Symmetric derivation of singlet correlations in a quaternionic 3-sphere model

TL;DR

This paper presents a geometric algebra-based derivation of singlet correlations within a quaternionic 3-sphere model, offering a local-realistic explanation that contrasts with Bell's flat space model and suggests potential experimental tests.

Contribution

It introduces a novel quaternionic 3-sphere model for singlet correlations, utilizing geometric algebra and quaternion properties to provide a local-realistic explanation.

Findings

Quaternionic 3-sphere model reproduces singlet correlations.

Spinorial sign changes in quaternions explain strong correlations.

Comparison with Bell's model highlights differences in probability expressions.

Abstract

Using the powerful language of geometric algebra, we present an observationally symmetric derivation of the strong correlations predicted by the entangled singlet state in a deterministic and locally causal model, usually also referred to as a local-realistic model, in which the physical space is assumed to be a quaternionic 3-sphere, or $S^3$, available as the spatial part of a solution of Einstein's field equations of general relativity, and compare it in quantitative detail with Bell's local-realistic model for the singlet correlations set within a flat Euclidean space ${\mathrm{I\!R}^3}$. Since the quantitatively detailed expressions of relative-angle-dependent probabilities of observing measurement outcomes for Bell's local model do not seem to have been fully articulated before, our novel analysis exploiting the non-commutative properties of quaternions, in addition to allowing the comparison with the quaternionic 3-sphere model, may also provide useful comparisons for other less compelling local-realistic models, such as those relying on retrocausality or superdeterminism. Apart from the conservation of zero spin angular momentum, the key attribute underlying the strong singlet correlations within $S^3$ in comparison with Bell's local model turns out to be the spinorial sign changes intrinsic to quaternions that constitute the 3-sphere. In addition, we also discuss anew a macroscopic experiment that can, in principle, test our 3-sphere hypothesis.