Exploring the Tomlin-Varadarajan quantum constraints in $U(1)^3$ loop quantum gravity: solutions and the Minkowski theorem

TL;DR

This paper explicitly solves anomaly-free quantum constraints in a loop quantum gravity model, revealing solutions related to Minkowski's theorem and diffeomorphism invariance, advancing understanding of quantum geometric states.

Contribution

It provides explicit solutions to Tomlin-Varadarajan constraints in a $U(1)^3$ loop quantum gravity model, connecting quantum states to classical polyhedral geometry.

Findings

Identified the subspace where constraints act convergently.

Found the complete set of solutions within this subspace.

Discovered solutions related to Minkowski's theorem and diffeomorphism invariance.

Abstract

We explicitly solved the anomaly-free quantum constraints proposed by Tomlin and Varadarajan for the weak Euclidean model of canonical loop quantum gravity, in a large subspace of the model's kinematic Hilbert space which is the space of the charge network states. In doing so, we first identified the subspace on which each of the constraints acts convergingly, and then by explicitly evaluating such actions we found the complete set of the solutions in the identified subspace. We showed that the space of solutions consists of two classes of states, with the first class having a property that involves the condition known from the Minkowski theorem on polyhedra, and the second class satisfying a weaker form of the spatial diffeomorphism invariance.