# How do you write a resolution of unity (the identity) in gauge theories?

Physics Asked on January 6, 2021

In, say, a quantum field theory of a single scalar field $$phi$$, it is common to write the identity as $${bf 1}=int{cal D}phi, |phiranglelangle phi|$$, a useful thing to do in various path integral manipulations. Here, the path integral is over configurations of $$phi$$ on some particular time slice. For instance, you could break up a vacuum-to-vacuum amplitude through such an insertion at $$t=0$$, as in:
$$langle 0|U(infty,-infty)|0rangle=int{cal D}phi,langle 0|U(infty,0) |phiranglelangle phi|U(0,-infty)|0rangle$$
in order to compute the full amplitude in terms of the individual matrix elements $$langle phi|U(0,-infty)|0rangle$$ and $$langle 0|U(infty,0) |phirangle$$, say.

How does the analogous thing work in gauge theories? In QED or YM, is it simply the naive thing, $${bf 1}=int{cal D}A_mu, |A_muranglelangle A_mu|$$, or something more complicated?

I feel like it should be something more complicated for the usual reasons that gauge-theory path integrals over-count gauge-equivalent configurations, but I’m not sure how exactly the naive result should be modified.

Do you just need to go through the usual Faddeev-Popov procedure for each insertion of unity in a gauge-theory path integral? That is, is the right answer something more like $${bf 1}=int{cal D}A_mu, |A_muranglelangle A_mu|delta(F[A])$$ for some gauge-fixing functional $$F[A]$$?

Edits: edited to give more detail, examples.