Dirac algebra
See also
Overview.
This section contains some explicit formulas used by FeynCalc when simpifying chains of Dirac matrices. Such relations can be found e.g. in Veltman’s Gammatrica or rederived by hand.
BMHV algebra
In the Breitenlohner-Maison-’t Hooft-Veltman scheme we are dealing with matrices in D, 4 and D−4 dimensions. Many identities of the BMHV algebra can be proven by by decomposing Dirac matrices into two pieces
dim(γμ)dim(γˉμ)dim(γ^μ)=D,=4,=D−4.γμgμνpμ=γˉμ+γ^μ,=gˉμν+g^μν,=pˉμ+p^μ.
The anticommuatators between Dirac matrices in different dimensions are given by
{γμ,γν}{γˉμ,γˉν}{γ^μ,γ^ν}{γˉμ,γ^ν}=2gμν,={γμ,γˉν}=2gˉμν,={γμ,γ^ν}=2g^μν,=0.
Notice that while γ5 anticommutes with all other Dirac matrices in 4 dimensions, it commutes with them in D−4 dimensions. However, in D dimensions the anticommutator is not zero
{γˉμ,γ5}{γμ,γ5}=[γ^μ,γ5]=0={γ^μ,γ5}=2γ^μγ5=2γ5γ^μ.
For the chiral projectors we obtain
PL/RγˉμPL/Rγ^μPL/RγμγμPL/R=γˉμPR/L=γ^μPL/R=γˉμPR/L+γ^μPL/R=γμPR/L∓γ^μγ5=PR/Lγˉμ+PL/Rγ^μ=PR/Lγμ∓γ5γ^μ
and
PL/R(γˉ⋅pˉ+m)PL/R(γˉ⋅p^+m)PL/R(γˉ⋅p+m)(γˉ⋅pˉ+m)PL/R(γˉ⋅p^+m)PL/R(γˉ⋅p+m)PL/R=γˉ⋅pˉPR/L+mPL/R=(γˉ⋅pˉ−m)PR/L+m=(γˉ⋅p^+m)PL/R=(γˉ⋅p^+m)PL/R+γˉ⋅pˉPR/L=(γˉ⋅pˉ−m)PR/L+γˉ⋅p^PL/R+m=PR/Lγˉ⋅pˉ+PL/Rm=PR/L(γˉ⋅pˉ−m)+m=PL/R(γˉ⋅p^+m)=PL/R(γˉ⋅p^+m)+PR/Lγˉ⋅pˉ=PR/L(γˉ⋅pˉ−m)+PL/Rγˉ⋅p^+m
Notice that package TRACER
resolves the redundancy of having γμ=γˉμ+γ^μ by eliminating γˉμ and offering a function that reintroduces it at the end of the calculation.
Contractions of Dirac matrices and vectors with the metric read
gμνγνgˉμνγˉνg^μνγ^νgˉμνγ^ν=γμ,=gμνγˉν=gˉμνγν=γˉμ,=gμνγ^ν=g^μνγν=γ^μ,=g^μνγˉν=0,gμνpνgˉμνpˉνg^μνp^νgˉμνp^ν=pμ,=gμνpˉν=gˉμνpν=pˉμ,=gμνp^ν=g^μνpν=p^μ,=g^μνpˉν=0.
Contractions of the metric with itself
gμνgνρgˉμνgˉνρg^μνg^νρgˉμνg^νρ=gρμ=gμνgˉνρ=gˉμνgνρ=gˉρμ=gμνg^νρ=g^μνgνρ=g^ρμ=g^μνgˉνρ=0,gμνgμνgˉμνgˉμνg^μνg^μνgˉμνg^μν=d,=gμνgˉμν=gˉμνgμν=4,=gμνg^μν=g^μνgμν=d−4,=g^μνgˉμν=0.
Contractions of Dirac matrices and vectors with themselves
γμγμγˉμγˉμγ^μγ^μγˉμγ^μ=D,=γμγˉμ=γˉμγμ=4,=γμγ^μ=γ^μγμ=D−4,=γ^μγˉμ=0,pμpμpˉμpˉμp^μp^μpˉμp^μ=p2,=pˉμpμ=pμpˉμ=pˉ2,=p^μpμ=pμp^μ=p^2,=p^μpˉμ=0.
