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I recently gave a presentation on gauge invariance in general relativity that was, in a small part based upon [https://arxiv.org/pdf/gr-qc/9910079v2.pdf]. In this the authors state (top of page 30);

General relativity is distinguished from other dynamical field theories by its invariance under active diffeomorphisms. Any theory can be made invariant under passive diffeomorphisms. Passive diffeomorphism invariance is a property of the formulation of a dynamical theory, while active diffeomorphism invariance is a property of the dynamical theory itself. Invariance under smooth displacements of the dynamical fields holds only in general relativity and in any general relativistic theory. It does not hold in QED, QCD, or any other theory on a fixed (flat or curved) background.

I received an excellent question relating to this which was: "How would you show that electromagnetism (for example) is not invariant under active diffeomorphisms?" I was pretty stumped by this question and would be very interested to hear what the answer is.

I know that to first order the active diffeomorphism transforms a tensor, $$Q\to\tilde{Q}(\tilde{x}_\tilde{P}=x_0)=[Q-\mathcal{L}_\xi Q](x_P=x_0)$$ where tildes denote the transformed point and coordinate system and $\xi$ generates the coordinate transformation $x\to\tilde{x}$ (or an exponential of the Lie derivative in non-infinitesimal form). So would I just need to perform this operation for the electromagnetic field tensor or for the EM gauge field, for example?

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  • $\begingroup$ Does this imply that the active diffeomorphism violates the U(1) symmetry? $\endgroup$
    – safesphere
    Commented Feb 1, 2018 at 16:14
  • $\begingroup$ QED works in flat space which has Lorentz symmetry. But in curved space, we don't have universal Lorentz transformation. It means that at each point we have Lorentz symmetry but not global ones. You can see directly that, in curved space form of Maxwell equation will change. You have a notion of covariant derivative not partial derivative. Simply partial derivative will not produce tensor quantity required by GCT. $\endgroup$
    – Hkw
    Commented Feb 2, 2018 at 14:14
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    $\begingroup$ In principle, we can write a covariant theory of Electromagnetism in curved space. It is very easy just replace $\eta$ with $g$ and partial derivative to the covariant derivative. In action write appropriate measure with $\sqrt{-g}$ and so on. $\endgroup$
    – Hkw
    Commented Feb 2, 2018 at 14:16

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The (active) diffeomorphism covariance of GR, is actually the background independence property of the theory (look at lemma 2.1 of https://arxiv.org/pdf/gr-qc/9403028.pdf). The property that the theory does not have any fixed structure when you want to start to solve the dynamical equations or when you want to write the action. The dynamics of anything in the action of GR is specified by the action of GR itself, including the metric. So here (active) diffeomorphism covariance really means that you have dynamical equations which are diffeomorphism covariant for any element of the theory. But this is not the case for any theory (like any field theory on fixed metric background) that has a fixed predetermined non-dynamical structure. That structure might by anything actually, but it is usually metric. Now, it is possible to write a field theory on fixed background say Minkowski spacetime in a covariant way (passive). But this is not covariant under active diffeomorphisms as there is no a diffeomorphism covariant dynamical equation (like Einstein equations) for the metric (metric is fixed). The isometries (subset of diffeomorphisms) of this fixed metric however are (global) symmetries of the theory and they are in fact used to defined energy and momentum via Noether theorem, but not all the diffeomorphisms are symmetries. You can make all of these concepts very precise mathematically using the language of covariant phase space.

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