Horizon wave function for single localized particles: GUP and quantum black-hole decay

A localized 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 Rela- tivity, if the particle’s energy density exceeds a certain thresh- old, it should be a black hole. To combine these two pictures, we introduce a horizon wave function determined by the par- ticle wave function in position space, which eventually yields the probability that the particle is a black hole. The existence of a minimum mass for black holes naturally follows, albeit not in the form of a sharp value around the Planck scale, but rather like a vanishing probability that a particle much lighter than the Planck mass may be a black hole. We also show that our construction entails an effective generalized uncertainty principle (GUP), simply obtained by adding the uncertainties coming from the two wave functions associated with a parti- cle. Finally, the decay of microscopic (quantum) black holes is also described in agreement with what the GUP predicts.