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I have seen various different definitions of a ghost field in the literature. For example, one can find many examples where ghosts are simply defined as any field with a negative sign in the kinetic term. However, some authors will claim that these are not ghosts if their energy density is positive, in which case they might be actually physical. There are, of course, other ghosts, such as Faddeev–Popov ghosts which arise in gauge theories.

What are the definitions of these different ghosts, and how are they related (if at all)?

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    $\begingroup$ I'm assuming you've already taken a look through en.wikipedia.org/wiki/Ghost_(physics) which includes quite a few examples? (From what I've seen its very context dependent what people mean) $\endgroup$
    – Eletie
    Dec 23, 2020 at 11:26
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    $\begingroup$ Some more useful resources: iopscience.iop.org/article/10.1088/1126-6708/2005/09/003/… this paper is about massive gravity but the introduction on ghosts is quite generic . iopscience.iop.org/article/10.1088/0143-0807/36/1/015009/… more of a specific look at ghosts in general $\endgroup$
    – Eletie
    Dec 23, 2020 at 11:33
  • $\begingroup$ I have read the wiki page, but the definitions are quite vague with no discussion on e.g. how a negative kinetic term could not be a ghost and in fact physical. Thanks for the references! $\endgroup$
    – Akoben
    Dec 23, 2020 at 11:37
  • $\begingroup$ Which some authors? Which page? $\endgroup$
    – Qmechanic
    Dec 23, 2020 at 11:41
  • $\begingroup$ @Akoben the paper on ghosts in massive gravity shows the case you mention here explicitly, see my other answer too. Hope this somewhat helps! $\endgroup$
    – Eletie
    Dec 24, 2020 at 0:01

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Just to add to the other answers, I wanted to give a perspective from ghosts found in modified gravity as opposed to quantum theory (which I think you may also be referring to as well). Here's a quote you might find useful:

We should be clear about the distinction between the kind of ghost that arises in certain modified gravity models and the Faddeev-Popov ghost used in the quantisation of non-abelian gauge theories. The latter is introduced in the path integral to absorb unphysical gauge degrees of freedom. It does not describe a physical particle and can only appear as an internal line in Feynman diagrams. In contrast, the ghosts that haunt modified gravity describe physical excitations and can appear as external lines in Feynman diagrams.

from the review paper Modifed Gravity and Cosmology, page 21 (a short section on ghosts in modified gravity). The ghosts - in this context - are not unstable by themselves but have a negative Hamiltonian, which causes instabilities when they couple to ordinary fields. The reason being (quote): with zero net energy one can indiscriminately excite both sectors, and this exchange of energy happens spontaneously already at classical level, see here for details.

References:
[1] Clifton, Timothy, et al. "Modified gravity and cosmology." Physics reports 513.1-3 (2012): 1-189. [2] Creminelli, Paolo, et al. "Ghosts in massive gravity." Journal of High Energy Physics 2005.09 (2005): 003.

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The general theory of the ghost structure in the Lagrangian field theory is rather a sophisticaed deal and is called Batalin–Vilkovisky (BV) formalism :

https://en.wikipedia.org/wiki/Batalin%E2%80%93Vilkovisky_formalism

https://ncatlab.org/nlab/show/BV-BRST+formalism

I would not dive into deep details, lacking knowledge and understadning of the general picture, but illustrate in a fashion, common for Yang-Mills theories.

On a high level, idea is appoximately the following. We have a gauge theory, where the supeficial number of degrees of freedom exceeds the actual, physical number of degrees of freedom. States, related by a certain gauge transformation are treated as identical.

In the process of calculation of certain correlators or a physical observable in order to avoid counting the same state multiple times, one imposes a gauge condition, which chooses from a numerous identified state a particular representative, specified by the condition (where $g$ are some parameters of the gauge transformation): $$ f(A_g) - G = 0 $$ Here $f$ can depend on $A$ and its derivatives $\partial A, \partial^2 A , \ldots$. In the path integral we make an insertion of identity: $$ 1 = \int D g \ \delta(f(A_g) - G) $$ Which, due to the properties of $\delta$-function is: $$ \sum_i \int D g \ \Delta_{\text{FP}}(g, A) \delta(A_g - A_{i, *}) $$ Where $\Delta_{\text{FP}}(g, A)$ is Fadeev-Popov determinant. Due to the property of the fermionic path integral: $$ \int d \bar{\psi} d \psi e^{-\bar{\psi} M \psi} = \det M $$ One can regard the determinant as the addition of the fermionic fields to the theory, and then perform perturbative computations with ghost propagators and vertices.

Several points are out of order:

  • Ghosts are not physically observed fields in the sense, that there are no such $| in \rangle $ and $| out \rangle$ states. One cannot measure on the experiment the decay rate of the ghost, or their $2 \rightarrow 2$ scattering.
  • In the perturbation theory ghost appear only in the loop diagrams.
  • For the case of Yang-Mills theory ghosts have fermionic statistic, but they can have a bosonic statistic, in case they are related to the fermionic degrees of freedom.
  • In more complicated theories there can be ghosts, corresponding to ghosts

For the gauging of relativity nice at simple to follows discussion is presented here http://www.damtp.cam.ac.uk/user/tong/string/five.pdf.

