Is it possible to use the Faddeev-Popov trick for discrete gauge symmetries? I was thinking of this previous question of mine, where I was trying to implement a path-integral over the half-line:
$$
Z=\int_{\varphi\ge0}\ \mathrm e^{iS[\varphi]}\mathrm d\varphi\tag1
$$
It seems to me that a possible approach is to integrate over all $\varphi$, but to regard the configurations $\varphi$ and $-\varphi$ as physically equivalent. In other words, we take the $\mathrm d\varphi$ to be unconstrained, but we introduce the gauge equivalence $\varphi\to-\varphi$; the orbits are of the form $\{\pm\varphi\}$, and a representative of each orbit is, for example, $+\varphi$. Integrating over one representative for each orbit brings us back to $(1)$.
Instead of eliminating the gauge redundancy explicitly (which would lead to the integral I don't know how to evaluate, $(1)$), we leave the gauge symmetry as is, but treat the resulting path-integral using the standard methods of gauge-theory. The problem is that I don't know how to implement the Faddeev-Popov trick for $\mathbb Z_2$-valued gauge fields or, more generally, fields over a discrete group.
Has this problem been analysed? (How) Can we implement the Faddeev-Popov trick for system with discrete gauge symmetries?
 A: The answer is no. This has, however, nothing to do with the symmetry being discrete. Rather, it is simply the statement that Faddee-Popov ghosts never eliminate a redundancy of charged "matter" fields. 
(There seems also to be some confusion on what gauge fixing does and what the ghosts do. From some text books one could get the impression that one has to eliminate the gauge redundancy in order to define the path integral. This is, however, incorrect. The thing that goes wrong on the gauge fields is that the Klein-Gordon operator has a nontrivial kernel and is hence not invertible. This is because if you apply a gauge transformation on a configuration in the kernel you get "another" configuration in the kernel. This is different from the situation of the matter fields, where the corresponding Klein-Gordon or Dirac operators have covariant derivatives in such that the above problem does not arise. What the gauge fixing and ghosts do for you is to make the Klein-Gordon operator for the gauge fields invertible on the set of the restricted gauge field configurations. That's why you get the gauge fixing parameter in the gauge field propagators only. That's it. They do not kill off phases of the matter fields in any way.)
Literature on Yang-Mills ghosts: I did not find any fully self-contained and clear treatment of these issues in freely available pdf files but something close to it in Timo Weigand notes. However, the discussion in Pokorski's book on "Gauge Field Theories" is IMHO clearer, yet not publicly available. Unfortunately, the otherwise great notes by Srednicki are not 100% helpful at this point. 
