Eps[a, b, c, d]
is the head of the totally antisymmetric (Levi-Civita) tensor. The a,b, ...
may have head LorentzIndex
or Momentum
.
When some indices of a Levi-Civita-tensor are contracted with 4-vectors, FeynCalc suppresses explicit dummy indices by putting those vectors into the corresponding index slots. For example, (accessible via LC[][p1,p2,p3,p4]
) correspond to .
Overview, EpsContract, EpsEvaluate, LC, LCD.
[LorentzIndex[\[Mu]], LorentzIndex[\[Nu]], LorentzIndex[\[Rho]], LorentzIndex[\[Sigma]]] Eps
[Momentum[p], LorentzIndex[\[Nu]], LorentzIndex[\[Rho]], LorentzIndex[\[Sigma]]] Eps
[b, a, c, d] // StandardForm
Eps
(*Eps[b, a, c, d]*)
[ExplicitLorentzIndex[0], ExplicitLorentzIndex[1], ExplicitLorentzIndex[2],
Eps[3]] ExplicitLorentzIndex
[LorentzIndex[\[Mu]], LorentzIndex[\[Nu]], LorentzIndex[\[Rho]], LorentzIndex[\[Sigma]]] *
Eps[LorentzIndex[\[Mu]], Momentum[Subscript[p, 1]]] Pair[LorentzIndex[\[Nu]], Momentum[Subscript[p, 2]]]*
Pair[LorentzIndex[\[Rho]], Momentum[Subscript[p, 3]]] Pair[LorentzIndex[\[Sigma]], Momentum[Subscript[p, 4]]]
Pair
[%] Contract
[LorentzIndex[\[Mu]], LorentzIndex[\[Nu]], LorentzIndex[\[Rho]], LorentzIndex[\[Sigma]]]
Eps
[% %] Contract
[LorentzIndex[\[Mu], D], LorentzIndex[\[Nu], D], LorentzIndex[\[Rho], D], LorentzIndex[\[Sigma], D]]
Eps
[% %] // Factor2 Contract
= -(I/24) LCD[\[Mu], \[Nu], \[Rho], \[Alpha]] . GAD[\[Mu], \[Nu], \[Rho], \[Alpha]] // FCI ex1
= -(I/24) LCD[\[Mu]', \[Nu]', \[Rho]', \[Alpha]'] . GAD[\[Mu]', \[Nu]', \[Rho]', \[Alpha]'] // FCI ex2
[ex1 . ex2] // Factor2
DiracSimplify
% /. D -> 4