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 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.
Causal Dynamical Triangulations (CDT) is a lattice theory where aspects of quantum gravity can be studied. Two-dimensional CDT can be solved analytically and the continuum (quantum) Hamiltonian obtained. In this article we show that this continuum Hamiltonian is the one obtained by quantizing two-dimensional projectable Horava-Lifshitz gravity.
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 Hořava. We show that the phase diagram of CDT bears a striking resemblance with the generic Lifshitz phase diagram appealed to by Hořava. We argue that CDT might provide a unifying nonperturbative framework for anisotropic as well as isotropic theories of quantum gravity.
We review some recent results from the causal dynamical triangulation (CDT) approach to quantum gravity. We review recent observations of dimensional reduction at a number of previously undetermined points in the parameter space of CDT, and discuss their possible relevance to the asymptotic safety scenario. We also present an updated phase diagram of CDT, discussing properties of a newly discovered phase and its possible relation to a signature change of the metric.
Causal Dynamical Triangulations (CDT) provide a non-perturbative formulation of Quantum Gravity assuming the existence of a global time foliation. In our earlier study we analyzed the effect of including $d$ copies of a massless scalar field in the two-dimensional CDT model with imaginary time. For $d > 1$ we observed the formation of a "blob", somewhat similar to that observed in four-dimensional CDT without matter. In the two-dimensional case the "blob" has a Hausdorff dimension $D_H=3$. In this paper we study the spectral dimension $D_S$ of the two-dimensional CDT-universe, both for $d = 0$…
We study matter with central charge $c >1$ coupled to two-dimensional (2d) quantum gravity, here represented as causal dynamical triangulations (CDT). 2d CDT is known to provide a regularization of (Euclidean) 2d Hořava-Lifshitz quantum gravity. The matter fields are massive Gaussian fields, where the mass is used to monitor the central charge $c$. Decreasing the mass we observe a higher order phase transition between an effective $c=0$ theory and a theory where $c>1$. In this sense the situation is somewhat similar to that observed for "standard" dynamical triangulations (DT) which provide a …
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.
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
CDT is an attempt to formulate a non-perturbative lattice theory of quantum gravity. We describe the phase diagram and analyse the phase transition between phase B and phase C (which is the analogue of the de Sitter phase observed for the spherical spatial topology). This transition is accessible to ordinary Monte Carlo simulations when the topology of space is toroidal. We find that the transition is most likely first order, but with unusual properties. The end points of the transition line are candidates for second order phase transition points where an UV continuum limit might exist.
We study the effective transfer matrix within the semiclassical and bifurcation phases of CDT quantum gravity. We find that for sufficiently large lattice volumes the kinetic term of the effective transfer matrix has a different sign in each of the two phases. We argue that this sign change can be viewed as a Wick rotation of the metric. We discuss the likely microscopic mechanism responsible for the bifurcation phase transition, and propose an order parameter that can potentially be used to determine the precise location and order of the transition. Using the effective transfer matrix we appr…
To facilitate the search for a continuum limit of causal dynamical triangulations, Ambjorn, Coumbe, Gizbert-Studnicki, and Jurkiewicz recently reported measurements of the lattice spacing as a function of the bare couplings. Although these authors' methods are technically sound, the conclusions that they draw from their analyses rest crucially on certain unstated assumptions. I elucidate these assumptions, and I argue that our current understanding of causal dynamical triangulations does not entail their justification.
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 ultravio…
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 …
We define multicritical CDT models of 2d quantum gravity and show that they are a special case of multicritical generalized CDT models obtained from the new scaling limit, the so-called "classical" scaling limit, of matrix models. The multicritical behavior agrees with the multicritical behavior of the so-called branched polymers.
