### 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 A_{D-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∝A_{D-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 language | English (US) |
---|---|

Article number | 044026 |

Pages (from-to) | 1-20 |

Number of pages | 20 |

Journal | Physical Review D - Particles, Fields, Gravitation and Cosmology |

Volume | 62 |

Issue number | 4 |

State | Published - Aug 15 2000 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

^{1}XR

^{+}and the mass spectrum of Schwarzschild black holes in D space-time dimensions.

*Physical Review D - Particles, Fields, Gravitation and Cosmology*,

*62*(4), 1-20. [044026].

}

^{1}XR

^{+}and the mass spectrum of Schwarzschild black holes in D space-time dimensions',

*Physical Review D - Particles, Fields, Gravitation and Cosmology*, vol. 62, no. 4, 044026, pp. 1-20.

**Group theoretical quantization of a phase space S ^{1}XR^{+} and the mass spectrum of Schwarzschild black holes in D space-time dimensions.** / Bojowald, Martin; Kastrup, H. A.; Schramm, F.; Strobl, T.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Group theoretical quantization of a phase space S1XR+ and the mass spectrum of Schwarzschild black holes in D space-time dimensions

AU - Bojowald, Martin

AU - Kastrup, H. A.

AU - Schramm, F.

AU - Strobl, T.

PY - 2000/8/15

Y1 - 2000/8/15

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=33750687940&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33750687940&partnerID=8YFLogxK

M3 - Article

VL - 62

SP - 1

EP - 20

JO - Physical Review D - Particles, Fields, Gravitation and Cosmology

JF - Physical Review D - Particles, Fields, Gravitation and Cosmology

SN - 0556-2821

IS - 4

M1 - 044026

ER -

^{1}XR

^{+}and the mass spectrum of Schwarzschild black holes in D space-time dimensions. Physical Review D - Particles, Fields, Gravitation and Cosmology. 2000 Aug 15;62(4):1-20. 044026.