ar X iv : h ep - t h / 02 01 10 4 v 1 1 5 Ja n 20 02 1 A Lorentzian cure for Euclidean troubles ∗

There is strong evidence coming from Lorentzian dynamical triangulations that the unboundedness of the gravitational action is no obstacle to the construction of a well-defined non-perturbative path integral. In a continuum approach, a similar suppression of the conformal divergence comes about as the result of a non-trivial path-integral measure. The progress of the last few years has established the method of Lorentzian dynamical trian-gulations (LDT) as a serious candidate for a non-perturbative theory of quantum gravity in four dimensions. In this approach one tries to define quantum gravity as the continuum limit of a statistical sum over Lorentzian dynamically triangu-lated space-times [1,2]. The transition amplitudes G with respect to discrete proper time t are given by sums G(τ 1 , τ 2 , t) = T,∂T =τ1∪τ2 1 C T e iS(T) (1) over inequivalent Lorentzian triangulations T with three-dimensional spatial boundary triangu-lations τ 1 and τ 2 , weighted by the gravitational Einstein action of T in Regge form, and including a discrete symmetry factor C T. Expression (1) looks unmanageable at first, but can be converted into a perfectly well-defined real * Talk presented by R. Loll. state sum by means of a non-perturbative Wick rotation which maps each Lorentzian triangula-tion uniquely to a Euclidean one,