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

A 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. By incorporating appropriate relative entropy for the random orientations of intrinsic angular momentum in the extended least action principle, the theory recovers the quantum formulation of electron spin. The two-level quantization of spin measurement is a natural mathematical consequence instead of a postulate. The formulation of 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 also derived successfully. Furthermore, we 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 correlation between the random orientations of the intrinsic angular momenta of the two electrons. Since the orientation is an intrinsic local property of electron, the correlation of orientations can be preserved even when the two electrons are remotely separated. Such a correlation can be manifested without causal effect. Owing to this orientation correlation, the Bell-CHSH inequality is shown to be violated in a Bell test. The standard quantum theory of electron spin can be considered as an ideal approximation of the present theory when certain conditions are taken to the limits. A potential experiment is proposed to test the difference between the present theory and the standard quantum theory. In a typical Bell test that confirms the violation of Bell-CHSH inequality, the theory suggests that by adding a sufficiently large time delay before Bob's measurement, the Bell-CHSH inequality can become non-violated.