Bouncing evolution in a model of loop quantum gravity

To understand the dynamics of loop quantum gravity, the deparametrized model of gravity coupled to a scalar field is studied in a simple case, where the graph underlying the spin network basis is one loop based at a single vertex. The Hamiltonian operator H^v is chosen to be graph-preserving, and the matrix elements of H^v are explicitly worked out in a suitable basis. The nontrivial Euclidean part H^vE of H^v is studied in details. It turns out that by choosing a specific symmetrization of H^vE, the dynamics driven by the Hamiltonian give a picture of bouncing evolution. Our result in the model of full loop quantum gravity gives a significant echo of the well-known quantum bounce in the symmetry-reduced model of loop quantum cosmology, which indicates a closed relation between singularity resolution and quantum geometry.