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I need to construct the following object

$\Gamma_{\mu} = \frac{1}{8} \omega_{\mu}^{ab}[\gamma_{a},\gamma_{b}]$

where $\omega_{\mu}^{ab}$ is object with space-time index $\mu$ and vierbeins index $a, \ b$, and the non-vanishing componnents are: $\omega_{x \ \ \ \ 0}^{ \ \ \ \ j} = \omega_{x \ \ \ \ j}^{ \ \ \ 0} = \frac{b}{a} \delta_{ij}$ and $\omega_{x \ \ \ k}^{\ \ \ j}=f \epsilon_{ijk} $, with $b, \ a, \ f$ time functions. My code for all of this:

Metric $g_{\mu \nu}$ and its inverse $g^{\mu \nu}$:

metric1 = {{a[t]^2, 0 , 0, 0}, {0, -a[t]^2, 0, 0}, {0, 0, -a[t]^2, 0}, {0, 0, 0, -a[t]^2}};
inversemetric1 = FullSimplify[Series[Inverse[metric1], {e, 0, 1}]];
cruzmetric = FullSimplify[Series[Table[Sum[inversemetric1[[μ, σ]]*metric1[[ν, σ]], {σ, 1, 4}], {μ, 1, 4}, {ν, 1, 4}], {e, 0, 1} ]];

etaAB[[a,b]] = $\eta^{a b}$ and inverseetaab[[a,b]]= $\eta_{a,b}$

etaAB = {{1, 0 , 0, 0}, {0, -1, 0, 0}, {0, 0, -1, 0}, {0, 0, 0, -1}};
inverseetaab = FullSimplify[Series[Inverse[etaAB], {e, 0, 1}]];

Vierbein: vierAmu[[a,μ]] = $e^{a}_{\mu}$, vieramu[[a,μ]] = $e^{a}_{\mu}$$e_{a \mu}$, vierAMU[[a,μ]] = $e^{a \mu}$

vierAmu = {{a[t], 0 , 0, 0}, {0, a[t], 0, 0}, {0, 0, a[t], 0}, {0, 0, 0, a[t]}};

vieramu = Expand[FullSimplify[Series[Table[Sum[inverseetaab[[a1, b]]*vierAmu[[b, μ]], {b, 1, 4}], {a1,1, 4}, {μ, 1, 4}], {e, 0, 1}]]];
vierAMU = Expand[FullSimplify[Series[Table[Sum[inversemetric1[[a1, b]]*vierAmu[[μ, b]], {b, 1, 4}], {a1, 1, 4}, {μ, 1, 4}], {e, 0, 1}]]];

$\omega_{\mu}^{ab} = e_{\nu }^a \frac{\partial e^{\text{b$\nu $}}}{\partial \mu }+e_{\nu }^a \Gamma _{\mu \rho }^{\nu } e^{\text{b$\rho $}} $. I divided it in two pieces, the first one is omegamuAB1[[μ,a,b]] = $e_{\nu}^a \Gamma_{\mu \rho }^{\nu }e^{b\rho}$, and omegamuAB2[[μ,a,b]]=$e_{\nu }^a \frac{\partial e^{b\nu}}{\partial \mu }$.

omegamuAB1 = FullSimplify[Series[Table[Sum[vierAmu[[a1,ν]]*vierAMU[[b1, ρ]]*Connecb[[ν, μ, ρ]], {ν, 1, 4}, {ρ, 1, 4}],{μ, 1, 4}, {a1, 1, 4}, {b1, 1, 4}], {e, 0, 1}]];

omegamuAB2 = FullSimplify[Series[Table[Sum[vierAmu[[a1, ν]]*D[vierAMU[[b1, ν]], crd[[μ]]], {ν, 1, 4}],{μ, 1, 4}, {a1, 1, 4}, {b1, 1, 4}], {e, 0, 1}]];

omegamuAB = FullSimplify[Series[Table[omegamuAB1[[μ, a1, b1]] + omegamuAB2[[μ, a1, b1]], {μ, 1, 4}, {a1, 1, 4}, {b1, 1, 4}], {e, 0, 1}]]; 

Connecb is a space-time connection with torsion, I made it by hand just for this example.:

 Connecb = = {{{Derivative[1][a][t]/a[t], 0, 0, 0}, {0, (a[t]*b[t])/a[t]^2, 0,  0}, {0, 0, (a[t] b[t])/a[t]^2, 0}, {0, 0, 0, (a[t] b[t])/a[t]^2}}, {{0, Derivative[1][a][t]/a[t], 0, 0}, {b[t]/a[t], 0, 0, 0}, {0, 0, 0, f[t]}, {0, 0, -f[t], 0}}, {{0, 0, Derivative[1][a][t]/a[t], 0}, {0, 0, 0, -f[t]}, {b[t]/a[t], 0, 0, 0}, {0, f[t], 0, 0}}, {{0, 0, 0, Derivative[1][a][t]/a[t]}, {0, 0, f[t], 0}, {0, -f[t], 0, 0}, {b[t]/a[t], 0, 0, 0}}}

