Canonical ``Loop'' Quantum Gravity and Spin Foam Models

TL;DR

This paper surveys how spin foam models provide a natural four-dimensional, background-independent extension of loop quantum gravity by interpreting the exponential of the Hamiltonian constraint as a sum over histories (spin foams) that parallels BF-type path integrals. It articulates the canonical structure of loop quantum gravity, including the auxiliary and diffeomorphism-invariant Hilbert spaces and the Thiemann formulation of the Hamiltonian constraint, and then presents a 3D prototype through the Ponzano-Regge-Turaev-Viro model, with a detailed account of spin-foam decorations and boundary functoriality. It then reviews four-dimensional topological BF models (Crane-Yetter-Ooguri) and gravity-like constrained models (Barrett-Crane), tying them to the Plebanski action and highlighting how gravity may emerge as a constrained BF theory. The paper concludes by identifying key open issues—triangulation dependence, operator choices for the constraints, and the precise relation between constraint quantization and spin-foam amplitudes—pointing to ongoing work to achieve a fully consistent spin-foam formulation of quantum gravity.

Abstract

The canonical ``loop'' formulation of quantum gravity is a mathematically well defined, background independent, non perturbative standard quantization of Einstein's theory of General Relativity. Some among the most meaningful results of the theory are: 1) the complete calculation of the spectrum of geometric quantities like the area and the volume and the consequent physical predictions about the structure of the space-time at the Plank scale; 2) a microscopical derivation of the Bekenstein-Hawking black-hole entropy formula. Unfortunately, despite recent results, the dynamical aspect of the theory (imposition of the Wheller-De Witt constraint) remains elusive. After a short description of the basic ideas and the main results of loop quantum gravity we show in which sence the exponential of the super Hamiltonian constraint leads to the concept of spin foam and to a four dimensional formulation of the theory. Moreover, we show that some topological field theories as the BF theory in 3 and 4 dimension admits a spin foam formulation. We argue that the spin-foams/spin-networks formalism it is the natural framework to discuss loop quantum gravity and topological field theory.

Figures (5)

• Figure 1: The graph $\gamma$, a possible extended-planar projection $\Gamma_{ex}$ and the graphical representation of a spin-network cylindrical function
• Figure 2: A potential first order contribution to the partition function $Z_{RR}[s',s;\sigma;1]$ of eq. (3) and the corresponding spin-network transition. A three dimensional situation is reported. From the point of view of a 4 dimensional theory this contribution correspond to a branched polyhedron dual to a singular triangulation.
• Figure 3: The local description of the standard polyhedron dual to a triangulation and a schematic representation of the duality. Note that in 3 dimension, a coloration of the edges of the triangulation determine a unique coloration of the faces of its dual 2-skeleton, and vice versa.
• Figure 4: A possible decomposition of an orientable Manifold in three components. This drawing has only a schematic purpose since we suppose that $\partial M_3 = - \Sigma_2$, $\partial M_1 = \Sigma_1$ and $\partial M_2 = \Sigma_2 \cup - \Sigma_1$.
• Figure 5: The elementary vertex of the BF theory. There are 5 nodes (with the associated intertwiner) and 10 links (with the associated representation). To the elementary vertex are indeed associated 15 colors.