Quantum black holes in bootstrapped Newtonian gravity

We analyse the classical configurations of a bootstrapped Newtonian potential generated by homogeneous spherically symmetric sources in terms of a quantum coherent state. We first compute how the mass and mean wavelength of these solutions scale in terms of the number of quanta in the coherent state. We then note that the classical relation between the ADM mass and the proper mass of the source naturally gives rise to a Generalised Uncertainty Principle for the size of the gravitational radius in the quantum theory. Consistency of the mass and wavelength scalings with this GUP requires the compactness remains at most of order one even for black holes, and the corpuscular predictions are thus recovered, with the quantised horizon area expressed in terms of the number of quanta in the coherent state. Our findings could be useful for analysing the classicalization of gravity in the presence of matter and the avoidance of singularities in the gravitational collapse of compact sources.