In quantum gravity, summing is refining

TL;DR

This paper explores the idea that in quantum gravity, refining the discretization of spacetime (foams) is mathematically equivalent to summing over all possible configurations, unifying two approaches used in other quantum field theories.

Contribution

It demonstrates that in quantum gravity, the processes of refining spinfoam discretizations and summing over configurations are equivalent under certain conditions, linking lattice refinement and path integral approaches.

Findings

Refining foams is equivalent to summing over them in quantum gravity.

Conditions involve cylindrical consistency and combinatorial factors related to diffeomorphism invariance.

Provides a unified perspective on discretization and summation in quantum gravity.

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.