 Infinitesimal transformation of a Dirac field

For $\Lambda =1+\omega$ with antisymmetric lowered parameters, the spin matrix is $S=1-(i/4){\omega }_{\mu \nu }{\sigma }^{\mu \nu }$, where ${\sigma }^{\mu \nu }=(i/2)[{\gamma }^{\mu },{\gamma }^{\nu }]$. At transformed coordinates this is the full component transformation. At fixed coordinates an additional orbital term $-{\omega }^{\mu }{}_{\nu }{x}^{\nu }{\partial }_{\mu }\Psi$ appears.