Revisiting the Simplicity Constraints and Coherent Intertwiners

TL;DR

This paper explores the application of the U(N) framework to impose simplicity constraints on intertwiners in loop quantum gravity, offering a new algebraic approach that accommodates the Immirzi parameter.

Contribution

It introduces a novel U(N) algebraic formulation for simplicity constraints and proposes coherent states that satisfy these constraints for arbitrary Immirzi parameters.

Findings

The simplicity constraints form a closed u(N) algebra.

U(N) coherent states can weakly solve all simplicity constraints.

The approach is applicable to the 4d Euclidean theory.

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.