Adjoint covariant derivative

On an adjoint-valued field $X$, the covariant derivative is $D_{\mu }X={\partial }_{\mu }X+[A_{\mu },X]$. If $X\mapsto gX{g}^{-1}$, then $D_{\mu }X\mapsto g(D_{\mu }X)g^{-1}$.

Choose the gauge covariant derivative $D_{\mu }={\partial }_{\mu }+ig{A}_{\mu }$ and matter transformation ${\varphi }^{\prime }=h\varphi$. Covariance $D_{\mu }^{\prime }{\varphi }^{\prime }=h{D}_{\mu }\varphi$ forces $A_{\mu }^{\prime }=h{A}_{\mu }{h}^{-1}+(i/g)({\partial }_{\mu }h)h^{-1}$. For $h=1-ig\omega +O({\omega }^2)$, expansion gives the displayed adjoint covariant derivative variation. With an orthonormal Hermitian Lie-algebra basis, $\delta A_{\mu }^a={\partial }_{\mu }{\omega }^a-g{f}^{abc}{A}_{\mu }^b{\omega }^c$. Thus Lorenz-gauge variation is $\delta (\partial \cdot A)=\partial \cdot \mathcal{D}\omega$. Choosing the Faddeev-Popov operator as $\mathcal{M}=-\partial \cdot \mathcal{D}$, with its field-independent sign absorbed in normalization, gives ghost action $-\bar{c}\partial \cdot \mathcal{D}c$; consistent propagator and vertex signs follow from this choice.

For Lie-algebra-valued fields and a constant invariant bilinear form on a Lie algebra, the connection terms in the displayed identity cancel. Indeed, invariance gives $\kappa ([A,Y],Z)+\kappa (Y,[A,Z])=0$. Ordinary integration by parts then yields $\int \kappa (D_{\mu }Y,Z)=-\int \kappa (Y,D_{\mu }Z)$ when boundary terms vanish, for example for compactly supported variations. This preserves covariance in variational calculations of Yang-Mills equations.