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.
QUANTUM GRAVITYFor the purposes of this presentation, quantum gravity will be defined as the non-perturbative quantization of the classical theory of general relativity, with and without the inclusion of other matter fields and interactions. Judging from our current knowledge of the fundamental laws of physics, it seems highly likely that at sufficiently large energies also gravitational interactions should be governed by quantum rather than by classical equations of motion. Quantum gravity -whose theoretical formulation is still elusive -should include a consistent description of local quantum phenomena in the presence of strong gravitational fields.It has been known for a long time that perturbative quantum gravity, based on a decompositionof the Lorentzian space-time metric g µν into the flat Minkowski metric η µν and a linear perturbation h µν (representing the degrees of freedom of a massless spin-2 graviton), leads to a nonrenormalizable field theory. Although this does not preclude the use of the perturbation series as an effective description of quantum gravity in the presence of an energy cut-off, it cannot serve as the definition of a fundamental theory. The ensuing need to quantize gravity nonperturbatively is not confined to field theory, but persists in string-theoretic formulations (where it is an unsolved problem as well). Imagine trying to obtain a quantum state representing a 4d Schwarzschild black hole with metricby superposing gravitonic excitations within string theory. However, because of the proportionality G ∼ g 2 str for Newton's constant G [1], this involves arbitrary powers of the string coupling g str , and is therefore an intrinsically nonperturbative construction.1.1. How do we quantize gravity nonperturbatively? The great success of lattice models in describing non-perturbative properties of QCD has for a long time been a motivation for applying discrete methods also in quantum gravity. I will be reporting on the status quo of path-integral ("covariant") lattice models for quantum gravity, and on how to make them more Lorentzian. However, it should be pointed out that there are other ways of tackling the problem, most notably, in a canonical continuum approach based...