On the Measure in Simplicial Gravity

Functional measures for lattice quantum gravity should agree with their continuum counterparts in the weak field, low momentum limit. After showing that the standard simplicial measure satisfies the above requirement, we prove that a class of recently proposed non-local measures for lattice gravity do not satisfy such a criterion, already to lowest order in the weak field expansion. We argue therefore that the latter cannot represent acceptable discrete functional measures for simplicial geometries.Comment: LaTeX, 15 pages, 2 figure

Gravitational Wilson Loop in Discrete Quantum Gravity

Results for the gravitational Wilson loop, in particular the area law for large loops in the strong coupling region, and the argument for an effective positive cosmological constant discussed in a previous paper, are extended to other proposed theories of discrete quantum gravity in the strong coupling limit. We argue that the area law is a generic feature of almost all non-perturbative lattice formulations, for sufficiently strong gravitational coupling. The effects on gravitational Wilson loops of the inclusion of various types of light matter coupled to lattice quantum gravity are discussed as well. One finds that significant modifications to the area law can only arise from extremely light matter particles. The paper ends with some general comments on possible physically observable consequences.Comment: 39 pages, 10 figure

Nonlocal Effective Field Equations for Quantum Cosmology

The possibility that the strength of gravitational interactions might slowly increase with distance, is explored by formulating a set of effective field equations, which incorporate the gravitational, vacuum-polarization induced, running of Newton's constant $G$. The resulting long distance (or large time) behaviour depends on only one adjustable parameter $\xi$, and the implications for the Robertson-Walker universe are calculated, predicting an accelerated power-law expansion at later times $t \sim \xi \sim 1/H$.Comment: 9 page

Gravitational Wilson Loop and Large Scale Curvature

In a quantum theory of gravity the gravitational Wilson loop, defined as a suitable quantum average of a parallel transport operator around a large near-planar loop, provides important information about the large-scale curvature properties of the geometry. Here we shows that such properties can be systematically computed in the strong coupling limit of lattice regularized quantum gravity, by performing a local average over rotations, using an assumed near-uniform measure in group space. We then relate the resulting quantum averages to an expected semi-classical form valid for macroscopic observers, which leads to an identification of the gravitational correlation length appearing in the Wilson loop with an observed large-scale curvature. Our results suggest that strongly coupled gravity leads to a positively curved (De Sitter-like) quantum ground state, implying a positive effective cosmological constant at large distances.Comment: 22 pages, 6 figure

Non-Perturbative Gravity and the Spin of the Lattice Graviton

The lattice formulation of quantum gravity provides a natural framework in which non-perturbative properties of the ground state can be studied in detail. In this paper we investigate how the lattice results relate to the continuum semiclassical expansion about smooth manifolds. As an example we give an explicit form for the lattice ground state wave functional for semiclassical geometries. We then do a detailed comparison between the more recent predictions from the lattice regularized theory, and results obtained in the continuum for the non-trivial ultraviolet fixed point of quantum gravity found using weak field and non-perturbative methods. In particular we focus on the derivative of the beta function at the fixed point and the related universal critical exponent $\nu$ for gravitation. Based on recently available lattice and continuum results we assess the evidence for the presence of a massless spin two particle in the continuum limit of the strongly coupled lattice theory. Finally we compare the lattice prediction for the vacuum-polarization induced weak scale dependence of the gravitational coupling with recent calculations in the continuum, finding similar effects.Comment: 46 pages, one figur

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

Discrete structures in gravity

Discrete approaches to gravity, both classical and quantum, are reviewed briefly, with emphasis on the method using piecewise-linear spaces. Models of 3-dimensional quantum gravity involving 6j-symbols are then described, and progress in generalising these models to four dimensions is discussed, as is the relationship of these models in both three and four dimensions to topological theories. Finally, the repercussions of the generalisations are explored for the original formulation of discrete gravity using edge-length variables.Comment: 30 pages, 4 figure

Renormalization Group Running of Newton's G: The Static Isotropic Case

Corrections are computed to the classical static isotropic solution of general relativity, arising from non-perturbative quantum gravity effects. A slow rise of the effective gravitational coupling with distance is shown to involve a genuinely non-perturbative scale, closely connected with the gravitational vacuum condensate, and thereby, it is argued, related to the observed effective cosmological constant. Several analogies between the proposed vacuum condensate picture of quantum gravitation, and non-perturbative aspects of vacuum condensation in strongly coupled non-abelian gauge theories are developed. In contrast to phenomenological approaches, the underlying functional integral formulation of the theory severely constrains possible scenarios for the renormalization group evolution of couplings. The expected running of Newton's constant $G$ is compared to known vacuum polarization induced effects in QED and QCD. The general analysis is then extended to a set of covariant non-local effective field equations, intended to incorporate the full scale dependence of $G$, and examined in the case of the static isotropic metric. The existence of vacuum solutions to the effective field equations in general severely restricts the possible values of the scaling exponent $\nu$.Comment: 61 pages, 3 figure

Quantum Gravity in Large Dimensions

Quantum gravity is investigated in the limit of a large number of space-time dimensions, using as an ultraviolet regularization the simplicial lattice path integral formulation. In the weak field limit the appropriate expansion parameter is determined to be $1/d$. For the case of a simplicial lattice dual to a hypercube, the critical point is found at $k_c/\lambda=1/d$ (with $k=1/8 \pi G$) separating a weak coupling from a strong coupling phase, and with $2 d^2$ degenerate zero modes at $k_c$. The strong coupling, large $G$, phase is then investigated by analyzing the general structure of the strong coupling expansion in the large $d$ limit. Dominant contributions to the curvature correlation functions are described by large closed random polygonal surfaces, for which excluded volume effects can be neglected at large $d$, and whose geometry we argue can be approximated by unconstrained random surfaces in this limit. In large dimensions the gravitational correlation length is then found to behave as $| \log (k_c - k) |^{1/2}$, implying for the universal gravitational critical exponent the value $\nu=0$ at $d=\infty$.Comment: 47 pages, 2 figure

Wheeler-DeWitt Equation in 2 + 1 Dimensions

The infrared structure of quantum gravity is explored by solving a lattice version of the Wheeler-DeWitt equations. In the present paper only the case of 2+1 dimensions is considered. The nature of the wavefunction solutions is such that a finite correlation length emerges and naturally cuts off any infrared divergences. Properties of the lattice vacuum are consistent with the existence of an ultraviolet fixed point in $G$ located at the origin, thus precluding the existence of a weak coupling perturbative phase. The correlation length exponent is determined exactly and found to be $\nu=6/11$. The results obtained so far lend support to the claim that the Lorentzian and Euclidean formulations belong to the same field-theoretic universality class.Comment: 56 pages, 7 figures, typos fixed, references adde