Dirac scalar bilinear ($\bar{\Psi }\Psi$)

The Dirac scalar bilinear is invariant under the simultaneous spinor and adjoint transformations because ${\bar{\Psi }}^{\prime }{\Psi }^{\prime }=\bar{\Psi }{S}^{-1}S\Psi$. It is the mass term of the Dirac action. At fixed coordinates it transforms as a scalar field, with an orbital transport term and no spin term.