FeynCalc manual (development version)

FCPartialD

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 μ\partial_{\mu }.

FCPartialD[LorentzIndex[mu ,D]] denotes the DD-dimensional μ\partial_{\mu }.

FCPartialD[CartesianIndex[i]] denotes i=i\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

See also

Overview, ExpandPartialD, LeftPartialD, LeftRightPartialD, RightPartialD.

Examples

QuantumField[A, {\[Mu]}] . LeftPartialD[\[Nu]] 
 
ex = ExpandPartialD[%]

Aμ.νA_{\mu }.\overleftarrow{\partial }_{\nu }

(νAμ)\left(\partial _{\nu }A_{\mu }\right)

ex // StandardForm

(*QuantumField[FCPartialD[LorentzIndex[\[Nu]]], A, LorentzIndex[\[Mu]]]*)
RightPartialD[{CartesianIndex[i], x}] . QuantumField[S, x] 
 
ex = ExpandPartialD[%]

{i,x}.Sx\vec{\partial }_{\{i,x\}}.S^x

({i,x}Sx)\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]

(Sx)\left(\nabla S^x\right)