Introduce Ghost Field to eliminate unphysical degrees of freedom in case of Photon Field

Question

In wikipedia's article about ghost fields is stated the following which requires a bit more clarification:

An example of the need of ghost fields is the photon, which is usually described by a four component vector potential $A_{\mu}$, even if light has only two allowed polarizations in the vacuum. To remove the unphysical degrees of freedom, it is necessary to enforce some restrictions; one way to do this reduction is to introduce some ghost field in the theory.

Question: Can this "toy example" be elaborated in more details, namely how to use a ghost field to eliminate unphysical degrees as indicated in quoted excerpt?
Indeed as also stated there the full power of applying ghost field techniques deploys in case one deals with non-Abelian fields, and so in case of four vector potential modeling the photons there is no necessity to introduce ghost fields as tool to kill redundant (=unphysical) degrees of freedom.

Indeed, usually (at least in all text book's I read on this topic) in this "simple case" of photon field it is handled more conventially by imposing additional equations (eg the Lorenz gauge ) to get rid of the redundant degrees.

Nevertheless I would like to see - for sake of didactical simplicity on this "toy example" of the photon field modeled by four potential (with only two physically non-redundant degrees of freedom= the two polarizations) - how instead alternatively the ghost field techniques could be applied here explicitly in order to remove the unphysical gauge degrees?

Could somebody elaborate how this approach is performed on this example? So far as I see in case of an Abelian theory the ghost construction (= adding new "ghost field term" to Lagrangian) gives nothing new since it is not interacting with original field and so can be more or less left out. But on the other hand above it is claimed that even for photon field (so Abelian) it could be used to eliminate unphysical degrees, so it provides "non useless" impact even in this Abelian case.

How to resolve these two seemingly contradicting each other statements?

Answer 1

Ghosts and the usual BRST formalism for gauge theories can be introduced in QED just as in non-abelian gauge theory. What one finds in the end is that the ghost fields $c,\bar{c}$ end up contributing to the Lagrangian only as

$\mathcal{L}_{\rm{ghost}} \approx \partial_{\mu}\bar{c}\partial^{\mu}c$

The ghosts are completely free and non-interacting. Thus one can drop them from the theory entirely, which is why QED is usually taught without ghosts and the BRST formalism to begin with.

Answer 2

TL;DR: Well, superficially it admittedly sounds contradictory that adding ghosts can remove unphysical DOF. And that is, at best, an incomplete picture.

1. Wikipedia is presumably alluding to the BRST formulation (sometimes called modern covariant quantization, partly because it is manifestly Lorentz covariant).

   Recall that in the Feynman path integral one should sum over all histories, or equivalently, all Feynman diagrams. In particular, there are Feynman loop diagrams, where virtual photons run in loops. Now it turns out that allowing ghosts to also run in loops has a cancelling effect because of a Feynman rule, which says that a Grassmann-odd loop carries an extra minus.

2. If we by hand were to remove 2 components of the 4-component photon field $A_{\mu}$ to get 2 physical polarizations, we would break manifest Lorentz symmetry.

   In contrast the BRST formulation is manifestly covariant and manifestly independent of the choice of gauge-fixing. But the price is to introduce auxiliary fields, such as e.g. Faddeev-Popov (FP) ghost and anti-ghost fields.

3. One may show that the ghosts decouple in a unitary gauge.

   The actual removal of 1 DOF is done with the help of a gauge-fixing condition. 1 DOF turns out to be non-propagating (i.e., it has no time-derivative in the action), leaving 2 propagating physical DOF.

4. If we restrict to Abelian gauge theories, the role of the FP ghost and anti-ghost is essentially just to construct the FP determinant; they don't remove physical DOF per se. The FP determinant is merely a measure factor in the path integral that ensures that the path integral doesn't depend on the choice of gauge fixing, cf. e.g. this related Phys.SE post.