= Gauge-covariant integration by parts
{title2=$\partial_\mu\kappa(Y,Z)=\kappa(D_\mu Y,Z)+\kappa(Y,D_\mu Z)$}
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>.
Back to article page
