Given a Lagrangian, how do I find the inverse propagator matrix?
For example,
$$ \begin{aligned} &\mathscr{L}_\pi=\mathrm{D}_0 \pi^* \mathrm{D}_0 \pi-\nabla \pi \nabla \pi^*-\mathrm{b}^2 \mu^2 \pi^* \pi-\frac{\mathrm{b}^2 \mu^2-\xi \mathrm{R}}{\mathrm{d}-1}\left(\pi+\pi^*\right)^2\\ \end{aligned} $$
With the covariant derivative $D_0=\partial_0-i b \mu $.
Leads to $$ \mathrm{D}^{-1}=\left(\begin{array}{cc} \frac{1}{2}\left(\nabla^2-\partial_0^2\right)-\frac{2\left(b^2 \mu^2-\xi \mathrm{R}\right)}{\mathrm{d}-1} & \mu \partial_0 \\ -\mu \partial_0 & \frac{1}{2}\left(\nabla^2-\partial_0^2\right) \end{array}\right) $$
for the inverse propagator of the real and imaginary part of $\pi$.
But that's maybe easier to understand with a simpler Lagrangian, if you have so.
