# Showing Electromagnetism (QED) is not invariant under active diffeomorphisms

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?

• Does this imply that the active diffeomorphism violates the U(1) symmetry? Commented Feb 1, 2018 at 16:14
• 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.
– Hkw
Commented Feb 2, 2018 at 14:14
• 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.
– Hkw
Commented Feb 2, 2018 at 14:16