We study a c = 2 conformal eld theory coupled to two-dimensional quantum gravity by means of dyna... more We study a c = 2 conformal eld theory coupled to two-dimensional quantum gravity by means of dynamical triangulations. We dene the geodesic distance r on the triangulated surface with N triangles, and show that dim[ r d H ] = dim[ N ], where the fractal dimension d H = 3:58 0:04. This result lends support to the conjecture d H = 2 1 = 1 , where n is the gravitational dressing exponent of a spin-less primary eld of conformal weight (n + 1 ; n + 1), and it disfavors the alternative prediction d H = 2= str. On the other hand, we nd dim[ l ] = dim[ r 2 ] with goodaccuracy, where l is the length of one of the boundaries of a circle with (geodesic) radius r, i.e. the length l has an anomalous dimension relative to the area of the surface. It is further shown that the spectral dimension d s = 1 : 9800:014 for the ensemble of (triangulated) manifolds used. The results are derived using nite size scaling and a very ecient recursive sampling technique known previously to work well for c = 2.
In this paper we review how a background independent non-pertur- bative regularization of quantum... more In this paper we review how a background independent non-pertur- bative regularization of quantum gravity, denoted causal dynamical triangulation (CDT), in the infrared leads to the standard minisuperspace effective action. We show how it is possible to study in detail the quantum fluctuations around the classical solution to the minisuperspace action and outline how one in principle might be able to study the quantum gravity theory in the sub-Planckian regime.
We show that there exists a divergent correlation length in 2d quantum gravity for the matter fie... more We show that there exists a divergent correlation length in 2d quantum gravity for the matter fields close to the critical point provided one uses the invariant geodesic distance as the measure of distance. The corresponding reparameterization invariant two-point functions satisfy all scaling relations known from the ordinary theory of critical phenomena and the KPZ exponents are determined by the power-like fall off of these two-point functions. The only difference compared to flat space is the appearance of a dynamically generated fractal dimension d_h in the scaling relations. We analyze numerically the fractal properties of space-time for Ising and three-states Potts model coupled to 2d dimensional quantum gravity using finite size scaling as well as small distance scaling of invariant correlation functions. Our data are consistent with d_h=4, but we cannot rule out completely the conjecture d_H = -2\alpha_1/\alpha_{-1}, where \alpha_{-n} is the gravitational dressing exponent o...
Proceedings of The XXVIII International Symposium on Lattice Field Theory — PoS(Lattice 2010)
We advocate lattice methods as the tool of choice to constructively define a backgroundindependen... more We advocate lattice methods as the tool of choice to constructively define a backgroundindependent 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.
We propose a theory of quantum gravity which formulates the quantum theory as a nonperturbative p... more We propose a theory of quantum gravity which formulates the quantum theory as a nonperturbative path integral, where each space–time history appears with the weight exp (iS EH ), with S EH the Einstein–Hilbert action of the corresponding causal geometry. The path integral is diffeomorphism-invariant (only geometries appear) and background-independent. The theory can be investigated by computer simulations, which show that a de Sitter universe emerges on large scales. This emergence is of an entropic, self-organizing nature, with the weight of the Einstein–Hilbert action playing a minor role. Also, the quantum fluctuations around this de Sitter universe can be studied quantitatively and remain small until one gets close to the Planck scale. The structures found to describe Planck-scale gravity are reminiscent of certain aspects of condensed-matter systems.
We propose a new, very ecient algorithm for sampling of random surfaces in the Monte Carlo simula... more We propose a new, very ecient algorithm for sampling of random surfaces in the Monte Carlo simulations, based on so-called baby universe surgery, i.e. cutting and pasting of baby universes. It drastically reduces slowing down as compared to the standard local ip algorithm, thereby allowing simulations of large random surfaces coupled to matter elds. As an example we i n v estigate the eciency of the algorithm for 2d simplicial gravity i n teracting with a one-component free scalar eld. The radius of gyration is the slowest mode in the standard local ip/shift algorithm. The use of baby universe surgery decreases the autocorrelation time by three order of magnitude for a random surface of 0:510 5 triangles, where it is found to be int = 150 31 sweeps.
