Unit Quaternions and the Bloch Sphere

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. ‡ Present address: The Graduate Center, CUNY, New York NY 10016. ar X iv :1 41 1. 49 99 v2 [ qu an tph ] 2 3 A pr 2 01 5 Unit Quaternions and the Bloch Sphere 2