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Consider the following paragraph taken from page 15 of Thomas Hartman's lecture notes on Quantum Gravity:

In gravity, local diffeomorphisms are gauge symmetries. They are redundancies. This means that local correlation functions like $\langle O_{1}(x_{1})\dots O_{n}(x_{n})\rangle$ are not gauge invariant, and so they are not physical observables. On the other hand, diffeomorphisms that reach infinity (like, say, a global translation) are physical symmetries - taking states in the Hilbert space to different states in the Hilbert space - so we get a physical observable by taking the insertion points to infinity. This defines the S-matrix, so it is sometimes said that ``The S-matrix is the only observable in quantum gravity.''


  1. Why does the fact that local diffeomorphisms are gauge symmetries mean that local correlation functions like $\langle O_{1}(x_{1})\dots O_{n}(x_{n})\rangle$ are not gauge invariant?

  2. Why do diffeomorphisms that reach infinity become global symmetries?

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In gravity theories gauge invariance is diffeomorphism invariance, so local operators are not gauge invariant because there are no preferred coordinates, you can always reparametrize them - choose another coordinates, and the operators will, in general, change.

As for 2, its just the way of saying that transformation that is applied to all the space (up to infinity) is a global transformation.

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