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The structure of quantum conformal superspace

  • Part II Global Structures
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Book cover Global Structure and Evolution in General Relativity

Part of the book series: Lecture Notes in Physics ((LNP,volume 460))

Abstract

For a compact connected orientable n-manifold M, n≥3, we study the structure of classical superspace S≡M/D, quantum superspace S 0≡M/D0, classical conformal superspace C≡M/P/D, and quantum conformal superspace C 0≡M/P/D0. The study of the structure of these spaces is motivated by questions involving reduction of the canonical Hamiltonian formulation of general relativity to a non-degenerate Hamiltonian formulation, and to questions involving both linearization stability and quantization of the gravitational field. We show that if the degree of symmetry of M is zero, then S, S 0, C, and C 0 are ilh-orbifolds. The case of most importance for general relativity is dimension n=3. In this case, for a broad class of 3-manifolds for which deg M=0, we show that quantum superspace S 0 and quantum conformal superspace C 0 are in fact ilh-manifolds. If M is a Haken 3-manifold with deg M=0, then quantum superspace and quantum conformal superspace are contractible ilh-manifolds. Under these circumstances, there are no Gribov ambiguities for the configuration spaces S 0 and C 0. Our results are also applicable to the problem of reduction of Einstein's vacuum equations, to linearization stability, and to the problem of quantization of the gravitational field. Our results can be used to reduce the canonical Hamiltonian formulation, together with its constraint equations, to an unconstrained Hamiltonian system. On the reduced phase space the canonical variables are free, or unconstrained, and carry complete information about the true degrees of freedom of the gravitational field. The structure of the reduced phase space is also of importance in understanding certain questions involving linearization stability and quantization of the gravitational field. For questions regarding linearization stability, the cotangent bundle of C plays a role in understanding the symplectic structure of the space of true degrees of freedom of the gravitational field. For questions regarding quantization, the space C 0 plays the role of the reduced configuration space for quantum gravity.

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Authors and Affiliations

  1. Department of Mathematics, University of California, 95064, Santa Cruz, California

    Arthur E. Fischer

  2. Departments of Mathematics and Physics, Yale University, 06511, New Haven, Connecticut

    Vincent Moncrief

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  1. Arthur E. Fischer
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  2. Vincent Moncrief
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Spiros Cotsakis Gary W. Gibbons

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Fischer, A.E., Moncrief, V. (1996). The structure of quantum conformal superspace. In: Cotsakis, S., Gibbons, G.W. (eds) Global Structure and Evolution in General Relativity. Lecture Notes in Physics, vol 460. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0103448

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  • DOI: https://doi.org/10.1007/BFb0103448

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