Coherent states, constraint classes, and area operators in the new spin-foam models

TL;DR

This paper analyzes two new spin-foam models, FKLS and flipped, focusing on their constraint structures and area spectra, revealing that the flipped model aligns with loop quantum gravity while FKLS differs.

Contribution

It clarifies the reformulated cross-simplicity constraints and their quantum imposition, showing FKLS's constraints are effectively first class, and compares area spectra across models.

Findings

FKLS model's constraints imply no restriction on states.

In the flipped model, the Hilbert space and area spectra match loop quantum gravity.

FKLS and Barrett-Crane models have different boundary Hilbert spaces and spectra.

Abstract

Recently, two new spin-foam models have appeared in the literature, both motivated by a desire to modify the Barrett-Crane model in such a way that the imposition of certain second class constraints, called cross-simplicity constraints, are weakened. We refer to these two models as the FKLS model, and the flipped model. Both of these models are based on a reformulation of the cross-simplicity constraints. This paper has two main parts. First, we clarify the structure of the reformulated cross-simplicity constraints and the nature of their quantum imposition in the new models. In particular we show that in the FKLS model, quantum cross-simplicity implies no restriction on states. The deeper reason for this is that, with the symplectic structure relevant for FKLS, the reformulated cross-simplicity constraints, in a certain relevant sense, are now \emph{first class}, and this causes the coherent state method of imposing the constraints, key in the FKLS model, to fail to give any restriction on states. Nevertheless, the cross-simplicity can still be seen as implemented via suppression of intertwiner degrees of freedom in the dynamical propagation. In the second part of the paper, we investigate area spectra in the models. The results of these two investigations will highlight how, in the flipped model, the Hilbert space of states, as well as the spectra of area operators exactly match those of loop quantum gravity, whereas in the FKLS (and Barrett-Crane) models, the boundary Hilbert spaces and area spectra are different.