Search

Being able to perform explicit computations in a nonperturbative, Planckian regime is key to understanding quantum gravity as a fundamental theory of gravity and spacetime. Rather than a variety of different approaches to quantum gravity, what we primarily need is a gravitational analogue of the highly successful lattice treatment of nonperturbative quantum chromodynamics. Unsurprisingly, however, lattice quantum gravity is not simple. The crucial insight that has finally led to success is to build the dynamical and Lorentzian nature of spacetime into the lattices from the outset. Lattice quantum gravity based on causal dynamical triangulations (CDT) puts this idea into practice and is producing new and exciting physical results from numerical experiments. This largely nontechnical account describes the challenges and achievements of modern lattice quantum gravity, which has opened an unprecedented computational window on quantum spacetime in a Planckian regime and is reshaping our understanding of what it means to "solve" quantum gravity. This methodology is well placed to unlock the physics of the early universe from first principles. Related topics discussed are the difference between lattice and discrete quantum gravity, and the role of spacetime emergence in the light of computational results., Comment: 20 pages. Dedicated to Chris Isham. To appear in D. Rickles, X. Arsiwalla, and H. Elshatlawy (eds.), Quantum Gravity and Computation (Routledge)

We re-examine the nonperturbative curvature properties of two-dimensional Euclidean quantum gravity, obtained as the scaling limit of a path integral over dynamical triangulations of a two-sphere, which lies in the same universality class as Liouville quantum gravity. The diffeomorphism-invariant observable that allows us to compare the averaged curvature of highly quantum-fluctuating geometries with that of classical spaces is the so-called curvature profile. A Monte Carlo analysis on three geometric ensembles, which are physically equivalent but differ by the inclusion of local degeneracies, leads to new insights on the influence of finite-size effects. After eliminating them, we find strong evidence that the curvature profile of 2D Euclidean quantum gravity is best matched by that of a classical round four-sphere, rather than the five-sphere found in previous work. Our analysis suggests the existence of a well-defined quantum Ricci curvature in the scaling limit., Comment: 38 pages, 18 figures

Correlation functions are ubiquitous tools in quantum field theory from both a fundamental and a practical point of view. However, up to now their use in theories of quantum gravity beyond perturbative and asymptotically flat regimes has been limited, due to difficulties associated with diffeomorphism invariance and the dynamical nature of geometry. We present an analysis of a manifestly diffeomorphism-invariant, nonperturbative two-point curvature correlator in two-dimensional Lorentzian quantum gravity. It is based on the recently introduced quantum Ricci curvature and uses a lattice regularization of the full path integral in terms of causal dynamical triangulations. We discuss some of the subtleties and ambiguities in defining connected correlators in theories of dynamical geometry, and provide strong evidence from Monte Carlo simulations that the connected two-point curvature correlator in 2D Lorentzian quantum gravity vanishes. This work paves the way for an analogous investigation in higher dimensions., Comment: 34 pages, 9 figures

Curvature is a key notion in General Relativity, characterizing the local physical properties of spacetime. By contrast, the concept of curvature has received scant attention in nonperturbative quantum gravity. One may even wonder whether in a Planckian regime meaningful notions of (quantum) curvature exist at all. Remarkably, recent work in quantum gravity using Causal Dynamical Triangulations (CDT) has demonstrated both the existence and usefulness of a new notion of quantum Ricci curvature (QRC), which relies neither on smooth structures nor on tensor calculus. This overview article recalls some classical notions related to curvature and parallel transport, as well as previous unsuccessful attempts to construct quantum curvature observables based on deficit angles and Wilson loops. It introduces the quasi-local QRC on piecewise flat triangulations, and describes its behaviour in a purely classical setting, its use in quantum observables, and currently known results in (C)DT quantum gravity in two and four dimensions. The QRC opens the door to a range of interesting physical observables that were previously out of reach, and will help to bridge the gap between the nonperturbative quantum theory and gravitational phenomena at lower energies., Comment: 38 pages, 17 figures, contribution to Handbook of Quantum Gravity (eds. C. Bambi, L. Modesto and I.L. Shapiro, Springer Singapore, expected in 2023)

