DiracReduce
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
- S:
DiracBasis[1]
- P:
DiracBasis[GA[5]]
- V:
DiracBasis[GA[mu]]
- A:
DiracBasis[GA[mu, 5]]
- T:
DiracBasis[DiracSigma[GA[mu, nu]]]
By default DiracBasis
is substituted to Identity
.
See also
Overview, Chisholm, DiracSimplify, EpsChisholm.
Examples
GA[\[Mu], \[Nu]]
DiracReduce[%]
γˉμ.γˉν
gˉμν−iσμν
DiracReduce
only works with Dirac matrices in 4 dimensions, D-dimensional matrices are ignored.
GAD[\[Mu], \[Nu]]
DiracReduce[%]
γμ.γν
γμ.γν
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)
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))
GA[\[Mu], \[Nu], \[Rho], \[Sigma]]
DiracReduce[%]
γˉμ.γˉν.γˉρ.γˉσ
−iγˉ5ϵˉμνρσ−iσρσgˉμν+iσνσgˉμρ−iσνρgˉμσ−iσμσgˉνρ+iσμρgˉνσ−iσμνgˉρσ+gˉμσgˉνρ−gˉμρgˉνσ+gˉμνgˉρσ
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 -> {}]
−igˉμνDiracBasis(DiracSigma(DiracBasis(γˉρ),DiracBasis(γˉσ)))+igˉμρDiracBasis(DiracSigma(DiracBasis(γˉν),DiracBasis(γˉσ)))−igˉμσDiracBasis(DiracSigma(DiracBasis(γˉν),DiracBasis(γˉρ)))−igˉνρDiracBasis(DiracSigma(DiracBasis(γˉμ),DiracBasis(γˉσ)))+igˉνσDiracBasis(DiracSigma(DiracBasis(γˉμ),DiracBasis(γˉρ)))−igˉρσDiracBasis(DiracSigma(DiracBasis(γˉμ),DiracBasis(γˉν)))−iDiracBasis(γˉ5)ϵˉμνρσ+DiracBasis(1)gˉμσgˉνρ−DiracBasis(1)gˉμρgˉνσ+DiracBasis(1)gˉμνgˉρσ