Classical dynamics for loop gravity: The two-vertex model

Abstract

The study of toy models in loop quantum gravity (LQG), defined as truncations of the full theory, is relevant to both the development of the LQG phenomenology, in cosmology and astrophysics, and the progress towards the resolution of the open issues of the theory, in particular the implementation of the dynamics. Here, we study the dynamics of spin network states of quantum geometry defined on the family of graphs consisting in 2 vertices linked by an arbitrary number of edges, or 2-vertex model in short. A symmetry reduced sector of this model -- to isotropic and homogeneous geometries -- was successfully studied in the past, where interesting cosmological insights were found. We now study the evolution of the classical trajectories for this system in the general case, for arbitrary number of edges with random initial configurations. We use the spinorial formalism and its clear interpretation of spin networks in terms of discrete twisted geometries, with the quantum 3d space made of superpositions of polyhedra glued together by faces of equal area. Remarkably, oscillatory and divergent regimes are found with a universal dependence on the coupling constants of the Hamiltonian and independent of the initial spinors or the number of edges. Furthermore, we explore the evolution of the associated polyhedra as well as their volumes and areas.