The code above works fine! Now I'll define the gamma matrices (using FeynCalc just to help with simplifications)

$\gamma_{0}$

g0 = {{0, 0, 1, 0}, {0, 0, 0, 1}, {1, 0, 0, 0}, {0, 1, 0, 0}}

$\gamma_{1}$

 g1 = {{0, 0, 0, -1}, {0, 0, -1, 0}, {0, 1, 0, 0}, {1, 0, 0, 0}}

$\gamma_{2}$

g2 = {{0, 0, 0, I}, {0, 0, -I, 0}, {0, -I, 0, 0}, {I, 0, 0, 0}}

$\gamma_{3}$

g3 = {{0, 0, -1, 0}, {0, 0, 0, 1}, {1, 0, 0, 0}, {0, -1, 0, 0}}

Problem 1#: How to put the gamma matrices into $\Gamma_{\mu}$ ?

Problem 2#: I'll create a covariant derivative with $\Gamma_{\mu}$ and then a Lagrangian density, so I need a correct contraction over my gamma matrices. By the ways I have tried, I'm getting a tensor $4 \times 4$ as Lagrangian.

Using a extreme lame trick:

G = {g0, g1, g2, g3};

It works just to get the commutators. Trying to create a $\Gamma_{\mu}$ (using $[\gamma_{a},\gamma_{b}]=2\gamma_{a}\gamma_{b}$)

Gammamu = FullSimplify[Series[Table[1/4*Sum[omegamuAB[[μ, a1, b1]](G[[a1]].G[[b1]]), {a1, 1, 4}, {b1, 1, 4}], {μ, 1, 4}], {e, 0, 1}]];

Constructing the covariant derivatives $D_{\mu}\varphi$:

Dercovphi = FullSimplify[Series[Table[D[(φ[t]), crd[[μ]]] - Gammamu[[μ]] (φ[t]), {μ, 1, 4}], {e, 0, 1}]];

I can see that I don't say to Mathematica sum up over gamma matrices "internal" index, but I don't know how to do it. By this way $D_{\mu}\varphi$ is $16 \times 16$ matrix. I'll need to make a $g^{\mu \nu}D_{\mu}\varphi D_{\nu}\varphi$ after it.

Let me add another example; The scalar $g^{\mu \nu}D_{\mu}\varphi D_{\nu}\varphi$ = prodder is:

prodder = FullSimplify[Series[Sum[Sum[(inversemetric1[[μ, ν]]* Dercovphi[[ν]].Dercovphic[[μ]]), {ν, 1, 4}], {μ, 1, 4}], {e, 0, 1}]]; 
Expand[DiracSimplify2[SeriesCoefficient[prodder, 0]]]/.f[t]->0

and the output is a $4 \times 4$ matrix:

$\left( \begin{array}{cccc} \frac{b^2 \varphi ^2}{a^4}+\frac{\left(\varphi '\right)^2}{a^2} & \frac{\left(\varphi '\right)^2}{a^2} & \frac{\left(\varphi '\right)^2}{a^2} & \frac{\left(\varphi '\right)^2}{a^2} \\ \frac{\left(\varphi '\right)^2}{a^2} & \frac{b^2 \varphi ^2}{a^4}+\frac{\left(\varphi '\right)^2}{a^2} & \frac{\left(\varphi '\right)^2}{a^2} & \frac{\left(\varphi '\right)^2}{a^2} \\ \frac{\left(\varphi '\right)^2}{a^2} & \frac{\left(\varphi '\right)^2}{a^2} & \frac{b^2 \varphi ^2}{a^4}+\frac{\left(\varphi '\right)^2}{a^2} & \frac{\left(\varphi '\right)^2}{a^2} \\ \frac{\left(\varphi '\right)^2}{a^2} & \frac{\left(\varphi '\right)^2}{a^2} & \frac{\left(\varphi '\right)^2}{a^2} & \frac{b^2 \varphi ^2}{a^4}+\frac{\left(\varphi '\right)^2}{a^2} \\ \end{array} \right)$

The correct scalar $g^{\mu \nu}D_{\mu}\varphi D_{\nu}\varphi$ must be:

$\frac{3 b^2 \varphi ^2}{4 a^4}+\frac{\left(\varphi '\right)^2}{a^2}$

Note: I'm using Series[,{e,0,1}] and SeriesCoef[] because it's a problem with first order perturbation, but for this thread it can be ignored.

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  • $\begingroup$ I suggest you add "using FeynCalc" or "with FeynCalc" in the title. This will help the search function and users in general $\endgroup$
    – magma
    Oct 26 '18 at 12:29
  • $\begingroup$ Thank you @magma $\endgroup$
    – Kamog
    Oct 27 '18 at 14:37

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