Past exam of the symmetry and relativity course of the University of Oxford / 2016 / 4 / b / Solution

In the massive particle's rest frame, $P=(m_{0}c,0)$ and the physical Pauli-Lubanski pseudovector is $W=(0,m_{0}c{\mathbf{s}}_{\mathrm{rest}})$, up to a common overall sign. Their Minkowski inner product is therefore zero. Since it is a Lorentz scalar,in every inertial frame, for any nonzero scalar normalization of $W$. This proves the requested orthogonality without a boosted component calculation.

The printed assertion that its particular normalization is always a unit vector needs qualification. For the displayed components $W=(\mathbf{s}\cdot \mathbf{p},E\mathbf{s}/c)$,Thus dividing by $m_{0}c|\mathbf{s}|$ is unit spacelike only when $\mathbf{s}\times \mathbf{p}=0$, or when $\mathbf{s}$ in that denominator is explicitly replaced by the proper rest-spin magnitude with correctly boosted components of $W$. As a counterexample to the literal displayed normalization, choose $\mathbf{s}\perp \mathbf{p}$ with nonzero momentum: it gives $U_W^2=-{\gamma }^2$, not $-1$. Orthogonality remains true, but it does not establish normalization.