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 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)
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Group theoretical quantization of a phase space S1XR+ and the mass spectrum of Schwarzschild black holes in D space-time dimensions. / Bojowald, Martin; Kastrup, H. A.; Schramm, F.; Strobl, T.
In: Physical Review D - Particles, Fields, Gravitation and Cosmology, Vol. 62, No. 4, 044026, 15.08.2000, p. 1-20.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.
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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 -