Lorentzian and Euclidean Quantum Gravity - Analytical and Numerical Results

TL;DR

The paper develops a covariant, non-perturbative framework for quantum gravity based on dynamical triangulations, solving 2d Lorentzian gravity analytically and clarifying its relation to Euclidean formulations. It shows how topology changes via baby universes reproduce Euclidean Liouville gravity and induce fractal space-time with Hausdorff dimension $d_h=4$, while Lorentzian gravity maintains $d_h=2$, highlighting gravity-matter coupling effects. The work then details the Euclidean DT approach in general d, including discretization, measures, and matter coupling, and emphasizes Monte Carlo methods and observables (minbu distributions, two-point functions, Hausdorff and spectral dimensions) used to probe geometry. In higher dimensions ($d>2$) the DT program remains largely numerical, with ergodic moves and finite-size scaling guiding the search for a viable continuum limit amid competing geometric phases. Overall, the article argues that Lorentzian DT is a promising route to a non-perturbative quantum gravity theory, while Euclidean DT provides a robust, exactly solvable 2d benchmark and a scalable numerical framework for higher-dimensional exploration.

Abstract

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.

Figures (14)

• Figure 1: The propagation of a spatial slice from step $t$ to step $t+1$. The ends of the strip should be joined to form a band with topology $S^1 \times [0,1]$.
• Figure 2: A "baby universe" created by a global pinching.
• Figure 3: Marking a vertex in the bulk of $W_\Lambda(X)$. The mark has a distance $T$ from the boundary loop, which itself has one marked vertex.
• Figure 4: A set of three moves which is ergodic in the class of two-dimensional triangulations of fixed topology. The first diagram shows the $(3,1)$-move and its inverse. The second diagram shows the $(2,2)$- or flip move. By itself, this move is ergodic in the class of triangulations of fixed volume $N_2$ and topology.
• Figure 5: The point-splitting moves constitute an alternative set of ergodic moves.