FeynCalc manual (development version)

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 DD, 44 and D4D-4 dimensions. Many identities of the BMHV algebra can be proven by by decomposing Dirac matrices into two pieces

dim(γμ)=D,dim(γˉμ)=4,dim(γ^μ)=D4.γμ=γˉμ+γ^μ,gμν=gˉμν+g^μν,pμ=pˉμ+p^μ. \begin{align} \begin{split} \dim(\gamma^\mu) &= D, \\ \dim(\bar{\gamma}^\mu) &= 4, \\ \dim(\hat{\gamma}^\mu) &= D-4 . \end{split} \begin{split} \gamma^\mu &= \bar{\gamma}^\mu +\hat{\gamma}^\mu, \\ g^{\mu \nu} &= \bar{g}^{\mu \nu} + \hat{g}^{\mu \nu}, \\ p^\mu & = \bar{p}^\mu + \hat{p}^\mu. \end{split} \end{align}

The anticommuatators between Dirac matrices in different dimensions are given by

{γμ,γν}=2gμν,{γˉμ,γˉν}={γμ,γˉν}=2gˉμν,{γ^μ,γ^ν}={γμ,γ^ν}=2g^μν,{γˉμ,γ^ν}=0. \begin{align} \{ \gamma^\mu, \gamma^\nu \} &= 2 g^{\mu \nu}, \\ \{ \bar{\gamma}^\mu, \bar{\gamma}^\nu \} &= \{ \gamma^\mu, \bar{\gamma}^\nu \} = 2 \bar{g}^{\mu \nu}, \\ \{ \hat{\gamma}^\mu, \hat{\gamma}^\nu \} &= \{ \gamma^\mu, \hat{\gamma}^\nu \} = 2 \hat{g}^{\mu \nu}, \\ \{ \bar{\gamma}^\mu, \hat{\gamma}^\nu \} &= 0. \end{align}

Notice that while γ5\gamma^5 anticommutes with all other Dirac matrices in 44 dimensions, it commutes with them in D4D-4 dimensions. However, in DD dimensions the anticommutator is not zero

{γˉμ,γ5}=[γ^μ,γ5]=0{γμ,γ5}={γ^μ,γ5}=2γ^μγ5=2γ5γ^μ. \begin{align} \{ \bar{\gamma}^\mu, \gamma^5 \} &=[ \hat{\gamma}^\mu, \gamma^5 ] = 0 \\ \{ \gamma^\mu, \gamma^5 \} & = \{ \hat{\gamma}^\mu, \gamma^5 \}= 2 \hat{\gamma}^\mu \gamma^5 = 2 \gamma^5 \hat{\gamma}^\mu. \end{align}

For the chiral projectors we obtain

PL/Rγˉμ=γˉμPR/LPL/Rγ^μ=γ^μPL/RPL/Rγμ=γˉμPR/L+γ^μPL/R=γμPR/Lγ^μγ5γμPL/R=PR/Lγˉμ+PL/Rγ^μ=PR/Lγμγ5γ^μ \begin{align} P_{L/R} \bar{\gamma}^\mu & = \bar{\gamma}^\mu P_{R/L} \\ P_{L/R} \hat{\gamma}^\mu &= \hat{\gamma}^\mu P_{L/R} \\ P_{L/R} \gamma^\mu & = \bar{\gamma}^\mu P_{R/L} + \hat{\gamma}^\mu P_{L/R} = \gamma^\mu P_{R/L} \mp \hat{\gamma}^\mu \gamma^5 \\ \gamma^\mu P_{L/R} & = P_{R/L} \bar{\gamma}^\mu + P_{L/R} \hat{\gamma}^\mu = P_{R/L} \gamma^\mu \mp \gamma^5 \hat{\gamma}^\mu \end{align}