There are not many tools to quantitatively monitor the emergence of classical geometric features from a quantum spacetime, whose microscopic structure may be a highly quantum-fluctuating "spacetime foam". To improve this situation, we introduce new quantum observables that allow us to measure the absolute and relative homogeneity of geometric properties of a nonperturbative quantum universe, as function of a chosen averaging scale. This opens a new way to compare results obtained in full quantum gravity to descriptions of the early universe that assume homogeneity and isotropy at the outset. Our construction is purely geometric and does not depend on a background metric. We illustrate the viability of the quantum homogeneity measures by a nontrivial application to two-dimensional Lorentzian quantum gravity formulated in terms of a path integral over Causal Dynamical Triangulations, and find some evidence of quantum inhomogeneity., Comment: 30 pages, 7 figures

Three-dimensional Lorentzian quantum gravity, expressed as the continuum limit of a nonperturbative sum over spacetimes, is tantalizingly close to being amenable to analytical methods, and some of its properties have been described in terms of effective matrix and other models. To gain a more detailed understanding of three-dimensional quantum gravity, we perform a numerical investigation of the nature of spatial hypersurfaces in three-dimensional Causal Dynamical Triangulations (CDT). We measure and analyze several quantum observables, the entropy exponent, the local and global Hausdorff dimensions, and the quantum Ricci curvature of the spatial slices, and try to match them with known continuum properties of systems of two-dimensional quantum geometry. Above the first-order phase transition of CDT quantum gravity, we find strong evidence that the spatial dynamics lies in the same universality class as two-dimensional Euclidean (Liouville) quantum gravity. Below the transition, the behaviour of the spatial slices does not match that of any known quantum gravity model. This may indicate the existence of a new type of two-dimensional quantum system, induced by the more complex nature of the embedding three-dimensional quantum geometry., Comment: 42 pages, 17 figures

Quantum gravity is the missing piece in our understanding of the fundamental interactions today. Given recent observational breakthroughs in gravity, providing a quantum theory for what lies beyond general relativity is more urgent than ever. However, the complex history of quantum gravity and the multitude of available approaches can make it difficult to get a grasp of the topic and its main challenges and opportunities. We provide a guided tour of quantum gravity in the form of 30 questions, aimed at a mixed audience of learners and practitioners. The issues covered range from basic motivational and background material to a critical assessment of the status quo and future of the subject. The emphasis is on structural issues and our current understanding of quantum gravity as a quantum field theory of dynamical geometry beyond perturbation theory. We highlight the identification of quantum observables and the development of effective numerical tools as critical to future progress., Comment: 63 pages, 3 figures, extended version of a set of lectures given by R. Loll at the School on Quantum Gravity Phenomenology in the Multi-Messenger Approach, Corfu, Greece, Sep 2021

Flatness -- the absence of spacetime curvature -- is a well-understood property of macroscopic, classical spacetimes in general relativity. The same cannot be said about the concepts of curvature and flatness in nonperturbative quantum gravity, where the microscopic structure of spacetime is not describable in terms of small fluctuations around a fixed background geometry. An interesting case are two-dimensional models of quantum gravity, which lack a classical limit and therefore are maximally "quantum". We investigate the recently introduced quantum Ricci curvature in CDT quantum gravity on a two-dimensional torus, whose quantum geometry could be expected to behave like a flat space on suitably coarse-grained scales. On the basis of Monte Carlo simulations we have performed, with system sizes of up to 600.000 building blocks, this does not seem to be the case. Instead, we find a scale-independent "quantum flatness", without an obvious classical analogue. As part of our study, we develop a criterion that allows us to distinguish between local and global, topological properties of the toroidal quantum system., Comment: 33 pages,16 figures

Building on the recently introduced notion of quantum Ricci curvature and motivated by considerations in nonperturbative quantum gravity, we advocate a new, global observable for curved metric spaces, the curvature profile. It is obtained by integrating the scale-dependent, quasi-local quantum Ricci curvature, and therefore also depends on a coarse-graining scale. To understand how the distribution of local, Gaussian curvature is reflected in the curvature profile, we compute it on a class of regular polygons with isolated conical singularities. We focus on the case of the tetrahedron, for which we have a good computational control of its geodesics, and compare its curvature profile to that of a smooth sphere. The two are distinct, but qualitatively similar, which confirms that the curvature profile has averaging properties which are interesting from a quantum point of view., Comment: 30 pages, 14 figures

