I'm considering a massive vector field in classical field theory.

With the Lagrangian density $$\mathscr{L}=-\frac{1}{4}V^{\mu\nu}V_{\mu\nu}+\frac{1}{2}m^2V^{\mu}V_{\mu}.$$ I want to prove from the Lorentz rotational generators that the Noether current is: $$J^{\mu}_{(\rho\sigma)}=(x_{\rho}T^{\mu}_{\sigma}-x_{\sigma}T^{\mu}_{\rho})+(V^{\mu}_{\rho}V_{\sigma}-V_{\mu}^{\sigma}V_{\rho})=L_{\rho\sigma}+S_{\rho\sigma}.$$ I have some problems getting the term spin $S_{\rho\sigma}$.


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