Massless longitudinal Pauli-Lubanski eigenvalues
= Massless longitudinal Pauli-Lubanski eigenvalues
{title2=$(W_0,W_3)=(kj_3,-kj_3)$}
With signature $(+---)$ and $\epsilon_{0123}=+1$, a momentum $(k,0,0,k)$ and <helicity> $j_3$ give lower components $W_0=kj_3$ and $W_3=-kj_3$. They obey $W_\mu P^\mu=k(W_0+W_3)=0$. These longitudinal identities require only the momentum and rotation-projection eigenvalues, not a statement about transverse little-group translations.