We investigate the quantum Ricci curvature, which was introduced in earlier work, in full, four-dimensional quantum gravity, formulated nonperturbatively in terms of Causal Dynamical Triangulations (CDT). A key finding of the CDT approach is the emergence of a universe of de Sitter-type, as evidenced by the successful matching of Monte Carlo measurements of the quantum dynamics of the global scale factor with a semiclassical minisuperspace model. An important question is whether the quantum universe exhibits semiclassicality also with regard to its more local geometric properties. Using the new quantum curvature observable, we examine whether the (quasi-)local properties of the quantum geometry resemble those of a constantly curved space. We find evidence that on sufficiently large scales the curvature behaviour is compatible with that of a four-sphere, thus strengthening the interpretation of the dynamically generated quantum universe in terms of a de Sitter space., Comment: 24 pages, 8 figures

Finding diffeomorphism-invariant observables to characterize the properties of gravity and spacetime at the Planck scale is essential for making progress in quantum gravity. The holonomy and Wilson loop of the Levi-Civita connection are potentially interesting ingredients in the construction of quantum curvature observables. Motivated by recent developments in nonperturbative quantum gravity, we establish new relations in three and four dimensions between the holonomy of a finite loop and certain curvature integrals over the surface spanned by the loop. They are much simpler than a gravitational version of the nonabelian Stokes' theorem, but require the presence of totally geodesic surfaces in the manifold, which follows from the existence of suitable Killing vectors. We show that the relations are invariant under smooth surface deformations, due to the presence of a conserved geometric flux., Comment: 36 pages, 5 figures; minor text changes, clarifying the role of diffeomorphism invariance; agrees with published version

A hallmark of non-perturbative theories of quantum gravity is the absence of a fixed background geometry, and therefore the absence in a Planckian regime of any notion of length or scale that is defined a priori. This has potentially far-reaching consequences for the application of renormalization group methods a la Wilson, which rely on these notions in a crucial way. We review the status quo of attempts in the Causal Dynamical Triangulations (CDT) approach to quantum gravity to find an ultraviolet fixed point associated with the second-order phase transitions observed in the lattice theory. Measurements of the only invariant correlator currently accessible, that of the total spatial three-volume, has not produced any evidence of such a fixed point. A possible explanation for this result is our incomplete and perhaps naive understanding of what constitutes an appropriate notion of (quantum) length near the Planck scale.

In this paper, which is of programmatic rather than quantitative nature, we aim to further delineate and sharpen the future potential of the LISA mission in the area of fundamental physics. Given the very broad range of topics that might be relevant to LISA, we present here a sample of what we view as particularly promising directions, based in part on the current research interests of the LISA scientific community in the area of fundamental physics. We organize these directions through a "science-first" approach that allows us to classify how LISA data can inform theoretical physics in a variety of areas. For each of these theoretical physics classes, we identify the sources that are currently expected to provide the principal contribution to our knowledge, and the areas that need further development. The classification presented here should not be thought of as cast in stone, but rather as a fluid framework that is amenable to change with the flow of new insights in theoretical physics., Comment: 22 pages, 1 figure, to appear in General Relativity and Gravitation

This topical review gives a comprehensive overview and assessment of recent results in Causal Dynamical Triangulations (CDT), a modern formulation of lattice gravity, whose aim is to obtain a theory of quantum gravity nonperturbatively from a scaling limit of the lattice-regularized theory. In this manifestly diffeomorphism-invariant approach one has direct, computational access to a Planckian spacetime regime, which is explored with the help of invariant quantum observables. During the last few years, there have been numerous new and important developments and insights concerning the theory's phase structure, the roles of time, causality, diffeomorphisms and global topology, the application of renormalization group methods and new observables. We will focus on these new results, primarily in four spacetime dimensions, and discuss some of their geometric and physical implications., Comment: 64 pages, 28 figures

Quantum Ricci curvature has been introduced recently as a new, geometric observable characterizing the curvature properties of metric spaces, without the need for a smooth structure. Besides coordinate invariance, its key features are scalability, computability and robustness. We demonstrate that these properties continue to hold in the context of nonperturbative quantum gravity, by evaluating the quantum Ricci curvature numerically in two-dimensional Euclidean quantum gravity, defined in terms of dynamical triangulations. Despite the well-known, highly nonclassical properties of the underlying quantum geometry, its Ricci curvature can be matched well to that of a five-dimensional round sphere., Comment: 15 pages, 7 figures

