arXiv subjects
Publications and source records attributed to R. Loll.
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.
It is well-known that the sum over topologies in quantum gravity is ill-defined, due to a super-exponential growth of the number of geometries as a function of the space-time volume, leading to a badly divergent gravitational path integral. Not even in dimension 2, where a non-perturbative quantum gravity theory can be constructed explicitly from a (regularized) path integral, has this problem found a satisfactory solution. -- In the present work, we extend a previous 2d Lorentzian path integral, regulated in terms of Lorentzian random triangulations, to include space-times with an arbitrary number of handles. We show that after the imposition of physically motivated causality constraints, the combined sum over geometries and topologies is well-defined and possesses a continuum limit which yields a concrete model of space-time foam in two dimensions.
Lorentzian simplicial quantum gravity is a non-perturbatively defined theory of quantum gravity which predicts a positive cosmological constant. Since the approach is based on a sum over space-time histories, it is perturbatively non-renormalizable even in three dimensions. By mapping the three-dimensional theory to a two-matrix model with ABAB interaction we show that both the cosmological and the (perturbatively) non-renormalizable gravitational coupling constant undergo additive renormalizations consistent with canonical quantization.
In these lecture notes, I describe the motivation behind a recent formulation of a non-perturbative gravitational path integral for Lorentzian (instead of the usual Euclidean) space-times, and give a pedagogical introduction to its main features. At the regularized, discrete level this approach solves the problems of (i) having a well-defined Wick rotation, (ii) possessing a coordinate-invariant cutoff, and (iii) leading to_convergent_ sums over geometries. Although little is known as yet about the existence and nature of an underlying continuum theory of quantum gravity in four dimensions, there are already a number of beautiful results in d=2 and d=3 where continuum limits have been found. They include an explicit example of the inequivalence of the Euclidean and Lorentzian path integrals, a non-perturbative mechanism for the cancellation of the conformal factor, and the discovery that causality can act as an effective regulator of quantum geometry.
We formulate a dynamically triangulated model of three-dimensional Lorentzian quantum gravity whose spatial sections are flat two-tori. It is shown that the combinatorics involved in evaluating the one-step propagator (the transfer matrix) is that of a set of vicious walkers on a two-dimensional lattice with periodic boundary conditions and that the entropy of the model scales exponentially with the volume. We also give explicit expressions for the Teichmüller parameters of the spatial slices in terms of the discrete parameters of the 3d triangulations, and reexpress the discretized action in terms of them. The relative simplicity and explicitness of this model make it ideally suited for an analytic study of the conformal-factor cancellation observed previously in Lorentzian dynamical triangulations and of its relation to alternative, reduced phase space quantizations of 3d gravity.
There is strong evidence coming from Lorentzian dynamical triangulations that the unboundedness of the gravitational action is no obstacle to the construction of a well-defined non-perturbative path integral. In a continuum approach, a similar suppression of the conformal divergence comes about as the result of a non-trivial path-integral measure.
The model of Lorentzian three-dimensional dynamical triangulations provides a non-perturbative definition of three-dimensional quantum gravity. The theory has two phases: a weak-coupling phase with quantum fluctuations around a ``semiclassical'' background geometry which is generated dynamically despite the fact that the formulation is explicitly background-independent, and a strong-coupling phase where ``classical'' space disintegrates into a foam of baby universes.
We uncover a surprising correspondence between a non-perturbative formulation of three-dimensional Lorentzian quantum gravity and a hermitian two-matrix model with ABAB-interaction. The gravitational transfer matrix can be expressed as the logarithm of a two-matrix integral, and we deduce from the known structure of the latter that the model has two phases. In the phase of weak gravity, well-defined two-dimensional universes propagate in proper time, whereas in the strong-coupling phase the spatial hypersurfaces disintegrate into many components connected by wormholes.
Fruitful ideas on how to quantize gravity are few and far between. In this paper, we give a complete description of a recently introduced non-perturbative gravitational path integral whose continuum limit has already been investigated extensively in d less than 4, with promising results. It is based on a simplicial regularization of Lorentzian space-times and, most importantly, possesses a well-defined, non-perturbative Wick rotation. We present a detailed analysis of the geometric and mathematical properties of the discretized model in d=3,4. This includes a derivation of Lorentzian simplicial manifold constraints, the gravitational actions and their Wick rotation. We define a transfer matrix for the system and show that it leads to a well-defined self-adjoint Hamiltonian. In view of numerical simulations, we also suggest sets of Lorentzian Monte Carlo moves. We demonstrate that certain pathological phases found previously in Euclidean models of dynamical triangulations cannot be realized in the Lorentzian case.
