Simplicial Euclidean and Lorentzian Quantum Gravity
Abstract
One can try to define the theory of quantum gravity as the sum over geometries. In two dimensions the sum over Euclidean geometries can be performed constructively by the method of dynamical triangulations. One can define a propertime propagator. This propagator can be used to calculate generalized HartleHawking amplitudes and it can be used to understand the the fractal structure of quantum geometry. In higher dimensions the philosophy of defining the quantum theory, starting from a sum over Euclidean geometries, regularized by a reparametrization invariant cut off which is taken to zero, seems not to lead to an interesting continuum theory. The reason for this is the dominance of singular Euclidean geometries. Lorentzian geometries with a global causal structure are less singular. Using the framework of dynamical triangulations it is possible to give a constructive definition of the sum over such geometries, In two dimensions the theory can be solved analytically. It differs from twodimensional Euclidean quantum gravity, and the relation between the two theories can be understood. In three dimensions the theory avoids the pathologies of threedimensional Euclidean quantum gravity. General properties of the fourdimensional discretized theory have been established, but a detailed study of the continuum limit in the spirit of the renormalization group and asymptotic safety is till awaiting.
 Publication:

General Relativity and Gravitation
 Pub Date:
 September 2002
 DOI:
 10.1142/9789812776556_0001
 arXiv:
 arXiv:grqc/0201028
 Bibcode:
 2002grg..conf....3A
 Keywords:

 General Relativity and Quantum Cosmology;
 High Energy Physics  Lattice;
 High Energy Physics  Theory
 EPrint:
 23 pages, 4 eps figures, Plenary talk GR16