Motivated by the search for geometric observables in nonperturbative quantum gravity, we define a notion of coarse-grained Ricci curvature. It is based on a particular way of extracting the local Ricci curvature of a smooth Riemannian manifold by comparing the distance between pairs of spheres with that of their centres. The quantum Ricci curvature is designed for use on non-smooth and discrete metric spaces, and to satisfy the key criteria of scalability and computability. We test the prescription on a variety of regular and random piecewise flat spaces, mostly in two dimensions. This enables us to quantify its behaviour for short lattices distances and compare its large-scale behaviour with that of constantly curved model spaces. On the triangulated spaces considered, the quantum Ricci curvature has good averaging properties and reproduces classical characteristics on scales large compared to the discretization scale., Comment: 43 pages, 27 figures

In the approach of Causal Dynamical Triangulations (CDT), quantum gravity is obtained as a scaling limit of a non-perturbative path integral over space-times whose causal structure plays a crucial role in the construction. After some general considerations about the relation between quantum gravity and cosmology, we examine which aspects of CDT are potentially interesting from a cosmological point of view, focussing on the emergence of a de Sitter universe in CDT quantum gravity., Comment: 14 pages, invited review for a special issue of Compte Rendus Physique

Causal Dynamical Triangulations (CDT), a candidate theory of nonperturbative quantum gravity in 4D, turns out to have a rich phase structure. We investigate the recently discovered bifurcation phase $C_b$ and relate some of its characteristics to the presence of singular vertices of very high order. The transition lines separating this phase from the "time-collapsed" $B$-phase and the de Sitter phase $C_{dS}$ are of great interest when searching for physical scaling limits. The work presented here sheds light on the mechanisms behind these transitions. First, we study how the $B$-$C_b$ transition signal depends on the volume-fixing implemented in the simulations, and find results compatible with the previously determined second-order character of the transition. The transition persists in a transfer matrix formulation, where the system's time extension is taken to be minimal. Second, we relate the new $C_b$-$C_{dS}$ transition to the appearance of singular vertices, which leads to a direct physical interpretation in terms of a breaking of the homogeneity and isotropy observed in the de Sitter phase when crossing from $C_{dS}$ to the bifurcation phase $C_b$., Comment: 32 pages, 17 figures

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) uniformly

We analyze different claims on the role of the coupling constant lambda in so-called lambda-R models, a minimal generalization of general relativity inspired by Horava-Lifshitz gravity. The dimensionless parameter lambda appears in the kinetic term of the Einstein-Hilbert action, leading to a one-parameter family of classical theories. Performing a canonical constraint analysis for closed spatial hypersurfaces, we obtain a result analogous to that of Bellorin and Restuccia, who showed that all non-projectable lambda-R models are equivalent to general relativity in the asymptotically flat case. However, the tertiary constraint present for closed boundary conditions assumes a more general form. We juxtapose this with an earlier finding by Giulini and Kiefer, who ruled out a range of lambda-R models by a physical, cosmological argument. We show that their analysis can be interpreted consistently within the projectable sector of Horava-Lifshitz gravity, thus resolving the apparent contradiction., Comment: 21 pages, no figures

We perform a first investigation of the coupling constant flow of the nonperturbative lattice model of four-dimensional quantum gravity given in terms of Causal Dynamical Triangulations (CDT). After explaining how standard concepts of lattice field theory can be adapted to the case of this background-independent theory, we define a notion of "lines of constant physics" in coupling constant space in terms of certain semiclassical properties of the dynamically generated quantum universe. Determining flow lines with the help of Monte Carlo simulations, we find that the second-order phase transition line present in this theory can be interpreted as a UV phase transition line if we allow for an anisotropic scaling of space and time., Comment: Typos corrected, 21 pages

We present a detailed analysis of a recently introduced version of Causal Dynamical Triangulations (CDT) that does not rely on a distinguished time slicing. Focussing on the case of 2+1 spacetime dimensions, we analyze its geometric and causal properties, present details of the numerical set-up and explain how to extract "volume profiles". Extensive Monte Carlo measurements of the system show the emergence of a de Sitter universe on large scales from the underlying quantum ensemble, similar to what was observed previously in standard CDT quantum gravity. This provides evidence that the distinguished time slicing of the latter is not an essential part of its kinematical set-up., Comment: 44 pages, 29 figures

