On the regularization of the constraints algebra of Quantum Gravity in 2+1 dimensions with non-vanishing cosmological constant

TL;DR

This paper investigates the quantization of 3D Riemannian gravity with positive cosmological constant using loop quantum gravity, revealing that regularization induces a deformation in the classical constraint algebra due to non-local connections.

Contribution

It demonstrates that standard regularization methods cause algebra deformation in 3D quantum gravity with Lambda>0, highlighting the impact of non-local connections on constraint algebra.

Findings

Regularization deforms the classical constraint algebra.

Deformation is proportional to curvature squared.

Non-locality of connections causes unavoidable anomalies.

Abstract

We use the mathematical framework of loop quantum gravity (LQG) to study the quantization of three dimensional (Riemannian) gravity with positive cosmological constant (Lambda>0). We show that the usual regularization techniques (successful in the Lambda=0 case and widely applied in four dimensional LQG) lead to a deformation of the classical constraint algebra (or anomaly) proportional to the local strength of the curvature squared. We argue that this is an unavoidable consequence of the non-local nature of generalized connections.