Linearized dynamics from the 4-simplex Regge action

TL;DR

This work derives a general Hessian $Q_{ijkl}$ for the Regge action on an arbitrary $n$-simplex and shows it has a single zero mode tied to the flat-simplice Gram determinant, clarifying its non-invertibility. A 3d lattice model demonstrates that when all tetrahedral contributions are summed and proper gauge fixing is applied (via a discretized harmonic gauge), the correct free graviton propagator with proper distance scaling and tensor structure emerges, identifying the zero mode as a diffeomorphism remnant. The general boundary formalism is used to connect these results to spinfoam graviton calculations, highlighting the boundary state’s gauge dependence and the need for consistent gauge fixing between boundary and bulk. Collectively, the results illuminate how Regge calculus encodes semiclassical gravity in spinfoams and outline a program for extending these insights to 4d, including the role of boundary data in fixing gauge and dynamics.

Abstract

We study the relation between the hessian matrix of the riemannian Reggae 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.