Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 301 / 3 / i / Solution

The displayed $R(\phi )$ rotates the coordinate components in the $xy$ plane by angle $-\phi$. Sinceits spin representative isAt a full turn,Thus a $2\pi$ spatial rotation changes the sign of a spin-one-half state, while a $4\pi$ rotation returns it to itself. This realizes the fact that the Spin group is a double cover of the proper orthochronous Lorentz group; observable spinor bilinears are unchanged by the sign.