SIMPLICIAL GEOMETRY AND QUANTUM GRAVITY

Introduction

Random geometries are very important in many areas of theoretical physics - from non-critical string theory to the fluctuations of biological membranes. They play a central role in attempts to quantize gravity using path integral techniques [1].

One particularly powerful approach to the study of such systems replaces a continuum (Euclidean) geometry by a finite simplicial approximation. The use of such simplicial manifolds or triangulations allows rigorous meaning to be given to a sum over geometries. In two dimensions certain of these triangulation sums have been performed exactly and have yielded considerable insight into the problem of defining non-perturbative string theories [2].

Generalizations of these triangulation models to three and four dimensions provide rather natural candidates for quantum gravity. In particular most of the usual problems relating to the unboundedness of the action and the ambiguities in the functional measure are avoided [3].

Furthermore, these models, being finite, are amenable to truly non-perturbative studies using numerical simulation. These methods, when combined with techniques drawn from the study of critical phenomena in statistical mechanics, give us a powerful set of tools for investigating these systems.

Current Projects

1.0 Real Space Renormalization Group

The real space renormalization group gives a powerful technique for probing the critical behavior of conventional statistical mechanical models. It is based on constructing an explicit transformation of the original degrees of freedom of a model associated with a change of scale. This can be pictured as generating a flow in the coupling constant space of the model's effective action. Fixed points of this transformation are associated with criticality, while the flow in the neighborhood of such a fixed point determines the critical exponents of the model.

Of course the very notion of a scale is problematic in continuum formulations of quantum gravity. In contrast within the dynamical triangulation approach we automatically have an invariant cut-off - the elementary link length. Further, we have recently demonstrated the emergence of a dynamical length scale in the model which characterizes the quantum geometry (Catterall, Thorleifsson, Bowick and John [4]). A simple fractal dimension relates this linear scale to the total volume of the triangulation.

In the light of this we have been experimenting with one particular `block-spin' transformation in which a given triangulation is replaced by a coarser mesh constructed from a subset of the fine lattice points. An explicit decimation scheme is used in which the local block curvature is exactly preserved. As a corollary of this it is clear that the distribution of geodesic paths on the blocked lattices is similar to the bare lattice. We regard this as a crucial requirement of any real space RG method for quantum gravity.

Preliminary studies in the case of pure gravity and gravity coupled to Ising spins have been encouraging (Catterall, Kogut and Renken [5], Thorleifsson and Catterall [6]). In particular the transformation appears to be driving the system towards a fixed point. Standard MCRG techniques allow us to extract the critical exponents associated to this fixed point and out results agree with the predictions of conformal field theory in several simple models. We are currently investigating extensions of these ideas to higher dimensional triangulated quantum gravity and are optimistic about substantial future progress.

2.0 Dynamical Triangulations in Four Dimensions

We have been active extending the ideas developed in two dimensions to models which incorporate a summation over four-dimensional triangulations. Here, little is known analytically and we must rely primarily on numerical simulations for guidance.

In this approach the partition function is simply a weighted sum over all possible ways of assembling elementary four-simplices into a simplicial manifold by associating them in pairs via their faces. A spherical topology is usually employed and the gluings restricted so that no two subsimplices are identical.

In analogy to two dimensions a set of ergodic, topology preserving `moves' are used to implement local changes of the triangulation (Catterall [7]). A Monte Carlo algorithm is then used to sample the dominant contributions to the partition function. Both ourselves (Catterall, Kogut and Renken [8]) and other groups have performed exploratory investigations of the phase structure of this model. All groups have seen evidence of a two phase structure; for small values of the inverse bare Newton constant the typical manifolds crumple to scales on the order of the cut-off - the mean curvature is large and negative. At large couplings the typical manifolds are extended structures with Hausdorff dimension two and positive curvature. A continuous phase transition appears to separate the two regions. There is thus the tantalizing possibility that in the vicinity of this critical point it may be possible to construct a continuum quantum theory of gravity.

However, many questions must first be addressed. Perhaps the most pressing of these relates to the existence of an exponential bound on the total number of triangulations of given volume. This feature is of the greatest importance - it allows a simple cosmological constant term in the discrete action to control the otherwise divergent sum over lattices. Unless the volume of the triangulation space grows at most exponentially with volume the partition is strictly divergent for all values of the bare lattice parameters. No continuum limit would then be possible.

The situation in dimensions greater than two is completely unknown - indeed it is easy to show that a factorial increase is generic if the topology is not sufficiently restricted. We have attempted to address this issue numerically by computing an estimate for the entropy of lattices as a function of mean volume in order to track the approach to the thermodynamic limit (Catterall, Kogut and Renken [9]). In three dimensions our data constitute the strongest evidence in favor of a bound - large transient finite size effects are seen but a rather robust extrapolation procedure seems to indicate the presence of a bound at large volumes. The data in four dimensions are much less conclusive - in the crumpled phase the evidence for a bound is much weaker and we have observed large finite size effects which do not admit any simple parameterization.

Most recently we have observed that the generic triangulations encountered in the four dimensional partition sum have a very singular structure: they are composed of two (linked) vertices which are shared by a very large number of fundamental simplex building blocks. Indeed, the simplex `coordination' of these two points increases linearly with the total volume of the triangulation [10]. It is possible to understand this structure using simple arguments based on local entropy considerations. It appears that the transition observed in four dimensions can be thought of as the breakdown of this singular structure as more vertices are forced into the triangulation with increasing gravitational coupling constant. We are investigating this in the context of other non-trivial measure terms for the triangulation.

Simulation Details

Monte Carlo simulation is used to generate a sequence of triangulations typical of those that dominate the partition function. The code is written in C in order to handle dynamic memory allocation; the fluctuations of the triangulation ensure our data structures are both random and dynamic as simplices are inserted and removed from the mesh. A structure of type SIMPLEX is defined which contains both the labels of the vertices making up the simplex and the addresses of neighbor simplices. Sequences of functions allocate and delete simplices and update the pointer fields of simplices neighbor to a move. We use a simple metropolis test in conjunction to local `moves' on the triangulation to evolve the lattice.

The total storage is quite modest: 0.75 MB for a V=10K simplex lattice in four dimensions. Currently we have data on lattices as large as 64K. The CPU time per attempted elementary move is approx at V=10K. Typical autocorrelation times in the critical region in D=4 are sweeps necessitating very long runs to compute critical behavior. Currently these runs take place on a variety of RISC workstation in Orlando, Illinois and Syracuse.

Click here to see our gallery of random surface pictures.

• References