Generating functions for black hole entropy in Loop Quantum Gravity

TL;DR

This paper develops systematic generating functions to solve combinatorial problems in black hole entropy calculations within Loop Quantum Gravity, facilitating analysis from microscopic to macroscopic regimes.

Contribution

It introduces a comprehensive set of generating functions that address various combinatorial aspects of black hole entropy in Loop Quantum Gravity, unifying different counting methods.

Findings

Provides generating functions for degeneracy related to the area spectrum

Offers solutions to the projection constraint

Enables extension to macroscopic black hole entropy analysis

Abstract

We introduce, in a systematic way, a set of generating functions that solve all the different combinatorial problems that crop up in the study of black hole entropy in Loop Quantum Gravity. Specifically we give generating functions for: The different sources of degeneracy related to the spectrum of the area operator, the solutions to the projection constraint, and the black hole degeneracy spectrum. Our methods are capable of handling the different countings proposed and discussed in the literature. The generating functions presented here provide the appropriate starting point to extend the results already obtained for microscopic black holes to the macroscopic regime --in particular those concerning the area law and the appearance of an effectively equidistant area spectrum.