Group theoretical quantization of a phase space S1XR+ and the mass spectrum of Schwarzschild black holes in D space-time dimensions

Martin Bojowald, H. A. Kastrup, F. Schramm, T. Strobl

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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 AD-2 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={(φ ∈ R mod 2 π, p∝AD-2 ∈ R+)} with the symplectic form dφ∧dp. 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 p 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.

Original languageEnglish (US)
Article number044026
Pages (from-to)1-20
Number of pages20
JournalPhysical Review D - Particles, Fields, Gravitation and Cosmology
Volume62
Issue number4
Publication statusPublished - Aug 15 2000

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All Science Journal Classification (ASJC) codes

  • Nuclear and High Energy Physics
  • Physics and Astronomy (miscellaneous)

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