The effective dynamics of weak coupling loop quantum gravity

TL;DR

This paper explores the weak coupling limit of loop quantum gravity, showing that it simplifies to a $U(1)^3$ gauge theory and that the effective dynamics in this limit are consistent with the original $SU(2)$ formulation.

Contribution

It introduces a parametrization of $SU(2)$ variables by $U(1)^3$ variables and derives the Hamiltonian operator for weak coupling loop quantum gravity.

Findings

Effective dynamics from $U(1)^3$ and $SU(2)$ formulations are consistent in the weak coupling limit.

The Hamiltonian operators' expectation values match classical expressions in coherent states.

The $U(1)^3$ formulation simplifies analysis of weak coupling loop quantum gravity.

Abstract

By taking the limit that Newton's Gravitational constant tends to zero, the weak coupling loop quantum gravity can be formulated as a $U(1)^3$ gauge theory instead of the original $SU(2)$ gauge theory. In this paper, a parametrization of the $SU(2)$ holonomy-flux variables by the $U(1)^3$ holonomy-flux variables is introduced, and the Hamiltonian operator based on this parametrization is obtained for the weak coupling loop quantum gravity. It is shown that the effective dynamics obtained from the coherent state path integrals in $U(1)^3$ and $SU(2)$ loop quantum gravity respectively are consistent to each other in the weak coupling limit, provided that the expectation values of the Hamiltonian operators on the coherent states in these two theories coincide with their classical expressions respectively.