Causal Dynamical Triangulations provide a non-perturbative regularization of a theory of quantum gravity. We describe how this approach connects with the asymptotic safety program and Ho\vrava-Lifshitz gravity theory, and present the most recent results from computer simulations., Comment: 23 pages, 5 figures. Based on plenary talk at MG13. To appear in the proceeding

Motivated by the search for new observables in nonperturbative quantum gravity, we consider Causal Dynamical Triangulations (CDT) in 2+1 dimensions with the spatial topology of a torus. This system is of particular interest, because one can study not only the global scale factor, but also global shape variables in the presence of arbitrary quantum fluctuations of the geometry. Our initial investigation focusses on the dynamics of the scale factor and uncovers a qualitatively new behaviour, which leads us to investigate a novel type of boundary conditions for the path integral. Comparing large-scale features of the emergent quantum geometry in numerical simulations with a classical minisuperspace formulation, we find partial agreement. By measuring the correlation matrix of volume fluctuations we succeed in reconstructing the effective action for the scale factor directly from the simulation data. Apart from setting the stage for the analysis of shape dynamics on the torus, the new set-up highlights the role of nontrivial boundaries and topology., Comment: 30 pages, 12 figures

We introduce a generalized version of the Causal Dynamical Triangulations (CDT) formulation of quantum gravity, in which the regularized, triangulated path integral histories retain their causal properties, but do not have a preferred proper-time foliation. An extensive numerical study of the associated nonperturbative path integral in 2+1 dimensions shows that it can nevertheless reproduce the emergence of an extended de Sitter universe on large scales, a key feature of CDT quantum gravity. This suggests that the preferred foliation normally used in CDT is not a crucial (albeit convenient) part of its background structure., Comment: 12 pages, 8 figures

The Causal Dynamical Triangulation model of quantum gravity (CDT) is a proposition to evaluate the path integral over space-time geometries using a lattice regularization with a discrete proper time and geometries realized as simplicial manifolds. The model admits a Wick rotation to imaginary time for each space-time configuration. Using computer simulations we determined the phase structure of the model and discovered that it predicts a de Sitter phase with a four-dimensional spherical semi-classical background geometry. The model has a transfer matrix, relating spatial geometries at adjacent (discrete lattice) times. The transfer matrix uniquely determines the theory. We show that the measurements of the scale factor of the (CDT) universe are well described by an effective transfer matrix where the matrix elements are labelled only by the scale factor. Using computer simulations we determine the effective transfer matrix elements and show how they relate to an effective minisuperspace action at all scales., Comment: 6 pages, 6 figures, contribution to the MULTIVERSE conference, Szczecin, Poland, September 2012

We argue that theories of quantum gravity constructed with the help of (Causal) Dynamical Triangulations have given us the most informative, quantitative models to date of quantum spacetime. Most importantly, these are derived dynamically from nonperturbative and background-independent quantum theories of geometry. In the physically relevant case of four spacetime dimensions, the ansatz of Causal Dynamical Triangulations produces - from a fairly minimal set of quantum field-theoretic inputs - an emergent spacetime which macroscopically looks like a de Sitter universe, and on Planckian scales possesses unexpected quantum properties. Important in deriving these results are a regularized version of the theory, in which the quantum dynamics is well defined, can be studied with the help of numerical Monte Carlo methods and extrapolated to infinite lattice volumes., Comment: 7 pages, 5 figures, submission to Multicosmofun '12, Szczecin

"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 pages

Causal Dynamical Triangulations (CDT) is a proposal for a theory of quantum gravity, which implements a path-integral quantization of gravity as the continuum limit of a sum over piecewise flat spacetime geometries. We use Monte Carlo simulations to analyse the phase transition lines bordering the physically interesting de Sitter phase of the four-dimensional CDT model. Using a range of numerical criteria, we present strong evidence that the so-called A-C transition is first order, while the B-C transition is second order. The presence of a second-order transition may be related to an ultraviolet fixed point of quantum gravity and thus provide the key to probing physics at and possibly beyond the Planck scale., Comment: 24 pages, 9 figures

