Four-dimensional Simplicial Quantum Gravity Research Articles

Matter Dependence of the String Susceptibility Exponent in Four-Dimensional Simplicial Quantum Gravity

The dependence of the number of vector and scalar fields on the string susceptibility exponent is evaluated numerically for four-dimensional simplicial quantum gravity using the grand canonical ensemble method. The numerical result indicates that the four-dimensional simplicial quantum gravity reproduces well predictions of the four-dimensional conformal gravity.

Clear evidence of a continuum theory of 4D Euclidean simplicial quantum gravity

Four-dimensional (4D) simplicial quantum gravity coupled to both scalar fields ( N X ) and gauge fields ( N A ) has been studied using Monte-Carlo simulations. The matter dependence of the string susceptibility exponent γ (4) is estimated. Furthermore, we compare our numerical results with Background-Metric-Indepenent (BMI) formulation conjectured to describe the quantum field theory of gravity in 4D. The numerical results suggest that the 4D simplicial quantum gravity is related to the conformal gravity in 4D. Therefore, we propose a phase structure in detail with adding both scalar and gauge fields and discuss the possibility and the property of a continuum theory of 4D Euclidean simplicial quantum gravity.

Phase Structure of Four-Dimensional Simplicial Quantum Gravity with a U(1) Gauge Field

The phase structure of four-dimensional simplicial quantum gravity coupled to U(1) gauge fields has been studied using Monte-Carlo simulations. The smooth phase is found in the intermediate region between the crumpled phase and the branched polymer phase. This new phase has a negative string susceptibility exponent, even if the number of vector fields (Nv) is 1. The phase transition between the crumpled phase and the smooth phase has been studied by a finite size scaling method. From the numerical results, we expect that this model (coupled to one gauge field) has a higher order phase transition than first order, which means the possibility to take the continuum limit at the critical point. Furthermore, we consider a modification of the balls-in-boxes model for a clear understanding of the relation between the numerical results and the analytical one.

Abelian gauge fields coupled to simplicial quantum gravity

We study the coupling of abelian gauge theories to four-dimensional simplicial quantum gravity. The gauge fields live on dual links. This is the correct formulation if we want to compare the effect of gauge fields on geometry with similar effects studied so far for scalar fields. It shows that gauge fields couple equally weakly to geometry as scalar fields, and it offers an understanding of the relation between measure factors and abelian gauge fields observed so-far.

Common structures in simplicial quantum gravity

Statistical properties of dynamically triangulated manifolds (DT mfds) in terms of the geodesic distance are studied numerically. The string susceptibility exponents for the boundary surfaces in three-dimensional DT mfds are measured numerically. For spherical boundary surfaces, we obtain a result consistent with the case of a two-dimensional spherical DT surface described by the matrix model. This gives a correct method to reconstruct two-dimensional random surfaces from three-dimensional DT mfds. Furthermore, a scaling property of the volume distribution of minimum neck baby universes is investigated numerically in the case of three and four dimensions, and we obtain a common scaling structure near to the critical points belonging to the strong coupling phase in both dimensions. We have evidence for the existence of a common fractal structure in three- and four-dimensional simplicial quantum gravity.

Simulating four-dimensional simplicial gravity using degenerate triangulations

We extend a model of four-dimensional simplicial quantum gravity to include degenerate triangulations in addition to combinatorial triangulations traditionally used. Relaxing the constraint that every 4-simplex is uniquely defined by a set of five distinct vertexes, we allow triangulations containing multiply connected simplexes and distinct simplexes defined by the same set of vertexes. We demonstrate numerically that including degenerated triangulations substantially reduces the finite-size effects in the model. In particular, we provide a strong numerical evidence for an exponential bound on the entropic growth of the ensemble of degenerate triangulations, and show that a discontinuous crumpling transition is already observed on triangulations of volume N_4 ~= 4000.

