FeynCalc manual (development version)

ExpandPartialD

ExpandPartialD[exp] expands noncommutative products of QuantumFields and partial differentiation operators in exp and applies the Leibniz rule.

By default the function assumes that there are no expressions outside of exp on which the derivatives inside exp could act. If this is not the case, please set the options LeftPartialD or RIghtPartialD to True.

See also

Overview, ExplicitPartialD, LeftPartialD, LeftRightPartialD, PartialDRelations, RightPartialD, LeftRightNablaD, LeftRightNablaD2, LeftNablaD, RightNablaD.

Examples

RightPartialD[\[Mu]] . QuantumField[A, LorentzIndex[\[Mu]]] . QuantumField[A, LorentzIndex[\[Nu]]] 
 
ExpandPartialD[%]

μ.Aμ.Aν\vec{\partial }_{\mu }.A_{\mu }.A_{\nu }

Aμ.(μAν)+(μAμ).AνA_{\mu }.\left(\partial _{\mu }A_{\nu }\right)+\left(\partial _{\mu }A_{\mu }\right).A_{\nu }

RightNablaD[i] . QuantumField[A, LorentzIndex[\[Mu]]] . QuantumField[A, LorentzIndex[\[Nu]]] 
 
ExpandPartialD[%]

i.Aμ.Aν\vec{\nabla }^i.A_{\mu }.A_{\nu }

Aμ.(iAν)(iAμ).Aν-A_{\mu }.\left(\partial _iA_{\nu }\right)-\left(\partial _iA_{\mu }\right).A_{\nu }

LeftRightPartialD[\[Mu]] . QuantumField[A, LorentzIndex[\[Nu]]] 
 
ExpandPartialD[%]

μ.Aν\overleftrightarrow{\partial }_{\mu }.A_{\nu }

12((μAν)μ.Aν)\frac{1}{2} \left(\left(\partial _{\mu }A_{\nu }\right)-\overleftarrow{\partial }_{\mu }.A_{\nu }\right)

LeftRightNablaD[i] . QuantumField[A, LorentzIndex[\[Nu]]] 
 
ExpandPartialD[%]

i.Aν\overleftrightarrow{\nabla }_i.A_{\nu }

12(i.Aν(iAν))\frac{1}{2} \left(\overleftarrow{\partial }_i.A_{\nu }-\left(\partial _iA_{\nu }\right)\right)

QuantumField[A, LorentzIndex[\[Mu]]] . (LeftRightPartialD[OPEDelta]^2) . QuantumField[A, 
    LorentzIndex[\[Rho]]] 
 
ExpandPartialD[%]

Aμ.Δ2.AρA_{\mu }.\overleftrightarrow{\partial }_{\Delta }^2.A_{\rho }

14(Aμ.(ΔΔAρ)2(ΔAμ).(ΔAρ)+(ΔΔAμ).Aρ)\frac{1}{4} \left(A_{\mu }.\left(\partial _{\Delta }\partial _{\Delta }A_{\rho }\right)-2 \left(\partial _{\Delta }A_{\mu }\right).\left(\partial _{\Delta }A_{\rho }\right)+\left(\partial _{\Delta }\partial _{\Delta }A_{\mu }\right).A_{\rho }\right)

8 LeftRightPartialD[OPEDelta]^3

8Δ38 \overleftrightarrow{\partial }_{\Delta }^3

ExplicitPartialD[%]

(ΔΔ)3\left(\vec{\partial }_{\Delta }-\overleftarrow{\partial }_{\Delta }\right){}^3

ExpandPartialD[%]

Δ.Δ.Δ+3Δ.Δ.Δ3Δ.Δ.Δ+Δ.Δ.Δ-\overleftarrow{\partial }_{\Delta }.\overleftarrow{\partial }_{\Delta }.\overleftarrow{\partial }_{\Delta }+3 \overleftarrow{\partial }_{\Delta }.\overleftarrow{\partial }_{\Delta }.\vec{\partial }_{\Delta }-3 \overleftarrow{\partial }_{\Delta }.\vec{\partial }_{\Delta }.\vec{\partial }_{\Delta }+\vec{\partial }_{\Delta }.\vec{\partial }_{\Delta }.\vec{\partial }_{\Delta }

LC[\[Mu], \[Nu], \[Rho], \[Tau]] RightPartialD[\[Alpha], \[Mu], \[Beta], \[Nu]] 
 
ExpandPartialD[%]

ϵˉμνρτα.μ.β.ν\bar{\epsilon }^{\mu \nu \rho \tau } \vec{\partial }_{\alpha }.\vec{\partial }_{\mu }.\vec{\partial }_{\beta }.\vec{\partial }_{\nu }

00

CLC[i, j, k] RightNablaD[i, j, k] 
 
ExpandPartialD[%]

ϵˉijki.j.k\bar{\epsilon }^{ijk} \vec{\nabla }^i.\vec{\nabla }^j.\vec{\nabla }^k

00

RightPartialD[CartesianIndex[i]] . QuantumField[S, x] 
 
% // ExpandPartialD

i.Sx\vec{\partial }_i.S^x

(iSx)\left(\partial _iS^x\right)

RightPartialD[{CartesianIndex[i], x}] . QuantumField[S, x] 
 
% // ExpandPartialD

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

({i,x}Sx)\left(\partial _{\{i,x\}}S^x\right)

By default the derivative won’t act on anything outside of the input expression. But it can be made to by setting the option RightPartialD to True

ExpandPartialD[RightPartialD[\[Mu]] . QuantumField[A, LorentzIndex[\[Mu]]] . QuantumField[A, LorentzIndex[\[Nu]]]]

Aμ.(μAν)+(μAμ).AνA_{\mu }.\left(\partial _{\mu }A_{\nu }\right)+\left(\partial _{\mu }A_{\mu }\right).A_{\nu }

ExpandPartialD[RightPartialD[\[Mu]] . QuantumField[A, LorentzIndex[\[Mu]]] . QuantumField[A, LorentzIndex[\[Nu]]], RightPartialD -> True]

Aμ.(μAν)+(μAμ).Aν+Aμ.Aν.μA_{\mu }.\left(\partial _{\mu }A_{\nu }\right)+\left(\partial _{\mu }A_{\mu }\right).A_{\nu }+A_{\mu }.A_{\nu }.\vec{\partial }_{\mu }

The same applies also to LeftPartialD

ExpandPartialD[QuantumField[A, LorentzIndex[\[Nu]]] . LeftNablaD[i]]

(iAν)-\left(\partial _iA_{\nu }\right)

ExpandPartialD[QuantumField[A, LorentzIndex[\[Nu]]] . LeftNablaD[i], LeftPartialD -> True]

(iAν)i.Aν-\left(\partial _iA_{\nu }\right)-\overleftarrow{\partial }_i.A_{\nu }