A Diffeomorphism invariance and gauge invariance

[spaghetti3451]

Consider the following paragraph taken from page 15 of Thomas Hartman's lecture notes (http://www.hartmanhep.net/topics2015/) 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.''

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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?

 

[haushofer]

To answer your first question: diff.invariance enforces the correlators to be constant, see e.g. Zee's Einstein Gravity... book.

 

[haushofer]

For your second question: have you read page 90?