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What is the correct series expansion for the $U(1)$ Faddeev-Popov ghosts?

I know that the $U(1)$ ghosts are only a phase such that they can be neglected in most cases but it turns out that this is not true in curved spaces even for $U(1)$ theories so please don't answer this...

In this thread Faddeev-Popov ghost propagator in canonical quantization I found that $c$ is hermitian and $\bar{c}$ anti-hermitian which makes sense since $\bar{c} = c^\dagger \gamma_0$.

But in the $U(1)$ case the ghost are Grassmann variables such that $\bar{c} = c^\dagger \gamma_0$ doesn't make sense does it?


For those willing to help me even more. I think that the source of my problem is a poor understanding of the Faddeev-Popov mechanism. More precisely, what happens when $\det(\square)$ is written as a path integral? What exactly do the $c$ and $\bar{c}$ fields mean? Why is it said that one is a ghost and the other an anti ghost?

When quantizing them I obtain $\{ c_k , \bar{c_{k'}}\} = -\delta(k-k')$ how does this tell us anything regarding the norm of these ghosts?

I read Peskin and Schroeder but they do not answer this question (or I missed it).

Finally, my sincere aplogies for this "all over the place" type question. I fail to pinpoint the exact sources of my confusion that's why my question is rather broad. I hope that someone more experiences can pinpoint it with the above information.

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  • $\begingroup$ What do you mean by "what happens" when the functional determinant is written as a path integral? $\endgroup$
    – JamalS
    Commented Apr 10, 2017 at 13:59
  • $\begingroup$ In general, what are the c and $\bar{c}$ fields ? Are they related or not ? I read that one is hermitian and one anti-hermitian and why ? What is the canonical commutation relation for these fields ? $\endgroup$
    – gertian
    Commented Apr 10, 2017 at 14:10
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    $\begingroup$ The claim $\overline{c}=c^\dagger\gamma_0$ is wrong, because it relates the antighost to the ghost, and these are completely different fields (the prefix anti- and the bar above $c$ don't have their usual meanings in this context). $\endgroup$
    – J.G.
    Commented Apr 10, 2017 at 15:09
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    $\begingroup$ @gertian That's only one convention. Kugo and Ojima 1979 is another important paper on the FP term, and scales so both fields are Hermitian. This requires an $i$ factor so the Lagrangian is Hermitian. $\endgroup$
    – J.G.
    Commented Apr 10, 2017 at 16:08
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    $\begingroup$ The paper: ci.nii.ac.jp/els/… $\endgroup$
    – J.G.
    Commented Apr 10, 2017 at 16:24

1 Answer 1

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As discussed in Kugo and Ojima 1979, "ghost is Hermitian, anti-ghost is anti-Hermitian" is just a convention, another being that both fields are Hermitian, which results in a factor of $i$ in the FP-ghost term so that the Lagrangian is still Hermitian. In their notation $c,\,\overline{c}$ are both Hermitian while $C:=c,\,\overline{C}:=i\overline{c}$ provide a half-Hermitian convention. Then $$\mathcal{L}_{FP}=-i\partial_\mu\overline{c}D^\mu c=-\partial_\mu\overline{C}D^\mu C.$$

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