Quantum Thermodynamics of Black Hole Horizons

Exploiting results recently proved in a technical paper (and some of them are reviewed herein in the language of theoretical physicists) we focus on quantization of the metric of a black hole restricted to the Killing horizon with universal radius r0. The metric is represented in a suitable manner after imposing spherical symmetry and, after restriction to the Killing horizon, it is quantized employing chiral currents. Two “components of the metric” are in fact quantized: one behaves as an affine scalar fields under changes of coordinates and the other is a proper scalar field. The symplectic group acts on both fields as subgroup of diffeomorphisms of the horizon and this action, in some cases depending on the choice of the vacuum state, can be implemented by means of a unitary group. If the reference state of the scalar field is not a vacuum state but a coherent state, spontaneous breaking of conformal symmetry arises and the state contains a Bose-Einstein condensate. In this case the order parameter fixes the actual size of the black hole with respect to r0. This state together with that associated with the affine scalar when restricted in a half horizon (the future boundary of the external region of the black hole) is recognized to be thermal (KMS) with respect to Schwarschild Killing time restricted to the horizon. The value of the order parameter individuates Hawking temperature as well. As a result it is found that the densities, energy and entropy of this state scales like the mass and the entropy of the black hole and they coincide with them provided the universal parameter r0 is fixed appropriately not depending on the size of the actual black hole.