Further evidence of a smooth phase in 4D simplicial quantum gravity

Four-dimensional (4D) simplicial quantum gravity coupled to U(1) gauge fields has been studied using Monte-Carlo simulations. A negative string susceptibility exponent is observed beyond the phase-transition point, even if the number of vector fields ( N V ) is 1. We find a scaling relation of the boundary volume distributions in this new phase. This scaling relation suggests a fractal structure similar to that of 2D quantum gravity. Furthermore, evidence of a branched polymer-like structure is suggested far into the weak-coupling region, even for N V > 1. As a result, we propose new phase structures and discuss the possibility of taking the continuum limit in a certain region between the crumpled and branched polymer phases.

Simplicial quantum gravity in the elongated phase

We discuss the elongated phase of four-dimensional simplicial quantum gravity by exploiting recent analytical results. In particular using Walkup’s theorem we prove that the dominating configurations in the elongated phase are treelike structures called “stacked spheres.” Such configurations can be mapped into branched polymers and baby universes arguments are used in order to analyze the critical behavior of theory in the weak coupling regime.

4d Simplicial quantum gravity: matter fields and the corresponding effective action

Four-dimensional simplicial quantum gravity is modified either by coupling it to U (1) gauge fields or by introducing a measure weighted by the orders of the triangles. Strong coupling expansion and Monte Carlo simulations are used. Although the two modifications of the standard pure-gravity model are apparently very distinct, they produce strikingly similar results, as far as the geometry of random manifolds is concerned. In particular, for an appropriate choice of couplings, the branched polymer phase is replaced by a crinkled phase, characterized by the susceptibility exponent γ <0 and the fractal dimension d H >2. The quasi-equivalence between the two models is exploited to get further insight into the extended phase diagram of the theory.

Scaling Behavior in 4D Simplicial Quantum Gravity

Scaling relations in four-dimensional simplicial quantum gravity are proposed using the concept of the geodesic distance. Based on the analogy of a loop length distribution in the two-dimensional case, the scaling relations of the boundary volume distribution in four dimensions are discussed in three regions: the strong-coupling phase, the critical point and the weak-coupling phase. In each phase a different scaling behavior is found.

Scaling in four-dimensional quantum gravity

We discuss scaling relations in four-dimensional simplicial quantum gravity. Using numerical results obtained with a new algorithm called “baby universe surgery” we study the critical region of the theory. The position of the phase transition is given with high accuracy and some critical exponents are measured. Their values prove that the transition is continuous. We discuss the properties of two distinct phases of the theory. For large values of the bare gravitational coupling constant the internal Hausdorff dimension is two (the elongated phase), and the continuum theory is that of so called branched polymers. For small values of the bare gravitational coupling constant the internal Hausdorff dimension seems to be infinite (the crumpled phase). We conjecture that this phase corresponds to a theory of topological gravity. At the transition point the Hausdorff dimension might be finite and larger than two. This transition point is a potential candidate for a non-perturbative theory of quantum gravity.

Two-point functions of four-dimensional simplicial quantum gravity

We investigate the interaction mechanism of pure quantum gravity in Regge discretization. We compute volume-volume and link-link correlation functions. In a preliminary analysis the forces turn out to be of Yukawa type, at least on our finite lattice being away from the continuum limit.

Four-dimensional quantum gravity by dynamical triangulations

The possibility of a sensible continuum limit of four-dimensional simplicial quantum gravity in the dynamical triangulation formulation is investigated. The existence of a phase transition with critical fluctuations in the curvature is established, and a number of indicators of the order of the transition computed. Judging from these results from lattices up to 16k simplices, the transition is of second order. The general appearance of the transition, as well as a preliminary estimate of the critical exponent for the susceptibility is in agreement with results obtained by Hamber in another formalism, as a strong indication of universality.

Observing 4d baby universes in quantum gravity

We measure the fractal structure of four-dimensional simplicial quantum gravity by identifying so-called baby universes. This allows an easy determination of the critical exponent γ connected to the entropy of four-dimensional manifolds.

Nonuniform measure in four-dimensional simplicial quantum gravity.