Causal Dynamical Triangulations (CDT) is a non-perturbative lattice approach to quantum gravity where one assumes space-time foliation into spatial hyper-surfaces of fixed topology. Most of the CDT results were obtained for the spatial topology of the 3-sphere. It was shown that CDT has rich phase structure, including the semiclassical phase consistent with Einstein's general relativity. Some of the phase transitions were found to be second (or higher) order which makes a possibility of taking continuum limit viable. Here we present new results of changing the spatial topology to that of the 3…
We study the geometries generated by two-dimensional causal dynamical triangulations (CDT) coupled to $d$ massless scalar fields. Using methods similar to those used to study four-dimensional CDT we show that there exists a $c=1$ "barrier", analogous to the $c=1$ barrier encountered in non-critical string theory, only the CDT transition is easier to be detected numerically. For $d\leq 1$ we observe time-translation invariance and geometries entirely governed by quantum fluctuations around the uniform toroidal topology put in by hand. For $d>1$ the effective average geometry is no longer toroid…
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.
We present a possibility of coupling a point-like, non-singular, mass distribution to four-dimensional quantum gravity in the nonperturbative setting of causal dynamical triangulations (CDT). In order to provide a point of comparison for the classical limit of the matter-coupled CDT model, we derive the spatial volume profile of the Euclidean Schwarzschild-de Sitter space glued to an interior matter solution. The volume profile is calculated with respect to a specific proper-time foliation matching the global time slicing present in CDT. It deviates in a characteristic manner from that of the …
We investigate the transition between the phases $B$ and $C_b$ observed in four-dimensional Causal Dynamical Triangulations (CDT). We find that the critical properties of CDT with toroidal spatial topology are the same as earlier observed in spherical spatial topology where the $B-C_b$ transition was found to be higher-order. This may have important consequences for the existence of the continuum limit of CDT, describing the perspective UV limit of quantum gravity, which potentially can be investigated in the toroidal model.
This article discusses the infrared and the (perspective) ultraviolet limits of four-dimensional Causal Dynamical Triangulations (CDT). CDT is a non-perturabtive and background-independent approach to quantization of Einstein's gravity, based on a lattice regularization of the quantum-gravitational path integral. It is shown that inside the, so-called, de Sitter phase one recovers a four-dimensional semiclassical universe, where behaviour of the scale factor is consistent with classical solutions of Einstein's field equations and fluctuations of the scale factor are very accurately described b…
Lattice formulations of gravity can be used to study non-perturbative aspects of quantum gravity. Causal Dynamical Triangulations (CDT) is a lattice model of gravity that has been used in this way. It has a built-in time foliation but is coordinate-independent in the spatial directions. The higher-order phase transitions observed in the model may be used to define a continuum limit of the lattice theory. Some aspects of the transitions are better studied when the topology of space is toroidal rather than spherical. In addition, a toroidal spatial topology allows us to understand more easily th…
We study the phase diagram of a one-dimensional balls-in-boxes (BIB) model that has been proposed as an effective model for the spatial-volume dynamics of (2+1)-dimensional causal dynamical triangulations (CDT). The latter is a statistical model of random geometries and a candidate for a nonperturbative formulation of quantum gravity, and it is known to have an interesting phase diagram, in particular including a phase of extended geometry with classical properties. Our results corroborate a previous analysis suggesting that a particular type of potential is needed in the BIB model in order to…
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) pa…
"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 typ…
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 transiti…
The phenomenon of scale dependent spectral dimension has attracted special interest in the quantum gravity community over the last eight years. It was first observed in computer simulations of the causal dynamical triangulation (CDT) approach to quantum gravity and refers to the reduction of the spectral dimension from 4 at classical scales to 2 at short distances. Thereafter several authors confirmed a similar result from different approaches to quantum gravity. Despite the contribution from different approaches, no analytical model was proposed to explain the numerical results as the continu…
Causal Dynamical Triangulations (CDT) is a non-perturbative quantisation of general relativity. Hořava-Lifshitz gravity on the other hand modifies general relativity to allow for perturbative quan- tisation. Past work has given rise to the speculation that Hořava-Lifshitz gravity might correspond to the continuum limit of CDT. In this paper we add another piece to this puzzle by applying the CDT quantisation prescription directly to Hořava-Lifshitz gravity in 2 dimensions. We derive the continuum Hamiltonian and we show that it matches exactly the Hamiltonian one derives from canonically quant…
We study the implications of the simplicity constraint in the spincube model of quantum gravity. By relating the edge-lengths to the integer areas of triangles, the simplicity constraint imposes very strong restrictions between them, ultimately leading to a requirement that all 4-simplices in the triangulation must be almost mutually identical. As a surprising and unexpected consequence of this property, one can obtain the CDT state sum as a special case of the spincube state sum. This relationship brings new insight into the long-standing problem of the relationship between the spinfoam appro…
Causal dynamical triangulations allows for a non perturbative approach to quantum gravity. In this article a solution for dimers coupled to CDT is presented and some of the conceptual problems that arise are reflected upon.
