Polytopes in all dimensional loop quantum gravity

TL;DR

This paper extends Lasserre's reconstruction algorithm to D-polytopes in all dimensions, enabling new geometric operators in loop quantum gravity that have expected semiclassical properties and consistent limits.

Contribution

It introduces a generalized shape space for D-polytopes and constructs new geometric operators using coherent intertwiners in all-dimensional loop quantum gravity.

Findings

Expressed areas of d-skeletons as functions of (D-1)-face data.

Proposed new geometric operators with semiclassical properties.

Achieved consistent semiclassical limits for the D-volume operator.

Abstract

The Lasserre's reconstruction algorithm is extended to the D-polytopes with the construction of their shape space. Thus, the areas of d-skeletons $(1\leq d\leq D)$ can be expressed as functions of the areas and normal bi-vectors of the (D-1)-faces of D-polytopes. As weak solutions of the simplicity constraints in all dimensional loop quantum gravity, the simple coherent intertwiners are employed to describe semiclassical D-polytopes. New general geometric operators based on D-polytopes are proposed by using the Lasserre's reconstruction algorithm and the coherent intertwiners. Such kind of geometric operators have expected semiclassical property by the definition. The consistent semiclassical limit with respect to the semiclassical D-polytopes can be obtained for the usual D-volume operator in all dimensional loop quantum gravity by fixing its undetermined regularization factor case by case.