= Cross-product convention for an adjoint covariant derivative
{title2=$[D_\mu,D_\nu]\Phi=eF_{\mu\nu}\times\Phi$}
For the ordinary three-dimensional cross product, $D_\mu\Phi=\partial_\mu\Phi+eA_\mu\times\Phi$ requires $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu+eA_\mu\times A_\nu$. This follows from $[A\times,B\times]=(A\times B)\times$. The same curvature then transforms covariantly under local rotations. Using a minus sign in its quadratic term while retaining the plus sign in $D$ breaks local gauge invariance: a pure gauge for $D$ has $F_+=0$ but generally $F_-=-2eA_\mu\times A_\nu\ne0$. Reversing both signs is an alternative consistent convention.
Back to article page
