DiracGammaExpand[exp] expands Dirac matrices contracted to linear combinations of 4-vectors. All DiracGamma[Momentum[a+b+ ...]] will be expanded to DiracGamma[Momentum[a]] + DiracGamma[Momentum[b]] + DiracGamma[Momentum[...]] .
Overview, DiracGamma, DiracGammaCombine, DiracSimplify, DiracTrick.
GS[q] . GS[p - q]
ex = DiracGammaExpand[%]\left(\bar{\gamma }\cdot \overline{q}\right).\left(\bar{\gamma }\cdot \left(\overline{p}-\overline{q}\right)\right)
\left(\bar{\gamma }\cdot \overline{q}\right).\left(\bar{\gamma }\cdot \overline{p}-\bar{\gamma }\cdot \overline{q}\right)
ex // StandardForm
(*DiracGamma[Momentum[q]] . (DiracGamma[Momentum[p]] - DiracGamma[Momentum[q]])*)DiracGammaExpand rewrites \gamma^{\mu } (p-q)_{\mu } as \gamma^{mu } p_{mu } - \gamma^{\mu } q_{\mu }.
The inverse operation is DiracGammaCombine.
GS[q] . (GS[p] - GS[q])
ex = DiracGammaCombine[%]\left(\bar{\gamma }\cdot \overline{q}\right).\left(\bar{\gamma }\cdot \overline{p}-\bar{\gamma }\cdot \overline{q}\right)
\left(\bar{\gamma }\cdot \overline{q}\right).\left(\bar{\gamma }\cdot \left(\overline{p}-\overline{q}\right)\right)
ex // StandardForm
(*DiracGamma[Momentum[q]] . DiracGamma[Momentum[p - q]]*)It is possible to perform the expansions only on Dirac matrices contracted with particular momenta.
c1 GAD[\[Mu]] . (GSD[p1 + p2] + m) . GAD[\[Nu]] + c2 GAD[\[Mu]] . (GSD[q1 + q2] + m) . GAD[\[Nu]]
DiracGammaExpand[%, Momentum -> {q1}]\text{c1} \gamma ^{\mu }.(m+\gamma \cdot (\text{p1}+\text{p2})).\gamma ^{\nu }+\text{c2} \gamma ^{\mu }.(m+\gamma \cdot (\text{q1}+\text{q2})).\gamma ^{\nu }
\text{c1} \gamma ^{\mu }.(m+\gamma \cdot (\text{p1}+\text{p2})).\gamma ^{\nu }+\text{c2} \gamma ^{\mu }.(m+\gamma \cdot \;\text{q1}+\gamma \cdot \;\text{q2}).\gamma ^{\nu }
If the input expression contains DiracSigma, DiracGammaExpand will expand Feynman slashes inside DiracSigma and call DiracSigmaExpand.
DiracSigma[GSD[p + q], GSD[r]]
DiracGammaExpand[%]\sigma ^{p+qr}
\sigma ^{pr}+\sigma ^{qr}
The call to DiracSigmaExpand can be inhibited by disabling the corresponding option.
DiracGammaExpand[DiracSigma[GSD[p + q], GSD[r]], DiracSigmaExpand -> False]\text{DiracSigma}(\gamma \cdot p+\gamma \cdot q,\gamma \cdot r)
Use DiracSimplify for noncommutative expansions with the corresponding simplifications.
DiracSimplify[GS[q] . (GS[p - q])]\left(\bar{\gamma }\cdot \overline{q}\right).\left(\bar{\gamma }\cdot \overline{p}\right)-\overline{q}^2
If simplifications are not required, you may also combine DiracGammaExpand with DotSimplify.
DotSimplify[DiracGammaExpand[GS[q] . (GS[p - q])]]\left(\bar{\gamma }\cdot \overline{q}\right).\left(\bar{\gamma }\cdot \overline{p}\right)-\left(\bar{\gamma }\cdot \overline{q}\right).\left(\bar{\gamma }\cdot \overline{q}\right)