Rotations, Quaternions, the Universal Covering Group, and the Electron Spin

Euler’s rotation formula (6.6) can be elegantly written by using Hamilton’s quaternions. This was discovered independently by Hamilton and Cayley in 1844, one year after Hamilton’s discovery of quaternions. In the language of quaternions, Euler’s rotation formula (6.6) reads elegantly as $$\fbox{$\mathbf{x}' = q\cdot \mathbf{x}\cdot q^{\dagger}, \qquad \mathbf{x}\in E_{3}.$}$$ Here, the given quaternion $$q: = \cos \tfrac{\varphi}{2} + \sin \tfrac{\varphi}{2} \; \mathbf{n} $$ contains the information about the rotation angle ϕ and the rotation axis vector n of length one. In particular, for the norm of the quaternion q we get $$|q| = \sqrt{ \cos^2 \tfrac{\varphi}{2} +\mathbf{n}^2\sin^2\tfrac{\varphi}{2}}=1.$$ Hence q∈U(1,ℍ).