(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Hi. This is not a homework question per se, but more of a personal question, but I thought I'd post it here.

I'm trying to prove the commutation relations of the Pauli-Lubanski pseudovector

\begin{equation}

W_\mu\equiv-\frac{1}{2} \epsilon_{\mu\nu\rho\sigma}J^{\nu\rho}P^\sigma

\end{equation}

with the Lorentz transformation generators \begin{equation}J^{\mu\nu}.\end{equation}

I'm supposed to find

\begin{equation}

\left[J_{\mu\nu},W_\rho\right]= \mathrm{i}\left( \eta_{\nu\rho}W_\mu-\eta_{\mu\rho}W_\nu\right)

\end{equation}

but I simply can't.

2. Relevant equations

Obviously I have to use

\begin{align}

\left[P_\mu,P_\nu\right]&= 0,\\

\left[P_\mu,J_{\nu\rho}\right]&= \mathrm{i}\left(\eta_{\mu\rho}P_\nu-\eta_{\mu\nu}P_\rho\right),\\

\left[J_{\mu\nu},J_{\rho\sigma}\right]&= \mathrm{i}\left(\eta_{\mu\rho}J_{\sigma\nu}- \eta_{\nu\rho}J_{\sigma\mu}- \eta_{\mu\sigma}J_{\rho\nu}+ \eta_{\nu\sigma}J_{\rho\mu}\right)

\end{align}

3. The attempt at a solution

My calculations gave

\begin{equation}

\left[J_{\kappa\lambda},W_\mu\right]= -i\left(\eta_{\tau\lambda} \epsilon_{\kappa\rho\mu\sigma} J^{\rho\tau}P^\sigma- \frac{1}{2} \eta_{\tau\lambda} \epsilon_{\kappa\rho\mu\sigma} J ^{\sigma\rho}P^\tau- \eta_{\tau\kappa} \epsilon_{\lambda\rho\mu\sigma} J^{\rho\tau}P^\sigma+ \frac{1}{2}\eta_{\tau\kappa} \epsilon_{\lambda\rho\mu\sigma} J^{\sigma\rho}P^\tau\right).

\end{equation}

I'm pretty confident this is correct, but in the meantime I don't see where to go when I get here.

Any help very much appreciated !

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# Pauli-Lubanski pseudovector commutation relations

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