The quantum space-time of c = −2 gravity

Nuclear Physics B - Proceedings Supplements, 1998

We couple c = ?2 matter to 2-dimensional gravity within the framework of dynamical triangulations. We use a very fast algorithm, special to the c = ?2 case, in order to test scaling of correlation functions de ned in terms of geodesic distance and we determine the fractal dimension dH with high accuracy. We nd dH = 3:58(4), consistent with a prediction coming from the study of di usion in the context of Liouville theory, and that the quantum space{time possesses the same fractal properties at all distance scales similarly to the case of pure gravity.

Physics Letters B, 1994

We investigate the fractal structure of 2d quantum gravity coupled to matter by measuring the distributions of so-called baby universes. We demonstrate that the method works well as long as c 1. For c > 1 it is not clear what distribution to expect. However, we observe strikingly similar distributions for various kinds of matter elds with the same c. This indicate that there might be some range of c > 1 where the central charge of the matter elds alone determines the fractal structure of gravity coupled to matter. The hypothesis that the string susceptibility = 1=3 is found to be compatible with the data for 1 < c 4.

Physics Letters B, 1998

We compute the intrinsic Hausdorff dimension of spacetime at the infrared fixed point of the quantum conformal factor in 4D gravity. The fractal dimension is defined by the appropriate covariant diffusion equation in four dimensions and is determined by the coefficient of the Gauss-Bonnet term in the trace anomaly to be generally greater than 4. In addition to being testable in simplicial simulations, this scaling behavior suggests a physical mechanism for the screening of the effective cosmological 'constant' and inverse Newtonian coupling at very large distance scales, which has implications for the dark matter content and large scale structure of the universe.

In this paper we calculated the spectral dimension of loop quantum gravity (LQG) using the scaling property of the area operator spectrum on spin-network states and using the scaling property of the volume and length operators on Gaussian states. We obtained that the spectral dimension of the spatial section runs from 1.5 to 3, and under particular assumptions from 2 to 3 across a 1.5 phase when the energy of a probe scalar field decreases from high to low energy in a fictitious time T . We calculated also the spectral dimension of space-time using the scaling of the area spectrum operator calculated on spin-foam models. The main result is that the effective dimension is 2 at the Planck scale and 4 at low energy. This result is consistent with two other approaches to non perturbative quantum gravity: causal dynamical triangulation and asymptotically safe quantum gravity. We studied the scaling properties of all the possible curvature invariants and we have shown that the singularity problem seems to be solved in the covariant formulation of quantum gravity in terms of spin-foam models. For a particular form of the scaling (or for a particular area operator spectrum) all the curvature invariants are regular also in the Trans-Planckian regime.

We show that there exists a divergent correlation length in 2d quantum gravity for the matter fields close to the critical point provided one uses the invariant geodesic distance as the measure of distance. The corresponding reparameterization invariant two-point functions satisfy all scaling relations known from the ordinary theory of critical phenomena and the KPZ exponents are determined by the power-like fall off of these two-point functions. The only difference compared to flat space is the appearance of a dynamically generated fractal dimension d_h in the scaling relations. We analyze numerically the fractal properties of space-time for Ising and three-states Potts model coupled to 2d dimensional quantum gravity using finite size scaling as well as small distance scaling of invariant correlation functions. Our data are consistent with d_h=4, but we cannot rule out completely the conjecture d_H = -2\alpha_1/\alpha_{-1}, where \alpha_{-n} is the gravitational dressing exponent o...

Physics Letters B, 1997

We study a c = ?2 conformal eld theory coupled to two-dimensional quantum gravity by means of dynamical triangulations. We de ne the geodesic distance r on the triangulated surface with N triangles, and show that dim r d H ] = dim N ], where the fractal dimension d H = 3:58 0:04. This result lends support to the conjecture d H = ?2 1 = ?1 , where ?n is the gravitational dressing exponent of a spin-less primary eld of conformal weight (n + 1; n + 1), and it disfavors the alternative prediction d H = ?2= str . On the other hand, we nd dim l ] = dim r 2 ] with good accuracy, where l is the length of one of the boundaries of a circle with (geodesic) radius r, i.e. the length l has an anomalous dimension relative to the area of the surface. It is further shown that the spectral dimension d s = 1:980 0:014 for the ensemble of (triangulated) manifolds used. The results are derived using nite size scaling and a very e cient recursive sampling technique known previously to work well for c = ?2.

Reports in Advances of Physical Sciences, 2023

Based on the smallest physical constant of the product of space interval, time interval, and energy, the fractal quantum gravity (FQG) theory has demonstrated that every particle or physical system consists of these smallest units in fractal structures. The general relativity is an approximation of the FQG equation when the quantum effect is negligible, while the quantum theory is an approximation of the FQG equation when the interaction between space, time, and energy is very weak or negligible. The stationary-action principle can be derived from the FQG equation. The mass range of possibly existing elementary particles and an accelerating expansion evolution model of the universe can be obtained through the FQG equation. This FQG equation satisfied almost all the requirements of a quantum gravity theory and there is no free constant needed in the FQG theory. It looks promising that the FQG theory may offer a novel way to calculate all the free constants in the Standard Model of particle physics and general relativity.

