Chisholm
Chisholm[exp]
substitutes products of three Dirac matrices or slashes by the Chisholm identity.
See also
Overview
Examples
GA[\[Mu], \[Nu], \[Rho]]
EpsChisholm[%]
γˉμ.γˉν.γˉρ
γˉμ.γˉν.γˉρ
Notice that the output contains dummy indices.
GA[\[Alpha], \[Beta], \[Mu], \[Nu]]
Chisholm[%]
γˉα.γˉβ.γˉμ.γˉν
iγˉα.γˉ$MU($22).γˉ5ϵˉβμν$MU($22)+γˉα.γˉνgˉβμ−γˉα.γˉμgˉβν+γˉα.γˉβgˉμν
Dummy Lorentz indices may also appear as FCGV.
SpinorVBar[p1, m1] . GA[\[Alpha], \[Beta], \[Mu], \[Nu]] . SpinorU[p2, m2]
Chisholm[%]
vˉ(p1,m1).γˉα.γˉβ.γˉμ.γˉν.u(p2,m2)
iϵˉβμν$MU($31)(φ(−p1,m1)).γˉα.γˉ$MU($31).γˉ5.(φ(p2,m2))+gˉβμ(φ(−p1,m1)).γˉα.γˉν.(φ(p2,m2))−gˉβν(φ(−p1,m1)).γˉα.γˉμ.(φ(p2,m2))+gˉμν(φ(−p1,m1)).γˉα.γˉβ.(φ(p2,m2))
Chisholm only works with Dirac matrices in 4 dimensions, D-dimensional objects are ignored.
Chisholm[GAD[\[Mu], \[Nu], \[Rho]]]
γμ.γν.γρ
Chisholm[GA[\[Alpha], \[Beta], \[Mu]]] . Chisholm[GA[\[Alpha], \[Beta], \[Mu]]]
DiracSimplify[%]
(iγˉ$MU($58).γˉ5ϵˉαβμ$MU($58)+γˉμgˉαβ−γˉβgˉαμ+γˉαgˉβμ).(iγˉ$MU($67).γˉ5ϵˉαβμ$MU($67)+γˉμgˉαβ−γˉβgˉαμ+γˉαgˉβμ)
16
Chisholm[GA[\[Alpha], \[Beta], \[Mu], \[Nu]]] . Chisholm[GA[\[Alpha], \[Beta], \[Mu], \[Nu]]]
DiracSimplify[%]
(iγˉα.γˉ$MU($81).γˉ5ϵˉβμν$MU($81)+γˉα.γˉνgˉβμ−γˉα.γˉμgˉβν+γˉα.γˉβgˉμν).(iγˉα.γˉ$MU($90).γˉ5ϵˉβμν$MU($90)+γˉα.γˉνgˉβμ−γˉα.γˉμgˉβν+γˉα.γˉβgˉμν)
−128
(γˉ⋅p).(γˉ⋅q).(γˉ⋅r)
−iγˉ$MU($116).γˉ5ϵˉ$MU($116)pqr+(p⋅q)γˉ⋅r−(p⋅r)γˉ⋅q+γˉ⋅p(q⋅r)
GA[\[Mu], \[Nu], \[Rho], \[Sigma], \[Tau], \[Kappa]]
Chisholm[%]
γˉμ.γˉν.γˉρ.γˉσ.γˉτ.γˉκ
igˉνργˉμ.γˉ$MU($125).γˉ5ϵˉκστ$MU($125)−igˉκσγˉμ.γˉ$MU($127).γˉ5ϵˉνρτ$MU($127)+igˉκτγˉμ.γˉ$MU($128).γˉ5ϵˉνρσ$MU($128)+igˉστγˉμ.γˉ$MU($129).γˉ5ϵˉκνρ$MU($129)−iγˉμ.γˉρ.γˉ5ϵˉκνστ+iγˉμ.γˉν.γˉ5ϵˉκρστ−γˉμ.γˉτgˉκσgˉνρ+γˉμ.γˉσgˉκτgˉνρ+γˉμ.γˉτgˉκρgˉνσ−γˉμ.γˉρgˉκτgˉνσ−γˉμ.γˉσgˉκρgˉντ+γˉμ.γˉρgˉκσgˉντ−γˉμ.γˉτgˉκνgˉρσ+γˉμ.γˉνgˉκτgˉρσ+γˉμ.γˉκgˉντgˉρσ+γˉμ.γˉσgˉκνgˉρτ−γˉμ.γˉνgˉκσgˉρτ−γˉμ.γˉκgˉνσgˉρτ−γˉμ.γˉρgˉκνgˉστ+γˉμ.γˉνgˉκρgˉστ+γˉμ.γˉκgˉνρgˉστ
Check the equality of the expressions before and after applying Chisholm
.
DiracSimplify[GA[\[Mu], \[Nu], \[Rho], \[Sigma], \[Tau], \[Kappa]] . GA[\[Mu], \[Nu], \[Rho], \[Sigma], \[Tau], \[Kappa]]]
−2048
DiracSimplify[Chisholm[GA[\[Mu], \[Nu], \[Rho], \[Sigma], \[Tau], \[Kappa]]] . Chisholm[GA[\[Mu], \[Nu], \[Rho], \[Sigma], \[Tau], \[Kappa]]]]
−2048
DiracReduce[GA[\[Mu], \[Nu], \[Rho], \[Sigma], \[Tau], \[Kappa]] . Chisholm[GA[\[Mu], \[Nu], \[Rho], \[Sigma], \[Tau], \[Kappa]]]]
−2048
Older FeynCalc versions had a function called Chisholm2
that acted on expressions like γμγνγ5. This functionality is now part of Chisholm
and can be activated by setting the option Mode
to 2
.
GA[\[Mu], \[Nu], 5]
Chisholm[%, Mode -> 2]
γˉμ.γˉν.γˉ5
21σ$MU($1022)$MU($1023)ϵˉμν$MU($1022)$MU($1023)+γˉ5gˉμν