Lattice quantum gravity and asymptotic safety

Abstract

We study the nonperturbative formulation of quantum gravity defined via Euclidean dynamical triangulations (EDT) in an attempt to make contact with Weinberg's asymptotic safety scenario. We find that a fine-tuning is necessary in order to recover semiclassical behavior. Such a fine-tuning is generally associated with the breaking of a target symmetry by the lattice regulator; in this case we argue that the target symmetry is the general coordinate invariance of the theory. After introducing and fine-tuning a non-trivial local measure term, we find no barrier to taking a continuum limit, and we find evidence that four-dimensional, semiclassical geometries are recovered at long distance scales in the continuum limit. We also find that the spectral dimension at short distance scales is consistent with 3/2, a value that could resolve the tension between asymptotic safety and the holographic entropy scaling of black holes. We argue that the number of relevant couplings in the continuum theory is one, once symmetry breaking by the lattice regulator is accounted for. Such a theory is maximally predictive, with no adjustable parameters. The cosmological constant in Planck units is the only relevant parameter, which serves to set the lattice scale. The cosmological constant in Planck units is of order one in the ultra-violet and undergoes renormalization group running to small values in the infrared. If these findings hold up under further scrutiny, the lattice may provide a nonperturbative definition of a renormalizable quantum field theory of general relativity with no adjustable parameters and a cosmological constant that is naturally small in the infrared.

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

“…Promising steps in this direction have been already done by means of diffusive processes, where the scale is fixed by the diffusion time, both in CDT [8] and in DT [20]. Here we propose that LB spectra and the observed scaling profiles may be helpful in this direction, and that a careful study of how such profiles change as a function of the bare parameters could provide useful information.…”

“…There are different methods to visualize a complex network, some of them already considered in previous studies (see, e.g., Ref. [20]), here we will briefly discuss only two of them: Laplace embedding [42] and spring embedding [43,44]. The former makes use of the eigenvectors associated to the smallest eigenvalues, which are already computed by solving the eigenvalue problem, while the latter is based on a mapping of the graph to a system of points connected by springs: as we are going to discuss, the two methods are strictly related, however spring embedding proves more useful to give an intuitive picture of the short-scale structures.…”

“…Indeed, the Weinberg conjecture of the asymptotic safety of the gravitational interaction [17] states that the existence of an UV non-Gaussian fixed point makes the theory well defined in the UV (i.e., renormalizable) but in a region of the phase diagram not accessible by perturbation theory. Various nonperturbative methods have been developed in the last decades to investigate this possibility, like functional renormalization group techniques [18] or the Monte Carlo simulations of standard Euclidean dynamical triangulations (DT) [19][20][21][22] or causal dynamical triangulations, the latter being the subject of this study.…”

Quantum gravity on foliated spacetimes: Asymptotically safe and sound

“…Moreover, a first step connecting the NGFP to the underlying conformal field theory appeared in [64], possible completions of the flow at low energy have been discussed in [17,24,38,40,41] and geometric arguments determining the scaling of Newton's constant at the NGFP have been forwarded in [65,66]. In parallel Monte Carlo approaches to quantum gravity including Causal Dynamical Triangulations [67][68][69][70][71][72][73][74][75], Euclidean Dynamical Triangulations [76][77][78][79][80] and Lattice Quantum Gravity [81,82] made vast progress towards constructing phase diagrams at the non-perturbative level. While it is conceivable that all of these approaches probe the same universal short distance physics, a unified picture has yet to emerge.…”

Renormalization group flow of coupled tensorial group field theories: Towards the Ising model on random lattices

“…Another important question which need to be addressed at this step is about the positivity of S V W given in (14). For this let us fix the index i to 1 and provide the following definition:…”