# Diffeomorphism invariance and correlation functions

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?