Spin Foams and Canonical Quantization

TL;DR

The work analyzes the mutual consistency of spin-foam and canonical loop-quantization approaches to quantum gravity across 3d and 4d. It demonstrates a complete, well-understood correspondence in 3d between Witten–Chern–Simons quantization (with Λ>0) and Turaev–Viro spin foams, and between Ponzano–Regge spin foams and the canonical LQG scalar product when Λ=0, including extensions with particles. In 4d, it reviews Lorentz-covariant boundary states via projected spin networks, compares central spin-foam proposals (BC, EPRL, FK) to canonical structures, and discusses how consistency with the canonical framework imposes constraints on the implementation of simplicity constraints, measures, and vertex amplitudes; it also presents a new 3d toy-model informing the role of secondary second-class constraints in obtaining correct dynamics. The analysis highlights where covariant and canonical quantizations align and where they diverge, offering concrete routes (e.g., incorporating secondary constraints into the path-integral measure) to achieve closer agreement and testable predictions. Overall, the paper emphasizes that a careful synthesis of path-integral and canonical perspectives—especially via refined constraint handling and boundary-state descriptions—is essential to progress toward a consistent, four-dimensional quantum theory of gravity.

Abstract

This review is devoted to the analysis of the mutual consistency of the spin foam and canonical loop quantizations in three and four spacetime dimensions. In the three-dimensional context, where the two approaches are in good agreement, we show how the canonical quantization à la Witten of Riemannian gravity with a positive cosmological constant is related to the Turaev-Viro spin foam model, and how the Ponzano-Regge amplitudes are related to the physical scalar product of Riemannian loop quantum gravity without cosmological constant. In the four-dimensional case, we recall a Lorentz-covariant formulation of loop quantum gravity using projected spin networks, compare it with the new spin foam models, and identify interesting relations and their pitfalls. Finally, we discuss the properties which a spin foam model is expected to possess in order to be consistent with the canonical quantization, and suggest a new model illustrating these results.

Figures (6)

• Figure 1: A spin foam defined on a two-complex interpolating between two boundary states ${\cal S}_1$ and ${\cal S}_2$. A section defines an intermediate state in the middle of the evolution.
• Figure 2: Decomposition of the manifold $\cal M$ into two pieces ${\cal M}_1$ and ${\cal M}_2$ on the left. On the right, $\overline{{\cal M}_i}$ is obtained as the connected sum of ${\cal M}_i$ with a three-ball.
• Figure 3: Examples of knots in $\mathbb{S}^3$. They coincide in the "southern hemisphere" of $\mathbb{S}^3$ but differ in the "northern hemisphere".
• Figure 4: The three "crossings" appearing in the definition of the skein relation.
• Figure 5: Graphical representation of the physical states on the sphere with four punctures labeled with $j=1/2$. On the top, we have given two different basis. The one on the left is described in terms of intertwiners ($j\in \{0,1\}$), while the one on the right has two different vectors (one of them being defined using the quantum $R$-matrix). In the middle, we have represented the quantum trace. At the bottom, we have given some properties related to the framing. The links have to be understood as ribbons, and therefore the twist has a non-trivial effect on the evaluation of the quantum scalar product. The phase is given by $\phi=-q^{-3/2}$.