In quantum gravity, summing is refining
Carlo Rovelli, Matteo Smerlak
Abstract
In perturbative QED, the approximation is improved by summing more Feynman graphs; in non-perturbative QCD, by refining the lattice. Here we observe that in quantum gravity the two procedures may well be the same. We outline the combinatorial structure of spinfoam quantum gravity, define the continuum limit, and show that under general conditions refining foams is the same as summing over them. The conditions bear on the cylindrical consistency of the spinfoam amplitudes and on the presence of appropriate combinatorial factors, related to the implementation of diffeomorphisms invariance. Intuitively, the sites of the lattice are points of space: these are themselves quanta of the gravitational field, and thus a lattice discretization is also a Feynman history of quanta.