and

PL/R(γˉpˉ+m)=γˉpˉPR/L+mPL/R=(γˉpˉm)PR/L+mPL/R(γˉp^+m)=(γˉp^+m)PL/RPL/R(γˉp+m)=(γˉp^+m)PL/R+γˉpˉPR/L=(γˉpˉm)PR/L+γˉp^PL/R+m(γˉpˉ+m)PL/R=PR/Lγˉpˉ+PL/Rm=PR/L(γˉpˉm)+m(γˉp^+m)PL/R=PL/R(γˉp^+m)(γˉp+m)PL/R=PL/R(γˉp^+m)+PR/Lγˉpˉ=PR/L(γˉpˉm)+PL/Rγˉp^+m \begin{align} P_{L/R} (\bar{\gamma} \cdot \bar{p} + m) &= \bar{\gamma} \cdot \bar{p} P_{R/L} + m P_{L/R} = (\bar{\gamma} \cdot \bar{p} - m) P_{R/L} + m \\ P_{L/R} (\bar{\gamma} \cdot \hat{p} + m) & = (\bar{\gamma} \cdot \hat{p} + m) P_{L/R} \\ P_{L/R} (\bar{\gamma} \cdot p + m) &= (\bar{\gamma} \cdot \hat{p} + m) P_{L/R} + \bar{\gamma} \cdot \bar{p} P_{R/L} = (\bar{\gamma} \cdot \bar{p} - m) P_{R/L} + \bar{\gamma} \cdot \hat{p} P_{L/R} + m \\ \nonumber \\ \nonumber %%%%%%%%%%%%%% (\bar{\gamma} \cdot \bar{p} + m) P_{L/R} &= P_{R/L} \bar{\gamma} \cdot \bar{p} + P_{L/R} m = P_{R/L} (\bar{\gamma} \cdot \bar{p} - m) + m \\ (\bar{\gamma} \cdot \hat{p} + m) P_{L/R} & = P_{L/R} (\bar{\gamma} \cdot \hat{p} + m) \\ (\bar{\gamma} \cdot p + m) P_{L/R} &= P_{L/R} ( \bar{\gamma} \cdot \hat{p} + m) + P_{R/L} \bar{\gamma} \cdot \bar{p} = P_{R/L} (\bar{\gamma} \cdot \bar{p} - m) + P_{L/R} \bar{\gamma} \cdot \hat{p} + m \end{align}

Notice that package TRACER resolves the redundancy of having γμ=γˉμ+γ^μ\gamma^\mu = \bar{\gamma}^\mu + \hat{\gamma}^\mu by eliminating γˉμ\bar{\gamma}^\mu and offering a function that reintroduces it at the end of the calculation.

Contractions of Dirac matrices and vectors with the metric read

gμνγν=γμ,gˉμνγˉν=gμνγˉν=gˉμνγν=γˉμ,g^μνγ^ν=gμνγ^ν=g^μνγν=γ^μ,gˉμνγ^ν=g^μνγˉν=0,gμνpν=pμ,gˉμνpˉν=gμνpˉν=gˉμνpν=pˉμ,g^μνp^ν=gμνp^ν=g^μνpν=p^μ,gˉμνp^ν=g^μνpˉν=0. \begin{align} \begin{split} g^{\mu \nu} \gamma_\nu &= \gamma^\mu, \\ \bar{g}^{\mu \nu} \bar{\gamma}_\nu &= g^{\mu \nu} \bar{\gamma}_\nu = \bar{g}^{\mu \nu} \gamma_\nu = \bar{\gamma}^\mu, \\ \hat{g}^{\mu \nu} \hat{\gamma}_\nu &= g^{\mu \nu} \hat{\gamma}_\nu = \hat{g}^{\mu \nu} \gamma_\nu = \hat{\gamma}^\mu, \\ \bar{g}^{\mu \nu} \hat{\gamma}_\nu &= \hat{g}^{\mu \nu} \bar{\gamma}_\nu = 0, \end{split} \begin{split} g^{\mu \nu} p_\nu &= p^\mu, \\ \bar{g}^{\mu \nu} \bar{p}_\nu &= g^{\mu \nu} \bar{p}_\nu = \bar{g}^{\mu \nu} p_\nu = \bar{p}^\mu, \\ \hat{g}^{\mu \nu} \hat{p}_\nu &= g^{\mu \nu} \hat{p}_\nu = \hat{g}^{\mu \nu} p_\nu = \hat{p}^\mu, \\ \bar{g}^{\mu \nu} \hat{p}_\nu &= \hat{g}^{\mu \nu} \bar{p}_\nu = 0. \end{split} \end{align}

Contractions of the metric with itself

gμνgνρ=gρμgˉμνgˉνρ=gμνgˉνρ=gˉμνgνρ=gˉρμg^μνg^νρ=gμνg^νρ=g^μνgνρ=g^ρμgˉμνg^νρ=g^μνgˉνρ=0,gμνgμν=d,gˉμνgˉμν=gμνgˉμν=gˉμνgμν=4,g^μνg^μν=gμνg^μν=g^μνgμν=d4,gˉμνg^μν=g^μνgˉμν=0. \begin{align} \begin{split} g^{\mu \nu} g_{\nu \rho} & = g^\mu_\rho \\ \bar{g}^{\mu \nu} \bar{g}_{\nu \rho} & =g^{\mu \nu} \bar{g}_{\nu \rho} = \bar{g}^{\mu \nu} g_{\nu \rho} = \bar{g}^\mu_\rho \\ \hat{g}^{\mu \nu} \hat{g}_{\nu \rho} & = g^{\mu \nu} \hat{g}_{\nu \rho} = \hat{g}^{\mu \nu} g_{\nu \rho}= \hat{g}^\mu_\rho \\ \bar{g}^{\mu \nu} \hat{g}_{\nu \rho} &= \hat{g}^{\mu \nu} \bar{g}_{\nu \rho} = 0, \end{split} \begin{split} g^{\mu \nu} g_{\mu \nu} & = d, \\ \bar{g}^{\mu \nu} \bar{g}_{\mu \nu} & = g^{\mu \nu} \bar{g}_{\mu \nu} = \bar{g}^{\mu \nu} g_{\mu \nu} = 4, \\ \hat{g}^{\mu \nu} \hat{g}_{\mu \nu} & = g^{\mu \nu} \hat{g}_{\mu \nu} = \hat{g}^{\mu \nu} g_{\mu \nu} = d-4, \\ \bar{g}^{\mu \nu} \hat{g}_{\mu \nu} &= \hat{g}^{\mu \nu} \bar{g}_{\mu \nu} = 0. \end{split} \end{align}

