FeynCalc manual (development version)

LCD

LCD[m, n, r, s] evaluates to D-dimensional \varepsilon^{m n r s} by virtue of applying FeynCalcInternal.

LCD[m,...][p, ...] evaluates to D-dimensional \epsilon ^{m \ldots \mu \ldots}p_{\mu \ldots} applying FeynCalcInternal.

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, \varepsilon^{p_1 p_2 p_3 p_4} (accessible via LCD[][p1,p2,p3,p4]) correspond to \varepsilon_{\mu \nu \rho \sigma} p_1^\mu p_2^\nu p_3^\rho p_4^\sigma.

See also

Overview, Eps, LC.

Examples

LCD[\[Mu], \[Nu], \[Rho], \[Sigma]]

\overset{\text{}}{\epsilon }^{\mu \nu \rho \sigma }

LCD[\[Mu], \[Nu], \[Rho], \[Sigma]] // FCI // StandardForm

(*Eps[LorentzIndex[\[Mu], D], LorentzIndex[\[Nu], D], LorentzIndex[\[Rho], D], LorentzIndex[\[Sigma], D]]*)

LCD[\[Mu], \[Nu]][p, q]

\overset{\text{}}{\epsilon }^{\mu \nu pq}

LCD[\[Mu], \[Nu]][p, q] // FCI // StandardForm

(*Eps[LorentzIndex[\[Mu], D], LorentzIndex[\[Nu], D], Momentum[p, D], Momentum[q, D]]*)

Factor2[Contract[LCD[\[Mu], \[Nu], \[Rho]][p] LCD[\[Mu], \[Nu], \[Rho]][q]]]

(1-D) (2-D) (3-D) (p\cdot q)

LCD[\[Mu], \[Nu], \[Rho], \[Sigma]] FVD[Subscript[p, 1], \[Mu]] FVD[Subscript[p, 2], \[Nu]] FVD[Subscript[p, 3], \[Rho]] FVD[Subscript[p, 4], \[Sigma]] 
 
Contract[%]

p_1{}^{\mu } p_2{}^{\nu } p_3{}^{\rho } p_4{}^{\sigma } \overset{\text{}}{\epsilon }^{\mu \nu \rho \sigma }

\overset{\text{}}{\epsilon }^{p_1p_2p_3p_4}