Direct expansion gives
$$
[D_\mu,D_\nu]\phi
=\{\partial_\mu A_\nu-\partial_\nu A_\mu+[A_\mu,A_\nu]\}\phi
=F_{\mu\nu}\phi.
$$
Since $D'_\mu(g\phi)=gD_\mu\phi$,
$$
F'_{\mu\nu}(g\phi)
=[D'_\mu,D'_\nu](g\phi)
=g[D_\mu,D_\nu]\phi
=gF_{\mu\nu}\phi.
$$
Therefore the non-Abelian <gauge field strength> transforms covariantly:
$$
F'_{\mu\nu}=gF_{\mu\nu}g^{-1}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 302 / 4 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-302/4/c/solution