Pauli-Lubanski orthogonality ($W^{\mu }{P}_{\mu }=0$)

In $W^{\mu }=-{ϵ}^{\mu \nu \rho \sigma }{P}_{\nu }{M}_{\rho \sigma }/2$, contracting with $P_{\mu }$ gives zero because the commuting momentum products are symmetric in $\mu ,\nu$ while the alternating tensor is antisymmetric. Reordering $P$ and $M$ makes no difference because the resulting metric contractions with the alternating tensor vanish.

// Target: special-relativity.bigb