LeftNablaD[i]
denotes acting to the left.
Overview, ExpandPartialD, FCPartialD, LeftRightNablaD, RightNablaD.
[A, LorentzIndex[\[Mu]]] . LeftNablaD[i]
QuantumField
= ExpandPartialD[%] ex
// StandardForm
ex
(*-QuantumField[FCPartialD[CartesianIndex[i]], A, LorentzIndex[\[Mu]]]*)
StandardForm[LeftNablaD[i]]
(*LeftNablaD[CartesianIndex[i]]*)
[A, LorentzIndex[\[Mu]]] . QuantumField[A, LorentzIndex[\[Nu]]] . LeftNablaD[i]
QuantumField
= ExpandPartialD[%] ex
// StandardForm
ex
(*-QuantumField[A, LorentzIndex[\[Mu]]] . QuantumField[FCPartialD[CartesianIndex[i]], A, LorentzIndex[\[Nu]]] - QuantumField[FCPartialD[CartesianIndex[i]], A, LorentzIndex[\[Mu]]] . QuantumField[A, LorentzIndex[\[Nu]]]*)