I'm wondering about the Faddeev-Popov method described in Peskin Schroeder and also on page 7 in this link.
What gives them the right to simply add the Gaussian $\omega$ and thus introduce the $\xi$ parameter? It seems so arbitrary to me.
Are there any rigorous derivation of the Faddeev-Popov method?
You are always allowed to introduce a new integration variable as long as its not its already being summed over. This might be more clear in discrete form:
\begin{align}
\int d x \, f (x) & \rightarrow \Delta x\sum _i \,f ( x _i ) \\
& = \big( N \Delta y\sum _j g ( y _j ) \big) \Delta x \sum _i f ( x _i ) \\
\end{align}
where $ N \Delta y\sum _j g ( y _j ) = 1 $ (note that it is very important this doesn't depend on $x$). Then,
\begin{align}
\int d x \, f (x)& \rightarrow N \Delta x \Delta y \sum _{i,j} f ( x _i ) g ( y _j )
\end{align}
The only difference with the Fadeev Poppov procedure is that now $ g ( y ) $ is also a function of a new unphysical parameter,$\xi$. In order to not change the value of the integral over $ f (x) $, the constant $ N $ needs to also change with $ \xi $.