2009

In this paper we perform the calculation of the spectral dimension of spacetime in 4d quantum gravity using the Barrett-Crane (BC) spinfoam model. We realize this considering a very simple decomposition of the 4d spacetime already used in the graviton propagator calculation and we introduce a boundary state which selects a classical geometry on the boundary. We obtain that the spectral dimension of the spacetime runs from $\approx 2$ to 4, across a $\approx 1.5$ phase, when the energy of a probe scalar field decreases from high $E \lesssim E_P/25$ to low energy. The spectral dimension at the Planck scale $E \approx E_P$ depends on the areas spectrum used in the calculation. For three different spectra $l_P^2 \sqrt{j(j+1)}$, $l_P^2 (2 j+1)$ and $l_P^2 j$ we find respectively dimension $\approx 2.31$, 2.45 and 2.08.

Journal of High Energy Physics, 1998

We show that the \time" t s de ned via spin clusters in the Ising model coupled to 2d gravity leads to a fractal dimension d h (s) = 6 of space-time at the critical point, as advocated by Ishibashi and Kawai. In the unmagnetized phase, however, this de nition of Hausdor dimension breaks down. Numerical measurements are consistent with these results. The same de nition leads to d h (s) = 16 at the critical point when applied to at space. The fractal dimension d h (s) is in disagreement with both analytical prediction and numerical determination of the fractal dimension d h (g), which is based on the use of the geodesic distance t g as \proper time". There seems to be no simple relation of the kind t s = t d h (g)=d h (s) g , as expected by dimensional reasons.

Chaos, Solitons & Fractals, 1999

No theory of four-dimensional quantum gravity exists as yet. In this situation the two-dimensional theory, which can be analyzed by conventional field-theoretical methods, can serve as a toy model for studying some aspects of quantum gravity. It represents one of the rare settings in a quantum-gravitational context where one can calculate quantities truly independent of any background geometry. We review recent progress in our understanding of 2d quantum gravity, and in particular the relation between the Euclidean and Lorentzian sectors of the quantum theory. We show that conventional 2d Euclidean quantum gravity can be obtained from Lorentzian quantum gravity by an analytic continuation only if we allow for spatial topology changes in the latter. Once this is done, one obtains a theory of quantum gravity where space-time is fractal: the intrinsic Hausdorff dimension of usual 2d Euclidean quantum gravity is four, and not two. However, certain aspects of quantum space-time remain two-dimensional, exemplified by the fact that its so-called spectral dimension is equal to two.

Elsevier SSRN

The paper looks at sub quantum imaginary mass as spinor fields which are associated with complex space time structures and fractal geometry at the quantum gravity interphases.Based on fractal space time geometry On a microscopic scale, time can be interpreted in terms of the appearance of particle-antiparticle pairs when observation energies become of the order of mc2,and at higher energy orders complex spacetime structure of imaginary masses are evidenced as sub quantum or Trans planckian structures,which have transitions to quantum gravity and classical relativistic phases through intrinsic fractal geometries

Acta Physica Polonica B, 2013

In these lectures we review our present understanding of the fractal structure of two-dimensional Euclidean quantum gravity coupled to matter.

In this paper we perform the calculation of the spectral dimension of the space-time in 3d quantum gravity using the dynamics of the Ponzano-Regge vertex (PR) and its quantum group generalization (Turaev-Viro model (TV) ). We realize this considering a very simple decomposition of the 3d space-time and introducing a boundary state which selects a classical geometry on the boundary. We obtain that the spectral dimension of the space-time runs from ≈ 2 to 3, across a ≈ 1.5 phase, when the energy of a probe scalar field decreases from high E EP to low energy. For the TV model the spectral dimension at hight energy increase with the value of the cosmological constant Λ. At low energy the presence of Λ does not change the spectral dimension.

Physical Review Letters, 2010

We propose a field theory which lives in fractal spacetime and is argued to be Lorentz invariant, power-counting renormalizable, ultraviolet finite, and causal. The system flows from an ultraviolet fixed point, where spacetime has Hausdorff dimension 2, to an infrared limit coinciding with a standard four-dimensional field theory. Classically, the fractal world where fields live exchanges energy momentum with the bulk with integer topological dimension. However, the total energy momentum is conserved. We consider the dynamics and the propagator of a scalar field. Implications for quantum gravity, cosmology, and the cosmological constant are discussed.

Nuclear Physics B, 1991

A two-dimensional quantum gravity is simulated by means of the dynamical triangulation model. The size of the lattice was up to hundred thousand triangles. Massively parallel simulations and recursive sampling were implemented independently and produced similar results. Wherever the analytical predictions existed, our results confirmed them. The cascade process of baby universes formulation a la Coleman-Hawking scenario in a two-dimensional case has been observed. We observed that there is a simple universal inclusive probability for a baby universe to appear. This anomalous branching of surfaces led to a rapid growth of the integral curvature inside a circle. The volume of a disk in the internal metric has been proven to grow faster than any power of radius. The scaling prediction for the mean square extent given by the Liouville theory has been confirmed. However, the naive expectation for the average Liouville lagrangian < f (p~6)2 > is about 1 order of magnitude different from the results. This apparently points out to some flaws in the current definition of a Liouville model.