LeftRightPartialD[mu] denotes \overleftrightarrow {\partial }_{\mu }, acting to the left and right.
ExplicitPartialD[LeftRightPartialD[mu]] gives 1/2 (RightPartialD[mu] - LeftPartialD[mu]).
Overview, ExplicitPartialD, ExpandPartialD, FCPartialD, LeftPartialD, LeftRightPartialD2, RightPartialD.
LeftRightPartialD[\[Mu]]
ExplicitPartialD[%]\overleftrightarrow{\partial }_{\mu }
\frac{1}{2} \left(\vec{\partial }_{\mu }-\overleftarrow{\partial }_{\mu }\right)
LeftRightPartialD[\[Mu]] . QuantumField[A, LorentzIndex[\[Nu]]]
ExpandPartialD[%]\overleftrightarrow{\partial }_{\mu }.A_{\nu }
\frac{1}{2} \left(\left.(\partial _{\mu }A_{\nu }\right)-\overleftarrow{\partial }_{\mu }.A_{\nu }\right)
QuantumField[A, LorentzIndex[\[Mu]]] . LeftRightPartialD[\[Nu]] . QuantumField[A, LorentzIndex[\[Rho]]]
ExpandPartialD[%]A_{\mu }.\overleftrightarrow{\partial }_{\nu }.A_{\rho }
\frac{1}{2} \left(A_{\mu }.\left(\left.(\partial _{\nu }A_{\rho }\right)\right)-\left(\left.(\partial _{\nu }A_{\mu }\right)\right).A_{\rho }\right)