Linking covariant and canonical LQG II: Spin foam projector

TL;DR

This paper investigates the connection between covariant and canonical Loop Quantum Gravity by analyzing spin foam models and their potential to define rigging maps, revealing limitations and proposing modifications for proper projection.

Contribution

It extends the definition of spin foam models to arbitrary boundary graphs and analyzes their role as rigging maps in LQG, identifying the need for a modified transfer matrix.

Findings

Elementary spin foam transfer matrix generates finite foams

The transfer matrix does not define a projector as is

A modified transfer matrix could serve as a rigging map

Abstract

In a seminal paper, Kaminski, Kisielowski an Lewandowski for the first time extended the definition of spin foam models to arbitrary boundary graphs. This is a prerequisite in order to make contact to the canonical formulation of Loop Quantum Gravity (LQG) whose Hilbert space contains all these graphs. This makes it finally possible to investigate the question whether any of the presently considered spin foam models yields a rigging map for any of the presently defined Hamiltonian constraint operators. In the analysis of this would-be spin foam rigging map we are able to identify an elementary spin foam transfer matrix that allows to generate any finite foam as a finite power of the transfer matrix. However, it transpires that the resulting object, as written, does not define a projector on the physical Hilbert space. This statement is independent of the concrete spin foam model and Hamiltonian constraint. Nevertehless, the transfer matrix potentially contains the necessary ingredient in order to construct a proper rigging map in terms of a modified transfer matrix.