U(N) invariant dynamics for a simplified Loop Quantum Gravity model

TL;DR

This paper proposes a new U(N)-based Hamiltonian dynamics for a simplified Loop Quantum Gravity model with two vertices, aiming to better understand homogeneous and isotropic states and their connection to Loop Quantum Cosmology.

Contribution

It introduces a U(N)-invariant Hamiltonian operator for a simplified LQG model, linking boundary deformations to cosmological states.

Findings

Constructed SU(2) invariant operators using the U(N) framework.

Defined a global U(N) symmetry to select homogeneous/isotropic states.

Proposed a Hamiltonian invariant under boundary area-preserving deformations.

Abstract

The implementation of the dynamics in Loop Quantum Gravity (LQG) is still an open problem. Here, we discuss a tentative dynamics for the simplest class of graphs in LQG: Two vertices linked with an arbitrary number of edges. We use the recently introduced U(N) framework in order to construct SU(2) invariant operators and define a global U(N) symmetry that will select the homogeneous/isotropic states. Finally, we propose a Hamiltonian operator invariant under area-preserving deformations of the boundary surface and we identify possible connections of this model with Loop Quantum Cosmology.