Revisiting the Simplicity Constraints and Coherent Intertwiners

TL;DR

The paper addresses how to implement the discrete simplicity constraints of Spin(4) in Euclidean spinfoam models without over-constraining the local geometry.It introduces a closed $\mathfrak{u}(N)$-algebra of observables within the $\mathrm{U}(N)$ framework to reformulate both diagonal and cross simplicity constraints and to construct coherent states that solve them weakly in the large-spin regime.A sequence of constraint sets—diagonal, cross, holomorphic, and $F$- or $F^\dagger$-based—are explored, culminating in a unified approach that treats all simplicity constraints on an equal footing via $\mathrm{U}(N)$ coherent states, for arbitrary Immirzi parameter $\gamma$.This framework provides a promising path to more flexible and semi-classical boundary states for spinfoam amplitudes and motivates future work on Lorentzian generalization, gluing to general triangulations, and probing extrinsic geometry in twisted geometries.

Abstract

In the context of loop quantum gravity and spinfoam models, the simplicity constraints are essential in that they allow to write general relativity as a constrained topological BF theory. In this work, we apply the recently developed U(N) framework for SU(2) intertwiners to the issue of imposing the simplicity constraints to spin network states. More particularly, we focus on solving them on individual intertwiners in the 4d Euclidean theory. We review the standard way of solving the simplicity constraints using coherent intertwiners and we explain how these fit within the U(N) framework. Then we show how these constraints can be written as a closed u(N) algebra and we propose a set of U(N) coherent states that solves all the simplicity constraints weakly for an arbitrary Immirzi parameter.