Coarse graining in spin foam models
Fotini Markopoulou
TL;DR
This work reframes the low-energy limit of spin foam models as a coarse-graining problem and proposes a Hopf-algebra-based renormalization framework to handle the unique challenges of spin foams, including background independence and sums over irregular lattices. By introducing partitioned spin foams and their weights, it builds a Kreimer-style Hopf algebra that encodes local block transformations via coproducts and an antipode, with an exact shrinking antipode $S_R^E$ implementing exact local coarse-graining. The key advance is embedding the renormalization group into the Hopf algebra, showing that standard RG flow can be recovered as a homogeneous sector of $S_R^E$, and outlining how approximate block transformations can be systematically incorporated through a generalized $S_R$. This approach provides a principled, scalable way to perform nonperturbative coarse-graining of spin foams and paves the way for analyzing fixed points and infrared behavior in quantum gravity models.
Abstract
We formulate the problem of finding the low-energy limit of spin foam models as a coarse-graining problem in the sense of statistical physics. This suggests that renormalization group methods may be used to find that limit. However, since spin foams are models of spacetime at Planck scale, novel issues arise: these microscopic models are sums over irregular, background-independent lattices. We show that all of these issues can be addressed by the recent application of the Kreimer Hopf algebra for quantum field theory renormalization to non-perturbative statistical physics. The main difference from standard renormalization group is that the Hopf algebra executes block transformations in parts of the lattice only but in a controlled manner so that the end result is a fully block-transformed lattice.