A Matrix Model Proposal for Quantum Gravity and the Quantum Mechanics of Black Holes

We propose a quantum mechanical theory of quantum spaces described by large $N$ noncommutative geometry as a model for quantum gravity. The model admits fuzzy sphere as static solution. Over the fuzzy geometry, the quantum mechanics of the fermions is given by a sum of oscillators with equal frequency. The energy state where exactly half of the Fermi sea is filled contains the maximal amount of degeneracy. This state of the fuzzy sphere obeys the mass-radius relation of a Schwarzschild black hole if the fuzzy sphere is identified with the black hole horizon. Moreover the set of states in the Fermi sea gives precisely the Bekenstein-Hawking entropy. We thus propose that quantum black holes are described by fuzzy spheres with a half-filled Fermi sea in our model. We also consider a system of two fuzzy spheres by embedding them as blocks in the matrix quantum mechanics. When the distance $r$ between the two fuzzy spheres is small, the total energy of the system can be computed using perturbation theory. We show that in the leading order of large $N$ limit, the interaction energy depends on $- G M_1 M_2$ exactly the manner as in Newton gravity. To reproduce the correct $r$ dependence in the long range, we expect the inclusion of large $N$ corrections and quantum effects will be needed.