Contractions of Dirac matrices and vectors with themselves

γμγμ=D,γˉμγˉμ=γμγˉμ=γˉμγμ=4,γ^μγ^μ=γμγ^μ=γ^μγμ=D4,γˉμγ^μ=γ^μγˉμ=0,pμpμ=p2,pˉμpˉμ=pˉμpμ=pμpˉμ=pˉ2,p^μp^μ=p^μpμ=pμp^μ=p^2,pˉμp^μ=p^μpˉμ=0. \begin{align} \begin{split} \gamma^\mu \gamma_\mu &= D, \\ \bar{\gamma}^\mu \bar{\gamma}_\mu &= \gamma^\mu \bar{\gamma}_\mu = \bar{\gamma}^\mu \gamma_\mu = 4, \\ \hat{\gamma}^\mu \hat{\gamma}_\mu &= \gamma^\mu \hat{\gamma}_\mu = \hat{\gamma}^\mu \gamma_\mu = D-4, \\ \bar{\gamma}^\mu \hat{\gamma}_\mu &= \hat{\gamma}^\mu \bar{\gamma}_\mu = 0, \end{split} \begin{split} p^\mu p_\mu &= p^2, \\ \bar{p}^\mu \bar{p}_\mu &= \bar{p}^\mu p_\mu = p^\mu \bar{p}_\mu = \bar{p}^2, \\ \hat{p}^\mu \hat{p}_\mu &= \hat{p}^\mu p_\mu = p^\mu \hat{p}_\mu = \hat{p}^2, \\ \bar{p}^\mu \hat{p}_\mu &= \hat{p}^\mu \bar{p}_\mu = 0. \end{split} \end{align}

Dirac slashes

γμpμ=γp,γˉμpˉμ=γˉμpμ=γμpˉμ=γˉpˉ,γ^μpμ=γ^μpμ=γμp^μ=γ^p^,γˉμp^μ=γ^μpˉμ=0. \begin{align} \begin{split} \gamma^\mu p_\mu &= \gamma \cdot p, \\ \bar{\gamma}^\mu \bar{p}_\mu &= \bar{\gamma}^\mu p_\mu = \gamma^\mu \bar{p}_\mu = \bar{\gamma} \cdot \bar{p}, \\ \hat{\gamma}^\mu p_\mu &= \hat{\gamma}^\mu p_\mu = \gamma^\mu \hat{p}_\mu = \hat{\gamma} \cdot \hat{p}, \\ \bar{\gamma}^\mu \hat{p}_\mu &= \hat{\gamma}^\mu \bar{p}_\mu = 0. \\ \end{split} \end{align}

Index pairs with one, two, three, four or five free indices

γμγνγμ=(d2)γν,γμγˉνγμ=(d2)γˉν,γμγ^νγμ=(d2)γ^ν,γˉμγˉνγˉμ=2γˉν,γˉμγνγˉμ=4γν+2γˉν,γˉμγ^νγˉμ=4γ^ν,γ^μγ^νγ^μ=(d6)γ^ν,γ^μγνγ^μ=(d4)γν+2γ^ν,γ^μγˉνγ^μ=(d4)γˉν, \begin{align} \begin{split} \gamma^\mu \gamma^\nu \gamma_\mu &= -(d-2) \gamma^\nu, \\ \gamma^\mu \bar{\gamma}^\nu \gamma_\mu &= -(d-2) \bar{\gamma}^\nu, \\ \gamma^\mu \hat{\gamma}^\nu \gamma_\mu &= -(d-2) \hat{\gamma}^\nu, \end{split} \quad \quad \begin{split} \bar{\gamma}^\mu \bar{\gamma}^\nu \bar{\gamma}_\mu &= -2 \bar{\gamma}^\nu, \\ \bar{\gamma}^\mu \gamma^\nu \bar{\gamma}_\mu &= -4 \gamma^\nu +2 \bar{\gamma}^\nu, \\ \bar{\gamma}^\mu \hat{\gamma}^\nu \bar{\gamma}_\mu &= -4 \hat{\gamma}^\nu, \\ \end{split} \quad \quad \quad \quad \begin{split} \hat{\gamma}^\mu \hat{\gamma}^\nu \hat{\gamma}_\mu &= -(d-6) \hat{\gamma}^\nu, \\ \hat{\gamma}^\mu \gamma^\nu \hat{\gamma}_\mu &= -(d-4) \gamma^\nu + 2\hat{\gamma}^\nu, \\ \hat{\gamma}^\mu \bar{\gamma}^\nu \hat{\gamma}_\mu &= -(d-4) \bar{\gamma}^\nu, \end{split} \end{align}