Asymptotic safety describes a scenario in which general relativity can be quantized as a conventional field theory, despite being nonrenormalizable when expanding it around a fixed background geometry. It is formulated in the framework of the Wilsonian renormalization group and relies crucially on the existence of an ultraviolet fixed point, for which evidence has been found using renormalization group equations in the continuum. "Causal Dynamical Triangulations" (CDT) is a concrete research program to obtain a nonperturbative quantum field theory of gravity via a lattice regularization, and represented as a sum over spacetime histories. In the Wilsonian spirit one can use this formulation to try to locate fixed points of the lattice theory and thereby provide independent, nonperturbative evidence for the existence of a UV fixed point. We describe the formalism of CDT, its phase diagram, possible fixed points and the "quantum geometries" which emerge in the different phases. We also argue that the formalism may be able to describe a more general class of Ho\v{r}ava-Lifshitz gravitational models., Comment: Review, 146 pages, many figures

The behaviour of baby universes has been an important ingredient in understanding and quantifying non-critical string theory or, equivalently, models of two-dimensional Euclidean quantum gravity coupled to matter. Within a regularized description based on dynamical triangulations, we amend an earlier conjecture by Jain and Mathur on the scaling behaviour of genus-$g$ surfaces containing particular baby universe `necks', and perform a nontrivial numerical check on our improved conjecture., Comment: 10 pages, 1 figure

Causal Dynamical Triangulations (CDT) are a concrete attempt to define a nonperturbative path integral for quantum gravity. We present strong evidence that the lattice theory has a second-order phase transition line, which can potentially be used to define a continuum limit in the conventional sense of nongravitational lattice theories., Comment: 10 pages, 3 figures

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 2010

Previous work has shown that the macroscopic structure of the theory of quantum gravity defined by causal dynamical triangulations (CDT) is compatible with that of a de Sitter universe. After emphasizing the strictly nonperturbative nature of this semiclassical limit we present a detailed study of the three-volume data, which allows us to re-confirm the de Sitter structure, exhibit short-distance discretization effects, and make a first detailed investigation of the presence of higher-order curvature terms in the effective action for the scale factor. Technically, we make use of a novel way of fixing the total four-volume in the simulations., Comment: 30 pages, 10 figures

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 book

In this short note we review a recently found formulation of two-dimensional causal quantum gravity defined through Causal Dynamical Triangulations and stochastic quantization. This procedure enables one to extract the nonperturbative quantum Hamiltonian of the random surface model including the sum over topologies. Interestingly, the generally fictitious stochastic time corresponds to proper time on the geometries., Comment: 5 pages, 2 figures, presented at XI Latin American Workshop on Nonlinear Phenomena, Buzios, 2009, accepted for publication in Journal of Physics: Conference Proceedings

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 Weltman

The theory of causal dynamical triangulations (CDT) attempts to define a nonperturbative theory of quantum gravity as a sum over space-time geometries. One of the ingredients of the CDT framework is a global time foliation, which also plays a central role in the quantum gravity theory recently formulated by Ho\v{r}ava. We show that the phase diagram of CDT bears a striking resemblance with the generic Lifshitz phase diagram appealed to by Ho\v{r}ava. We argue that CDT might provide a unifying nonperturbative framework for anisotropic as well as isotropic theories of quantum gravity., Comment: 17 pages, 3 figures. Typos corrected, a few remarks added.

A universe much like the (Euclidean) de Sitter space-time appears as background geometry in the causal dynamical triangulation (CDT) regularization of quantum gravity. We study the geometry of such universes which appear in the path integral as a function of the bare coupling constants of the theory., Comment: 19 pages, 7 figures. Typos corrected. Conclusions unchanged

We show that proper time, when defined in the quantum theory of 2d gravity, becomes identical to the stochastic time associated with the stochastic quantization of space. This observation was first made by Kawai and collaborators in the context of 2d Euclidean quantum gravity, but the relation is even simpler and more transparent in he context of 2d gravity formulated in the framework of CDT (causal dynamical triangulations)., Comment: 30 pages, Talk presented at the meeting "Foundations of Space and Time", Cape Town, 10-14 August 2009. To appear in the proceedings, CUP

