Coupling of spacetime atoms and spin foam renormalisation from group field theory

TL;DR

This work targets the coupling of spacetime atoms (4-simplices) in spin foam models by embedding an extra normal variable into a five-argument group field theory, thereby enabling cross-simplice coupling that controls locality. It introduces a generalized Barrett-Crane model with a tunable coupling implemented via a heat-kernel function ${ m K}^{eta,L}(G)$, allowing interpolation between the standard Barrett-Crane amplitudes and a strongly coupled flat (BF-like) regime as $eta$ varies. A further advancement promotes the coupling parameter to a dynamical variable $eta$ in a new type of GFT, yielding a diffusion-type kinetic term and derivative interactions, which reframes renormalisation-group ideas as classical equations of motion within the theory. The paper outlines a program to study renormalisation, fixed points, and coarse-grained behavior—potentially clarifying the emergence of semiclassical physics and the role of diffeomorphism-like Ward identities in background-independent spin foam models, while also accommodating particle insertions and numerical simulations.

Abstract

We study the issue of coupling among 4-simplices in the context of spin foam models obtained from a group field theory formalism. We construct a generalisation of the Barrett-Crane model in which an additional coupling between the normals to tetrahedra, as defined in different 4-simplices that share them, is present. This is realised through an extension of the usual field over the group manifold to a five argument one. We define a specific model in which this coupling is parametrised by an additional real parameter that allows to tune the degree of locality of the resulting model, interpolating between the usual Barrett-Crane model and a flat BF-type one. Moreover, we define a further extension of the group field theory formalism in which the coupling parameter enters as a new variable of the field, and the action presents derivative terms that lead to modified classical equations of motion. Finally, we discuss the issue of renormalisation of spin foam models, and how the new coupled model can be of help regarding this.