γμγνγργμ=(d4)γνγρ+4gνρI,γμγˉνγˉργμ=(d4)γˉνγˉρ+4gˉνρI,γμγ^νγ^ργμ=(d4)γ^νγ^ρ+4g^νρI,γˉμγˉνγˉργˉμ=4gˉνρ,γˉμγνγργˉμ=4γνγρ2γˉνγρ+2γˉργν,γˉμγ^νγ^ργˉμ=4γ^νγ^ρ,γ^μγ^νγ^ργ^μ=(d8)γ^νγ^ρ+4g^νρI,γ^μγνγργ^μ=(d4)γνγρ2γ^νγρ+2γ^ργν,γ^μγˉνγˉργ^μ=(d4)γˉνγˉρ, \begin{align} \gamma^\mu \gamma^\nu \gamma^\rho \gamma_\mu &= (d-4) \gamma^\nu \gamma^\rho + 4 g^{\nu \rho} I, \\ \gamma^\mu \bar{\gamma}^\nu \bar{\gamma}^\rho \gamma_\mu &= (d-4) \bar{\gamma}^\nu \bar{\gamma}^\rho + 4 \bar{g}^{\nu \rho} I, \\ \gamma^\mu \hat{\gamma}^\nu \hat{\gamma}^\rho\gamma_\mu &= (d-4) \hat{\gamma}^\nu \hat{\gamma}^\rho + 4 \hat{g}^{\nu \rho} I, \\ \bar{\gamma}^\mu \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}_\mu &= 4 \bar{g}^{\nu \rho}, \\ \bar{\gamma}^\mu \gamma^\nu \gamma^\rho \bar{\gamma}_\mu &= 4 \gamma^\nu \gamma^\rho - 2 \bar{\gamma}^\nu \gamma^\rho + 2\bar{\gamma}^\rho \gamma^\nu, \\ \bar{\gamma}^\mu \hat{\gamma}^\nu \hat{\gamma}^\rho \bar{\gamma}_\mu &= 4 \hat{\gamma}^\nu \hat{\gamma}^\rho, \\ \hat{\gamma}^\mu \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}_\mu &= (d-8) \hat{\gamma}^\nu \hat{\gamma}^\rho + 4 \hat{g}^{\nu \rho} I, \\ \hat{\gamma}^\mu \gamma^\nu \gamma^\rho \hat{\gamma}_\mu &= (d-4) \gamma^\nu \gamma^\rho - 2 \hat{\gamma}^\nu \gamma^\rho + 2\hat{\gamma}^\rho \gamma^\nu, \\ \hat{\gamma}^\mu \bar{\gamma}^\nu \bar{\gamma}^\rho \hat{\gamma}_\mu &= (d-4) \bar{\gamma}^\nu \bar{\gamma}^\rho, \end{align}

γμγνγργσγμ=(d4)γνγργσ2γσγργνγμγˉνγˉργˉσγμ=(d4)γˉνγˉργˉσ2γˉσγˉργˉνγμγ^νγ^ργ^σγμ=(d4)γ^νγ^ργ^σ2γ^σγ^ργ^νγˉμγˉνγˉργˉσγˉμ=2γˉσγˉργˉνγˉμγνγργσγˉμ=4γνγργσ+2γˉνγργσ2γˉργνγσ+2γˉσγνγρ,γˉμγ^νγ^ργ^σγˉμ=4γ^νγ^ργ^σ,γ^μγ^νγ^ργ^σγ^μ=(d8)γ^νγ^ργ^σ2γ^σγ^ργ^νγ^μγνγργσγ^μ=(d4)γνγργσ+2γ^νγργσ2γ^ργνγσ+2γ^σγνγρ,γ^μγˉνγˉργˉσγ^μ=(d4)γˉνγˉργˉσ, \begin{align} \gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma \gamma_\mu &= -(d-4) \gamma^\nu \gamma^\rho \gamma^\sigma -2 \gamma^\sigma \gamma^\rho \gamma^\nu \\ \gamma^\mu \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \gamma_\mu &=-(d-4) \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma -2 \bar{\gamma}^\sigma \bar{\gamma}^\rho \bar{\gamma}^\nu \\ \gamma^\mu \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \gamma_\mu &= -(d-4) \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma -2 \hat{\gamma}^\sigma \hat{\gamma}^\rho \hat{\gamma}^\nu \\ \bar{\gamma}^\mu \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \bar{\gamma}_\mu &= -2 \bar{\gamma}^\sigma \bar{\gamma}^\rho \bar{\gamma}^\nu \\ \bar{\gamma}^\mu \gamma^\nu \gamma^\rho \gamma^\sigma \bar{\gamma}_\mu &= -4 \gamma^\nu \gamma^\rho \gamma^\sigma + 2 \bar{\gamma}^\nu \gamma^\rho \gamma^\sigma - 2\bar{\gamma}^\rho \gamma^\nu \gamma^\sigma + 2 \bar{\gamma}^\sigma \gamma^\nu \gamma^\rho, \\ \bar{\gamma}^\mu \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \bar{\gamma}_\mu &= - 4 \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma , \\ \hat{\gamma}^\mu \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}_\mu &= -(d-8) \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma -2 \hat{\gamma}^\sigma \hat{\gamma}^\rho \hat{\gamma}^\nu \\ \hat{\gamma}^\mu \gamma^\nu \gamma^\rho \gamma^\sigma \hat{\gamma}_\mu &= -(d-4) \gamma^\nu \gamma^\rho \gamma^\sigma + 2 \hat{\gamma}^\nu \gamma^\rho \gamma^\sigma - 2\hat{\gamma}^\rho \gamma^\nu \gamma^\sigma + 2 \hat{\gamma}^\sigma \gamma^\nu \gamma^\rho, \\ \hat{\gamma}^\mu \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \hat{\gamma}_\mu &= -(d-4) \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma, \end{align}

