Dynamics for a simple graph using the U(N) framework for loop quantum gravity

TL;DR

This paper explores a simplified model of loop quantum gravity dynamics on a two-vertex graph, revealing a U(N) symmetry, proposing a quantum Hamiltonian, and developing a classical effective dynamics framework.

Contribution

It introduces a U(N) symmetric model for LQG on a basic graph, proposing a quantum Hamiltonian and a classical effective dynamics approach.

Findings

Identification of a global U(N) symmetry in the model

Proposal of a quantum Hamiltonian operator for the reduced sector

Development of a classical effective dynamics using spinor representation

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 find an interesting global U(N) symmetry in this model that selects the homogeneous/isotropic sector. Then, we propose a quantum Hamiltonian operator for this reduced sector. Finally, we introduce the spinor representation for LQG in order to propose a classical effective dynamics for this model.