Linearized dynamics from the 4-simplex Regge action

We study the relation between the Hessian matrix of the Riemannian Regge action on a 4-simplex and linearized quantum gravity. We give an explicit formula for the Hessian as a function of the geometry, and show that it has a single zero mode. We then use a 3D lattice model to show that (i) the zero mode is a remnant of the continuum diffeomorphism invariance, and (ii) we recover the complete free graviton propagator in the continuum limit. The results help clarify the structure of the boundary state needed in the recent calculations of the graviton propagator in loop quantum gravity, and, in particular, its role in fixing the gauge.