Spin Theory Based on the Extended Least Action Principle and Information Metrics: Quantization, Entanglement, and Bell Test With Time Delay

TL;DR

This paper develops a quantum theory of electron spin using an extended least action principle combined with information metrics, explaining quantization, entanglement, and Bell test violations through a new physical model.

Contribution

It introduces a novel formulation of electron spin based on relative entropy and the extended least action principle, providing explanations for quantization and entanglement phenomena.

Findings

Quantization of electron spin emerges as a mathematical limit of the model.

The model successfully reproduces measurement probabilities and the Schrödinger-Pauli equation.

Bell-CHSH inequality violation depends on time delay in the proposed Bell test experiment.

Abstract

Quantum theory of electron spin is developed here based on the extended least action principle and assumptions of intrinsic angular momentum of an electron with random orientations. The novelty of the formulation is the introduction of relative entropy for the random orientations of intrinsic angular momentum when extremizing the total actions. Applying recursively this extended least action principle, we show that the quantization of electron spin is a mathematical consequence when the order of relative entropy approaches a limit. In addition, the formulation of the measurement probability when a second Stern-Gerlach apparatus is rotated relative to the first Stern-Gerlach apparatus, and the Schr\"{o}dinger-Pauli equation, are recovered successfully. Furthermore, the principle allows us to provide an intuitive physical model and formulation to explain the entanglement phenomenon between two electron spins. In this model, spin entanglement is the consequence of the correlation between the random orientations of the intrinsic angular momenta of the two electrons. Since spin orientation is an intrinsic local property of the electron, the correlation of spin orientations can be preserved and manifested even when the two electrons are remotely separated. The entanglement of a spin singlet state is represented by two joint probability density functions that reflect the orientation correlation. Using these joint probability density functions, we prove that the Bell-CHSH inequality is violated in a Bell test. To test the validity of the spin-entanglement model, a Bell test experiment with time delay is proposed. As the time delay increases, we predict that the Bell-CHSH inequality changes from being violated to non-violated.