Singular Vertices in the Strong Coupling Phase of Four--Dimensional Simplicial Gravity

We study four--dimensional simplicial gravity through numerical simulation with special attention to the existence of singular vertices, in the strong coupling phase, that are shared by abnormally large numbers of four--simplices. The second order phase transition from the strong coupling phase into the weak coupling phase could be understood as the disappearance of the singular vertices. We also change the topology of the universe from the sphere to the torus.Comment: 10 pages, six PostScript figures; figures are also available at http://hep-th.phys.s.u-tokyo.ac.jp/~izubuchi/paper/4dqg

Critical Behavior of Dynamically Triangulated Quantum Gravity in Four Dimensions

We performed 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 perturbative side. The curvature susceptibility diverges with the scaling index $-.6$. 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.Comment: 12pg

Gauge Invariance in Simplicial Gravity

The issue of local gauge invariance in the simplicial lattice formulation of gravity is examined. We exhibit explicitly, both in the weak field expansion about flat space, and subsequently for arbitrarily triangulated background manifolds, the exact local gauge invariance of the gravitational action, which includes in general both cosmological constant and curvature squared terms. We show that the local invariance of the discrete action and the ensuing zero modes correspond precisely to the diffeomorphism invariance in the continuum, by carefully relating the fundamental variables in the discrete theory (the edge lengths) to the induced metric components in the continuum. We discuss mostly the two dimensional case, but argue that our results have general validity. The previous analysis is then extended to the coupling with a scalar field, and the invariance properties of the scalar field action under lattice diffeomorphisms are exhibited. The construction of the lattice conformal gauge is then described, as well as the separation of lattice metric perturbations into orthogonal conformal and diffeomorphism part. The local gauge invariance properties of the lattice action show that no Fadeev-Popov determinant is required in the gravitational measure, unless lattice perturbation theory is performed with a gauge-fixed action, such as the one arising in the lattice analog of the conformal or harmonic gauges.Comment: LaTeX, 68 pages, 24 figure

Entropy of random coverings and 4D quantum gravity

We discuss the counting of minimal geodesic ball coverings of $n$-dimensional riemannian manifolds of bounded geometry, fixed Euler characteristic and Reidemeister torsion in a given representation of the fundamental group. This counting bears relevance to the analysis of the continuum limit of discrete models of quantum gravity. We establish the conditions under which the number of coverings grows exponentially with the volume, thus allowing for the search of a continuum limit of the corresponding discretized models. The resulting entropy estimates depend on representations of the fundamental group of the manifold through the corresponding Reidemeister torsion. We discuss the sum over inequivalent representations both in the two-dimensional and in the four-dimensional case. Explicit entropy functions as well as significant bounds on the associated critical exponents are obtained in both cases.Comment: 54 pages, latex, no figure

Discrete approaches to quantum gravity in four dimensions

The construction of a consistent theory of quantum gravity is a problem in theoretical physics that has so far defied all attempts at resolution. One ansatz to try to obtain a non-trivial quantum theory proceeds via a discretization of space-time and the Einstein action. I review here three major areas of research: gauge-theoretic approaches, both in a path-integral and a Hamiltonian formulation, quantum Regge calculus, and the method of dynamical triangulations, confining attention to work that is strictly four-dimensional, strictly discrete, and strictly quantum in nature.Comment: 33 pages, invited contribution to Living Reviews in Relativity; the author welcomes any comments and suggestion