shows that correcting this step yields Tsirelson bounds of $\pm 2\sqrt{2}$, aligning with quantum predictions and negating the necessity of nonlocality. The critique extends to GHZ and Hardy variants, arguing that their apparent contradictions depend on illegitimate multiplicative compatibility assumptions for non-commuting observables. A contextual hidden-variable framework, including a quaternionic 3-sphere geometry, is presented as a constructive local-realistic model that reproduces quantum correlations. Collectively, the work reframes Bell-test implications, attributing violations to the non-additivity of non-commuting observables rather than to nonlocality, with potential foundational and interpretational impact for hidden-variable theories.

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.

Figures (1)

• Figure 1: In an EPR-Bohm-type experiment, a spin-less fermion -- such as a neutral pion -- is assumed to decay from a source into an electron-positron pair, as depicted. Then, measurements of the spin components of each separated fermion are performed at space-like separated observation stations ${\mathbf{1}}$ and ${\mathbf{2}}$, obtaining binary results $\mathscr{A}=\pm1$ and $\mathscr{B}=\pm1$ along directions ${\mathbf a}$ and ${\mathbf b}$. The conservation of spin momentum dictates that the total spin of the system remains zero during its free evolution. After Ref. IEEE-1.