Fractal Structure in Two-Dimensional Quantum Regge Calculus

TL;DR

This paper investigates the fractal geometry of two-dimensional quantum gravity using Regge calculus and Monte Carlo simulations, comparing results with analytical continuum models to assess its validity as a regularization method.

Contribution

It provides the first detailed comparison between Monte Carlo simulations of quantum Regge calculus and analytical continuum scaling functions in two dimensions.

Findings

Correct scaling observed for loops attached to baby universes with scale-invariant measure.

Data for baby universe loops converge to universal scaling function as triangles increase.

No scaling behavior observed for loops attached to the mother universe at current sizes.

Abstract

We study the fractal structure of the surface in two-dimensional quantum Regge calculus by performing Monte Carlo simulation with up to 200,000 triangles. The result can be compared with the universal scaling function obtained analytically in the continuum limit of dynamical triangulation, which provides us with a definite criterion whether Regge calculus serves as a proper regularization of quantum gravity. When the scale-invariant measure is taken as the measure of the link-length integration, we observe the correct scaling behavior in the data for the type of loop attached to a baby universe. The data seem to converge to the universal scaling function as the number of triangles is increased. The data for the type of loop attached to the mother universe, on the other hand, shows no scaling behavior up to the present size.