Spectrum of the Laplace-Beltrami operator and the phase structure of causal dynamical triangulations

Abstract

We propose a new method to characterize the different phases observed in the nonperturbative numerical approach to quantum gravity known as causal dynamical triangulations. The method is based on the analysis of the eigenvalues and the eigenvectors of the Laplace-Beltrami operator computed on the triangulations: it generalizes previous works based on the analysis of diffusive processes and proves capable of providing more detailed information on the geometric properties of the triangulations. In particular, we apply the method to the analysis of spatial slices, showing that the different phases can be characterized by a new order parameter related to the presence or absence of a gap in the spectrum of the Laplace-Beltrami operator, and deriving an effective dimensionality of the slices at the different scales. We also propose quantities derived from the spectrum that could be used to monitor the running to the continuum limit around a suitable critical point in the phase diagram, if any is found.

Quantum gravity from causal dynamical triangulations: a review

“…Computational efficiency of CDT has been improving steadily since. At the same time, a number of individuals and groups have independently written CDT code from scratch and run simulations [48][49][50], finding consistent results. 5…”

“…Recent developments regarding spectral observables in CDT include measure-ments of the spectral dimension away from the "canonical" phase space point (κ 0 , ∆) = (2.2, 0.6), in phases C dS and A [91,92], the suggestion to investigate the function D S (σ) on all scales, generalising a corresponding study of CDT in three dimensions [90], and a comprehensive analysis of the spectral properties of three-dimensional spatial slices in four-dimensional CDT quantum gravity [50].…”

“…Noteworthy in the context of spectral observables is a recent attempt to tap into the potentially rich information contained in the eigenvalues and eigenvectors of the Laplace-Beltrami operator on spatial slices with topology S 3 in CDT [50]. Which properties of the spectrum lend themselves to ensemble averaging and are ultimately related to geometric observables in the continuum is largely unexplored territory.…”

“…. , can be ordered according to size, with λ 0 = 0 and λ 1 denoting the smallest nonvanishing eigenvalue, one can for example monitor the distribution of a particular eigenvalue with label n 0 , where [50].] the outcome will in general depend on both the location in phase space and the spatial volume of the slice.…”

“…For the slices of B-type this implies the existence of a spectral gap λ 1 and a general shift to larger values for eigenvalues λ n of the same order n. The jumps between large and small values are particularly apparent for the low-lying part of the spectrum, and become somewhat less pronounced as n increases. The suggestion in [50], further elaborated in [96], is to approach the C b -C dS phase transition from inside the bifurcation phase C b , and use the discrete set of eigenvalues λ n associated with the slices of type B, which one expects to vanish as the transition line is approached, as order parameters. In the exploratory study [96], the coupling κ 0 is kept fixed as ∆ is increased towards the phase transition.…”