CovariantFieldDerivative[f[x], x, {li1, li2, ...}] is a covariant derivative of f[x] with respect to space-time variables x and with Lorentz indices li1, li2, .... CovariantFieldDerivative has only typesetting definitions by default. The user is must supply his/her own definition of the actual function.
Overview, CovariantD, ExpandPartialD, FieldDerivative.
CovariantFieldDerivative[QuantumField[A, {\[Mu]}][x], x, {\[Mu]}]\text{\textit{$\mathcal{D}$}}_{\mu }\left(A_{\mu }(x)\right)