Discrete Gravity in One Dimension

TL;DR

This paper explores a one-dimensional quantum gravity model using Regge calculus, analyzing lattice diffeomorphisms, scalar field coupling, measure renormalization, and spectrum properties, providing exact solutions and renormalization insights.

Contribution

It introduces an exact scalar functional integral in 1D quantum gravity with Regge discretization and examines measure renormalization and spectrum properties.

Findings

Scalar field fluctuations do not renormalize the cosmological constant in 1D.

Exact solutions for the scalar functional integral are obtained.

Spectrum analysis shows differences with Poissonian edge length distributions.

Abstract

A model for quantum gravity in one (time) dimension is discussed, based on Regge's discrete formulation of gravity. The nature of exact continuous lattice diffeomorphisms and the implications for a regularized gravitational measure are examined. After introducing a massless scalar field coupled to the edge lengths, the scalar functional integral is performed exactly on a finite lattice, and the ensuing change in the measure is determined. It is found that the renormalization of the cosmological constant due to the scalar field fluctuations vanishes identically in one dimension. A simple decimation renormalization group transformation is performed on the partition function and the results are compared with the exact solution. Finally the properties of the spectrum of the scalar Laplacian are compared with results obtained for a Poissonian distribution of edge lengths.