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

Abstract

We propose and analytically solve a simple Euclidean model for quantum gravity in one dimension. The geometry is parametrized by angles associated with a circle of fixed radius $\ell$, allowing the use of the normalized Haar measure of U(1) and character expansions. In the case of an open curve, the continuum limit is trivial: the size of the universe is infinite, independently of the value of the cosmological constant. In the closed case, however, the continuum limit is nontrivial, resulting in a finite-size universe for positive cosmological constant. The presence of a scalar field slightly modifies this result while preserving its behavior. We also discuss possible generalizations of the model to higher dimensions.

arXiv comment9 pages, 5 figures