Inspired by Verlinde's idea, some modified versions of entropic gravity have appeared in the literature. Extending them in a unified formalism, we derive the generalized gravitational equations accordingly. From gravitational equations, the energy-momentum conservation law and cosmological equations are investigated. The covariant conservation law of energy-momentum tensor severely constrains viable modifications of entropic gravity. A discrepancy arises when two independent methods are applied to the homogeneous isotropic universe, posing a serious challenge to modified models of entropic gra…
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 show…
It is shown that generalized CDT, the two-dimensional theory of quantum gravity, constructed as a scaling limit from so-called causal dynamical triangulations, can be obtained from a cubic matrix model. It involves taking a new scaling limit of matrix models, which is more natural from a classical point of view.
If we assume that the source of thermodynamic system, $ρ$ and $p$, are also the source of gravity, thermal quantities, such as entropy, temperature, and chemical potential, can induce effects of gravity, or gravity can induce thermal effects. We find only for systems with constant temperature and zero chemical potential, gravity can be seen as an entropic force. The case for Newtonian approximation is discussed.
We interpret the Brans-Dicke gravity from entropic viewpoint. We first apply the Verlinde's entropic formalism in the Einstein frame, then perform the conformal transformation which connects the Einstein frame to the Jordan frame. The transformed result yields the equation of motion of the Brans-Dicke theory in the Jordan frame.
A lattice regularization for the $2$d projectable Horava-Lifshitz (HL) quantum gravity is known to be the $2$d causal dynamical triangulations (CDT), and the $2$d CDT can be generalized so as to include all possible genus contributions non-perturbatively. We show that in the context of HL gravity, effects coming from such a non-perturbative sum over topologies can be successfully taken into account, if we quantize the $2$d projectable HL gravity with a simple bi-local wormhole interaction. This conference paper is based on the article, Phys. Lett. B 816 (2021), 136205.
We consider a freely falling holographic screen for the Schwarzschild and Reissner-Nordström black holes and evaluate the entropic force à la Verlinde. When the screen crosses the event horizon, the temperature of the screen agrees to the Hawking temperature and the entropic force gives rise to the surface gravity for both of the black holes.
This work presents a study of the possibility for extending the well-known results of E.Verlinde concerning the entropic nature of gravity to the ultraviolet region (Planck's energies) and also the derivation of quantum corrections to Einstein Equations
We argue that neutron interference experiments and experiments on gravitational bound states of neutron unambiguously disprove entropic origin of gravitation. The criticism expressed in a recent paper [arXiv:1104.4650] concerning our arguments against entropic gravity is shown to be invalid.
We discuss the algorithmic problem of minimal coupling gauge fields of the Yang-Mills type to Quantum Gravity in the approach known as Causal Dynamical Triangulations (CDT) as a step towards studying, ultimately, systems of gravity coupled with bosonic and fermionic matter. We first describe the algorithm for general dimensions and gauge groups and then focus on the results obtained from simulations of 2d CDT coupled to gauge fields with U{(1)} and SU{(2)} gauge groups, where we studied both observables related to gravity and gauge fields, and compared them with analogous simulations in the st…
A new entropic gravity inspired derivation of general relativity from thermodynamics is presented. This generalizes, within Einstein gravity, the "Thermodynamics of Spacetime" approach by T. Jacobson, which relies on the Raychaudhuri evolution equation. Here the rest of the first law of thermodynamics is incorporated by using the Damour-Navier-Stokes equation, known from the membrane paradigm for describing fluid dynamics on the horizon.
