这是一份liverpool利物浦大学PHYS493的成功案例

The expression $\mathcal{L}{0}(G)$ may be chosen in the form of the componentwise sum: $$ \mathcal{L}{0}(G)=\sum_{a} \mathcal{L}{0}\left(G{a}\right)
$$
which represents the sum of the quadratic nondegenerate forms to be analogous to the electromagnetic field Lagrangian
$$
\mathcal{L}=-\frac{1}{4}\left(\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}\right)^{2}-\frac{1}{2 \xi} \partial_{\mu} A^{\mu} \partial_{\nu} A^{\nu}
$$
Using the same methods as in the case of the electromagnetic field, we get the expression for the gluon propagator
$$
D_{a b \mu \nu}^{c}(k)=-\delta_{a b}\left[g_{\mu \nu}+(\xi-1) \frac{k_{\mu} k_{\nu}}{k^{2}}\right] \frac{1}{k^{2}} .
$$

PHYS493 COURSE NOTES ：

It is more convenient to use real Grassmann fields $\rho$ and $\sigma$ to be defined as
$$
\omega_{a}=\frac{1}{\sqrt{2}}\left(\rho_{a}+i \sigma_{a}\right), \quad \omega_{a}^{\dagger}=\frac{1}{\sqrt{2}}\left(\rho_{a}-i \sigma_{a}\right)
$$
rather than the complex ghost fields $\omega_{a}$ and $\omega_{a}^{\dagger}$. Making use of the anticommutativity property of Grassmann fields
$$
\rho_{a}^{2}=\sigma_{a}^{2}=0, \quad \rho_{a} \sigma_{b}=-\sigma_{b} \rho_{a}
$$
we obtain
$$
\mathcal{L}{F P G}=-i \sum\left(\partial{\mu} \rho_{a}(x)\right)\left[\delta_{a b} \partial^{\mu}-g_{s} f_{a b c} G_{c}^{\mu}(x)\right] \sigma_{b}(x)=-i \sum\left(\partial_{\mu} \rho_{a}(x)\right) D^{\mu} \sigma_{a}(x)
$$