ShiftPartialD
ShiftPartialD[exp, {FCPartialD[i1], FCPartialD[i2], ...}, field]
uses integration-by-parts identities to shift the derivatives of QuantumField
s such, that a term containing derivatives with indices i1, i2, ...
acting on field
is eliminated from the final expression.
Notice that one must explicitly specify the type of the indices, e.g. by writing FCPartialD[LorentzIndex[mu]]
or FCPartialD[CartesianIndex[i]]
. Furthermore, the function always assumes that the surface term vanishes.
Often, when dealing with large expressions one would to integrate by parts only certain terms but not every term containing given fields and derivatives. In such situation one can specify a filter function via the option Select
.
See also
Overview , ExplicitPartialD , ExpandPartialD , QuantumField
Examples
exp1 = QuantumField[ QuarkFieldPsiDagger, PauliIndex[ di1]] . RightPartialD[ CartesianIndex[ i
]] . QuantumField[ \ [ Phi]] . RightPartialD[ CartesianIndex[ j ]] . QuantumField[ QuarkFieldPsi, PauliIndex[ di2]]
ψ † di1 . ∂ ⃗ i . ϕ . ∂ ⃗ j . ψ di2 \psi ^{\dagger \;\text{di1}}.\vec{\partial }_i.\phi .\vec{\partial }_j.\psi ^{\text{di2}} ψ † di1 . ∂ i . ϕ . ∂ j . ψ di2
ψ † di1 . ϕ . ( ∂ i ∂ j ψ di2 ) + ψ † di1 . ( ∂ i ϕ ) . ( ∂ j ψ di2 ) \psi ^{\dagger \;\text{di1}}.\phi .\left(\partial _i\partial _j\psi ^{\text{di2}}\right)+\psi ^{\dagger \;\text{di1}}.\left(\partial _i\phi \right).\left(\partial _j\psi ^{\text{di2}}\right) ψ † di1 . ϕ . ( ∂ i ∂ j ψ di2 ) + ψ † di1 . ( ∂ i ϕ ) . ( ∂ j ψ di2 )
ShiftPartialD[ exp1, { FCPartialD[ CartesianIndex[ i ]]}, QuarkFieldPsi, FCVerbose -> - 1 ]
− ( ∂ i ψ † di1 ) . ϕ . ( ∂ j ψ di2 ) -\left(\partial _i\psi ^{\dagger \;\text{di1}}\right).\phi .\left(\partial _j\psi ^{\text{di2}}\right) − ( ∂ i ψ † di1 ) . ϕ . ( ∂ j ψ di2 )
This expression vanishes if one integrates by parts the term containing ∂ μ A ν \partial_\mu A_\nu ∂ μ A ν
exp2 = QuantumField[ GaugeField, LorentzIndex[ nu]] . QuantumField[ FCPartialD[ LorentzIndex[ mu]], FCPartialD[ LorentzIndex[ mu]],
FCPartialD[ LorentzIndex[ rho]], FCPartialD[ LorentzIndex[ rho]], FCPartialD[ LorentzIndex[ nu]], FCPartialD[ LorentzIndex[ tau]],
GaugeField, LorentzIndex[ tau]] + QuantumField[ FCPartialD[ LorentzIndex[ mu]], GaugeField, LorentzIndex[ nu]] . QuantumField[ FCPartialD[ LorentzIndex[ rho]],
FCPartialD[ LorentzIndex[ rho]], FCPartialD[ LorentzIndex[ mu]], FCPartialD[ LorentzIndex[ nu]], FCPartialD[ LorentzIndex[ tau]], GaugeField, LorentzIndex[ tau]]
A nu . ( ∂ mu ∂ mu ∂ rho ∂ rho ∂ nu ∂ tau A tau ) + ( ∂ mu A nu ) . ( ∂ rho ∂ rho ∂ mu ∂ nu ∂ tau A tau ) A_{\text{nu}}.\left(\partial _{\text{mu}}\partial _{\text{mu}}\partial _{\text{rho}}\partial _{\text{rho}}\partial _{\text{nu}}\partial _{\text{tau}}A_{\text{tau}}\right)+\left(\partial _{\text{mu}}A_{\text{nu}}\right).\left(\partial _{\text{rho}}\partial _{\text{rho}}\partial _{\text{mu}}\partial _{\text{nu}}\partial _{\text{tau}}A_{\text{tau}}\right) A nu . ( ∂ mu ∂ mu ∂ rho ∂ rho ∂ nu ∂ tau A tau ) + ( ∂ mu A nu ) . ( ∂ rho ∂ rho ∂ mu ∂ nu ∂ tau A tau )
By default ShiftPartialD
will also apply IBPs to the other term, which is not useful here
ShiftPartialD[ exp2, { FCPartialD[ LorentzIndex[ mu]]}, GaugeField]
