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.

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