Simplicity constraints: a 3d toy-model for Loop Quantum Gravity

TL;DR

This paper introduces a 3D toy model with simplicity constraints to explore quantum imposition of these constraints, aiming to inform the development of covariant Loop Quantum Gravity.

Contribution

It presents a simplified 3D model with similar constraints, demonstrating a quantum theory via gauge-unfixing that aligns with standard 3D Euclidean quantum gravity.

Findings

Classical 3D model with simplicity constraints defined.

Quantum theory constructed through gauge-unfixing.

Model as a potential step towards covariant Loop Quantum Gravity.

Abstract

In Loop Quantum Gravity, tremendous progress has been made using the Ashtekar-Barbero variables. These variables, defined in a gauge-fixing of the theory, correspond to a parametrization of the solutions of the so-called simplicity constraints. Their geometrical interpretation is however unsatisfactory as they do not constitute a space-time connection. It would be possible to resolve this point by using a full Lorentz connection or, equivalently, by using the self-dual Ashtekar variables. This leads however to simplicity constraints or reality conditions which are notoriously difficult to implement in the quantum theory. We explore in this paper the possibility of imposing such constraints at the quantum level in the context of canonical quantization. To do so, we define a simpler model, in 3d, with similar constraints by extending the phase space to include an independent vielbein. We define the classical model and show that a precise quantum theory by gauge-unfixing can be defined out of it, completely equivalent to the standard 3d euclidean quantum gravity. We discuss possible future explorations around this model as it could help as a stepping stone to define full-fledged covariant Loop Quantum Gravity.