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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?

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  • $\begingroup$ The em field has four components. Zero charge is removes one component. Charge conservation implies the Lorenz condition so that two polarisation directions remain. I have a paper on this in a reviewed mainstream journal. $\endgroup$
    – my2cts
    Commented Dec 8, 2023 at 13:30
  • $\begingroup$ @my2cts: yes sure, that's the usual - let me can it "standard"- strategy to impose additional equations eliminating the redundant degrees (eg the Lorentz gauge condition as you said; compare also with 3rd paragraph above). But my concern here is about how to use the techniques involving the introduction of ghost fields there to eliminate the redundant degrees. As you showed one can do it differently adding additional restricting equations, but it's not the ghost field approach I'm interested in, right? $\endgroup$
    – user267839
    Commented Dec 8, 2023 at 13:43
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    $\begingroup$ Check the Scholarpedia article on Fadeev Popov ghosts $\endgroup$
    – Mauricio
    Commented Dec 8, 2023 at 14:46
  • $\begingroup$ @user267839 My point is that there is no need for ghost fields in electromagnetism. Also it is not standard to consider the Lorenz gauge (L.V. Lorenz) a consequence of charge-current conservation as it flies against electromagnetic gauge invariance. $\endgroup$
    – my2cts
    Commented Dec 8, 2023 at 20:15
  • $\begingroup$ @Mauricio: so far I understand the explanation in the linked excerpt correctly, the ghost field contribute an additional term to the Lagrangian which in case of Abelian theory is nor coupled interactively to the field, so can be essentially leaved out, or not? (...so also CStarAlgebra's answer). How does it help in case of the photon to get rid of the two unphysical degrees as claimed in the queted text? $\endgroup$
    – user267839
    Commented Dec 8, 2023 at 21:33

2 Answers 2

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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.

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  • $\begingroup$ but then as far as I understand your explanations correctly applying this ghost field techniques in QED is useless due to non interaction of gauge field in the added term, right? On the other hand the quoted statement on the photon field $A_{\mu}$ claims that to remove the unphysical degrees of freedom one way to do this is to introduce some ghost field in the theory. So seemingly ghost fields can be applied effectively in QED to remove the unphys degrees. Could you resolve my confusion about if the ghost fields provide a useful method (how?) in case of QED in light of this quoted statement $\endgroup$
    – user267839
    Commented Dec 9, 2023 at 13:51
  • $\begingroup$ because seemingly we facing two statements contradicting in logical since each other; on one hand that for QED ( or more general an Abelian theory) the additional ghost fields do not contribute something new, on the other hand that their impact eliminate the redundant/ unphys degrees, so it does solve the problem. So do they contribute an auxilary effect in QED or not? $\endgroup$
    – user267839
    Commented Dec 9, 2023 at 14:01
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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.

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  • $\begingroup$ let me try to rephrase the philosophy of this BRST philosophy to see if I understaand it correctly on the QED as "toy example": so say we start with the free Largrangian $\mathcal{L}(A_{\mu})$ and facing the remarked problem that $A_{\mu})$ has two unphysical degrees. The first thing that we do we add to the Lagrangian the new fixing term $\mathcal{L}_{\rm{fix}} \approx k \cdot \partial^{\mu} A_{\mu}$ but facing now that this new Lagrangian would be not not Lorentz invariant, and the crutial idea is that now we add one more term $\endgroup$
    – user267839
    Commented Dec 9, 2023 at 16:55
  • $\begingroup$ $\mathcal{L}_{\rm{ghost}} \approx \partial_{\mu}\bar{c}\partial^{\mu}c$ ( ... if theory non Abelian there is one more term) with new fields $c, \bar{c}$ s "price to pay" such that now $\mathcal{L}(A_{\mu})+ \mathcal{L}_{\rm{fix}}+ \mathcal{L}_{\rm{ghost}}$ satisfies two two things: $\mathcal{L}_{\rm{fix}}$ kills redundant degrees and $\mathcal{L}_{\rm{ghost}}$ gives back the Lorentz invariance. $\endgroup$
    – user267839
    Commented Dec 9, 2023 at 16:59
  • $\begingroup$ Is this roughly the idea how gauge fields in sense of BRST provide a way to get rid of unphysical degees? Could you check briefly if I rephrased the rough idea behind it correctly? Or did I misunderstood the idea? $\endgroup$
    – user267839
    Commented Dec 9, 2023 at 17:05
  • $\begingroup$ I updated the answer. $\endgroup$
    – Qmechanic ♦
    Commented Dec 9, 2023 at 17:58
  • $\begingroup$ Thank you! One nitpick: what do you mean in last paragraph by that 1 dof turns out to be non-propagating? What does this imply for further handling of the theory? $\endgroup$
    – user267839
    Commented Dec 9, 2023 at 18:10

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