In recent years several approaches to quantum gravity have found evidence for a scale dependent spectral dimension of space-time varying from four at large scales to two at small scales of order of the Planck length. The first evidence came from numerical results on four-dimensional causal dynamical triangulations (CDT) [Ambjorn et al., Phys. Rev. Lett. 95 (2005) 171]. Since then little progress has been made in analytically understanding the numerical results coming from the CDT approach and showing that they remain valid when taking the continuum limit. Here we argue that the spectral dimens…
It has been suggested that the Newtonian gravitational force may emerge as an entropic force from a holographic microscopic theory. In this framework, the possibility is reconsidered that Newton's gravitational coupling constant G can be derived from the fundamental constants of the underlying microscopic theory.
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 dime…
We consider the controversial hypothesis that gravity is an entropic force that has its origin in the thermodynamics of holographic screens. Several key aspects of entropic gravity are discussed. In particular, we revisit and elaborate on our criticism of the recent claim that entropic gravity fails to explain observations involving gravitationally-bound quantum states of neutrons in the GRANIT experiment and gravitationally induced quantum interference. We argue that the analysis leading to this claim is troubled by a misinterpretation concerning the relation between the microstates of a holo…
A heuristic entropic-gravity derivation has previously been given of the gravitational two-body force of modified Newtonian dynamics (MOND). Here, it is shown that also another characteristic of MOND can be recovered, namely, the external field effect (implying a violation of the Strong Equivalence Principle). In fact, the derivation gives precisely the modified Poisson equation which Bekenstein and Milgrom proposed as a consistent nonrelativistic theory of MOND.
In this study, we investigate the effects of the one and two-parameters deformed systems on the Friedmann equations of the Friedmann-Robertson-Walker (FRW) universe in the context of the entropic gravity approach. We give simplified forms for the deformed Unruh temperature and Einstein field equations for three different deformed systems. Based on these compact equations, we derive the Friedmann equations with the effective gravitational and cosmological constant.
The entropic gravity scenario recently proposed by Erik Verlinde reproduced the Newton's law of purely classical gravity yet the key assumptions of this approach all have quantum mechanical origins. This is atypical for emergent phenomena in physics, where the underlying, more fundamental physics often reveals itself as corrections to the leading classical behavior. So one naturally wonders: where is hbar hiding in entropic gravity? To address this question, we first revisit the idea of holographic screen as well as entropy and its variation law in order to obtain a self-consistent approach to…
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 ti…
In this letter, we use the Parikh-Wilczek tunneling model of Hawking radiation to illustrate that a reformulation of Verlinde's entropic gravity is needed to derive the Newton's law for a temperature-varying screen, demanded by the conservation of energy. Furthermore, the entropy stored in the holographic screen is shown to be additive and its temperature dependence can be obtained.
We argue that experiments with ultra-cold neutrons in the gravitational field of Earth disprove recent speculations on the entropic origin of gravitation.
Erik Verlinde's theory of entropic gravity [arXiv:1001.0785], postulating that gravity is not a fundamental force but rather emerges thermodynamically, has garnered much attention as a possible resolution to the quantum gravity problem. Some have ruled this theory out on grounds that entropic forces are by nature noisy and entropic gravity would therefore display far more decoherence than is observed in ultra-cold neutron experiments. We address this criticism by modeling linear gravity acting on small objects as an open quantum system. In the strong coupling limit, when the model's unitless f…
I present a new derivation of Hawking radiation by applying the Crooks fluctuation theorem to the laws of black hole thermodynamics. Then, by analogy with the quantum fluctuation theorem, this allows one to identify microstates contributing to the black hole entropy. These microstates have an evolution operator on the horizon between initial and final states that is related to the entropic gravity proposal. Their full calculation perhaps requires a deeper understanding of nonperturbative Hawking radiation.
