Lorentzian and Euclidean Quantum Gravity - Analytical and Numerical Results

TL;DR

This paper reviews covariant lattice approaches to quantum gravity, emphasizing dynamical triangulations in 2D and higher dimensions, highlighting analytical solutions in 2D and numerical simulations in Euclidean and Lorentzian signatures.

Contribution

It provides a comprehensive overview of analytical and numerical methods in dynamical triangulations, including the relation between Lorentzian and Euclidean quantum gravity models.

Findings

Explicit solutions in 2D quantum gravity models

Development of Monte Carlo techniques for Euclidean gravity simulations

Potential of Lorentzian dynamical triangulations as a non-perturbative quantum gravity theory

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.