DiracReduce[exp] reduces all 4-dimensional Dirac matrices in exp to the standard basis (S, P, V, A, T) using the Chisholm identity.
In the result the basic Dirac structures can be wrapped with a head DiracBasis, that is
DiracBasis[1]DiracBasis[GA[5]]DiracBasis[GA[mu]]DiracBasis[GA[mu, 5]]DiracBasis[DiracSigma[GA[mu, nu]]]By default DiracBasis is substituted to Identity.
Overview, Chisholm, DiracSimplify, EpsChisholm.
GA[\[Mu], \[Nu]]
DiracReduce[%]\bar{\gamma }^{\mu }.\bar{\gamma }^{\nu }
\bar{g}^{\mu \nu }-i \sigma ^{\mu \nu }
DiracReduce only works with Dirac matrices in 4 dimensions, D-dimensional matrices are ignored.
GAD[\[Mu], \[Nu]]
DiracReduce[%]\gamma ^{\mu }.\gamma ^{\nu }
\gamma ^{\mu }.\gamma ^{\nu }
SpinorUBar[Subscript[p, 1], Subscript[m, 1]] . GA[\[Mu], \[Nu], \[Rho]] . SpinorV[Subscript[p, 2], Subscript[m, 2]]
DiracReduce[%]\bar{u}\left(p_1,m_1\right).\bar{\gamma }^{\mu }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\rho }.v\left(p_2,m_2\right)
i \bar{\epsilon }^{\mu \nu \rho \;\text{\$MU}(\text{\$31})} \left(\varphi (\overline{p}_1,m_1)\right).\bar{\gamma }^{\text{\$MU}(\text{\$31})}.\bar{\gamma }^5.\left(\varphi (-\overline{p}_2,m_2)\right)+\bar{g}^{\mu \nu } \left(\varphi (\overline{p}_1,m_1)\right).\bar{\gamma }^{\rho }.\left(\varphi (-\overline{p}_2,m_2)\right)-\bar{g}^{\mu \rho } \left(\varphi (\overline{p}_1,m_1)\right).\bar{\gamma }^{\nu }.\left(\varphi (-\overline{p}_2,m_2)\right)+\bar{g}^{\nu \rho } \left(\varphi (\overline{p}_1,m_1)\right).\bar{\gamma }^{\mu }.\left(\varphi (-\overline{p}_2,m_2)\right)
GA[\[Mu], \[Nu], \[Rho], \[Sigma]]
DiracReduce[%]\bar{\gamma }^{\mu }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\rho }.\bar{\gamma }^{\sigma }
-i \bar{\gamma }^5 \bar{\epsilon }^{\mu \nu \rho \sigma }-i \sigma ^{\rho \sigma } \bar{g}^{\mu \nu }+i \sigma ^{\nu \sigma } \bar{g}^{\mu \rho }-i \sigma ^{\nu \rho } \bar{g}^{\mu \sigma }-i \sigma ^{\mu \sigma } \bar{g}^{\nu \rho }+i \sigma ^{\mu \rho } \bar{g}^{\nu \sigma }-i \sigma ^{\mu \nu } \bar{g}^{\rho \sigma }+\bar{g}^{\mu \sigma } \bar{g}^{\nu \rho }-\bar{g}^{\mu \rho } \bar{g}^{\nu \sigma }+\bar{g}^{\mu \nu } \bar{g}^{\rho \sigma }
Do some checks of the results
DiracSimplify[GA[\[Mu], \[Nu], \[Rho], \[Sigma]] . GA[\[Mu], \[Nu], \[Rho], \[Sigma]]]-128
DiracSimplify[DiracReduce[GA[\[Mu], \[Nu], \[Rho], \[Sigma]]] . DiracReduce[GA[\[Mu], \[Nu], \[Rho], \[Sigma]]]]-128
We may also keep the head DiracBasis in the final result
DiracReduce[GA[\[Mu], \[Nu], \[Rho], \[Sigma]], FinalSubstitutions -> {}]-i \bar{g}^{\mu \nu } \;\text{DiracBasis}\left(\text{DiracSigma}\left(\text{DiracBasis}\left(\bar{\gamma }^{\rho }\right),\text{DiracBasis}\left(\bar{\gamma }^{\sigma }\right)\right)\right)+i \bar{g}^{\mu \rho } \;\text{DiracBasis}\left(\text{DiracSigma}\left(\text{DiracBasis}\left(\bar{\gamma }^{\nu }\right),\text{DiracBasis}\left(\bar{\gamma }^{\sigma }\right)\right)\right)-i \bar{g}^{\mu \sigma } \;\text{DiracBasis}\left(\text{DiracSigma}\left(\text{DiracBasis}\left(\bar{\gamma }^{\nu }\right),\text{DiracBasis}\left(\bar{\gamma }^{\rho }\right)\right)\right)-i \bar{g}^{\nu \rho } \;\text{DiracBasis}\left(\text{DiracSigma}\left(\text{DiracBasis}\left(\bar{\gamma }^{\mu }\right),\text{DiracBasis}\left(\bar{\gamma }^{\sigma }\right)\right)\right)+i \bar{g}^{\nu \sigma } \;\text{DiracBasis}\left(\text{DiracSigma}\left(\text{DiracBasis}\left(\bar{\gamma }^{\mu }\right),\text{DiracBasis}\left(\bar{\gamma }^{\rho }\right)\right)\right)-i \bar{g}^{\rho \sigma } \;\text{DiracBasis}\left(\text{DiracSigma}\left(\text{DiracBasis}\left(\bar{\gamma }^{\mu }\right),\text{DiracBasis}\left(\bar{\gamma }^{\nu }\right)\right)\right)-i \;\text{DiracBasis}\left(\bar{\gamma }^5\right) \bar{\epsilon }^{\mu \nu \rho \sigma }+\text{DiracBasis}(1) \bar{g}^{\mu \sigma } \bar{g}^{\nu \rho }-\text{DiracBasis}(1) \bar{g}^{\mu \rho } \bar{g}^{\nu \sigma }+\text{DiracBasis}(1) \bar{g}^{\mu \nu } \bar{g}^{\rho \sigma }