Four-dimensional Euclidean quantum gravity has been studied as a discrete model based on dynamical triangulations by Ambjorn and Jurkiewicz and by Agishtein and Migdal. We discuss a particular implementation of a Monte Carlo simulation of simplicial quantum gravity. As an application we introduce a nonuniform measure and examine its effect on simple aspects of the mathematical geometry. We find that the transition region from the hot to the cold phase is shifted and that the criticality of the transition changes

Measure of four-dimensional simplicial quantum gravity

Four-dimensional euclidean quantum gravity has been studied as a discrete model based on dynamical triangulations by Ambjørn and Jurkiewicz [1] and by Agishtein and Migdal [2]. We present results for a non-uniform measure which indicate that the criticality of the phase transition depends on the measure.

Critical behavior of dynamically triangulated quantum gravity in four dimensions

We performed a detailed study of the phase transition region in four-dimensional simplicial quantum gravity, using the dynamical triangulation approach. The phase transition between the gravity and antigravity phases turned out to be asymmetrical, so that we observed the scaling laws only when the Newton constant approached the critical value from the perturbative side. The curvature susceptibility diverges with the scaling index - 0.6. The physical (i.e. measured with heavy particle propagation) Hausdorff dimension of the manifolds, which is 2.3 in the gravity phase and 4.6 in the antigravity phase, turned out to be 4 at the critical point, within the measurement accuracy. These facts indicate the existence of the continuum limit in four-dimensional euclidean quantum gravity.

SIMULATIONS OF FOUR-DIMENSIONAL SIMPLICIAL QUANTUM GRAVITY AS DYNAMICAL TRIANGULATION

Four-Dimensional Simplicial Quantum Gravity is simulated using the dynamical triangulation approach. We studied simplicial manifolds of spherical topology and found the critical line for the cosmological constant as a function of the gravitational one, separating the phases of opened and closed Universe. When the bare cosmological constant approaches this line from above, the four-volume grows: we reached about 5×104 simplexes, which proved to be sufficient for the statistical limit of infinite volume. However, for the genuine continuum theory of gravity, the parameters of the lattice model should be further adjusted to reach the second order phase transition point, where the correlation length grows to infinity. We varied the gravitational constant, and we found the first order phase transition, similar to the one found in three-dimensional model, except in 4D the fluctuations are rather large at the transition point, so that this is close to the second order phase transition. The average curvature in cutoff units is large and positive in one phase (gravity), and small negative in another (antigravity). We studied the fractal geometry of both phases, using the heavy particle propagator to define the geodesic map, as well as with the old approach using the shortest lattice paths. The heavy propagator geodesic appeared to be much smoother, so that the scaling laws were found, corresponding to finite fractal dimensions: D+~2.3 in the gravity phase and D−~4.6 in the antigravity phase. Similar, but somewhat lower numbers were obtained from the heat kernel singularity. The influence of the αR2 terms in 2, 3 and 4 dimensions is discussed.

Phases of four-dimensional simplicial quantum gravity.

The phase diagram and critical exponents for pure simplicial quantum gravity (Regge calculus) in four dimensions are discussed. In the small-$G$ phase, where $G$ is the bare Newton's constant, the simplices are collapsed and no continuum limit exists. In the large-$G$ phase the ground state appears to be well behaved, and the curvature goes to zero continuously as the critical value of $G$ is approached. Fluctuations in the curvature diverge at the critical point, while volume fluctuations remain finite. The critical exponents at the transition are estimated, and appear to be independent of the strength of the higher-derivative coupling $a$. With the lattice analogue of the DeWitt gravitational measure and for large enough $G$, the lattice higher-derivative theories ($a&gt;0$) and the reflection-positive pure Regge theory ($a=0$) appear to belong to the same phase for large enough $G$, which would suggest a common, unitary quantum continuum limit.

Three- and four-dimensional simplicial quantum gravity

The Application of numerical methods to the dynamical triangulation formulation of quantum gravity is discussed and the phase diagram for three - and four-dimensional simplicial quantum gravity is outlined. The main part of the paper presents the (still preliminary) results in four dimensions.