We present the multi-matrix models that are the generating functions for branched covers of the c... more We present the multi-matrix models that are the generating functions for branched covers of the complex projective line ramified over n fixed points z i , i = 1,. .. , n, (generalized Grotendieck's dessins d'enfants) of fixed genus, degree, and the ramification profiles at two points, z 1 and z n. We take a sum over all possible ramifications at other n − 2 points with the fixed length of the profile at z 2 and with the fixed total length of profiles at the remaining n − 3 points. All these models belong to a class of hypergeometric Hurwitz models thus being tau functions of the Kadomtsev-Petviashvili (KP) hierarchy. In the case described above, we can present the obtained model as a chain of matrices with a (nonstandard) nearest-neighbor interaction of the type tr M i M −1 i+1. We describe the technique for evaluating spectral curves of such models, which opens the possibility of applying the topological recursion for developing 1/N 2-expansions of these model. These spectral curves turn out to be of an algebraic type.
We study numerically the classical behaviour of the Skyrme model coupled to an SU(2)L gauge field... more We study numerically the classical behaviour of the Skyrme model coupled to an SU(2)L gauge field. We find that if the parameters of the model are chosen in such a way that the skyrmion mass is smaller than a critical value M~m, the skyrmion is classically stable. For other values of the parameters there are no soliton solutions to the classical field equations, i.e. skyrmions are classically unstable. We calculate the critical values of the parameters and the critical mass of the skyrmion. We find that the transition between these two classical regimes is a first-order one. We comment on the implications of our results to technicolour theories.
By explicit construction, we show that one can in a simple way introduce and measure gravitationa... more 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 setup 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.
In two space-time dimensions, there is a theory of Lorentzian quantum gravity which can be define... more In two space-time dimensions, there is a theory of Lorentzian quantum gravity which can be defined by a rigorous, non-perturbative path integral and is inequivalent to the well-known theory of (Euclidean) quantum Liouville gravity. It has a number of appealing features: i) its quantum geometry is non-fractal, ii) it remains consistent when coupled to matter, even beyond the c=1 barrier, iii) it is closer to canonical quantization approaches than previous path-integral formulations, and iv) its construction generalizes to higher dimensions.
We show that for 1+1 dimensional Causal Dynamical Triangulations (CDT) coupled to 4 massive scala... more We show that for 1+1 dimensional Causal Dynamical Triangulations (CDT) coupled to 4 massive scalar fields one can construct an effective transfer matrix if the masses squared is larger than or equal to 0.05. The properties of this transfer matrix can explain why CDT coupled to matter can behave completely different from "pure" CDT. We identify the important critical exponent in the effective action, which may determine the universality class of the model.
Causal Dynamical Triangulations (CDT) provide a non-perturbative formulation of Quantum Gravity a... more 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 DH = 3. In this paper, we study the spectral dimension DS of the two-dimensional CDT-universe, both for d = 0 (pure gravity) and d = 4. We show that in both cases the spectral dimension is consistent with DS = 2.
We review some recent attempts to extract information about the nature of quantum gravity, with a... more We review some recent attempts to extract information about the nature of quantum gravity, with and without matter, by quantum field theoretical methods. More specifically, we work within a covariant lattice approach where the individual space-time geometries are constructed from fundamental simplicial building blocks, and the path integral over geometries is approximated by summing over a class of piece-wise linear geometries. This method of "dynamical triangulations" is very powerful in 2d, where the regularized theory can be solved explicitly, and gives us more insights into the quantum nature of 2d space-time than continuum methods are presently able to provide. It also allows us to establish an explicit relation between the Lorentzian-and Euclidean-signature quantum theories. Analogous regularized gravitational models can be set up in higher dimensions. Some analytic tools exist to study their state sums, but, unlike in 2d, no complete analytic solutions have yet been constructed. However, a 1 The method of dynamical triangulations was introduced in the context of string theory and 2d quantum gravity in [3, 4, 5], and subsequently extended to higher-dimensional Euclidean quantum gravity [6, 7]. An extensive review covering the developments up to 1996 can be found in the book [8]. A more recent summary is contained in [9], while the review [10] deals with a variety of lattice approaches to four-dimensional quantum gravity, including dynamical triangulations. The use of dynamical-triangulations methods in Lorentzian gravity was pioneered in [11, 12, 13].
Journal of Physics A: Mathematical and Theoretical, 2014
We introduce a restricted hard dimer model on a random causal triangulation that is exactly solva... more We introduce a restricted hard dimer model on a random causal triangulation that is exactly solvable and generalizes a model recently proposed by Atkin and Zohren [16]. We show that the latter model exhibits unusual behaviour at its multicritical point; in particular, its Hausdorff dimension equals 3 and not 3/2 as would be expected from general scaling arguments. When viewed as a special case of the generalized model introduced here we show that this behaviour is not generic and therefore is not likely to represent the true behaviour of the full dimer model on a random causal triangulation.
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Papers by Jan Ambjorn