We have recently introduced a discrete model of Lorentzian quantum gravity, given as a regularized non-perturbative state sum over simplicial Lorentzian space-times, each possessing a unique Wick rotation to Euclidean signature. We investigate here the phase structure of the Wick-rotated path integral in three dimensions with the aid of computer simulations. After fine-tuning the cosmological constant to its critical value, we find a whole range of the gravitational coupling constant $k_0$ for which the functional integral is dominated by non-degenerate three-dimensional space-times. We therefore have a situation in which a well-defined ground state of extended geometry is generated dynamically from a non-perturbative state sum of fluctuating geometries. Remarkably, its macroscopic scaling properties resemble those of a semi-classical spherical universe. Measurements so far indicate that $k_0$ defines an overall scale in this extended phase, without affecting the physics of the continuum limit. These findings provide further evidence that discrete {\it Lorentzian} gravity is a promising candidate for a non-trivial theory of quantum gravity.
A well-defined regularized path integral for Lorentzian quantum gravity in three and four dimensions is constructed, given in terms of a sum over dynamically triangulated causal space-times. Each Lorentzian geometry and its associated action have a unique Wick rotation to the Euclidean sector. All space-time histories possess a distinguished notion of a discrete proper time. For finite lattice volume, the associated transfer matrix is self-adjoint and bounded. The reflection positivity of the model ensures the existence of a well-defined Hamiltonian. The degenerate geometric phases found previously in dynamically triangulated Euclidean gravity are not present. The phase structure of the new Lorentzian quantum gravity model can be readily investigated by both analytic and numerical methods.
Starting from the space of Lorentzian metrics, we examine the full gravitational path integral in 3 and 4 space-time dimensions. Inspired by recent results obtained in a regularized, dynamically triangulated formulation of Lorentzian gravity, we gauge-fix to proper-time coordinates and perform a non-perturbative ``Wick rotation'' on the physical configuration space. Under certain assumptions about the behaviour of the partition function under renormalization, we find that the divergence due to the conformal modes of the metric is cancelled non-perturbatively by a Faddeev-Popov determinant contributing to the effective measure. We illustrate some of our claims by a 3d perturbative calculation.
Just as for non-abelian gauge theories at strong coupling, discrete lattice methods are a natural tool in the study of non-perturbative quantum gravity. They have to reflect the fact that the geometric degrees of freedom are dynamical, and that therefore also the lattice theory must be formulated in a background-independent way. After summarizing the status quo of discrete covariant lattice models for four-dimensional quantum gravity, I describe a new class of discrete gravity models whose starting point is a path integral over Lorentzian (rather than Euclidean) space-time geometries. A number of interesting and unexpected results that have been obtained for these dynamically triangulated models in two and three dimensions make discrete Lorentzian gravity a promising candidate for a non-trivial theory of quantum gravity.
We investigate the phase diagram of non-perturbative three-dimensional Lorentzian quantum gravity with the help of Monte Carlo simulations. The system has a first-order phase transition at a critical value $k_0^c$ of the bare inverse gravitational coupling constant $k_0$. For $k_0 > k_0^c$ the system reduces to a product of uncorrelated Euclidean 2d gravity models and has no intrinsic interest as a model of 3d gravity. For $k_0 < k_0^c$, extended three-dimensional geometries dominate the functional integral despite the fact that we perform a sum over geometries and no particular background is distinguished at the outset. Furthermore, all systems with $k_0 < k_0^c$ have the same continuum limit. A different choice of $k_0$ corresponds merely to a redefinition of the overall length scale.
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 great advantage of our approach is the fact that it is well-suited for numerical simulations. In the second part of this review we describe the relevant Monte Carlo techniques, as well as some of the physical results that have been obtained from the simulations of Euclidean gravity. We also explain why the Lorentzian version of dynamical triangulations is a promising candidate for a non-perturbative theory of quantum gravity.
Starting from 2D Euclidean quantum gravity, we show that one recovers 2D Lorentzian quantum gravity by removing all baby universes. Using a peeling procedure to decompose the discrete, triangulated geometries along a one-dimensional path, we explicitly associate with each Euclidean space-time a (generalized) Lorentzian space-time. This motivates a map between the parameter spaces of the two theories, under which their propagators get identified. In two dimensions, Lorentzian quantum gravity can therefore be viewed as a ``renormalized'' version of Euclidean quantum gravity.
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.
In an extension of earlier work we investigate the behaviour of two-dimensional Lorentzian quantum gravity under coupling to a conformal field theory with c>1. This is done by analyzing numerically a system of eight Ising models (corresponding to c=4) coupled to dynamically triangulated Lorentzian geometries. It is known that a single Ising model couples weakly to Lorentzian quantum gravity, in the sense that the Hausdorff dimension of the ensemble of two-geometries is two (as in pure Lorentzian quantum gravity) and the matter behaviour is governed by the Onsager exponents. By increasing the amount of matter to 8 Ising models, we find that the geometry of the combined system has undergone a phase transition. The new phase is characterized by an anomalous scaling of spatial length relative to proper time at large distances, and as a consequence the Hausdorff dimension is now three. In spite of this qualitative change in the geometric sector, and a very strong interaction between matter and geometry, the critical exponents of the Ising model retain their Onsager values. This provides evidence for the conjecture that the KPZ values of the critical exponents in 2d Euclidean quantum gravity are entirely due to the presence of baby universes. Lastly, we summarize the lessons learned so far from 2d Lorentzian quantum gravity.