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 (Formula presented) 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 (Formula presented) 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 (Formula presented) with the symplectic form (Formula presented) As the action of the group (Formula presented) on that manifold is transitive, effective and Hamiltonian, it can be used for a group theoretical quantization of the system. The area operator (Formula presented) for the horizon corresponds to the generator of the compact subgroup (Formula presented) and becomes quantized accordingly: The positive discrete series of the irreducible unitary representations of the group (Formula presented) yields an (horizon) area spectrum (Formula presented) where (Formula presented) characterizes the representation and (Formula presented) the number of area quanta. If one employs the unitary representations of the universal covering group of (Formula presented) the number k can take any fixed positive real value (Formula presented) parameter). The unitary representations of the positive discrete series provide concrete Hilbert spaces for quantum Schwarzschild black holes.
|Original language||English (US)|
|Number of pages||1|
|Journal||Physical Review D - Particles, Fields, Gravitation and Cosmology|
|State||Published - Jan 1 2000|
All Science Journal Classification (ASJC) codes
- Nuclear and High Energy Physics
- Physics and Astronomy (miscellaneous)