Applying the following IBP relation: { ( ∂ mu A nu ) . ( ∂ rho ∂ rho ∂ mu ∂ nu ∂ tau A tau ) → − A nu . ( ∂ mu ∂ mu ∂ rho ∂ rho ∂ nu ∂ tau A tau ) } \text{Applying the following IBP relation: }\left\{\left(\partial _{\text{mu}}A_{\text{nu}}\right).\left(\partial _{\text{rho}}\partial _{\text{rho}}\partial _{\text{mu}}\partial _{\text{nu}}\partial _{\text{tau}}A_{\text{tau}}\right)\to -A_{\text{nu}}.\left(\partial _{\text{mu}}\partial _{\text{mu}}\partial _{\text{rho}}\partial _{\text{rho}}\partial _{\text{nu}}\partial _{\text{tau}}A_{\text{tau}}\right)\right\} Applying the following IBP relation: { ( ∂ mu A nu ) . ( ∂ rho ∂ rho ∂ mu ∂ nu ∂ tau A tau ) → − A nu . ( ∂ mu ∂ mu ∂ rho ∂ rho ∂ nu ∂ tau A tau ) }
Applying the following IBP relation: { A nu . ( ∂ mu ∂ mu ∂ rho ∂ rho ∂ nu ∂ tau A tau ) → − 2 ( ∂ mu A nu ) . ( ∂ rho ∂ rho ∂ mu ∂ nu ∂ tau A tau ) − ( ∂ mu ∂ mu A nu ) . ( ∂ rho ∂ rho ∂ nu ∂ tau A tau ) } \text{Applying the following IBP relation: }\left\{A_{\text{nu}}.\left(\partial _{\text{mu}}\partial _{\text{mu}}\partial _{\text{rho}}\partial _{\text{rho}}\partial _{\text{nu}}\partial _{\text{tau}}A_{\text{tau}}\right)\to -2 \left(\partial _{\text{mu}}A_{\text{nu}}\right).\left(\partial _{\text{rho}}\partial _{\text{rho}}\partial _{\text{mu}}\partial _{\text{nu}}\partial _{\text{tau}}A_{\text{tau}}\right)-\left(\partial _{\text{mu}}\partial _{\text{mu}}A_{\text{nu}}\right).\left(\partial _{\text{rho}}\partial _{\text{rho}}\partial _{\text{nu}}\partial _{\text{tau}}A_{\text{tau}}\right)\right\} Applying the following IBP relation: { A nu . ( ∂ mu ∂ mu ∂ rho ∂ rho ∂ nu ∂ tau A tau ) → − 2 ( ∂ mu A nu ) . ( ∂ rho ∂ rho ∂ mu ∂ nu ∂ tau A tau ) − ( ∂ mu ∂ mu A nu ) . ( ∂ rho ∂ rho ∂ nu ∂ tau A tau ) }
− A nu . ( ∂ mu ∂ mu ∂ rho ∂ rho ∂ nu ∂ tau A tau ) − 2 ( ∂ mu A nu ) . ( ∂ rho ∂ rho ∂ mu ∂ nu ∂ tau A tau ) − ( ∂ mu ∂ mu A nu ) . ( ∂ rho ∂ rho ∂ nu ∂ tau A tau ) -A_{\text{nu}}.\left(\partial _{\text{mu}}\partial _{\text{mu}}\partial _{\text{rho}}\partial _{\text{rho}}\partial _{\text{nu}}\partial _{\text{tau}}A_{\text{tau}}\right)-2 \left(\partial _{\text{mu}}A_{\text{nu}}\right).\left(\partial _{\text{rho}}\partial _{\text{rho}}\partial _{\text{mu}}\partial _{\text{nu}}\partial _{\text{tau}}A_{\text{tau}}\right)-\left(\partial _{\text{mu}}\partial _{\text{mu}}A_{\text{nu}}\right).\left(\partial _{\text{rho}}\partial _{\text{rho}}\partial _{\text{nu}}\partial _{\text{tau}}A_{\text{tau}}\right) − A nu . ( ∂ mu ∂ mu ∂ rho ∂ rho ∂ nu ∂ tau A tau ) − 2 ( ∂ mu A nu ) . ( ∂ rho ∂ rho ∂ mu ∂ nu ∂ tau A tau ) − ( ∂ mu ∂ mu A nu ) . ( ∂ rho ∂ rho ∂ nu ∂ tau A tau )
Using a suitable filter function we can readily achieve the desired result
ShiftPartialD[ exp2, { FCPartialD[ LorentzIndex[ mu]]}, GaugeField, Select ->
Function [ x , FreeQ [ x , QuantumField[ GaugeField, LorentzIndex[ nu]]]]]
Applying the following IBP relation: { ( ∂ mu A nu ) . ( ∂ rho ∂ rho ∂ mu ∂ nu ∂ tau A tau ) → − A nu . ( ∂ mu ∂ mu ∂ rho ∂ rho ∂ nu ∂ tau A tau ) } \text{Applying the following IBP relation: }\left\{\left(\partial _{\text{mu}}A_{\text{nu}}\right).\left(\partial _{\text{rho}}\partial _{\text{rho}}\partial _{\text{mu}}\partial _{\text{nu}}\partial _{\text{tau}}A_{\text{tau}}\right)\to -A_{\text{nu}}.\left(\partial _{\text{mu}}\partial _{\text{mu}}\partial _{\text{rho}}\partial _{\text{rho}}\partial _{\text{nu}}\partial _{\text{tau}}A_{\text{tau}}\right)\right\} Applying the following IBP relation: { ( ∂ mu A nu ) . ( ∂ rho ∂ rho ∂ mu ∂ nu ∂ tau A tau ) → − A nu . ( ∂ mu ∂ mu ∂ rho ∂ rho ∂ nu ∂ tau A tau ) }
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