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 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 H1 of scalar functions on piecewise flat manifolds and compare them with corresponding results obtained by using the dual graph representation. We show that, except for nonpathological 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 for this discrepancy and discuss its possible implications on the definition of generic observables built from the dual graph representation.

Automated summary

We explore the use of dual graph representation in causal dynamical triangulations (CDT) to construct observables and propose a new representation based on finite element methods (FEM). By applying FEM techniques, we extract the low-lying spectrum of the Laplace-Beltrami operator on piecewise flat manifolds in the Sobolev space H1 and compare it with the results obtained using the dual graph representation. We find that, except for nonpathological cases in two dimensions, the dual graph spectrum and spectral dimension do not generally agree with the ones obtained from the LB operator on the continuous space, both quantitatively and qualitatively. We analyze the reasons for this discrepancy and discuss its potential impact on defining generic observables constructed from the dual graph representation.