Horizon Quantum Mechanics: a hitchhiker's guide to quantum black holes

TL;DR

This paper explores a quantum framework for black holes using horizon wave-functions, focusing on spherically symmetric sources to determine their black hole probabilities.

Contribution

It introduces a quantum mechanical operator for the gravitational radius and applies the horizon wave-function formalism to analyze quantum black holes.

Findings

Quantum sources with mass near the fundamental scale can be probabilistically identified as black holes.

The horizon wave-function formalism provides a new way to quantify black hole properties in quantum mechanics.

The approach bridges classical and quantum descriptions of black holes through a simplified minisuperspace model.

Abstract

It is congruous with the quantum nature of the world to view the space-time geometry as an emergent structure that shows classical features only at some observational level. One can thus conceive the space-time manifold as a purely theoretical arena, where quantum states are defined, with the additional freedom of changing coordinates like any other symmetry. Observables, including positions and distances, should then be described by suitable operators acting on such quantum states. In principle, the top-down (canonical) quantisation of Einstein-Hilbert gravity falls right into this picture, but is notoriously very involved. The complication stems from allowing all the classical canonical variables that appear in the (presumably) fundamental action to become quantum observables acting on the "superspace" of all metrics, regardless of whether they play any role in the description of a specific physical system. On can instead revisit the more humble "minisuperspace" approach and choose the gravitational observables not simply by imposing some symmetry, but motivated by their proven relevance in the (classical) description of a given system. In particular, this review focuses on compact, spherically symmetric, quantum mechanical sources, in order to determine the probability they are black holes rather than regular particles. The gravitational radius is therefore lifted to the status of a quantum mechanical operator acting on the "horizon wave-function", the latter being determined by the quantum state of the source. This formalism is then applied to several sources with a mass around the fundamental scale, which are viewed as natural candidates of quantum black holes.