Why can't Faddeev-Popov ghosts be replaced with bosons?

Question

Faddeev-Popov ghosts are introduced in the quantization of Yang-Mills theory to absorb the Faddeev-Popov determinant into the action, $$\det \Delta_{\text{FP}} = \int \mathcal{D} \bar{c} \mathcal{D} c \, e^{i \int dx \, \bar{c}^a(x) (\Delta_{\text{FP}} c(x))^a}.$$ Here, $c$ is a Lorentz scalar, and it must be fermionic, as if we used a bosonic variable $\psi$ instead, we would get the determinant in the denominator rather than the numerator, $$\frac{1}{\det \Delta_{\text{FP}}} \propto \int \mathcal{D} \bar{\psi} \mathcal{D} \psi \, e^{i \int dx \, \bar{\psi}^a(x) (\Delta_{\text{FP}} \psi(x))^a}.$$ So we have to accept a violation of spin-statistics to quantize Yang-Mills.

But a friend of mine came up with a simple alternative: simply note that $$\det \Delta_{\text{FP}} = \frac{1}{\det \Delta_{\text{FP}}^{-1}} \propto \int \mathcal{D} \bar{\psi} \mathcal{D} \psi \, e^{i \int dx \, \bar{\psi}^a(x) (\Delta_{\text{FP}}^{-1} \psi(x))^a}.$$ Note that the inverse exists because the determinant is nonzero; if the determinant were zero the whole path integral would be zero and we wouldn't be able to do anything, ghosts or not.

Is there anything wrong with this method? Does it lead to some complications down the line? If not, why are the spin-statistics violating Faddeev-Popov ghosts typically used instead, when this setup looks much nicer?

Answer 1

Note that the inverse of $\Delta_{FP}$ is the Green function $G_{FP}(x,y)$ obeying:

$$ \Delta_{FP}|_x G_{FP}(x,y)=\delta(x-y) $$

so, the action that you wrote down is non-local, should be of the form:

$$ S_{FP}^{bosonic}=i\int dx \int dy\, \bar{\psi}^{a}(x)G_{FP}(x,y) \psi^{a}(y) $$

this action is non-local, since we cannot get away with just one integral in space-time. Since we are dealing with local theories, the presence of non-local interactions make the computations very complicated with lots of cancellations that are obscure in this formalism. Those cancellations should happen since the theory is local, so all the non-local terms must conspire to produce a local theory at the very end.

Also, it is better to have the ghosts as fermions since they will shows up in loops with the opposite sign of the pure gauge loops, erasing them. In this formalism of bosonic ghosts is not clear how this cancellation will be happening, but of course will be happen since there is no inconsistency over here. There will be extra cancellations that are obscure in this formalism.