R. Loll

The Spectral Dimension of the Universe is Scale Dependent

We measure the spectral dimension of universes emerging from nonperturbative quantum gravity, defined through state sums of causal triangulated geometries. While four-dimensional on large scales, the quantum universe appears two-dimensional at short distances. We conclude that quantum gravity may be “selfrenormalizing” at the Planck scale, by virtue of a mechanism of dynamical dimensional reduction.

Emergence of a 4D world from causal quantum gravity

Causal Dynamical Triangulations in four dimensions provide a background-independent definition of the sum over geometries in nonperturbative quantum gravity, with a positive cosmological constant. We present evidence that a macroscopic four-dimensional world emerges from this theory dynamically.

Reconstructing the universe

We provide detailed evidence for the claim that nonperturbative quantum gravity, defined through state sums of causal triangulated geometries, possesses a large-scale limit in which the dimension of spacetime is four and the dynamics of the volume of the universe behaves semiclassically. This is a first step in reconstructing the universe from a dynamical principle at the Planck scale, and at the same time provides a nontrivial consistency check of the method of causal dynamical triangulations. A closer look at the quantum geometry reveals a number of highly nonclassical aspects, including a dynamical reduction of spacetime to two dimensions on short scales and a fractal structure of slices of constant time.

Non-perturbative Lorentzian Quantum Gravity, Causality and Topology Change

Abstract We formulate a non-perturbative lattice model of two-dimensional Lorentzian quantum gravity by performing the path integral over geometries with a causal structure. The model can be solved exactly at the discretized level. Its continuum limit coincides with the theory obtained by quantizing 2d continuum gravity in proper-time gauge, but it disagrees with 2d gravity defined via matrix models or Liouville theory. By allowing topology change of the compact spatial slices (i.e. baby universe creation), one obtains agreement with the matrix models and Liouville theory.

Dynamically Triangulating Lorentzian Quantum Gravity

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 < 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.

Nonperturbative quantum gravity

Abstract 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 Hořava–Lifshitz gravitational models.

Discrete approaches to quantum gravity in four-dimensions

AbstractThe construction of a consistent theory of quantum gravity is a problem in theoretical physics that has so far defied all attempts at resolution. One ansatz to try to obtain a non-trivial quantum theory proceeds via a discretization of space-time and the Einstein action. I review here three major areas of research: gauge-theoretic approaches, both in a path-integral and a Hamiltonian formulation; quantum Regge calculus; and the method of dynamical triangulations, confining attention to work that is strictly four-dimensional, strictly discrete, and strictly quantum in nature.

A non-perturbative Lorentzian path integral for gravity

We construct a well-defined regularized path integral for Lorentzian quantum gravity in terms of dynamically triangulated causal space-times. Each Lorentzian geometry and its action have a unique Wick rotation to the Euclidean sector. All space-time histories possess a distinguished notion of a discrete proper time and, for finite lattice volume, the associated transfer matrix is self-adjoint, bounded, and strictly positive. The degenerate geometric phases found in dynamically triangulated Euclidean gravity are not present.

Planckian Birth of a Quantum de Sitter Universe

We show that the quantum universe emerging from a nonperturbative, Lorentzian sum over geometries can be described with a high accuracy by a four-dimensional de Sitter spacetime. By a scaling analysis involving Newtons constant, we establish that the linear size of the quantum universes under study is in between 17 and 28 Planck lengths. Somewhat surprisingly, the measured quantum fluctuations around the de Sitter universe in this regime are to good approximation still describable semiclassically. The numerical evidence presented comes from a regularization of quantum gravity in terms of causal dynamical triangulations.

Nonperturbative quantum de Sitter universe

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.