Dirac slashes
γμpμγˉμpˉμγ^μpμγˉμp^μ=γ⋅p,=γˉμpμ=γμpˉμ=γˉ⋅pˉ,=γ^μpμ=γμp^μ=γ^⋅p^,=γ^μpˉμ=0.
Index pairs with one, two, three, four or five free indices
γμγνγμγμγˉνγμγμγ^νγμ=−(d−2)γν,=−(d−2)γˉν,=−(d−2)γ^ν,γˉμγˉνγˉμγˉμγνγˉμγˉμγ^νγˉμ=−2γˉν,=−4γν+2γˉν,=−4γ^ν,γ^μγ^νγ^μγ^μγνγ^μγ^μγˉνγ^μ=−(d−6)γ^ν,=−(d−4)γν+2γ^ν,=−(d−4)γˉν,
γμγνγργμγμγˉνγˉργμγμγ^νγ^ργμγˉμγˉνγˉργˉμγˉμγνγργˉμγˉμγ^νγ^ργˉμγ^μγ^νγ^ργ^μγ^μγνγργ^μγ^μγˉνγˉργ^μ=(d−4)γνγρ+4gνρI,=(d−4)γˉνγˉρ+4gˉνρI,=(d−4)γ^νγ^ρ+4g^νρI,=4gˉνρ,=4γνγρ−2γˉνγρ+2γˉργν,=4γ^νγ^ρ,=(d−8)γ^νγ^ρ+4g^νρI,=(d−4)γνγρ−2γ^νγρ+2γ^ργν,=(d−4)γˉνγˉρ,
γμγνγργσγμγμγˉνγˉργˉσγμγμγ^νγ^ργ^σγμγˉμγˉνγˉργˉσγˉμγˉμγνγργσγˉμγˉμγ^νγ^ργ^σγˉμγ^μγ^νγ^ργ^σγ^μγ^μγνγργσγ^μγ^μγˉνγˉργˉσγ^μ=−(d−4)γνγργσ−2γσγργν=−(d−4)γˉνγˉργˉσ−2γˉσγˉργˉν=−(d−4)γ^νγ^ργ^σ−2γ^σγ^ργ^ν=−2γˉσγˉργˉν=−4γνγργσ+2γˉνγργσ−2γˉργνγσ+2γˉσγνγρ,=−4γ^νγ^ργ^σ,=−(d−8)γ^νγ^ργ^σ−2γ^σγ^ργ^ν=−(d−4)γνγργσ+2γ^νγργσ−2γ^ργνγσ+2γ^σγνγρ,=−(d−4)γˉνγˉργˉσ,
γμγνγργσγτγμγμγˉνγˉργˉσγˉτγμγμγ^νγ^ργ^σγ^τγμγˉμγˉνγˉργˉσγˉτγˉμγˉμγνγργσγτγˉμγˉμγ^νγ^ργ^σγ^τγˉμγ^μγ^νγ^ργ^σγ^τγ^μγ^μγνγργσγτγ^μγ^μγˉνγˉργˉσγˉτγ^μ=(d−4)γνγργσγτ+2γσγργνγτ+2γτγνγργσ=(d−4)γˉνγˉργˉσγˉτ+2γˉσγˉργˉνγˉτ+2γˉτγˉνγˉργˉσ=(d−4)γ^νγ^ργ^σγ^τ+2γ^σγ^ργ^νγ^τ+2γ^τγ^νγ^ργ^σ=2γˉσγˉργˉνγˉτ+2γˉτγˉνγˉργˉσ=4γνγργσγτ−2γˉνγργσγτ+2γˉργνγσγτ−2γˉσγνγργτ+2γˉτγνγργσ,=4γ^νγ^ργ^σγ^τ,=(d−8)γ^νγ^ργ^σγ^τ+2γ^σγ^ργ^νγ^τ+2γ^τγ^νγ^ργ^σ=(d−4)γνγργσγτ−2γ^νγργσγτ+2γ^ργνγσγτ−2γ^σγνγργτ+2γ^τγνγργσ,=(d−4)γˉνγˉργˉσγˉτ,
γμγνγργσγτγκγμ=γμγˉνγˉργˉσγˉτγˉκγμ=γμγ^νγ^ργ^σγ^τγ^κγμ=γˉμγˉνγˉργˉσγˉτγˉκγˉμγˉμγνγργσγτγκγˉμ=γˉμγ^νγ^ργ^σγ^τγ^κγˉμ=γ^μγ^νγ^ργ^σγ^τγ^κγ^μ=γ^μγνγργσγτγκγ^μ=γ^μγˉνγˉργˉσγˉτγˉκγ^μ=−(d−4)γνγργσγτγκ−2γσγργνγτγκ−2γτγνγργσγκ+2γκγνγργσγτ−(d−4)γˉνγˉργˉσγˉτγˉκ−2γˉσγˉργˉνγˉτγˉκ−2γˉτγˉνγˉργˉσγˉκ+2γˉκγˉνγˉργˉσγˉτ−(d−4)γ^νγ^ργ^σγ^τγ^κ−2γ^σγ^ργ^νγ^τγ^κ−2γ^τγ^νγ^ργ^σγ^κ+2γ^κγ^νγ^ργ^σγ^τ=2γˉτγˉσγˉργˉνγˉκ+2γˉκγˉνγˉργˉσγˉτ=−2γˉκγˉτγˉσγˉργˉν−4γνγργσγτγκ+2γˉνγργσγτγκ−2γˉργνγσγτγκ+2γˉσγνγργτγκ−2γˉτγνγργσγκ+2γˉκγνγργσγτ−4γ^νγ^ργ^σγ^τγ^κ,−(d−8)γ^νγ^ργ^σγ^τγ^κ−2γ^σγ^ργ^νγ^τγ^κ−2γ^τγ^νγ^ργ^σγ^κ+2γ^κγ^νγ^ργ^σγ^τ−(d−4)γνγργσγτγκ+2γ^νγργσγτγκ−2γ^ργνγσγτγκ+2γ^σγνγργτγκ−2γ^τγνγργσγκ+2γ^κγνγργσγτ,−(d−4)γˉνγˉργˉσγˉτγˉκ,
Index contractions
If the first and the last matrix are in different dimensions, we can always write them as
γμ…γˉμγμ…γ^μγˉμ…γ^μ=γˉμ…γμ=γˉμ…γˉμ,=γ^μ…γμ=γ^μ…γ^μ,=γ^μ…γˉμ=0.
For general index pairs we have that
γμγˉν1…γˉνnγμγμγ^ν1…γ^νnγμγˉμγ^ν1…γ^νnγˉμγ^μγˉν1…γˉνnγ^μ=γˉμγˉν1…γˉνnγˉμ+(−1)n(D−4)γˉν1…γˉνn,=γ^μγ^ν1…γ^νnγ^μ+4(−1)nγ^ν1…γ^νn,=4(−1)nγ^ν1…γ^νn,=(D−4)(−1)nγˉν1…γˉνn,
This means that if we have a general formula for γμγν1…γνnγμ in D dimensions, we can easily obtain 7 of 9 possible combinations of dimensions. The other two cases are special and related with each other
γˉμγν1…γνnγˉμ=γμγν1…γνnγμ−γ^μγν1…γνnγ^μ.
If we know γˉμγν1…γνnγˉμ we can easily compute γ^μγν1…γνnγ^μ and vice versa.
For purely four dimensional chains it is known that if the number of the Dirac matrices between the index pair is odd, then
γˉμγˉν1…γˉν2n+1γˉμ=−2γˉν2n+1…γˉν1.
This can be trivially generalized to even chains, i.e.
γˉμγˉν1…γˉν2n+1γˉργˉμ=2γˉν2n+1…γˉν1γˉρ+2γˉργˉν1…γˉν2n+1
Following formula for γμγν1…γνnγμ in d dimensions and n≥3 is given in Veltman’s Gammatrica
γμγν1…γνnγμ=(d−4)(−1)nγν1…γνn+2(−1)nγν3γν2γν1γν4…γνn+2j=4∑m(−1)n−jγνjγν1…γνj−1γνj+1…γνn.
Another useful and more compact formula for the same expression with n≥2 was derived my R. Mertig
γμγν1…γνnγμ=(−1)n{(d−2n)γν1…γνn−4i=1∑l−1j=i+1∑l(−1)j−iγν1…γνi−1γνi+1…γνj−1γνj+1…γνngμiμj}