Dynamics for a simple graph using the U(N) framework for loop quantum   gravity
  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. 
Enrique F. Borja, Jacobo Diaz-Polo, Laurent Freidel, I\~naki Garay,   Etera R. Livine
gr-qc