On the distribution of the eigenvalues of the area operator in loop quantum gravity

TL;DR

This paper analyzes the eigenvalue distribution of the area operator in loop quantum gravity, introducing a Laplace transform method to accurately determine its asymptotic behavior for large areas.

Contribution

It develops a novel Laplace transform-based method to accurately analyze the eigenvalue distribution of the area operator in loop quantum gravity, surpassing previous integer partition approximations.

Findings

Integer partition approximations are insufficient for large areas.

The Laplace transform method provides accurate eigenvalue distribution estimates.

The approach is valid for any area and useful for asymptotic analysis.

Abstract

We study the distribution of the eigenvalues of the area operator in loop quantum gravity concentrating on the part of the spectrum relevant for isolated horizons. We first show that the approximations relying on integer partitions are not sufficient to obtain the asymptotic behaviour of the eigenvalue distribution for large areas. We then develop a method, based on Laplace transforms, that provides a very accurate solution to this problem. The representation that we get is valid for any area and can be used to obtain its asymptotics in the large area limit.