 Active Lorentz transformation of a vector field

An active transformation keeps the coordinate argument fixed and changes a contravariant vector field by $A^{\prime \mu }(x)={\Lambda }^{\mu }{}_{\nu }{A}^{\nu }({\Lambda }^{-1}x)$. Infinitesimally, $\delta A^{\mu }=\alpha {\omega }^{\mu }{}_{\nu }{A}^{\nu }-\alpha {\omega }^{\rho }{}_{\sigma }{x}^{\sigma }{\partial }_{\rho }{A}^{\mu }$. For a Lorentz scalar density, the spin-index term is absent, giving $\delta \mathcal{L}=-\alpha {\omega }^{\rho }{}_{\sigma }{x}^{\sigma }{\partial }_{\rho }\mathcal{L}$.