[QUE/QFT-03004] QFT-PROBLEM

Define Pauli Lubanski operator \(W_\sigma\) by \[ W_\sigma = -\frac{1}{2} \epsilon_{\mu\nu\lambda\sigma} M^{\mu\nu} P^\sigma\] where \(P^\mu\) is energy momentum four vector and \(M^{\mu\nu}\) angular momentum tensor. Prove the following relations

• \(W^\sigma P^\sigma =0\)
• \(\big[W^\sigma , P^\mu\big] =0 \)
• \(W_\sigma\) is a four vector, {\it i.e.} \(\big[M_{\mu\nu}, W_\sigma\big] = -i(W_\mu g_{\nu\sigma}-W_\nu g_{\mu\sigma})\)
• \( \big[W_\lambda, W_\sigma \big] = i\epsilon_{\lambda\sigma\alpha\beta}W^\alpha P^\beta\)
• \(P_\mu P^\mu\) and \(W^\sigma W_\sigma\) are Cashimir invariants of the Poincare group,  they commute with all the ten generators of the Poincare group.
• Prove that \(W^2 = -\frac{1}{2} M_{\mu\nu} M^{\mu\nu} P^2 + M_{\mu\sigma}M^{\nu\sigma} P^\mu P_\sigma \)