[QUE/QFT-03003] QFT-PROBLEM

It is given that the generators \(M^{\mu\nu}\) of Lorentz group satisfy the commutation relations \begin{eqnarray}\nonumber [M^{\mu\nu} , P^\sigma] &=& i(g^{\sigma\mu} P^\nu - g^{\sigma\nu} P^\mu )\\\nonumber [M^{\mu\nu} , M ^{\rho\sigma}] &=& i(M^{\mu\rho} g^{ \nu\sigma} + M^{\nu\sigma} g^{\mu\rho} - M^{\nu\rho} g^{\mu\sigma} - M^{\mu\sigma} g^{\nu\rho} ) \end{eqnarray}

• Compute the commutator \(\big[ \epsilon^{\mu\nu\lambda\sigma} M_{\mu\nu}M_{\lambda\sigma}, M_{\alpha\beta}\big].\)
• Define \(K_i= M_{i0}\) and \(J_i=-\frac{1}{2}\epsilon_{imn}M^{mn}\) and prove the following commutation relations. \begin{eqnarray}\nonumber \Big[ J_i, P_k\Big]= -\epsilon_{ik\ell} P_ell, &\qquad& \big[J_i,P_0\big] =0 \\{}\nonumber \Big[ K_i, P_k\Big]= - P_0 g_{ik}, &\qquad& \big[K_i,P_0\big] =-iP_0 . \end{eqnarray}
• The commutators of operators \(\vec{J}, \vec{K}\) are given by \begin{eqnarray}\nonumber \big[J_\ell,J_m\big] &=& i\epsilon_{\ell m n }J_n\\\nonumber \big[J_\ell,J_m\big] &=& i\epsilon_{\ell m n }K_n\\\nonumber \big[K_\ell,K_m\big] &=& -i\epsilon_{\ell m n }J_n. \end{eqnarray}
• Show that \(\vec{j}^{\pm}\) defined by \(j^{(\pm)}_m = \frac{1}{2}\big(J_m\pm i K_m\big)\) obey angular momentum commutations relations \begin{eqnarray}\nonumber \big[j^{(\pm)}_\ell, j^{(\pm)}_m\big] &=& i \epsilon_{\ell m n }j^{(\pm)}_n\\\nonumber \big[j^{(+)}_\ell, j^{(-)}_m\big] &=&0. \end{eqnarray}