γμγνγργσγτγμ=(d4)γνγργσγτ+2γσγργνγτ+2γτγνγργσγμγˉνγˉργˉσγˉτγμ=(d4)γˉνγˉργˉσγˉτ+2γˉσγˉργˉνγˉτ+2γˉτγˉνγˉργˉσγμγ^νγ^ργ^σγ^τγμ=(d4)γ^νγ^ργ^σγ^τ+2γ^σγ^ργ^νγ^τ+2γ^τγ^νγ^ργ^σγˉμγˉνγˉργˉσγˉτγˉμ=2γˉσγˉργˉνγˉτ+2γˉτγˉνγˉργˉσγˉμγνγργσγτγˉμ=4γνγργσγτ2γˉνγργσγτ+2γˉργνγσγτ2γˉσγνγργτ+2γˉτγνγργσ,γˉμγ^νγ^ργ^σγ^τγˉμ=4γ^νγ^ργ^σγ^τ,γ^μγ^νγ^ργ^σγ^τγ^μ=(d8)γ^νγ^ργ^σγ^τ+2γ^σγ^ργ^νγ^τ+2γ^τγ^νγ^ργ^σγ^μγνγργσγτγ^μ=(d4)γνγργσγτ2γ^νγργσγτ+2γ^ργνγσγτ2γ^σγνγργτ+2γ^τγνγργσ,γ^μγˉνγˉργˉσγˉτγ^μ=(d4)γˉνγˉργˉσγˉτ, \begin{align} \gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\tau \gamma_\mu &= (d-4) \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\tau + 2 \gamma^\sigma \gamma^\rho \gamma^\nu \gamma^\tau + 2 \gamma^\tau \gamma^\nu \gamma^\rho \gamma^\sigma \\ \gamma^\mu \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \bar{\gamma}^\tau \gamma_\mu &= (d-4) \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \bar{\gamma}^\tau + 2 \bar{\gamma}^\sigma \bar{\gamma}^\rho \bar{\gamma}^\nu \bar{\gamma}^\tau + 2 \bar{\gamma}^\tau \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \\ \gamma^\mu \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}^\tau \gamma_\mu &= (d-4) \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}^\tau +2 \hat{\gamma}^\sigma \hat{\gamma}^\rho \hat{\gamma}^\nu \hat{\gamma}^\tau + 2\hat{\gamma}^\tau \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \\ \bar{\gamma}^\mu \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \bar{\gamma}^\tau \bar{\gamma}_\mu &= 2 \bar{\gamma}^\sigma \bar{\gamma}^\rho \bar{\gamma}^\nu \bar{\gamma}^\tau + 2 \bar{\gamma}^\tau \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \\ \bar{\gamma}^\mu \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\tau \bar{\gamma}_\mu &= 4 \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\tau \\ &- 2 \bar{\gamma}^\nu \gamma^\rho \gamma^\sigma \gamma^\tau + 2\bar{\gamma}^\rho \gamma^\nu \gamma^\sigma \gamma^\tau - 2 \bar{\gamma}^\sigma \gamma^\nu \gamma^\rho \gamma^\tau + 2 \bar{\gamma}^\tau \gamma^\nu \gamma^\rho \gamma^\sigma, \\ \bar{\gamma}^\mu \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}^\tau \bar{\gamma}_\mu &= 4 \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}^\tau , \\ \hat{\gamma}^\mu \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}^\tau \hat{\gamma}_\mu &= (d-8) \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}^\tau + 2 \hat{\gamma}^\sigma \hat{\gamma}^\rho \hat{\gamma}^\nu \hat{\gamma}^\tau + 2 \hat{\gamma}^\tau \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \\ \hat{\gamma}^\mu \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\tau \hat{\gamma}_\mu &= (d-4) \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\tau \\ &- 2 \hat{\gamma}^\nu \gamma^\rho \gamma^\sigma \gamma^\tau + 2\hat{\gamma}^\rho \gamma^\nu \gamma^\sigma \gamma^\tau - 2 \hat{\gamma}^\sigma \gamma^\nu \gamma^\rho \gamma^\tau + 2 \hat{\gamma}^\tau \gamma^\nu \gamma^\rho \gamma^\sigma, \\ \hat{\gamma}^\mu \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \bar{\gamma}^\tau \hat{\gamma}_\mu &= (d-4) \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \bar{\gamma}^\tau, \end{align}

