New Variables for Classical and Quantum Gravity in all Dimensions III. Quantum Theory

TL;DR

This paper extends Loop Quantum Gravity quantization techniques to a new connection formulation of higher-dimensional General Relativity, addressing challenges in implementing the simplicity constraint.

Contribution

It generalizes LQG methods to higher dimensions and develops a novel approach to handle the off-diagonal simplicity constraints.

Findings

Successful quantization of the new connection formulation in all dimensions.

Generalization of LQG tools to higher-dimensional gravity.

Resolution strategies for the off-diagonal simplicity constraints.

Abstract

We quantise the new connection formulation of D+1 dimensional General Relativity developed in our companion papers by Loop Quantum Gravity (LQG) methods. It turns out that all the tools prepared for LQG straightforwardly generalise to the new connection formulation in higher dimensions. The only new challenge is the simplicity constraint. While its "diagonal" components acting at edges of spin network functions are easily solved, its "off-diagonal" components acting at vertices are non trivial and require a more elaborate treatment.