Massless limit of a transverse canonical-spin label ($u_x=\sqrt{E}(0,1,1,0{)}^T$)

Boost a massive rest Dirac spinor labelled by $\xi =(1,1{)}^T/\sqrt{2}$ along the third axis and take its massless limit. In the Weyl representation the result is the displayed equal superposition of helicities, not an eigenvector of the fixed transverse rotation generator ${\Sigma }_1/2$. For momentum $(E,0,0,E)$ the positive-energy Hamiltonian is $E{\gamma }^0{\gamma }^3$, which anticommutes with ${\Sigma }_1=\mathrm{diag}({\sigma }_1,{\sigma }_1)$. A nonzero positive-energy solution cannot also be a ${\Sigma }_1$ eigenvector. Thus a transported canonical spin label must be distinguished from literal transverse spin at zero mass; longitudinal helicity remains a good quantum number.