Lattice Models of Quantum Gravity

Standard Regge Calculus provides an interesting method to explore quantum gravity in a non-perturbative fashion but turns out to be a CPU-time demanding enterprise. One therefore seeks for suitable approximations which retain most of its universal features. The $Z_2$-Regge model could be such a desired simplification. Here the quadratic edge lengths $q$ of the simplicial complexes are restricted to only two possible values $q=1+\epsilon\sigma$, with $\sigma=\pm 1$, in close analogy to the ancestor of all lattice theories, the Ising model. To test whether this simpler model still contains the essential qualities of the standard Regge Calculus, we study both models in two dimensions and determine several observables on the same lattice size. In order to compare expectation values, e.g. of the average curvature or the Liouville field susceptibility, we employ in both models the same functional integration measure. The phase structure is under current investigation using mean field theory and numerical simulation.Comment: 4 pages, 1 figure

Positivity in Lorentzian Barrett-Crane Models of Quantum Gravity

The Barrett-Crane models of Lorentzian quantum gravity are a family of spin foam models based on the Lorentz group. We show that for various choices of edge and face amplitudes, including the Perez-Rovelli normalization, the amplitude for every triangulated closed 4-manifold is a non-negative real number. Roughly speaking, this means that if one sums over triangulations, there is no interference between the different triangulations. We prove non-negativity by transforming the model into a ``dual variables'' formulation in which the amplitude for a given triangulation is expressed as an integral over three copies of hyperbolic space for each tetrahedron. Then we prove that, expressed in this way, the integrand is non-negative. In addition to implying that the amplitude is non-negative, the non-negativity of the integrand is highly significant from the point of view of numerical computations, as it allows statistical methods such as the Metropolis algorithm to be used for efficient computation of expectation values of observables.Comment: 13 page

Towards a spin foam model description of black hole entropy

We propose a way to describe the origin of black hole entropy in the spin foam models of quantum gravity. This stimulates a new way to study the relation of spin foam models and loop quantum gravity.Comment: 5 pages, 1 figur

Continuous point symmetries in Group Field Theories

We discuss the notion of symmetries in non-local field theories characterized by integro-differential equations of motion, from a geometric perspective. We then focus on Group Field Theory (GFT) models of quantum gravity and provide a general analysis of their continuous point symmetry transformations, including the generalized conservation laws following from them

Relating Covariant and Canonical Approaches to Triangulated Models of Quantum Gravity

In this paper explore the relation between covariant and canonical approaches to quantum gravity and $BF$ theory. We will focus on the dynamical triangulation and spin-foam models, which have in common that they can be defined in terms of sums over space-time triangulations. Our aim is to show how we can recover these covariant models from a canonical framework by providing two regularisations of the projector onto the kernel of the Hamiltonian constraint. This link is important for the understanding of the dynamics of quantum gravity. In particular, we will see how in the simplest dynamical triangulations model we can recover the Hamiltonian constraint via our definition of the projector. Our discussion of spin-foam models will show how the elementary spin-network moves in loop quantum gravity, which were originally assumed to describe the Hamiltonian constraint action, are in fact related to the time-evolution generated by the constraint. We also show that the Immirzi parameter is important for the understanding of a continuum limit of the theory.Comment: 28 pages, 10 figure