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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?

user267839
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  • 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. – my2cts Dec 08 '23 at 13:30
  • @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? – user267839 Dec 08 '23 at 13:43
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    Check the Scholarpedia article on Fadeev Popov ghosts – Mauricio Dec 08 '23 at 14:46
  • @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. – my2cts Dec 08 '23 at 20:15
  • @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? – user267839 Dec 08 '23 at 21:33
  • @my2cts: yes in case of qed ghost fields are not really neccessary. So we can work there without it, yes. But the motivation of this question is how it would help to kill the unphysical degrees of freedom if we would apply it to this situation. – user267839 Dec 08 '23 at 22:12

2 Answers2

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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.

CStarAlgebra
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  • 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 – user267839 Dec 09 '23 at 13:51
  • 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? – user267839 Dec 09 '23 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.

Qmechanic
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  • 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 – user267839 Dec 09 '23 at 16:55
  • $\mathcal{L}{\rm{ghost}} \approx \partial{\mu}\bar{c}\partial^{\mu}c$ ( ... 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. – user267839 Dec 09 '23 at 16:59
  • 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? – user267839 Dec 09 '23 at 17:05
  • I updated the answer. – Qmechanic Dec 09 '23 at 17:58
  • 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? – user267839 Dec 09 '23 at 18:10
  • I updated the answer. – Qmechanic Dec 09 '23 at 18:15
  • One another point: Do I understand it correctly that your formulation "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. " is refering to the "naive" attempt what we would obtain if we would only add $\mathcal{L}{\rm{fix}} \approx k \cdot \partial^{\mu} A{\mu}$ to the free $\mathcal{L}(A_{\mu})$? Is that what you technically mean there by the "by hand removing of 2 components"? – user267839 Dec 09 '23 at 23:55
  • Or do you mean it 'literally' by considering only 2 of the given 4 components of $A_{\mu}$? In other words, on which "level" you perform the thought experiment on "naive removing procedure" of the two unphys components of field $A$ you are refering to it's deficiency there? On the level of the field $A_{\mu}$ itself by literally dropping out two unphysical components, or the "level of the Lagrangian" by adding this gauge fixing term, "but not touching $A_{\mu}$ itself"? – user267839 Dec 10 '23 at 00:09
  • So in summary one should say that the quoted phrase " To remove the unphysical degrees of freedom, [...] one way to do this reduction is to introduce some ghost field in the theory. " is as stated there not correct. Instead one should say that there are always the gauge fixing conditions which remove redundant dof's, but the ghost fields assure that the action integral of the modified Lagrangian by this gauge fixing terms is keeped Lorentz invariant, thats the point, right? – user267839 Dec 10 '23 at 00:26
  • I updated the answer. – Qmechanic Dec 10 '23 at 08:38
  • What I still not complete got is your point 2: "If we by hand were to remove 2 components of the 4-component photon field $A_{\mu} $ [...] " : How is this "by hand removal of these 2 unphys dof's" modification is intended to be performed explicitly at that stage? (...which indeed a posteriori turns out to be insufficient since not Lorentz invariant). By literally considering only 2 of 4 components of $A$ or by adding a gauge fixing term to the Lagrangian? So on level of field $A$ directly or on level of it's Lagrangian? – user267839 Dec 10 '23 at 11:31