Minimal electromagnetic coupling of a Dirac field
= Minimal electromagnetic coupling of a Dirac field
{title2=$\mathcal L_{\rm int}=-qA_\mu\bar\Psi\gamma^\mu\Psi$}
Replace the derivative in the <Dirac action> by the <gauge covariant derivative> $D_\mu=\partial_\mu+iqA_\mu$. The interaction is then $-qA_\mu\bar\Psi\gamma^\mu\Psi$. With $\Psi\mapsto e^{-iq\alpha}\Psi$ and $A_\mu\mapsto A_\mu+\partial_\mu\alpha$, the covariant derivative transforms in the same way as the spinor, proving local <gauge invariance>.