Bouncing evolution in a model of loop quantum gravity

Abstract

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 ${\hat{H}}_v$ is chosen to be graph-preserving, and the matrix elements of ${\hat{H}}_v$ are explicitly worked out in a suitable basis. The non-trivial Euclidean part ${\hat{H}}_v^E$ of ${\hat{H}}_v$ is studied in details. It turns out that by choosing a specific symmetrization of ${\hat{H}}_v^E$, the dynamics driven by the Hamiltonian gives 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.

27 pages, 7 figures