Discrete Lorentzian Quantum Gravity
R. Loll
TL;DR
The paper argues that non-perturbative quantum gravity benefits from a Lorentzian, causally structured discretization rather than Euclidean methods, circumventing the problematic Wick rotation in a dynamical spacetime. It introduces Lorentzian dynamical triangulations, with a partition function over causal triangulations and a unique Wick rotation to Euclidean geometries, yielding a distinct universality class and well-defined transfer matrices in low dimensions. In 2d, the model is exactly solvable, featuring a ground-state geometry with $d_H=2$ and robust matter exponents; in 3d, a new intermediate phase exhibits an extended ground-state universe, unlike Euclidean DT where degenerate phases dominate. These results support Lorentzian DT as a promising framework for a non-perturbative theory of quantum gravity, though the 4d case with propagating degrees of freedom remains to be established.
Abstract
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.