Spin-Foam Models and the Physical Scalar Product

TL;DR

This work constructs a bridge between canonical Loop Quantum Gravity and covariant Spin-Foam models in four dimensions by expressing Spin-Foam vertex amplitudes as physical scalar products of kinematical spin-network states. It provides a universal integral representation for the vertex and then builds explicit projector-like operators P for BF, Barrett–Crane, EPR, and Freidel–Krasnov models, showing that their vertex amplitudes arise as matrix elements of P between boundary spin-network states. The approach clarifies how covariant amplitudes encode the canonical constraints and highlights both the potential and limits of interpreting Spin-Foam vertices as physical inner products. The results illuminate a concrete, model-by-model path to relate the covariant and canonical quantizations of Euclidean gravity and suggest directions for extending the construction to more general spin networks and to GNS-type formulations.

Abstract

This paper aims at clarifying the link between Loop Quantum Gravity and Spin-Foam models in four dimensions. Starting from the canonical framework, we construct an operator P acting on the space of cylindrical functions Cyl($Γ$), where $Γ$ is the 4-simplex graph, such that its ma- trix elements are, up to some normalization factors, the vertex amplitude of Spin-Foam models. The Spin-Foam models we are considering are the topological model, the Barrett-Crane model and the Engle-Pereira-Rovelli model. The operator P is usually called the "projector" into physical states and its matrix elements gives the physical scalar product. Therefore, we relate the physical scalar product of Loop Quantum Gravity to vertex amplitudes of some Spin-Foam models. We discuss the possibility to extend the action of P to any cylindrical functions on the space manifold.

Figures (10)

• Figure 1: Illustration of the 1-tetrahedron state $\tau_1$ on the left and the 4-tetrahedron state $\tau_4$ on the right. Vertices, labelled by $i\in\{0,5\}$, are colored with intertwiners $\omega_i$ and edges $\ell_{ij}$ with representations $I_{ij}$. The 4 free ends are colored with representations $I_{1i}$.
• Figure 2: This picture is a graphical representation of the integrand in the formula (\ref{['Vema']}) defining the vertex amplitude. Each line are doubled because it carries a representation of $SU(2)\times SU(2)$ and the single lines in the pair colored with $(I,J)$ are colored by $I$ and $J$ separately. Furthermore, the single lines are endowed with bullets that represent the insertion of $SU(2)$ group elements: the small ones are associated to diagonal elements $u_i \in SU(2)$ whereas the big ones are associated to spherical elements $x_ix_j^{-1}\in S^3$. The vectors $v_i$ are represented by boxes and they are contracted with the free ends of the graph.
• Figure 3: Structure of the node $i=1$. Four pairs of edges are attached at each node of the graph: each edge are colored with a $SU(2)$ representation. The bullets illustrate the inclusions of $SU(2)$ variables $u_i$ or $S^3$ variables $x_ix_j^{-1}$. Notice that, in the $SU(2)\times SU(2)$ formulation, each pair of lines is associated to the element $(g_L,g_R)$, $g_L$ corresponding to the left line and $g_R$ to the right one.
• Figure 4: The three canonical basis of the space of 4-valent intertwiners. The intermediate channel is endowed with the representation $\alpha$.
• Figure 5: Pictorial representation of a 15j symbol: vertices are labelled by representations $\omega_i$ and edges by representations $I_{ij}$.