LeftNablaD[i] denotes \overleftarrow{\nabla}_{i} acting to the
left.
Overview, ExpandPartialD, FCPartialD, LeftRightNablaD, RightNablaD.
QuantumField[A, LorentzIndex[\[Mu]]] . LeftNablaD[i]
ex = ExpandPartialD[%]A_{\mu }.\overleftarrow{\nabla }^i
-\left(\partial _iA_{\mu }\right)
ex // StandardForm
(*-QuantumField[FCPartialD[CartesianIndex[i]], A, LorentzIndex[\[Mu]]]*)StandardForm[LeftNablaD[i]]
(*LeftNablaD[CartesianIndex[i]]*)QuantumField[A, LorentzIndex[\[Mu]]] . QuantumField[A, LorentzIndex[\[Nu]]] . LeftNablaD[i]
ex = ExpandPartialD[%]A_{\mu }.A_{\nu }.\overleftarrow{\nabla }^i
-A_{\mu }.\left(\partial _iA_{\nu }\right)-\left(\partial _iA_{\mu }\right).A_{\nu }
ex // StandardForm
(*-QuantumField[A, LorentzIndex[\[Mu]]] . QuantumField[FCPartialD[CartesianIndex[i]], A, LorentzIndex[\[Nu]]] - QuantumField[FCPartialD[CartesianIndex[i]], A, LorentzIndex[\[Mu]]] . QuantumField[A, LorentzIndex[\[Nu]]]*)