For $U(x)\in SU(3)$ and $A_\mu=A_\mu^aT^a$, choose $D_\mu=\partial_\mu-igA_\mu$. The <Yang-Mills gauge transformation> is
$$
A'_\mu=UA_\mu U^{-1}-\frac{i}{g}(\partial_\mu U)U^{-1},
$$
which ensures $D'_\mu(U\Phi)=U D_\mu\Phi$. Since
$$
[D_\mu,D_\nu]=-igF_{\mu\nu},
$$
conjugating this commutator immediately gives
$$
\boxed{F'_{\mu\nu}=UF_{\mu\nu}U^{-1}}.
$$
Thus the <gauge field strength> is gauge covariant.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 305 / 3 / b / i / Solution

 Solution

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