Causal dynamical triangulations (CDT) can be used as a regularization of quantum gravity. In two dimensions the theory can be solved anlytically, even before the cut-off is removed and one can study in detail how to take the continuum limit. We show how the CDT theory is related to Euclidean 2d quantum gravity (Liouville quantum gravity), how it can be generalized and how this generalized CDT model has a string field theory representation as well as a matrix model representationof a new kind, and finally how it examplifies the possibility that time in quantum gravity might be the stochastic time related to the branching of space into baby universes., Comment: Lectures presented at the 49th Cracow School of Theoretical Physics, "Non-Perturbative Gravity and Quantum Chromodynamics", Zakopane May 31-June 10, 2009. To appear in Acta Physica Polonica B 40 (2009) 1001-1031

We show that the noncritical string field theory developed from two-dimensional quantum gravity in the framework of causal dynamical triangulations can be viewed as arising through a stochastic quantization. This requires that the proper time appearing in the string field theory be identified with the stochastic time of the stochastic formulation. The framework of stochastic quantization gives rise to a natural nonperturbative quantum Hamiltonian, which incorporates a sum over all spacetime topologies. We point out that the external character of stochastic time is a feature that pertains more generally to the proper time or distance appearing in nonperturbative correlation functions in quantum gravity., Comment: 15 pages

Inspired by previous work in 2+1 dimensional quantum gravity, which found evidence for a discretization of time in the quantum theory, we reexamine the issue for the case of pure Lorentzian gravity with vanishing cosmological constant and spatially compact universes of genus larger than 1. Taking as our starting point the Chern-Simons formulation with Poincare gauge group, we identify a set of length variables corresponding to space- and timelike distances along geodesics in three-dimensional Minkowski space. These are Dirac observables, that is, functions on the reduced phase space, whose quantization is essentially unique. For both space- and timelike distance operators, the spectrum is continuous and not bounded away from zero., Comment: 29 pages, 18 figures

By explicitly allowing for topology to change as a function of time, two-dimensional quantum gravity defined through causal dynamical triangulations gives rise to a new continuum string field theory. Within a matrix-model formulation we show that -- rather remarkably -- the associated sum over all genera can be performed in closed form, leading to a nonperturbative definition of CDT string field theory. We also obtain explicit formulas for the n-loop correlation functions. Our construction exhibits interesting parallels with previous, purely Euclidean treatments., Comment: 15 pages

We review a recently discovered continuum limit for the one-matrix model which describes "causal" two-dimensional quantum gravity. The behaviour of the quantum geometry in this limit is different from the quantum geometry of Euclidean two-dimensional quantum gravity defined by taking the "standard" continuum limit of the one-matrix model. Geodesic distance and time scale with canonical dimensions in this new limit, contrary to the situation in Euclidean two-dimensional quantum gravity. Remarkably, whenever we compare, the known results of (generalized) causal dynamical triangulations are reproduced exactly by the one-matrix model. We complement previous results by giving a geometrical interpretation of the new model in terms of a generalization of the loop equation of Euclidean dynamical triangulations. In addition, we discuss the time evolution of the quantum geometry., Comment: 10 pages, 4 figures, Presented at "The 48th Cracow School of Theoretical Physics: Aspects of Duality", June 13-22, 2008, Zakopane, Poland

We define a new scaling limit of matrix models which can be related to the method of causal dynamical triangulations (CDT) used when investigating two-dimensional quantum gravity. Surprisingly, the new scaling limit of the matrix models is also a matrix model, thus explaining why the recently developed CDT continuum string field theory (arXiv:0802.0719) has a matrix-model representation (arXiv:0804.0252)., Comment: 17 pages, 2 figures

The dynamical generation of a four-dimensional classical universe from nothing but fundamental quantum excitations at the Planck scale is a long-standing challenge to theoretical physicists. A candidate theory of quantum gravity which achieves this goal without invoking exotic ingredients or excessive fine-tuning is based on the nonperturbative and background-independent technique of Causal Dynamical Triangulations. We demonstrate in detail how in this approach a macroscopic de Sitter universe, accompanied by small quantum fluctuations, emerges from the full gravitational path integral, and how the effective action determining its dynamics can be reconstructed uniquely from Monte Carlo data. We also provide evidence that it may be possible to penetrate to the sub-Planckian regime, where the Planck length is large compared to the lattice spacing of the underlying regularization of geometry., Comment: Article unchanged. Line added in acknowledgment