Group theoretical quantization of a phase space S1×R+ and the mass spectrum of Schwarzschild black holes in D space-time dimensions

M. Bojowald, H. A. Kastrup, F. Schramm, and T. Strobl
Phys. Rev. D 62, 044026 – Published 24 July 2000
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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 M(>0) 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 AD2 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 S={(φRmod2π,pAD2R+)} with the symplectic form dφdp. As the action of the group SO(1,2) 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 SO(2) and becomes quantized accordingly: The positive discrete series of the irreducible unitary representations of the group SO(1,2) yields an (horizon) area spectrum (k+n), where k=1,2,, characterizes the representation and n=0,1,2,, the number of area quanta. If one employs the unitary representations of the universal covering group of SO(1,2), the number k can take any fixed positive real value (θ 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

Authors & Affiliations

M. Bojowald*, H. A. Kastrup, F. Schramm, and T. Strobl

  • Institute for Theoretical Physics, RWTH Aachen, D-52056 Aachen, Germany

  • *Email address: bojowald@physik.rwth-aachen.de
  • Email address: kastrup@physik.rwth-aachen.de
  • Email address: pth@tpi.uni-jena.de

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Vol. 62, Iss. 4 — 15 August 2000

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