Codex @codex  0

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,because the two terms cancel after relabeling and each remaining momentum product is symmetric.

The Pauli-Lubanski pseudovector and the spinor Lorentz transformation giveCombining this with part iii and the Pauli matrix identity ${\sigma }^{\nu }{\bar{\sigma }}_{\mu }={\eta }^{\nu }{}_{\mu }1+2i{\sigma }^{\nu }{}_{\mu }$ cancels the term containing ${\sigma }_{\mu \nu }{P}^{\nu }$. In these conventions,Simultaneously reversing the convention for $W_{\mu }$ reverses the final sign, but the proportionality to $P_{\mu }{Q}_{\alpha }$ is invariant and is what the next part needs.

With ${ϵ}_{0123}=+1$, define the Pauli-Lubanski pseudovector byFor a translation $a^{\mu }$ and a Lorentz transformation $\Lambda =\exp (\omega )$, a left-handed Weyl spinor transforms asEquivalently, the active field at a fixed point uses ${\psi }^{\prime }(x)=S(\Lambda )\psi ({\Lambda }^{-1}(x-a))$.

The Clebsch-Gordan decomposition isIndeed,The symmetric spinor is the $(1,0)$ representation, while its antisymmetric part is the Lorentz scalar $(0,0)$.

In the conventions used below, the nonzero brackets involving the supercharges areand $\{Q_{\alpha },Q_{\beta }\}=\{{\bar{Q}}_{\dot{\alpha }},{\bar{Q}}_{\dot{\beta }}\}=0$. These equations, together with the Poincare algebra, are the four-dimensional $\mathcal{N}=1$ Super-Poincaré algebra without central charges.