Spin-Foam Models and the Physical Scalar Product

TL;DR

This paper establishes a connection between Loop Quantum Gravity and Spin-Foam models by constructing an operator that links the physical scalar product to vertex amplitudes, clarifying their relationship in four dimensions.

Contribution

It constructs an operator on cylindrical functions whose matrix elements correspond to Spin-Foam vertex amplitudes, linking the physical scalar product to Spin-Foam models in four dimensions.

Findings

The operator P acts as a projector into physical states.

Matrix elements of P relate to Spin-Foam vertex amplitudes.

The approach applies to several Spin-Foam models, including Barrett-Crane and EPRL.

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($\Gamma$), where $\Gamma$ 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.