Gauge covariance of the charged Dirac equation
= Gauge covariance of the charged Dirac equation
{title2=$D'_\mu\Psi'=e^{-iq\alpha}D_\mu\Psi$}
The derivative of the local phase in $\Psi\mapsto e^{-iq\alpha}\Psi$ cancels the added gradient in the electromagnetic potential. Thus $D'_\mu\Psi'=e^{-iq\alpha}D_\mu\Psi$, and the charged <Dirac equation> maps solutions to solutions. The same cancellation makes its action gauge invariant.