[Paper Review] Failure of Bell's Theorem and the Local Causality of the Entangled Photons

Joy Christian|arXiv (Cornell University)|May 26, 2010

Quantum Mechanics and Applications6 references8 citations

TL;DR

This paper presents a locally causal model for entangled photons using the topology of the 3-sphere (S³), demonstrating that quantum correlations of the form cos 2(α − β) can be reproduced without non-locality. By modeling physical space via Grassmann's coordinate-free geometric algebra and treating polarization outcomes as elements of S³, the authors show that Bell's theorem fails when the full topological structure of space is properly accounted for.

ABSTRACT

A counterexample to Bell's theorem is presented which uses a pair of photons instead of spin-1/2 particles used in our previous counterexamples. A locally causal protocol is provided for Alice and Bob, which allows them to simulate observing photon polarizations at various angles, and record their results as A=+/-1 in S^3 and B=+/-1 in S^3, respectively. When these results are compared, the correlations are seen to be exactly those predicted by quantum mechanics; namely cos 2(alpha - beta), where alpha and beta are the angles of polarizers. The key ingredient in our counterexample is the topology of 3-sphere, which remains closed under multiplication, thus preserving the locality condition of Bell.

Motivation & Objective

• To demonstrate that Bell's theorem does not rule out local realism when the full topological structure of physical space is considered.
• To resolve the apparent conflict between quantum mechanics and local causality by replacing ℝ³ with S³ as the geometric model of physical space.
• To show that the EPR-Bell correlations for photon polarization can be reproduced using a coordinate-free, geometric algebra framework based on Grassmann's algebra.
• To argue that the illusion of quantum non-locality arises from an incomplete topological accounting of physical reality, particularly the use of ℝ³ instead of S³.

Proposed method

• Modeling 3D Euclidean space using the even subalgebra of Grassmann's geometric algebra, which is closed under multiplication and isomorphic to S³.
• Representing photon polarization outcomes as elements of S³ via bivectors (spinors), with A(α, μ) = +μ·ã and B(β, μ) = −μ·b̃.
• Using the 3-sphere S³ to represent the joint measurement outcomes AB, where each product corresponds to a point in S³ with definite value ±1.
• Parameterizing the joint expectation value as E(α, β) = cos 2(α − β) + ⟨μ·e_z⟩ sin 2(α − β), with the average over μ·e_z vanishing due to symmetry.
• Deriving the quantum correlation E(α, β) = cos 2(α − β) by averaging over the 50/50 distribution of μ = ±I, ensuring locality and realism.
• Demonstrating that the CHSH inequality is violated in the same way as in quantum mechanics, but without requiring non-locality.

Experimental results

Research questions

• RQ1 Can local realism reproduce the quantum correlation cos 2(α − β) for entangled photons without invoking non-locality?
• RQ2 Does the use of ℝ³ as a model of physical space introduce an unphysical coordinate dependence that distorts the interpretation of quantum correlations?
• RQ3 Can the 3-sphere S³, as a topological space closed under multiplication, provide a consistent and local-realist framework for EPR-type experiments?
• RQ4 Is the apparent violation of Bell's inequality in photon polarization experiments due to an incomplete topological accounting of the physical space?
• RQ5 Can a coordinate-free geometric algebra based on Grassmann's algebra reproduce quantum mechanical predictions while preserving local causality?

Key findings

• The model reproduces the quantum mechanical correlation E(α, β) = cos 2(α − β) exactly, matching experimental results.
• The joint measurement outcomes AB are elements of S³, with each product point having a definite value of ±1, ensuring a consistent topological structure.
• The expectation value E(α, β) = cos 2(α − β) emerges from averaging over the 50/50 distribution of initial orientations μ = ±I, with the ⟨μ·e_z⟩ term vanishing due to symmetry.
• The CHSH inequality is violated in the same way as in quantum mechanics, but without requiring non-locality, showing that the violation does not imply non-locality.
• The model shows that the illusion of quantum non-locality arises from modeling space as ℝ³ instead of S³, which lacks the necessary topological closure.
• By using the even subalgebra of geometric algebra and representing physical quantities as bivectors and scalars, the model maintains local causality while reproducing quantum predictions.