Abstract
The symplectic reduction of pure spherically symmetric (Schwarzschild) classical gravity in D space-time dimensions yields a two-dimensional phase space of observables consisting of the mass and a canonically conjugate (Killing) time variable T. Imposing (mass-dependent) periodic boundary conditions in time on the associated quantum-mechanical plane waves which represent the Schwarzschild system in the period just before or during the formation of a black hole yields an energy spectrum of the hole which realizes the old Bekenstein postulate that the quanta of the horizon are multiples of a basic area quantum. In the present paper it is shown that the phase space of such Schwarzschild black holes in D space-time dimensions is symplectomorphic to a symplectic manifold with the symplectic form As the action of the group on that manifold is transitive, effective and Hamiltonian, it can be used for a group theoretical quantization of the system. The area operator for the horizon corresponds to the generator of the compact subgroup and becomes quantized accordingly: The positive discrete series of the irreducible unitary representations of the group yields an (horizon) area spectrum where characterizes the representation and the number of area quanta. If one employs the unitary representations of the universal covering group of the number k can take any fixed positive real value parameter). The unitary representations of the positive discrete series provide concrete Hilbert spaces for quantum Schwarzschild black holes.
- Received 26 August 1999
DOI:https://doi.org/10.1103/PhysRevD.62.044026
©2000 American Physical Society