Local Quantum Mechanical Prediction of the Singlet State Using Geometric Algebra
Carl F. Diether
TL;DR
The paper addresses whether the quantum mechanical prediction for the singlet-state correlations in the EPR-Bohm setup can be obtained from a strictly local model using geometric-algebra-based measurement functions. It develops explicit local functions A({\bf a}, {\bf s1}) and B({\bf b}, {\bf s2}) under 3-sphere/quaternionic topology, applies a product-of-limits argument, and analytically derives ${\cal E}({\bf a},{\bf b}) = -{\bf a}\cdot{\bf b}$, independent of separation. The result is validated by a Mathematica simulation that reproduces the same $- {\bf a}\cdot {\bf b}$ correlation, reinforcing a locally causal interpretation within this formalism. The work contributes to ongoing discussions about locality, Bell inequalities, and the interpretation of entanglement, by illustrating how geometric-algebraic structures can yield standard quantum correlations without invoking nonlocal signaling.
Abstract
We deduce the quantum mechanical prediction of $-{\bf a}\cdot{\bf b}$ for the singlet spin state employing local measurement functions following Bell's approach. This result represents the quantum mechanical expectation value for the joint measurement of spin projections in the singlet state. And is equal to the negative cosine of the angle between vectors {\bf a} and {\bf b}. Our derivation is corroborated through a computational simulation conducted in the Mathematica programming environment using geometric algebra.