Spectral analysis of causal dynamical triangulations via finite element method

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