Past exam of the theoretical physics II course of the University of Cambridge / 2014 / Paper 1 / 6 / b / Solution

At first order write $\alpha =1+iz$ and $\beta =x+iy$ with real $x,y,z$. The constraint forces the real part of the change in $\alpha$ to vanish. ThenThus the three supplied generators span the tangent space over the reals. Multiplying their matrices givesFor example, $m_1={\sigma }_x/2$, $m_2={\sigma }_y/2$, $m_3=i{\sigma }_z/2$ makes the noncompact signs explicit.