Spectral Analysis of Causal Dynamical Triangulations via Finite Element Method

Abstract

We examine the dual graph representation of simplicial manifolds in Causal Dynamical Triangulations (CDT) as a mean to build observables, and propose a new representation based on the Finite Element Methods (FEM). In particular, with the application of FEM techniques, we extract the (low-lying) spectrum of the Laplace-Beltrami (LB) operator on the Sobolev space $H^1$ of scalar functions on piecewise flat manifolds, and compare them with corresponding results obtained by using the dual graph representation. We show that, besides for non-pathological cases in two dimensions, the dual graph spectrum and spectral dimension do not generally agree, neither quantitatively nor qualitatively, with the ones obtained from the LB operator on the continuous space. We analyze the reasons of this discrepancy and discuss its possible implications on the definition of generic observables built from the dual graph representation.