Horizon Quantum Mechanics of Generalized Uncertainty Principle Black Holes

We study the Horizon Wavefunction (HWF) description of a generalized uncertainty principle inspired metric that admits sub-Planckian black holes, where the black hole mass $m$ is replaced by $M = m\left( 1 + \fracβ{2} \frac{M_{\rm Pl}^2}{m^2} \right)$. Considering the case of a wave-packet shaped by a Gaussian distribution, we compute the HWF and the probability ${\cal {P}}_{BH}$ that the source is a (quantum) black hole, i.e., that it lies within its horizon radius. The case $β<0$ is qualitatively similar to the standard Schwarzschild case, and the general shape of ${\cal {P}}_{BH}$ is maintained when decreasing the free parameter, but shifted to reduce the probability for the particle to be a black hole accordingly. The probability grows with increasing mass slowly for more negative $β$, and drops to 0 for a minimum mass value. The scenario differs in significantly for increasing $β>0$, where a minimum in ${\cal {P}}_{BH}$ is encountered, thus meaning that every particle has some probability of decaying to a black hole. Furthermore, for sufficiently large $β$ we find that every particle is a quantum black hole, in agreement with the intuitive effect of increasing $β$, which creates larger $M$ and $R_{H}$ terms. This is likely due to a "dimensional reduction" feature of the model, where the black hole characteristics for sub-Planckian black holes mimic those in $(1+1)$-dimensions and the horizon size grows as $R_H \sim M^{-1}$.

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