Gauge-field transformation law
= Gauge-field transformation law
For $D_\mu=\partial_\mu+A_\mu$ and a matter field transforming as $\phi\mapsto g\phi$, covariance of $D_\mu\phi$ requires
$$
A_\mu\mapsto gA_\mu g^{-1}-(\partial_\mu g)g^{-1}.
$$
= Gauge-field transformation law
For $D_\mu=\partial_\mu+A_\mu$ and a matter field transforming as $\phi\mapsto g\phi$, covariance of $D_\mu\phi$ requires
$$
A_\mu\mapsto gA_\mu g^{-1}-(\partial_\mu g)g^{-1}.
$$