γμγνγργσγτγκγμ=(d4)γνγργσγτγκ2γσγργνγτγκ2γτγνγργσγκ+2γκγνγργσγτγμγˉνγˉργˉσγˉτγˉκγμ=(d4)γˉνγˉργˉσγˉτγˉκ2γˉσγˉργˉνγˉτγˉκ2γˉτγˉνγˉργˉσγˉκ+2γˉκγˉνγˉργˉσγˉτγμγ^νγ^ργ^σγ^τγ^κγμ=(d4)γ^νγ^ργ^σγ^τγ^κ2γ^σγ^ργ^νγ^τγ^κ2γ^τγ^νγ^ργ^σγ^κ+2γ^κγ^νγ^ργ^σγ^τγˉμγˉνγˉργˉσγˉτγˉκγˉμ=2γˉτγˉσγˉργˉνγˉκ+2γˉκγˉνγˉργˉσγˉτ=2γˉκγˉτγˉσγˉργˉνγˉμγνγργσγτγκγˉμ=4γνγργσγτγκ+2γˉνγργσγτγκ2γˉργνγσγτγκ+2γˉσγνγργτγκ2γˉτγνγργσγκ+2γˉκγνγργσγτγˉμγ^νγ^ργ^σγ^τγ^κγˉμ=4γ^νγ^ργ^σγ^τγ^κ,γ^μγ^νγ^ργ^σγ^τγ^κγ^μ=(d8)γ^νγ^ργ^σγ^τγ^κ2γ^σγ^ργ^νγ^τγ^κ2γ^τγ^νγ^ργ^σγ^κ+2γ^κγ^νγ^ργ^σγ^τγ^μγνγργσγτγκγ^μ=(d4)γνγργσγτγκ+2γ^νγργσγτγκ2γ^ργνγσγτγκ+2γ^σγνγργτγκ2γ^τγνγργσγκ+2γ^κγνγργσγτ,γ^μγˉνγˉργˉσγˉτγˉκγ^μ=(d4)γˉνγˉργˉσγˉτγˉκ, \begin{align} \gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\tau \gamma^\kappa \gamma_\mu =& -(d-4) \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\tau \gamma^\kappa \\ &- 2 \gamma^\sigma \gamma^\rho \gamma^\nu \gamma^\tau \gamma^\kappa - 2 \gamma^\tau \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\kappa + 2\gamma^\kappa \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\tau \\ \gamma^\mu \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \bar{\gamma}^\tau \bar{\gamma}^\kappa \gamma_\mu =& -(d-4) \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \bar{\gamma}^\tau \bar{\gamma}^\kappa \\ & - 2 \bar{\gamma}^\sigma \bar{\gamma}^\rho \bar{\gamma}^\nu \bar{\gamma}^\tau \bar{\gamma}^\kappa - 2 \bar{\gamma}^\tau \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \bar{\gamma}^\kappa + 2\bar{\gamma}^\kappa \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \bar{\gamma}^\tau\\ \gamma^\mu \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}^\tau \hat{\gamma}^\kappa \gamma_\mu =& -(d-4) \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}^\tau \hat{\gamma}^\kappa \\ & - 2 \hat{\gamma}^\sigma \hat{\gamma}^\rho \hat{\gamma}^\nu \hat{\gamma}^\tau \hat{\gamma}^\kappa - 2 \hat{\gamma}^\tau \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}^\kappa + 2\hat{\gamma}^\kappa \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}^\tau \\ \bar{\gamma}^\mu \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \bar{\gamma}^\tau \bar{\gamma}^\kappa \bar{\gamma}_\mu & = 2 \bar{\gamma}^\tau \bar{\gamma}^\sigma \bar{\gamma}^\rho \bar{\gamma}^\nu \bar{\gamma}^\kappa + 2 \bar{\gamma}^\kappa \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \bar{\gamma}^\tau = - 2 \bar{\gamma}^\kappa \bar{\gamma}^\tau \bar{\gamma}^\sigma \bar{\gamma}^\rho \bar{\gamma}^\nu \\ \bar{\gamma}^\mu \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\tau \gamma^\kappa \bar{\gamma}_\mu =& -4 \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\tau \gamma^\kappa + 2 \bar{\gamma}^\nu \gamma^\rho \gamma^\sigma \gamma^\tau \gamma^\kappa - 2\bar{\gamma}^\rho \gamma^\nu \gamma^\sigma \gamma^\tau \gamma^\kappa \\ &+ 2 \bar{\gamma}^\sigma \gamma^\nu \gamma^\rho \gamma^\tau \gamma^\kappa - 2 \bar{\gamma}^\tau \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\kappa + 2\bar{\gamma}^\kappa \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\tau \\ \bar{\gamma}^\mu \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}^\tau \hat{\gamma}^\kappa \bar{\gamma}_\mu =& -4 \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}^\tau \hat{\gamma}^\kappa , \\ \hat{\gamma}^\mu \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}^\tau \hat{\gamma}^\kappa \hat{\gamma}_\mu =& -(d-8) \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}^\tau \hat{\gamma}^\kappa \\ &- 2 \hat{\gamma}^\sigma \hat{\gamma}^\rho \hat{\gamma}^\nu \hat{\gamma}^\tau \hat{\gamma}^\kappa - 2 \hat{\gamma}^\tau \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}^\kappa + 2 \hat{\gamma}^\kappa \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}^\tau \\ \hat{\gamma}^\mu \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\tau \gamma^\kappa \hat{\gamma}_\mu =& -(d-4) \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\tau \gamma^\kappa + 2 \hat{\gamma}^\nu \gamma^\rho \gamma^\sigma \gamma^\tau \gamma^\kappa - 2\hat{\gamma}^\rho \gamma^\nu \gamma^\sigma \gamma^\tau \gamma^\kappa \\ & + 2 \hat{\gamma}^\sigma \gamma^\nu \gamma^\rho \gamma^\tau \gamma^\kappa - 2 \hat{\gamma}^\tau \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\kappa +2 \hat{\gamma}^\kappa \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\tau, \\ \hat{\gamma}^\mu \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \bar{\gamma}^\tau \bar{\gamma}^\kappa \hat{\gamma}_\mu =& -(d-4) \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \bar{\gamma}^\tau \bar{\gamma}^\kappa, \end{align}

