Closing the Loop: from EPRL-FK spinfoams to Regge dynamics

Abstract: The EPRL-FK spinfoam model in its semiclassical regime at fixed discretization faces a big limitation: its dominant configurations describe only flat geometries. We ask what additional condition is needed for the variational principle to reproduce the equations of motion of Regge calculus, the standard discretization of General Relativity. We recast the EPRL-FK model in wedge-holonomy variables and identify geometric closure as the missing condition. We prove that imposing geometric closure as an additional constraint alongside local flatness allows us to reconstruct curved geometries. On a regular, nondegenerate Lorentzian branch with spacelike tetrahedra, modulo gauge, they define a constraint surface equivalent to the space of length-Regge geometries. The wedge action restricted to this surface becomes the Regge action, and its tangent variations yield the length-Regge equations. Finally, we propose a way to implement geometric closure directly in the spinfoam amplitude. With this additional constraint, the EPRL-FK model is therefore equivalent to Regge calculus in its semiclassical regime.