# Difference between gauge invariance and diffeomorphism invariance

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?

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.

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.

• I thought that two manifolds that are related by a diffeomorphism represent the same physical situation though? I.e. If the universe is described by some manifold $M$ with metric $g_{\mu\nu}$ and matter fields $\psi$, and $\phi :M\rightarrow M$ is a diffeomorphism, then the sets $(M, g_{\mu\nu}, \psi)$ and $(M, (\phi^{\ast}g)_{\mu\nu}, \phi^{\ast}\psi)$ describe the same physical situation?! ... – user35305 Jan 11 '17 at 22:33
• @user35305 Yes, that is what I said, a diffeomorphism still describes the same manifold. My point is it is not background independent in the sense you can go from $M$ with $g_{\mu\nu}$ to a $\Sigma$ with $h_{\mu\nu}$ not related by some $\phi$. – JamalS Jan 11 '17 at 22:36
• Ah, apologies, I was a bit too liberal in my wording there, by "background independence" I had meant in the case of related by diffeomorphisms. Regarding the gauge vs diffeo stuff, what confuses me is that in the literature a clear distinction seems to be made between diffeomorphism invariance and gauge invariance without really stating what this is? For example, I have seen papers constructing gravity as a gauge theory, which to me seems to suggest that it wasn't before (although it clear was already diffeomorphism invariant before)?! – user35305 Jan 11 '17 at 22:37
• @user35305 It's simply a different, more QFT-like way of viewing gravity. – JamalS Jan 11 '17 at 22:52
• Ah ok. I guess they naively seem like different "entities" since one seems to take place on the base manifold, which induces changes in the scalar/vector/tensor fields involved in the theory (diffeomorphisms), whereas the the other (gauge transformations) seems to take place in the fibre bundle over some base manifold (that is kept fixed) in which the vector/tensor fields themselves are directly transformed with the base space kept fixed. – user35305 Jan 11 '17 at 23:00