What is the meaning of diffeomorphism invariance? I have read a bunch of papers and I see "diffeomorphism invariance" and I cannot understand how it works.
For instance, in asymptotic safe quantum gravity, we make 2 assumptions:

*

*Diffeomorphism invariance


*The metric carries the fundamental symmetry of space-time
 A: Typically, although not always, quantum gravity is considered with a Wick rotated action. Imagine a general Euclidean Einstein - Hilbert action (just the simplest (S= R - Lambda), it enters the partition function. Depending on whether you have a continous (integral) or a discrete (sum) model the form of the path integral changes. In the continous form, you integrate over a measure term (D[g(x)] ), in the discrete form you you have some parameter ( let's say T) that you some over, and then a normalisation factor dependent! on the measure term also enters the partition function (let's call it 1/C[T] )
In both cases, you count all spacetime points and all possible configurations of the metrics. In the integral form this measure term is shown typically by big D (instead of the typical small d, signaling the differential/ infinitesimal step in calculus). In the discrete case 1/C[T] is the size of the automorphism group of the given configuration (T).
Why is it important? Well, actually physics lies on this measure term, this is what is truly important. The action is just a parameterization, which is a guess, placed there by hand. We parametrize the model with some parameters/couplings (guess and hope its right), then integrate over all "diffeomorphism invariant configurations", because several "seemingly different" configurations are actually related  to the same physical situation, and not normalising properly would ruin the idea behind the path integral formalism.
In the Euclidean models these factors are just Boltzmann weights, but they are  extremely important, as true physics lies under them.
