Because the translations commute, $P^\sigma P^\mu$ is symmetric in $(\sigma,\mu)$, whereas the Levi-Civita tensor in the <Pauli-Lubanski pseudovector> is antisymmetric, so $W_\mu P^\mu=0$. Moreover,
$$
[W_\mu,P^\tau]
=\frac12\epsilon_{\mu\nu\rho\sigma}
(\eta^{\rho\tau}P^\nu-\eta^{\nu\tau}P^\rho)P^\sigma=0,
$$
because the two terms cancel after relabeling and each remaining momentum product is symmetric.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 302 / 3 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-302/3/c/solution