Quantum conformal superspace

Details Quantum conformal superspace Quantum conformal superspace

Feb 1996

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1996gregr..28..221f&link_type=abstract

General Relativity and Gravitation, Volume 28, Issue 2, pp.221-237

Computer Science

16

Scientific paper

For a compact connected orientablen-manifold M, n ≥ 3, we study the structure ofclassical superspace {S} equiv {{M} {/ {{M} {D}}} } {D}}, quantum superspace {S}_0 equiv {{M} {/ {{M} {{D}_0 }}} } {{D}_0}}, classical conformal superspace {C} equiv {{( {{{M} {/ {M} {P}}} {P}}} )} {/ {{( {{{M} {left/ {M} {P}}} } {P}}} )} {D}}} } {D}}, and quantum conformal superspace {C}_0 equiv {{( {{{M} {/ {M} {P}}} } {P}}} )} {/ {{( {{{M} {/ {M} {P}}} {P}}} )} {{D}_0 }}} {D}_0 }}. The study of the structure of these spaces is motivated by questions involving reduction of the usual canonical Hamiltonian formulation of general relativity to a non-degenerate Hamiltonian formulation, and to questions involving the quantization of the gravitational field. We show that if the degree of symmetry ofM is zero, thenS,S 0,C, andC 0 are ilh orbifolds. The case of most importance for general relativity is dimension n=3. In this case, assuming that the extended Poincaré conjecture is true, we show that quantum superspaceS 0 and quantum conformal superspace C 0 are in fact ilh-manifolds. If, moreover, M is a Haken manifold, then quantum superspace and quantum conformal superspace arecontractible ilh-manifolds. In this case, there are no Gribov ambiguities for the configuration spacesS 0 andC 0. Our results are applicable to questions involving the problem of thereduction of Einstein's vacuum equations and to problems involving quantization of the gravitational field. For the problem of reduction, one searches for a way to reduce the canonical Hamiltonian formulation together with its constraint equations to an unconstrained Hamiltonian system on a reduced phase space. For the problem of quantum gravity, the spaceC 0 will play a natural role in any quantization procedure based on the use of conformal methods and the reduced Hamiltonian formulation.

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