= Normalization of a Pauli-Lubanski spin direction
The physical massive-particle <Pauli-Lubanski pseudovector> obeys $W\cdot P=0$ and $W^2=-m^2c^2|\mathbf s_{\mathrm{rest}}|^2$ for classical rest-frame <spin>. Orthogonality by itself does not establish this normalization. In particular the components $W=(\mathbf s\cdot\mathbf p,(E/c)\mathbf s)$ give $W^2=-m^2c^2|\mathbf s|^2-|\mathbf s\times\mathbf p|^2$; dividing by $mc|\mathbf s|$ is a unit spacelike normalization only when the cross product vanishes.
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