FCPartialD[ind] denotes a partial derivative of a field. It is an internal object that may appear only inside a QuantumField.
FCPartialD[LorentzIndex[mu]] denotes \partial_{\mu }.
FCPartialD[LorentzIndex[mu ,D]] denotes the D-dimensional \partial_{\mu }.
FCPartialD[CartesianIndex[i]] denotes \partial^{i} = - \nabla^i.
If you need to specify a derivative with respect to a particular variable it also possible to use FCPartialD[{LorentzIndex[mu],y}] or FCPartialD[{CartesianIndex[i],x}]although this notation is still somewhat experimental
Overview, ExpandPartialD, LeftPartialD, LeftRightPartialD, RightPartialD.
QuantumField[A, {\[Mu]}] . LeftPartialD[\[Nu]]
ex = ExpandPartialD[%]A_{\mu }.\overleftarrow{\partial }_{\nu }
\left(\partial _{\nu }A_{\mu }\right)
ex // StandardForm
(*QuantumField[FCPartialD[LorentzIndex[\[Nu]]], A, LorentzIndex[\[Mu]]]*)RightPartialD[{CartesianIndex[i], x}] . QuantumField[S, x]
ex = ExpandPartialD[%]\vec{\partial }_{\{i,x\}}.S^x
\left(\partial _{\{i,x\}}S^x\right)
ex // StandardForm
(*QuantumField[FCPartialD[{CartesianIndex[i], x}], S, x]*)FCPartialD also accepts FCGV symbols as arguments, which can be sometimes useful to make the final expression look nicer.
QuantumField[FCPartialD[FCGV["\[Del]"]], S, x]\left(\nabla S^x\right)