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The original (and most fundamental) definition of a ghost field was one for which, when it is quantized, the particle-like excitations do not have positive norm. One of the first places this came up was in attempts to canonically quantize the electromagnetic field in a relativistically-invariant way. If you look at the (Feynman-gauge*) propagator for the electromagnetic field, $$\langle0|T\left[A_{\mu}(x)A_{\nu}(y)\right]|0\rangle=\int\frac{d^{4}q}{(2\pi)^{4}}\frac{-ig_{\mu\nu}}{q^{2}+i\varepsilon}e^{-iq\cdot(x-y)},$$ this looks just like four massless Klein-Gordon propagators, along with an overall factor of $-g_{\mu\nu}$, which is $+1$ for $\mu=\nu$ spacelike, but for $\mu=\nu=0$, there is an overall factor of $-1$. This can be traced back to that fact that, if we expand $A_{0}$ in creation and annihilation operators, the timelike polarization state $|k,\hat{0}\rangle=a^{(0)}_{k}|0\rangle$ created by a creation operator $a^{(0)}_{k}$ in the mode expansion of $A_{0}$ as negative norm, $\langle0|a^{(0)\dagger}_{k}a^{(0)}_{k}|0\rangle<0.$ These ghosts appear in the Gupta-Bleuler quantization method, and demonstrating that the negative norms states are never produced physically is a necessary part of the formalism.

By choosing a different basis of polarization states for the quantization of the electromagnetic field, it is possible to get different kinds of ghosts. If (when $\vec{q}$ is along the $z$-direction), instead of the timelike polarization vector $\epsilon_{(0)}^{\mu}=\hat{0}=[1,0,0,0]$ and the longitudinal spacelike vector $\epsilon_{(3)}^{\mu}=[0,0,0,1]$, we use a basis containing $$\epsilon_{(\pm)}^{\mu}=\frac{1}{\sqrt{2}}[1,0,0,\pm1],$$ we find that states created with these polarizations have zero norm. The zero-norm ghosts are in many ways easier to deal with; in particular, these lightlike polarization vectors can be used to show how the effects of the timelike $\epsilon_{(0)}^{\mu}$ and longitudinal $\epsilon_{(3)}^{\mu}$ cancel out in such a way as to make the overall time evolution unitary.

The other examples in which ghosts arise also involve the creation of excitations with negative or zero norm. This is simplest to see with the Fadeev-Popov ghosts used in quantization of gauge theories. The ghost fields in this case are Lorentz scalars but are quantized according to Fermi-Dirac statistics, with the sign differences resulting in the creation of negative-norm states.

For a scalar field theory with a Lagrangian density like $${\cal L}=-\frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}-\frac{\lambda}{4!}\phi^{4},$$ the negative sign in the first term is a problem. If the states are to have positive energy when the theory is quantized with this ${\cal L}$, then it is necessary for the $\phi$ creation operators to, once again, create states with negative norm. If the creation operators are the usual ones, then the unconventional negative sign in front of the kinetic term will lead to negative energies.

However, in a theory like this, there is a trade-off possible. Since the equations of motion derived from ${\cal L}$ and another Lagrange density $${\cal L}'=-{\cal L}=\frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi+\frac{1}{2}m^{2}\phi^{2}+\frac{\lambda}{4!}\phi^{4}$$ are exactly the same, we can use this Lagrangian instead. Now the particle states will have positive kinetic energies when the theory is quantized according to the usual method, but the potential energy is now unbounded below, since we have switched $V(\phi)\rightarrow-V(\phi)$. The negative norm states have been traded for an instability; which formulation is more useful to use will depend on the circumstances.

*With care, this can be generalized to other gauges, with $-i(g_{\mu\nu}-\alpha k_{\mu}k_{\nu}/k^{2})$ in the numerator of the propagator.

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In the context of Yang-Mills, the ghosts are auxiliary fields introduced only as a gauge fixing mechanism (they have no physical motivation, as far as I am aware). They are just a tool to introduce a gauge fixing condition, since integration on the space of all distinct connections (connections not related by a gauge transformation) is very difficult.

This may be a good starting point: understanding ghosts as auxiliary fields arising from some mathematical manipulation. However, I am not familiar with their usage in other areas, in which they might have a different meaning.

Anyway, I think you need to clarify what you understand by physical.

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  • $\begingroup$ In modified gravity I think they have a different meaning, e.g. they show up in the classical solutions for EFT's and generally plague the theories with unphysical solutions and instabilities, etc. This might be closer to what OP is referring to? $\endgroup$
    – Eletie
    Dec 23, 2020 at 12:09
  • $\begingroup$ I am not familiar with modified gravity. Do you know the origin of those instabilities? In the context of Yang-Mills, I do not think the ghosts are physical either e.g. there are no external lines (Feynmann diagrams) with ghosts (if that is a criterion for a something to be physical). $\endgroup$
    – JGBM
    Dec 23, 2020 at 17:57
  • $\begingroup$ I realised I mis-spoke with the 'unphysical' comment in the mod-GR context. I should've said the exact opposite. I added an answer with the mod-GR perspective with some useful references though. $\endgroup$
    – Eletie
    Dec 23, 2020 at 18:29
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    $\begingroup$ Thanks for your answer. I think that is the answer OP was looking for. $\endgroup$
    – JGBM
    Dec 23, 2020 at 21:27

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