The weak coupling theory of all dimensional loop quantum gravity

TL;DR

This paper develops a weak coupling version of all-dimensional loop quantum gravity by mapping the $SO(D+1)$ phase space to an Abelian $U(1)^{D(D+1)/2}$ phase space, enabling analysis of weak coupling properties.

Contribution

It establishes a symplectic-morphism between $SO(D+1)$ and Abelian $U(1)^{D(D+1)/2}$ phase spaces, generalizing key operators and states for weak coupling analysis in all-dimensional LQG.

Findings

Generalized Gaussian, simplicity, diffeomorphism, and scalar constraints to Abelian $U(1)^{D(D+1)/2}$ LQG.

Constructed heat-kernel coherent states peaked at weak coupling regions.

Provided a new perspective for studying weak coupling properties of all-dimensional LQG.

Abstract

The weak coupling loop quantum theory with Abelian gauge group provides us a new perspective to study the weak coupling properties of LQG. In this paper, the weak coupling theory of all dimensional loop quantum gravity is established based on a symplectic-morphism between the $SO(D+1)$ holonomy-flux phase space and the $U(1)^{\frac{D(D+1)}{2}}$ holonomy-flux phase space. More explicitly, the Gaussian, simplicity, diffeomorphism and scalar constraint operators in $SO(D+1)$ loop quantum gravity will be generalized to the $U(1)^{\frac{D(D+1)}{2}}$ loop quantum theory based on the symplectic-morphism, and the $U(1)^{\frac{D(D+1)}{2}}$ loop quantum theory equipped with these constraint operators gives the weak coupling $U(1)^{\frac{D(D+1)}{2}}$ loop quantum gravity, with the corresponding Hilbert space is composed by the $U(1)^{\frac{D(D+1)}{2}}$ heat-kernel coherent states which are peaked at the weak coupling region of the $U(1)^{\frac{D(D+1)}{2}}$ holonomy-flux phase space.