Failure of Bell’s Theorem and the Local Causality of the Entangled Photons

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 ∈ S and B = ± 1 ∈ S, respectively. When these results are compared, the correlations are seen to be exactly those predicted by quantum mechanics; namely cos 2(α− β), where α and β 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.