The Spin Foam Approach to Quantum Gravity

TL;DR

The paper surveys the spin foam program as a covariant, background‑independent path integral realization of loop quantum gravity, focusing on the progression from BF theory to gravity via linear simplicity constraints. It develops both Riemannian and Lorentzian EPRL–FK models, analyzes their amplitudes, and examines their semiclassical limits in relation to Regge gravity, while also addressing 3D gravity as a tractable testbed. Key issues discussed include the measure, discreteness and discretization independence, coupling to matter, cosmological constant, and potential group field theory formulations, along with conceptual questions about the interpretation of spin foams as gauge histories rather than spacetime geometries. The review highlights how the new models aim to reconcile canonical LQG with a meaningful covariant dynamics, and it outlines the ongoing challenges in achieving a fully consistent continuum limit and a robust semiclassical regime. Overall, the work underscores both the promise and the technical obstacles in deriving a predictive quantum theory of gravity from spin foams.

Abstract

This article reviews the present status of the spin foam approach to the quantization of gravity. Special attention is payed to the pedagogical presentation of the recently introduced new models for four dimensional quantum gravity. The models are motivated by a suitable implementation of the path integral quantization of the Plebanski formulation of gravity on a simplicial regularization. The article also includes a self-contained treatment of the 2+1 gravity. The simple nature of the latter provides the basis and a perspective for the analysis of both conceptual and technical issues that remain open in four dimensions.

Figures (18)

• Figure 1: The larger cone represents the light-cone at a point according to the ad hoc background $\eta_{ab}$. The smaller cones are a cartoon representation of the fluctuations of the true gravitational field represented by $g_{ab}$. Gravitons respect the causal structure of the un physical metric to all orders in perturbation theory.
• Figure 2: On the left: the geometry of phase space in gauge theories. On the right: the quantization path of LQG (continuous arrows).
• Figure 3: Spin-network state: At 3-valent nodes the intertwiner is uniquely specified by the corresponding spins. At 4 or higher valent nodes an intertwiner has to be specified. Choosing an intertwiner corresponds to decompose the $n$-valent node in terms of 3-valent ones adding new virtual links (dashed lines) and their corresponding spins. This is illustrated explicitly in the figure for the two 4-valent nodes.
• Figure 4: A typical transition generated by the action of the scalar constraint
• Figure 5: A typical path in a path integral version of loop quantum gravity is given by a series of transitions through different spin-network states representing a state of 3-geometries. Nodes and links in the spin network evolve into 1-dimensional edges and faces. New links are created and spins are reassigned at vertexes (emphasized on the right). The 'topological' structure is provided by the underlying 2-complex while the geometric degrees of freedom are encoded in the labeling of its elements with irreducible representations and intertwiners.