A Simple Model for Quantum Gravity: the one-dimensional case

We propose and solve mathematically a simple euclidean model for quantum gravity in one dimension. In the case of an open curve, the continuum limit is trivial, that is, the size of the universe is infinite, independently of the value of the cosmological constant. Nevertheless, it is nontrivial in the closed case, resulting in a finite size universe for positive cosmological constant. The presence of a scalar field slightly changes this last result, regardless of the mass. We also discuss a generalization of our model for higher dimensions.