Active Lorentz transformation of a vector field
= Active Lorentz transformation of a vector field
An active transformation keeps the coordinate argument fixed and changes a contravariant vector field by $A'^\mu(x)=\Lambda^\mu{}_{\nu}A^\nu(\Lambda^{-1}x)$. 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 $\delta\mathcal L=-\alpha\omega^\rho{}_{\sigma}x^\sigma\partial_\rho\mathcal L$.