Linearized dynamics from the 4-simplex Regge action

TL;DR

This paper analyzes the Hessian matrix of the 4-simplex Regge action, revealing its relation to linearized quantum gravity, the role of a zero mode, and how it reproduces the graviton propagator in the continuum limit.

Contribution

It provides an explicit formula for the Hessian, identifies its zero mode as a remnant of diffeomorphism invariance, and connects these findings to the graviton propagator in loop quantum gravity.

Findings

Hessian has a single zero mode related to gauge invariance

The zero mode is a remnant of continuum diffeomorphism invariance

Reproduction of the graviton propagator in the continuum limit

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.