Unit Quaternions and the Bloch Sphere

TL;DR

This paper explores representing spin-1/2 quantum states using unit quaternions, simplifying Bloch sphere rotations and revealing new symmetries and potential extensions to hidden variables or larger gauge spaces.

Contribution

It introduces a quaternionic map for Bloch sphere rotations, providing a new perspective and analysis tools for spin-1/2 states, including symmetry insights and extensions.

Findings

Quaternionic representation simplifies dynamical equations.

Reveals large broken symmetry in the quaternionic framework.

Proposes expansion to 'second order qubits' with implications for gauge freedom.

Abstract

The spinor representation of spin-1/2 states can equally well be mapped to a single unit quaternion, yielding a new perspective despite the equivalent mathematics. This paper first demonstrates a useable map that allows Bloch-sphere rotations to be represented as quaternionic multiplications, simplifying the form of the dynamical equations. Left-multiplications generally correspond to non-unitary transformations, providing a simpler (essentially classical) analysis of time-reversal. But the quaternion viewpoint also reveals a surprisingly large broken symmetry, as well as a potential way to restore it, via a natural expansion of the state space that has parallels to second order fermions. This expansion to "second order qubits" would imply either a larger gauge freedom or a natural space of hidden variables.