Quaternions, Spinors and the Hopf Fibration: Hidden Variables in Classical Mechanics

TL;DR

This paper explores the mathematical structures of quaternions, spinors, and the Hopf fibration, revealing how the global phase acts as a hidden variable in classical mechanics through geometric and topological insights.

Contribution

It introduces the global phase as a natural hidden variable in classical mechanics, connecting quaternionic rotations, the Hopf fibration, and phase quantization.

Findings

Global phase is quantized in integer multiples of 2π.

Hopf fibration links 4D quaternion space to 3D sphere.

Hidden variables emerge from geometric phases in classical systems.

Abstract

Rotations in 3 dimensional space are equally described by the SU(2) and SO(3) groups. These isomorphic groups generate the same 3D kinematics using different algebraic structures of the unit quaternion. The Hopf Fibration is a projection between the hypersphere $\mathbb{S}^3$ of the quaternion in 4D space, and the unit sphere $\mathbb{S}^2$ in 3D space. Great circles in $\mathbb{S}^3$ are mapped to points in $\mathbb{S}^2$ via the 6 Hopf maps, and are illustrated via the stereographic projection. The higher and lower dimensional spaces are connected via the $\mathbb{S}^1$ fibre bundle which consists of the global, geometric and dynamic phases. The global phase is quantized in integer multiples of $2\pi$ and presents itself as a natural hidden variable of Classical Mechanics.