FeynCalc manual (development version)

DiracReduce

DiracReduce[exp] reduces all 44-dimensional Dirac matrices in exp to the standard basis (S,P,V,A,T)(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

By default DiracBasis is substituted to Identity.

See also

Overview, Chisholm, DiracSimplify, EpsChisholm.

Examples

GA[\[Mu], \[Nu]] 
 
DiracReduce[%]

γˉμ.γˉν\bar{\gamma }^{\mu }.\bar{\gamma }^{\nu }

gˉμνiσμν\bar{g}^{\mu \nu }-i \sigma ^{\mu \nu }

DiracReduce only works with Dirac matrices in 44 dimensions, DD-dimensional matrices are ignored.

GAD[\[Mu], \[Nu]] 
 
DiracReduce[%]

γμ.γν\gamma ^{\mu }.\gamma ^{\nu }

γμ.γν\gamma ^{\mu }.\gamma ^{\nu }

SpinorUBar[Subscript[p, 1], Subscript[m, 1]] . GA[\[Mu], \[Nu], \[Rho]] . SpinorV[Subscript[p, 2], Subscript[m, 2]] 
 
DiracReduce[%]

uˉ(p1,m1).γˉμ.γˉν.γˉρ.v(p2,m2)\bar{u}\left(p_1,m_1\right).\bar{\gamma }^{\mu }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\rho }.v\left(p_2,m_2\right)

iϵˉμνρ  $MU($31)(φ(p1,m1)).γˉ$MU($31).γˉ5.(φ(p2,m2))+gˉμν(φ(p1,m1)).γˉρ.(φ(p2,m2))gˉμρ(φ(p1,m1)).γˉν.(φ(p2,m2))+gˉνρ(φ(p1,m1)).γˉμ.(φ(p2,m2))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[%]

γˉμ.γˉν.γˉρ.γˉσ\bar{\gamma }^{\mu }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\rho }.\bar{\gamma }^{\sigma }

iγˉ5ϵˉμνρσiσρσgˉμν+iσνσgˉμρiσνρgˉμσiσμσgˉνρ+iσμρgˉνσiσμνgˉρσ+gˉμσgˉνρgˉμρgˉνσ+gˉμνgˉρσ-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]]]

128-128

DiracSimplify[DiracReduce[GA[\[Mu], \[Nu], \[Rho], \[Sigma]]] . DiracReduce[GA[\[Mu], \[Nu], \[Rho], \[Sigma]]]]

128-128

We may also keep the head DiracBasis in the final result

DiracReduce[GA[\[Mu], \[Nu], \[Rho], \[Sigma]], FinalSubstitutions -> {}]

igˉμν  DiracBasis(DiracSigma(DiracBasis(γˉρ),DiracBasis(γˉσ)))+igˉμρ  DiracBasis(DiracSigma(DiracBasis(γˉν),DiracBasis(γˉσ)))igˉμσ  DiracBasis(DiracSigma(DiracBasis(γˉν),DiracBasis(γˉρ)))igˉνρ  DiracBasis(DiracSigma(DiracBasis(γˉμ),DiracBasis(γˉσ)))+igˉνσ  DiracBasis(DiracSigma(DiracBasis(γˉμ),DiracBasis(γˉρ)))igˉρσ  DiracBasis(DiracSigma(DiracBasis(γˉμ),DiracBasis(γˉν)))i  DiracBasis(γˉ5)ϵˉμνρσ+DiracBasis(1)gˉμσgˉνρDiracBasis(1)gˉμρgˉνσ+DiracBasis(1)gˉμνgˉρσ-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 }