MPS-RR 2000-2

January 2000

# Lorentzian and Euclidean Quantum Gravity - Analytical and Numerical Results

by:

### J. Jurkiewicz, R. Loll

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.

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This paper has now been published in *NATO Sci. Ser. C Math. Phys. Sci., 556, Kluwer Acad. Publ., Dordrecht, 2000*