We introduce simple generic models of surface dynamics in loop quantum gravity (LQG). A quantum surface is defined as a set of elementary patches of area glued together. We provide it with an extra structure of locality (nearest neighbors), thought of as induced by the whole spin network state defining the 3d bulk geometry around the quantum surface. Here, we focus on classical surface dynamics, using a spinorial description of surface degrees of freedom. We introduce two classes of dynamics, to be thought as templates for future investigation of LQG dynamics with the dynamics of quantum black holes in mind. The first defines global dynamics of the closure defect of the surface, with two basic toy models, either a dissipative dynamics relaxing towards the closure constraint or a Hamiltonian dynamics precessing the closure defect. The second class of dynamics describes the isolated regime, when both area and closure defect are conserved throughout the evolution. The surface dynamics is implemented through $\mathrm{U}(N)$ transformations and generalizes to a Bose-Hubbard Hamiltonian with a local quadratic potential interaction. We briefly discuss the implications of modeling the quantum black hole dynamics by a surface Bose-Hubbard model.

• Received 7 March 2017

DOI:https://doi.org/10.1103/PhysRevD.95.124038