On the Statistical Mechanics of Quantum Geometry

This thesis presents an original derivation of the Bekenstein-Hawking formula for the entropy of a black hole within the framework of loop quantum gravity. This derivation differs from preceding ones in that it models the black hole as a grand canonical ensemble and makes use of a recently introduced quasi-local energy operator. It is shown that the statistical mechanics of the model reduces to that of a simple non-interacting gas of distinguishable particles with spin. For temperatures low in comparison with the Planck temperature and boundaries large in comparison with the Planck area, the entropy of the system is shown to be proportional to area (with a logarithmic correction), providing a simple derivation of the Bekenstein-Hawking result (for a certain choice of the Immirzi parameter). Also in this limit, the quantum geometry on the boundary forms a "condensate" in the lowest energy level (j = 1/2). Finally, we relate our description, in terms of the grand canonical ensemble, to previous geometric entropy calculations, which made use of area ensembles.