Generalized Spinfoams

TL;DR

This paper extends spinfoam models to a generalized framework using non-triangulated complexes, incorporating a new quantum number and analyzing both Euclidean and Lorentzian cases, with implications for quantum geometry.

Contribution

It introduces a generalized spinfoam model based on arbitrary 2-cell complexes, including a new quantum number and analyzing its implications for quantum geometry.

Findings

Recovered KKL amplitude for r_f=0 in Euclidean case

Boundary states admit polyhedral geometric interpretation

Introduced a new quantum number r_f affecting solutions

Abstract

We reconsider the spinfoam dynamics that has been recently introduced, in the generalized Kaminski-Kisielowski-Lewandowski (KKL) version where the foam is not dual to a triangulation. We study the Euclidean as well as the Lorentzian case. We show that this theory can still be obtained as a constrained BF theory satisfying the simplicity constraint, now discretized on a general oriented 2-cell complex. This constraint implies that boundary states admit a (quantum) geometrical interpretation in terms of polyhedra, generalizing the tetrahedral geometry of the simplicial case. We also point out that the general solution to this constraint (imposed weakly) depends on a quantum number r_f in addition to those of loop quantum gravity. We compute the vertex amplitude and recover the KKL amplitude in the Euclidean theory when r_f=0. We comment on the eventual physical relevance of r_f, and the formal way to eliminate it.