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

Understand

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.