Index contractions

If the first and the last matrix are in different dimensions, we can always write them as

γμγˉμ=γˉμγμ=γˉμγˉμ,γμγ^μ=γ^μγμ=γ^μγ^μ,γˉμγ^μ=γ^μγˉμ=0. \begin{align} \gamma^\mu \dots \bar{\gamma}_\mu & = \bar{\gamma}^\mu \dots \gamma_\mu = \bar{\gamma}^\mu \dots \bar{\gamma}_\mu, \\ \gamma^\mu \dots \hat{\gamma}_\mu & = \hat{\gamma}^\mu \dots \gamma_\mu = \hat{\gamma}^\mu \dots \hat{\gamma}_\mu, \\ \bar{\gamma}^\mu \dots \hat{\gamma}_\mu & = \hat{\gamma}^\mu \dots \bar{\gamma}_\mu = 0. \end{align}

For general index pairs we have that

γμγˉν1γˉνnγμ=γˉμγˉν1γˉνnγˉμ+(1)n(D4)γˉν1γˉνn,γμγ^ν1γ^νnγμ=γ^μγ^ν1γ^νnγ^μ+4(1)nγ^ν1γ^νn,γˉμγ^ν1γ^νnγˉμ=4(1)nγ^ν1γ^νn,γ^μγˉν1γˉνnγ^μ=(D4)(1)nγˉν1γˉνn, \begin{align} \gamma^\mu \bar{\gamma}^{\nu_1} \dots \bar{\gamma}^{\nu_n} \gamma_\mu &= \bar{\gamma}^\mu \bar{\gamma}^{\nu_1} \dots \bar{\gamma}^{\nu_n} \bar{\gamma}_\mu + (-1)^n (D-4) \bar{\gamma}^{\nu_1} \dots \bar{\gamma}^{\nu_n}, \\ \gamma^\mu \hat{\gamma}^{\nu_1} \dots \hat{\gamma}^{\nu_n} \gamma_\mu &= \hat{\gamma}^\mu \hat{\gamma}^{\nu_1} \dots \hat{\gamma}^{\nu_n} \hat{\gamma}_\mu + 4 (-1)^n \hat{\gamma}^{\nu_1} \dots \hat{\gamma}^{\nu_n}, \\ \bar{\gamma}^\mu \hat{\gamma}^{\nu_1} \dots \hat{\gamma}^{\nu_n} \bar{\gamma}_\mu &= 4 (-1)^n \, \hat{\gamma}^{\nu_1} \dots \hat{\gamma}^{\nu_n}, \\ \hat{\gamma}^\mu \bar{\gamma}^{\nu_1} \dots \bar{\gamma}^{\nu_n} \hat{\gamma}_\mu &= (D-4) (-1)^n \, \bar{\gamma}^{\nu_1} \dots \bar{\gamma}^{\nu_n}, \end{align}

