# Discrete spectrum of the quantum Reissner - Nordstr\"om geometry

## TL;DR

This paper explores the quantum properties of the Reissner-Nordström black hole by constructing a mini-superspace model, identifying classical integrals of motion, and analyzing the spectrum of quantum observables related to mass and charge.

## Contribution

It introduces a quantum framework for the Reissner-Nordström geometry, linking horizon presence to the discreteness of the spectrum of quantum observables.

## Key findings

- Spectrum of observables can be fully discrete with horizons.
- Orthogonal basis of quantum states constructed for each case.
- Connection established between classical horizons and quantum spectra.

## Abstract

We start from a static, spherically symmetric space-time in the presence of an electrostatic field and construct the mini-superspace Lagrangian that reproduces the well known Reissner - Nordstr\"om solution. We identify the classical integrals of motion that are to be mapped to quantum observables and which are associated with the mass and charge. Their eigenvalue equations are used as supplementary conditions to the Wheeler-DeWitt equation and a link is provided between the existence of an horizon and to whether the spectrum of the observables is fully discrete or not. For each case we provide an orthonormal basis of states as emerges through the process of canonical quantization.