Difference between gauge invariance and diffeomorphism invariance

Question

What is the difference between gauge invariance and diffeomorphism invariance?

The two seem very similar, but is the distinction between them that a gauge transformation changes the field variables of the given theory, but has no effect on the coordinates on the underlying manifold (the background spacetime remains "fixed"). Whereas a diffeomorphism is a mapping between different manifolds. Thus, gauge invariance of a theory is when such gauge transformations leave the theory unchanged, and diffeomorphism invariance of a theory is when a diffeomorphism leaves the theory unchanged (expresses the background independence of the theory)?!

What I find confusing the most is that in general relativity, under a(n) (active) diffeomorphism $\phi:M\rightarrow M$, the relevant quantities are also transformed, i.e. $R\rightarrow\phi^{\ast}R$, $g_{\mu\nu}\rightarrow(\phi^{\ast}g)_{\mu\nu}$, etc... such that the theory is invariant under such transformations. (In essence, it is a statement of the background independence of the theory.) And then, in a gauge theory such as QED one has a local gauge transformation $A^{\mu}\rightarrow A^{\mu}(x)+\partial^{\mu}\Lambda(x)$ that leaves the theory invariant. In this case, the vector fields are transformed, but the underlying geometry remains fixed.

Is the point that a diffeomorphism is a mapping between manifolds, whereas a gauge transformation is a mapping between vector fields in the overlying tangent bundle of a given manifold?

Answer 1

Diffeomorphism invariance is an example of a gauge invariance, but not all gauge invariances are diffeomorphisms and moreover often gauge invariances of theories are much more restrictive than allowing any coordinate transformation.

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Given two manifolds, $M$ and $N$, a diffeomorphism is a differentiable map $f : M \to N$ that is bijective with a continuous inverse. Physically, for a theory like general relativity, diffeomorphisms are coordinate transformations, $x^\mu \to y^\mu(x)$ which induce a change in the metric. Since changing coordinates should not change the physics, we would expect diffeomorphism invariance.

Diffeomorphism invariance is a gauge symmetry, and as such we do impose gauge-fixing conditions in general relativity, to take (some) of these into account, such as for example de Donder gauge.

I do not like to think of diffeomorphism invariance as background independence, since changing to a completely different manifold does affect the physics. Rather, I see it as independence of how we choose to set up a coordinate system to measure distances.

On the other hand, invariance under say a Weyl transformation, $g_{ab} \to \Omega(x)g_{ab}$ means the theory does have a sort of background independence, at least up to those in the same conformal class.

It should also be noted that, just as we can view, say $A_\mu$ of electromagnetism as a section of a bundle, we can also view the metric on $E$ as a global section of $(S^2 E)^* \subset (E \otimes E)^* $ for a bundle $\pi : E \to M$. Note in this construction not all sections correspond to metrics, but there is the possibility to construct such a bundle.