# Adding stuff to the path integral (Faddeev-Popov method)

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?

• Minor comment to the post (v4): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot. Mar 1, 2014 at 17:02

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 $\xi$.