# Commutator of gamma matrices?

Is the matrix $\sigma_{\mu\nu}$

$$\sigma_{\mu\nu} = \frac{i}{2} [\gamma_\mu, \gamma_\nu]$$

defined in FeynRules, FeynCalc or any similar packages? I know that $\gamma_\mu$ is Ga[mu] in FeynRules.

• PS it won't let me enclose my equation in $$for latex... Nov 13 '17 at 0:51 • Not sure what you did wrong; enclosing in $$ works, as you might now see. Nov 13 '17 at 3:23
• Funny, it said that my question contained code that wasn't formatted properly and wouldn't let me post. The preview looked fine. Nov 13 '17 at 3:25

The $\gamma$ matrices are built-in, but undocumented, as InternalDiracGammaMatrix[]. Their indexing is also a bit different from the wiki page:

Table[InternalDiracGammaMatrix[k, "Basis" -> "Dirac"] // MatrixForm, {k, 4}]


$$\{\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \\ \end{pmatrix}, \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ \end{pmatrix}, \begin{pmatrix} 0 & 0 & 0 & -i \\ 0 & 0 & i & 0 \\ 0 & i & 0 & 0 \\ -i & 0 & 0 & 0 \\ \end{pmatrix}, \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \end{pmatrix}\}$$

(Other possible settings for "Basis" include "Chiral" (the default) and "Majorana".)

Thus, you can implement the commutator like this:

Options[diracCommutator] = Options[InternalDiracGammaMatrix];
diracCommutator[p_, q_, opts : OptionsPattern[]] :=
With[{pm = InternalDiracGammaMatrix[p, opts], qm = InternalDiracGammaMatrix[q, opts]},
I (pm.qm - qm.pm)/2]


and then you can do e.g.

diracCommutator[2, 4, "Basis" -> "Dirac"] // MatrixForm


$$\begin{pmatrix} 0 & i & 0 & 0 \\ -i & 0 & 0 & 0 \\ 0 & 0 & 0 & i \\ 0 & 0 & -i & 0 \\ \end{pmatrix}$$

• This is without using FeynCalc, right? Nov 13 '17 at 7:11
• Nope, it's built-in, but undocumented. Nov 13 '17 at 7:12

In FeynCalc there is DiracSigma

DiracSigma[GA[mu], GA[nu]]


with

?DiracSigma


DiracSigma[a, b] stands for I/2*(a . b - b . a) in 4 dimensions. a and b must have Head DiracGamma, DiracMatrix or DiracSlash. Only antisymmetry is implemented.

However, you cannot have explicit Dirac indices attached to the Dirac matrices. This is something planned for the future (which is needed to use FeynCalc with QGraf)

The matrix $\sigma_{\mu\nu}$ is given by Sig[mu, nu] in FeynRules, a related package, and defined

Sig[mu_,nu_,ss1_,ss2_]->I/2 TensDot[Ga[mu].Ga[nu]][ss1,ss2]-I/2TensDot[Ga[nu].Ga[mu]][ss1,ss2]},\[Infinity],Heads->True];


in Interfaces/FeynArtsInterface.m.

In Package-X, the object DiracS (σ`) represents the commutator of Dirac gamma matrices.