On the diffeomorphism commutators of lattice quantum gravity

We show that the algebra of discretized spatial diffeomorphism constraints in Hamiltonian lattice quantum gravity closes without anomalies in the limit of small lattice spacing. The result holds for arbitrary factor-ordering and for a variety of different discretizations of the continuum constraints, and thus generalizes an earlier calculation by Renteln.Comment: 16 pages, Te

Putting a cap on causality violations in CDT

The formalism of causal dynamical triangulations (CDT) provides us with a non-perturbatively defined model of quantum gravity, where the sum over histories includes only causal space-time histories. Path integrals of CDT and their continuum limits have been studied in two, three and four dimensions. Here we investigate a generalization of the two-dimensional CDT model, where the causality constraint is partially lifted by introducing weighted branching points, and demonstrate that the system can be solved analytically in the genus-zero sector.Comment: 17 pages, 4 figure

Counting a black hole in Lorentzian product triangulations

We take a step toward a nonperturbative gravitational path integral for black-hole geometries by deriving an expression for the expansion rate of null geodesic congruences in the approach of causal dynamical triangulations. We propose to use the integrated expansion rate in building a quantum horizon finder in the sum over spacetime geometries. It takes the form of a counting formula for various types of discrete building blocks which differ in how they focus and defocus light rays. In the course of the derivation, we introduce the concept of a Lorentzian dynamical triangulation of product type, whose applicability goes beyond that of describing black-hole configurations.Comment: 42 pages, 11 figure

Emergence of a 4D World from Causal Quantum Gravity

Causal Dynamical Triangulations in four dimensions provide a background-independent definition of the sum over geometries in nonperturbative quantum gravity, with a positive cosmological constant. We present evidence that a macroscopic four-dimensional world emerges from this theory dynamically.Comment: 11 pages, 3 figures; some short clarifying comments added; final version to appear in Phys. Rev. Let

The moduli space of isometry classes of globally hyperbolic spacetimes

This is the last article in a series of three initiated by the second author. We elaborate on the concepts and theorems constructed in the previous articles. In particular, we prove that the GH and the GGH uniformities previously introduced on the moduli space of isometry classes of globally hyperbolic spacetimes are different, but the Cauchy sequences which give rise to well-defined limit spaces coincide. We then examine properties of the strong metric introduced earlier on each spacetime, and answer some questions concerning causality of limit spaces. Progress is made towards a general definition of causality, and it is proven that the GGH limit of a Cauchy sequence of $\mathcal{C}^{\pm}_{\alpha}$, path metric Lorentz spaces is again a $\mathcal{C}^{\pm}_{\alpha}$, path metric Lorentz space. Finally, we give a necessary and sufficient condition, similar to the one of Gromov for the Riemannian case, for a class of Lorentz spaces to be precompact.Comment: 29 pages, 9 figures, submitted to Class. Quant. Gra

The gravitational path integral and trace of the diffeomorphisms

I give a resolution of the conformal mode divergence in the Euclidean gravitational path-integral by isolating the trace of the diffeomorphisms and its contribution to the Faddeev-Popov measure.Comment: 20 pgs

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

Area spectrum in Lorentz covariant loop gravity

We use the manifestly Lorentz covariant canonical formalism to evaluate eigenvalues of the area operator acting on Wilson lines. To this end we modify the standard definition of the loop states to make it applicable to the present case of non-commutative connections. The area operator is diagonalized by using the usual shift ambiguity in definition of the connection. The eigenvalues are then expressed through quadratic Casimir operators. No dependence on the Immirzi parameter appears.Comment: 12 pages, RevTEX; improved layout, typos corrected, references added; changes in the discussion in sec. IIIB and

(Broken) Gauge Symmetries and Constraints in Regge Calculus

We will examine the issue of diffeomorphism symmetry in simplicial models of (quantum) gravity, in particular for Regge calculus. We find that for a solution with curvature there do not exist exact gauge symmetries on the discrete level. Furthermore we derive a canonical formulation that exactly matches the dynamics and hence symmetries of the covariant picture. In this canonical formulation broken symmetries lead to the replacements of constraints by so--called pseudo constraints. These considerations should be taken into account in attempts to connect spin foam models, based on the Regge action, with canonical loop quantum gravity, which aims at implementing proper constraints. We will argue that the long standing problem of finding a consistent constraint algebra for discretized gravity theories is equivalent to the problem of finding an action with exact diffeomorphism symmetries. Finally we will analyze different limits in which the pseudo constraints might turn into proper constraints. This could be helpful to infer alternative discretization schemes in which the symmetries are not broken.Comment: 32 pages, 15 figure

A proposal for analyzing the classical limit of kinematic loop gravity

We analyze the classical limit of kinematic loop quantum gravity in which the diffeomorphism and hamiltonian constraints are ignored. We show that there are no quantum states in which the primary variables of the loop approach, namely the SU(2) holonomies along {\em all} possible loops, approximate their classical counterparts. At most a countable number of loops must be specified. To preserve spatial covariance, we choose this set of loops to be based on physical lattices specified by the quasi-classical states themselves. We construct ``macroscopic'' operators based on such lattices and propose that these operators be used to analyze the classical limit. Thus, our aim is to approximate classical data using states in which appropriate macroscopic operators have low quantum fluctuations. Although, in principle, the holonomies of `large' loops on these lattices could be used to analyze the classical limit, we argue that it may be simpler to base the analysis on an alternate set of ``flux'' based operators. We explicitly construct candidate quasi-classical states in 2 spatial dimensions and indicate how these constructions may generalize to 3d. We discuss the less robust aspects of our proposal with a view towards possible modifications. Finally, we show that our proposal also applies to the diffeomorphism invariant Rovelli model which couples a matter reference system to the Hussain Kucha{\v r} model.Comment: Replaced with substantially revised versio