2
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ What do you mean by "what happens" when the functional determinant is written as a path integral? $\endgroup$ – JamalS Apr 10 '17 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 Apr 10 '17 at 14:10
  • 2
    $\begingroup$ 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. Apr 10 '17 at 15:09
  • 1
    $\begingroup$ @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. Apr 10 '17 at 16:08
  • 1
    $\begingroup$ The paper: ci.nii.ac.jp/els/… $\endgroup$ – J.G. Apr 10 '17 at 16:24
3
$\begingroup$

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 $$\mathcal{L}_{FP}=-i\partial_\mu\overline{c}D^\mu c=-\partial_\mu\overline{C}D^\mu C.$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.