What is the Schwarzschild radius of a quantum mechanical particle?

TL;DR

This paper introduces a horizon wave-function approach to unify quantum mechanics and general relativity, providing a probabilistic framework for black hole formation from quantum particles and deriving a generalized uncertainty principle.

Contribution

It proposes a novel horizon wave-function method linking quantum wave-functions to black hole probabilities, revealing a minimum mass and a generalized uncertainty principle.

Findings

Derived the probability that a quantum particle is a black hole.

Established a minimum mass threshold for black hole formation.

Showed the emergence of a generalized uncertainty principle from quantum and gravitational uncertainties.

Abstract

A localised particle in Quantum Mechanics is described by a wave packet in position space, regardless of its energy. However, from the point of view of General Relativity, if the particle's energy density exceeds a certain threshold, it should be a black hole. In order to combine these two pictures, we introduce a horizon wave-function determined by the position wave-function, which yields the probability that the particle is a black hole. The existence of a (fuzzy) minimum mass for black holes naturally follows, and we also show that our construction entails an effective Generalised Uncertainty Principle simply obtained by adding the uncertainties coming from the two wave-functions.