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

TL;DR

This paper introduces and solves a simple one-dimensional Euclidean quantum gravity model, revealing different behaviors for open and closed universes and exploring effects of scalar fields and higher-dimensional generalizations.

Contribution

It presents a mathematically solvable one-dimensional quantum gravity model with novel insights into universe size behavior and potential extensions to higher dimensions.

Findings

Open universe size is infinite regardless of cosmological constant.

Closed universe size is finite for positive cosmological constant.

Scalar fields do not significantly alter the universe size in the closed case.

Abstract

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.