Invariant Correlations in Simplicial Gravity

Some first results are presented regarding the behavior of invariant correlations in simplicial gravity, with an action containing both a bare cosmological term and a lattice higher derivative term. The determination of invariant correlations as a function of geodesic distance by numerical methods is a difficult task, since the geodesic distance between any two points is a function of the fluctuating background geometry, and correlation effects become rather small for large distances. Still, a strikingly different behavior is found for the volume and curvature correlation functions. While the first one is found to be negative definite at large geodesic distances, the second one is always positive for large distances. For both correlations the results are consistent in the smooth phase with an exponential decay, turning into a power law close to the critical point at $G_c$. Such a behavior is not completely unexpected, if the model is to reproduce the classical Einstein theory at distances much larger than the ultraviolet cutoff scale.Comment: 27 pages, conforms to published versio

On the I=2 channel pi-pi interaction in the chiral limit

An approximate local potential for the residual pi+ pi+ interaction is computed. We use an O(a**2) improved action on a coarse 9x9x9x13 lattice with approximately a=0.4fm. The results present a continuation of previous work: Increasing the number of gauge configurations and quark propagators we attempt extrapolation of the pi+ pi+ potential to the chiral limit.Comment: LATTICE98(spectrum) LaTeX2e, 3 pages, 3 eps figure

In-plane deformation of a triangulated surface model with metric degrees of freedom

Using the canonical Monte Carlo simulation technique, we study a Regge calculus model on triangulated spherical surfaces. The discrete model is statistical mechanically defined with the variables $X$, $g$ and $\rho$, which denote the surface position in ${\bf R}^3$, the metric on a two-dimensional surface $M$ and the surface density of $M$, respectively. The metric $g$ is defined only by using the deficit angle of the triangles in {$M$}. This is in sharp contrast to the conventional Regge calculus model, where {$g$} depends only on the edge length of the triangles. We find that the discrete model in this paper undergoes a phase transition between the smooth spherical phase at $b to infty$ and the crumpled phase at $b to 0$, where $b$ is the bending rigidity. The transition is of first-order and identified with the one observed in the conventional model without the variables $g$ and $\rho$. This implies that the shape transformation transition is not influenced by the metric degrees of freedom. It is also found that the model undergoes a continuous transition of in-plane deformation. This continuous transition is reflected in almost discontinuous changes of the surface area of $M$ and that of $X(M)$, where the surface area of $M$ is conjugate to the density variable $\rho$.Comment: 13 pages, 7 figure

Two-body spectra of pseudoscalar mesons with an O(a2)O(a^2)--improved lattice action using Wilson fermions

We extend our calculations with the second-order tree-level and tadpole improved next-nearest-neighbor action to meson-meson systems. Correlation matrices built from interpolating fields representing two pseudoscalar mesons (pion-pion) with relative momenta p are diagonalized, and the mass spectrum is extracted. Link variable fuzzing and operator smearing at both sinks and sources is employed. Calculations are presented for two values of the hopping parameter. The spectrum is used to discuss the residual interaction in the meson-meson system.Comment: 3 pages, 4 EPS figures, Poster presented at "Lattice'97", to appear in the proceeding

Quantizing Horava-Lifshitz Gravity via Causal Dynamical Triangulations

We extend the discrete Regge action of causal dynamical triangulations to include discrete versions of the curvature squared terms appearing in the continuum action of (2+1)-dimensional projectable Horava-Lifshitz gravity. Focusing on an ensemble of spacetimes whose spacelike hypersurfaces are 2-spheres, we employ Markov chain Monte Carlo simulations to study the path integral defined by this extended discrete action. We demonstrate the existence of known and novel macroscopic phases of spacetime geometry, and we present preliminary evidence for the consistency of these phases with solutions to the equations of motion of classical Horava-Lifshitz gravity. Apparently, the phase diagram contains a phase transition between a time-dependent de Sitter-like phase and a time-independent phase. We speculate that this phase transition may be understood in terms of deconfinement of the global gravitational Hamiltonian integrated over a spatial 2-sphere.Comment: 24 pages; 10 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

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

Exact Renormalization Group and Running Newtonian Coupling in Higher Derivative Gravity

We discuss exact renormalization group (RG) in $R^2$-gravity using effective average action formalism. The truncated evolution equation for such a theory on De Sitter background leads to the system of nonperturbative RG equations for cosmological and gravitational coupling constants. Approximate solution of these RG equations shows the appearence of antiscreening and screening behaviour of Newtonian coupling what depends on higher derivative coupling constants.Comment: Latex file, 9 page

The Color--Flavor Transformation of induced QCD

The Zirnbauer's color-flavor transformation is applied to the $U(N_c)$ lattice gauge model, in which the gauge theory is induced by a heavy chiral scalar field sitting on lattice sites. The flavor degrees of freedom can encompass several `generations' of the auxiliary field, and for each generation, remaining indices are associated with the elementary plaquettes touching the lattice site. The effective, color-flavor transformed theory is expressed in terms of gauge singlet matrix fields carried by lattice links. The effective action is analyzed for a hypercubic lattice in arbitrary dimension. We investigate the corresponding d=2 and d=3 dual lattices. The saddle points equations of the model in the large-$N_c$ limit are discussed.Comment: 24 pages, 6 figures, to appear in Int. J. Mod. Phys.

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