albertanattachen/hopf_fibration

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

Quaternions, Spinors and the Hopf Fibration

- Geometric phase of a parallel tranported tangent vector, and the Stereographic projection of the S2 path

Overview

matlab and python scripts used to create the figures in the article:

• ”Quaternions, Spinors and the Hopf Fibration: Hidden Variables in Classical Mechanics.” arXiv:1601.02569 November (2021)

@Article{OSullivan21,
  author = {Brian O'Sullivan},
  title = {Quaternions, Spinors and the Hopf Fibration: Hidden Variables in Classical Mechanics},
  booktitle = {arXiv:1601.02569},
  year = {2021}
}

About

The Hopf Fibration is a fundamental example of a fibre bundle. Named after Heinz Hopf who discovered (1931) a many-to-one continuous function (or "map") from the 3-sphere onto the 2-sphere such that each distinct point of the 2-sphere is mapped from a distinct great circle of the 3-sphere.

This article details the 6 Hopf projections between the 3-sphere and 2-sphere and illustrates their stereographic projections. Most notably the S1 fibre bundle connecting the 3-sphere and 2-sphere, consisting of the global geometric and dynamics phases, is derived in it's closed form for the first time.

Stereographic projection of the 6 Hopf maps