Lattice quantum gravity - an update

We advocate lattice methods as the tool of choice to constructively define a background-independent theory of Lorentzian quantum gravity and explore its physical properties in the Planckian regime. The formulation that arguably has most furthered our understanding of quantum gravity (and of various pitfalls present in the nonperturbative sector) uses dynamical triangulations to regularize the nonperturbative path integral over geometries. Its Lorentzian version in terms of Causal Dynamical Triangulations (CDT) - in addition to having a definite quantum signature on short scales - has been shown to reproduce important features of the classical theory on large scales. This article recaps the most important developments in CDT of the last few years for the physically relevant case of four spacetime dimensions, and describes its status quo at present.Comment: 14 pages, 8 figures, write-up of plenary talk at Lattice 2010, Villasimius, Sardegna, Italy, 14-19 June 201

Spectral Dimension of the Universe

We measure the spectral dimension of universes emerging from nonperturbative quantum gravity, defined through state sums of causal triangulated geometries. While four-dimensional on large scales, the quantum universe appears two-dimensional at short distances. We conclude that quantum gravity may be "self-renormalizing" at the Planck scale, by virtue of a mechanism of dynamical dimensional reduction.Comment: 10 pages, 1 figure, added referenc

The Universe from Scratch

A fascinating and deep question about nature is what one would see if one could probe space and time at smaller and smaller distances. Already the 19th-century founders of modern geometry contemplated the possibility that a piece of empty space that looks completely smooth and structureless to the naked eye might have an intricate microstructure at a much smaller scale. Our vastly increased understanding of the physical world acquired during the 20th century has made this a certainty. The laws of quantum theory tell us that looking at spacetime at ever smaller scales requires ever larger energies, and, according to Einstein's theory of general relativity, this will alter spacetime itself: it will acquire structure in the form of "curvature". What we still lack is a definitive Theory of Quantum Gravity to give us a detailed and quantitative description of the highly curved and quantum-fluctuating geometry of spacetime at this so-called Planck scale. - This article outlines a particular approach to constructing such a theory, that of Causal Dynamical Triangulations, and its achievements so far in deriving from first principles why spacetime is what it is, from the tiniest realms of the quantum to the large-scale structure of the universe.Comment: 31 pages, 5 figures; review paper commissioned by Contemporary Physics and aimed at a wider physics audience; minor beautifications, coincides with journal versio

Causal Dynamical Triangulations and the Quest for Quantum Gravity

Quantum Gravity by Causal Dynamical Triangulation has over the last few years emerged as a serious contender for a nonperturbative description of the theory. It is a nonperturbative implementation of the sum-over-histories, which relies on few ingredients and initial assumptions, has few free parameters and - crucially - is amenable to numerical simulations. It is the only approach to have demonstrated that a classical universe can be generated dynamically from Planckian quantum fluctuations. At the same time, it allows for the explicit evaluation of expectation values of invariants characterizing the highly nonclassical, short-distance behaviour of spacetime. As an added bonus, we have learned important lessons on which aspects of spacetime need to be fixed a priori as part of the background structure and which can be expected to emerge dynamically.Comment: To appear in "Foundations of Space and Time", Cambridge Univ. Press, eds. G. Ellis, J. Murugan, A Weltma

Quantum Gravity, or The Art of Building Spacetime

The method of four-dimensional Causal Dynamical Triangulations provides a background-independent definition of the sum over geometries in quantum gravity, in the presence of a positive cosmological constant. We present the evidence accumulated to date that a macroscopic four-dimensional world can emerge from this theory dynamically. Using computer simulations we observe in the Euclidean sector a universe whose scale factor exhibits the same dynamics as that of the simplest mini-superspace models in quantum cosmology, with the distinction that in the case of causal dynamical triangulations the effective action for the scale factor is not put in by hand but obtained by integrating out {\it in the quantum theory} the full set of dynamical degrees of freedom except for the scale factor itself.Comment: 22 pages, 6 figures. Contribution to the book "Approaches to Quantum Gravity", ed. D. Oriti, Cambridge University Pres

Quantum Gravity via Causal Dynamical Triangulations

"Causal Dynamical Triangulations" (CDT) represent a lattice regularization of the sum over spacetime histories, providing us with a non-perturbative formulation of quantum gravity. The ultraviolet fixed points of the lattice theory can be used to define a continuum quantum field theory, potentially making contact with quantum gravity defined via asymptotic safety. We describe the formalism of CDT, its phase diagram, and the quantum geometries emerging from it. We also argue that the formalism should be able to describe a more general class of quantum-gravitational models of Horava-Lifshitz type.Comment: To appear in "Handbook of Spacetime", Springer Verlag. 31 page

CDT---an Entropic Theory of Quantum Gravity

In these lectures we describe how a theory of quantum gravity may be constructed in terms of a lattice formulation based on so-called causal dynamical triangulations (CDT). We discuss how the continuum limit can be obtained and how to define and measure diffeomorphism-invariant correlators. In four dimensions, which has our main interest, the lattice theory has an infrared limit which can be identified with de Sitter spacetime. We explain why this infrared property of the quantum spacetime is nontrivial and due to "entropic" effects encoded in the nonperturbative path integral measure. This makes the appearance of the de Sitter universe an example of true emergence of classicality from microscopic quantum laws. We also discuss nontrivial aspects of the UV behaviour, and show how to investigate quantum fluctuations around the emergent background geometry. Finally, we consider the connection to the asymptotic safety scenario, and derive from it a new, conjectured scaling relation in CDT quantum gravity.Comment: 49 pages, many figures. Lectures presented at the "School on Non-Perturbative Methods in Quantum Field Theory" and the "Workshop on Continuum and Lattice Approaches to Quantum Gravity", Sussex, September 15th-19th 2008 . To appear as a contribution to a Springer Lecture Notes in Physics boo

3D Lorentzian Quantum Gravity from the asymmetric ABAB matrix model

The asymmetric ABAB-matrix model describes the transfer matrix of three-dimensional Lorentzian quantum gravity. We study perturbatively the scaling of the ABAB-matrix model in the neighbourhood of its symmetric solution and deduce the associated renormalization of three-dimensional Lorentzian quantum gravity.Comment: 21 pages, typo in references correcte

Wilson loops in CDT quantum gravity

By explicit construction, we show that one can in a simple way introduce and measure gravitational holonomies and Wilson loops in lattice formulations of nonperturbative quantum gravity based on (Causal) Dynamical Triangulations. We use this set-up to investigate a class of Wilson line observables associated with the world line of a point particle coupled to quantum gravity, and deduce from their expectation values that the underlying holonomies cover the group manifold of SO(4) uniforml