Lattice gravity and random surfaces

Abstract

I review recent progress in simplicial quantum gravity in three and four dimensions, in particular new results on the phase structure of modified models of dynamical triangulations, the application of a strong-coupling expansion, and the benefits provided by including degenerate triangulations. In addition, I describe some recent numerical and analytical results on anisotropic crystalline membranes. Simplicial Quantum Gravity in D > 2The task of formulating a consistent theory of quantum gravity in four dimensions, hence unifying the two pillars of modern physics, general relativity and quantum mechanics, is a formidable one and still unresolved. Many different approaches have been tried (see e.g. Ref.[1]), some have led to a deeper understanding of the problem but, as of yet, none can claim much success. In this article I will review the status, and recent progress, of one such approach, namely the attempt to make sense of an Euclidean path-integral quantization of general relativity via a discretization known as dynamical triangulations.In Euclidean quantum gravity a path-integral is written as a formal sum over D-dimensional Euclidean geometries (metrics g µν )weighted by the Einstein-Hilbert actionG is the Newton's constant, Λ the cosmological constant and R the scalar curvature. However, this path-integral formulation faces some sever problems and unresolved questions:• the theory is non-renormalizable, • the action is unbounded from below, 1

Gibbs and quantum discrete spaces

“…Numerical modelling, see [20], shows the appearance of similar phases in two-dimensional gravity with spins. That is why various explanations and analogs are of interest.…”

Spin models on random lattices

“…The uctuating geometry is supposed to realize the functional integration over the degrees of freedom of the metric tensor and can be generated in a Monte Carlo (MC) procedure by link-ip moves applied to tri-valent ( 3 ) planar graphs [31][32][33][34][35]. In the simplest case matter ÿelds are approximated by spins on the vertices of these dynamical graphs.…”

New analytical results on anisotropic membranes

“…The results are compared with the uncontrolled Flory estimate, which is usually a good approximation. (1) 0.500 0.72 0.750 4 0.54 (2) 0.600 0.76 0.818 3 0.62 (2) 0.750 0.80 0.900 Table 1 Final results for critical exponents. ν is the Flory exponent and ζ is the roughness exponent.…”

“…The statistical properties of D-dimensional objects embedded in d-dimensional space have been the subject of intense analytical and numerical work in the last ten years. An introduction to the problem, as well as an update with some recent results has been already presented in M. Bowick's talk [1] and in the plenary talk [2]. These studies are of direct experimental interest for cases such as (D = 2, d = 3) (membranes) or (D = 1, d = 3) (polymers) (see [1]).…”