Spin-Foams for All Loop Quantum Gravity

TL;DR

This paper extends the spin-foam models in loop quantum gravity to a more general, diffeomorphism-invariant framework that includes all spin-network states and allows for knotting and linking histories, broadening the scope of the theory.

Contribution

It generalizes the simplicial spin-foam models to arbitrary linear 2-cell spin-foams, encompassing all spin-network states and knotting/linking histories in loop quantum gravity.

Findings

Framework admits all LQG spin-network states.

Establishes a 1-1 correspondence between SU(2) and EPRL intertwiners for certain parameters.

Generalizes vertex structure and amplitudes to arbitrary valency.

Abstract

The simplicial framework of Engle-Pereira-Rovelli-Livine spin-foam models is generalized to match the diffeomorphism invariant framework of loop quantum gravity. The simplicial spin-foams are generalized to arbitrary linear 2-cell spin-foams. The resulting framework admits all the spin-network states of loop quantum gravity, not only those defined by triangulations (or cubulations). In particular the notion of embedded spin-foam we use allows to consider knotting or linking spin-foam histories. Also the main tools as the vertex structure and the vertex amplitude are naturally generalized to arbitrary valency case. The correspondence between all the SU(2) intertwiners and the SU(2)$\times$SU(2) EPRL intertwiners is proved to be 1-1 in the case of the Barbero-Immirzi parameter $|\gamma|\ge 1$, unless the co-domain of the EPRL map is trivial and the domain is non-trivial.