Spin-Foams for All Loop Quantum Gravity

TL;DR

This work develops a generalized, diffeomorphism-invariant spin-foam framework that extends beyond simplicial discretizations to arbitrary linear 2-cell complexes, enabling embedded, knotted histories for all loop quantum gravity spin networks. It defines a consistent spin-foam trace and boundary-encoded vertex structure, and constructs simple intertwiners (BC and EPRL) within this scheme, proving an injectivity result for $|\gamma|\ge 1$. The approach unifies LQG with spin-foam histories without privileging triangulations, preserves knotting degrees of freedom, and provides a clear amplitude prescription $A(\kappa,\rho,\iota) = \Big(\prod_f \mathrm{dim}(\rho_f)\Big) \mathrm{Tr}(\kappa,\rho,\iota)$. It also clarifies the behavior of the theory under key limits of the Barbero–Immirzi parameter, connecting BC, EPRL, and BF-type regimes and ensuring a robust path to semiclassical analysis within the diffeomorphism-invariant setting.

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 $|γ|\ge 1$, unless the co-domain of the EPRL map is trivial and the domain is non-trivial.

Figures (8)

• Figure 1: A spin-network.
• Figure 2: a) A history of a spin-network. b) The initial, and, respectively, final spin-network.
• Figure 3: Cells of the complex.
• Figure 4: a) Faces are coloured by irreducible representations of $G$. b), c) Edges are coloured with invariants. It is convenient to assign $\iota_e^\dagger$ to the beginning of $e$ and $\iota_e$ to the end of $e$.
• Figure 5: a) Spin-foam with boundary (in the bottom). b) Induced spin-network on the boundary.