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

TL;DR

This paper analyzes two recently proposed spin-foam models (the flipped model and the FKLS model) that modify Barrett-Crane constraints via a reformulated cross-simplicity condition. By examining both the classical discretization and quantum kinematics, and by applying coherent-state techniques alongside master-constraint reasoning, it shows that the flipped model reproduces the loop quantum gravity boundary Hilbert space and area spectrum, while the FKLS model leaves the entire $SO(4)$ intertwiner space unconstrained, yielding a boundary space and area spectra that differ from LQG. The work clarifies how the constraint class (first vs second) affects the effectiveness of coherent-state imposition and highlights that, in FKLS, cross-simplicity influences dynamics rather than states. The results underscore that only the flipped model aligns with LQG at the kinematical and spectral levels, while FKLS (and BC) diverge in key structural aspects, with implications for semiclassical limits and future Lorentzian extensions.

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.