Survey of Hopf fibrations and rotation conventions in mathematics and physics

TL;DR

This paper provides a unifying framework for various Hopf fibrations and reconciles two common rotation representations in mathematics and physics, enhancing understanding across disciplines.

Contribution

It introduces a unifying framework that connects different Hopf fibration versions and clarifies rotation conventions used in mathematics and physics.

Findings

Unified understanding of Hopf fibrations.

Reconciliation of quaternion and Bloch sphere rotation conventions.

Clarifies mathematical and physical rotation representations.

Abstract

We present a unifying framework for understanding several different versions of the Hopf fibration, and use this framework to reconcile two methods of representing rotations of 3-space by unitary matrices--the mathematician's convention based on quaternion algebra, and the physicist's convention based on the Bloch sphere.