Obtaining the propagator for the Faddeev-Popov (FP) ghosts from the path integral language is straightforward. It is simply
$$\langle T(c(x) \bar c(y))\rangle~=~\int\frac{d^4 p}{(2\pi)^4}\frac{i e^{-ip.(x-y)}}{p^2-\xi m^2+i\epsilon}.$$
But I am unable to properly derive it following canonical quantization route.
The problem is that for anti-commuting fields, the time-ordered product is defined as
$$\langle T(c(x) \bar c(y))\rangle=\theta(x^0-y^0)\langle c(x) \bar c(y)\rangle-\theta(y^0-x^0)\langle \bar c(y) c(x)\rangle$$
with a minus sign between the two terms. This minus sign is preventing me from closing the contours in the correct way to obtain the expression in my first equation.
The only way I can save this is to say that FP ghosts are special, and their time-ordered product is defined with a plus sign instead of the minus sign. Is this legitimate? What is the right way to get the Ghost propagator following canonical quantization route?
Answer
The solution to this problem comes from the sneaky fact (Kugo, 1978) that while the FP ghost field is hermitian $c^\dagger (x) = c(x)$, the anti-ghost field is anti-hermitian $\bar c^\dagger (x)=-\bar c (x)$ .
As a result, the plane wave expansion for the ghost/anti-ghost fields (Becchi, 2008), Scholarpedia are:
$$ c^a(x)={1 \over(2 \pi)^{3/2}} \int_{k_0= | \vec k|} {d \vec k \over 2 k_0}( \gamma^a( \vec k)e^{-ik \cdot x}+ (\gamma^a)^\dagger( \vec k)e^{ik \cdot x})$$ $$\bar c^a(x)={1 \over(2 \pi)^{3/2}} \int_{k_0= | \vec k|} {d \vec k \over 2 k_0}( \bar \gamma^a( \vec k)e^{-ik \cdot x}- (\bar\gamma^a)^\dagger( \vec k)e^{ik \cdot x}) \ , $$ with a minus sign between the two terms in mode expansion for the anti-ghost field.
Thus, when evaluating the time ordered correlator (propagator), the minus sign in the plane-wave expansion compensates the minus sign in the definition of the time-ordering shown in my question above. Thus, I am able to derive the standard Feynman propagator for the FP ghost field.
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