Solution
= Solution
In the rest frame $P^\mu=(m,0,0,0)$, antisymmetry gives $W_0=0$. Taking $\epsilon_{0123}=+1$ and $J_3=M^{12}$ gives
$$
W_3=-mJ_3.
$$
Hence
$$
W_0|j,j_3\rangle=0,
\qquad
W_3|j,j_3\rangle=-m j_3|j,j_3\rangle.
$$
The sign of the second eigenvalue reverses if the opposite Levi-Civita convention is chosen.