This means that if we have a general formula for γμγν1γνnγμ\gamma^\mu \gamma^{\nu_1} \dots \gamma^{\nu_n} \gamma_\mu in DD dimensions, we can easily obtain 7 of 9 possible combinations of dimensions. The other two cases are special and related with each other

γˉμγν1γνnγˉμ=γμγν1γνnγμγ^μγν1γνnγ^μ. \begin{align} \bar{\gamma}^\mu \gamma^{\nu_1} \dots \gamma^{\nu_n} \bar{\gamma}_\mu = \gamma^\mu \gamma^{\nu_1} \dots \gamma^{\nu_n} \gamma_\mu - \hat{\gamma}^\mu \gamma^{\nu_1} \dots \gamma^{\nu_n} \hat{\gamma}_\mu. \end{align}

If we know γˉμγν1γνnγˉμ\bar{\gamma}^\mu \gamma^{\nu_1} \dots \gamma^{\nu_n} \bar{\gamma}_\mu we can easily compute γ^μγν1γνnγ^μ\hat{\gamma}^\mu \gamma^{\nu_1} \dots \gamma^{\nu_n} \hat{\gamma}_\mu 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

γˉμγˉν1γˉν2n+1γˉμ=2γˉν2n+1γˉν1. \begin{align} \bar{\gamma}^\mu \bar{\gamma}^{\nu_1} \dots \bar{\gamma}^{\nu_{2n+1}} \bar{\gamma}_\mu & = -2 \bar{\gamma}^{\nu_{2n+1}} \dots \bar{\gamma}^{\nu_1}. \end{align}

This can be trivially generalized to even chains, i.e.

γˉμγˉν1γˉν2n+1γˉργˉμ=2γˉν2n+1γˉν1γˉρ+2γˉργˉν1γˉν2n+1 \begin{align} \bar{\gamma}^\mu \bar{\gamma}^{\nu_1} \dots \bar{\gamma}^{\nu_{2n+1}} \bar{\gamma}^\rho \bar{\gamma}_\mu & = 2 \bar{\gamma}^{\nu_{2n+1}} \dots \bar{\gamma}^{\nu_1} \bar{\gamma}^\rho + 2 \bar{\gamma}^\rho \bar{\gamma}^{\nu_1} \dots \bar{\gamma}^{\nu_{2n+1}} \end{align}

Following formula for γμγν1γνnγμ\gamma^\mu \gamma^{\nu_1} \dots \gamma^{\nu_n} \gamma_\mu in d dimensions and n3n\geq 3 is given in Veltman’s Gammatrica

γμγν1γνnγμ=(d4)(1)nγν1γνn+2(1)nγν3γν2γν1γν4γνn+2j=4m(1)njγνjγν1γνj1γνj+1γνn. \begin{align} \gamma^\mu \gamma^{\nu_1} \dots \gamma^{\nu_n} \gamma_\mu &= (d-4)(-1)^n \gamma^{\nu_1} \dots \gamma^{\nu_n} + 2(-1)^n \gamma^{\nu_3} \gamma^{\nu_2} \gamma^{\nu_1} \gamma^{\nu_4}\dots \gamma^{\nu_n} \\ & + 2 \sum_{j=4}^m (-1)^{n-j} \gamma^{\nu_j} \gamma^{\nu_1} \dots \gamma^{\nu_{j-1}} \gamma^{\nu_{j+1}} \dots \gamma^{\nu_{n}}. \end{align}

Another useful and more compact formula for the same expression with n2n \geq 2 was derived my R. Mertig

γμγν1γνnγμ=(1)n{(d2n)γν1γνn4i=1l1j=i+1l(1)jiγν1γνi1γνi+1γνj1γνj+1γνngμiμj} \begin{align} \gamma^\mu \gamma^{\nu_1} \dots \gamma^{\nu_n} \gamma_\mu &= (-1)^n \biggl \{ (d-2n) \gamma^{\nu_1} \dots \gamma^{\nu_n} \\ & -4 \sum_{i=1}^{l-1} \sum_{j=i+1}^{l} (-1)^{j-i} \gamma^{\nu_1} \dots \gamma^{\nu_{i-1}} \gamma^{\nu_{i+1}} \dots \gamma^{\nu_{j-1}} \gamma^{\nu_{j+1}} \dots \gamma^{\nu_n} g_{{\mu_i} {\mu_j}} \biggr \} \end{align}