FeynCalc manual (development version)

LC

LC[m, n, r, s] evaluates to 4-dimensional \varepsilon^{m n r s} by virtue of applying FeynCalcInternal.

LC[m,...][p, ...] evaluates to 4-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 LC[][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, LCD.

Examples

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

\bar{\epsilon }^{\mu \nu \rho \sigma }

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

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

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

\bar{\epsilon }^{\mu \nu \overline{p}\overline{q}}

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

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

Contract[LC[\[Mu], \[Nu], \[Rho]][p] LC[\[Mu], \[Nu], \[Rho]][q]] 

-6 \left(\overline{p}\cdot \overline{q}\right)

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

\overline{p}_1{}^{\mu } \overline{p}_2{}^{\nu } \overline{p}_3{}^{\rho } \overline{p}_4{}^{\sigma } \bar{\epsilon }^{\mu \nu \rho \sigma }

\bar{\epsilon }^{\overline{p}_1\overline{p}_2\overline{p}_3\overline{p}_4}