DiracSigmaExpand[exp] applies linearity to the arguments of DiracSigma.
Overview, DiracGamma, DiracSigma.
DiracSigma[GSD[p] + GSD[q], GSD[r]]
ex = % // DiracSigmaExpand\text{DiracSigma}(\gamma \cdot p+\gamma \cdot q,\gamma \cdot r)
\sigma ^{pr}+\sigma ^{qr}
ex // FCE // StandardForm
(*DiracSigma[GSD[p], GSD[r]] + DiracSigma[GSD[q], GSD[r]]*)Notice that DiracSigmaExpand does not expand Dirac matrices contracted to linear combinations of 4-vectors by default.
DiracSigma[GSD[p + q], GSD[r]]
DiracSigmaExpand[%]\sigma ^{p+qr}
\sigma ^{p+qr}
If such expansions are required, use the option DiracGammaExpand.
DiracSigmaExpand[DiracSigma[GSD[p + q], GSD[r]], DiracGammaExpand -> True]\sigma ^{pr}+\sigma ^{qr}
The option Momentum allows us to perform more fine-grained expansions of DiracSigma.
DiracSigma[GSD[p], GSD[r] + GSD[t]] + DiracSigma[GSD[l] + GSD[n], GSD[p]]
DiracSigmaExpand[%, Momentum -> {r}]\text{DiracSigma}(\gamma \cdot l+\gamma \cdot n,\gamma \cdot p)+\text{DiracSigma}(\gamma \cdot p,\gamma \cdot r+\gamma \cdot t)
\text{DiracSigma}(\gamma \cdot l+\gamma \cdot n,\gamma \cdot p)+\sigma ^{pr}+\sigma ^{pt}