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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Arnowitt, R, Deser, S, and Misner, C (1962), The dynamics of general relativity, in Gravitation: an introduction to current research, L. Witten, editor, John Wiley and Sons, Inc., New York.
Balachandran, A P (1989), Classical topology and quantum phases: quantum mechanics, in Geometrical and Algebraic Aspects of Nonlinear Field Theory, S. De Filippo, M. Marinaro, G. Marmo, and G. Vilasi, editors, Elsevier Science Publishers.
Bourguignon, J-P (1975), Une stratification de l'espace des structures riemanniennes, Compositio Math. 30, 1–41.
Bröcker, T, and Jänich, K (1982), Introduction to Differential Topology, Cambridge University Press, Cambridge.
Davis, J F, and Milgram, R J (1985), A Survey of the Spherical Space Form Problem, Mathematical Reports, Volume 2, Part 2, Harwood Academic Publishers, New York.
Davis, M, and Morgan, J (1984), Finite group actions on homotopy 3-spheres, in The Smith Conjecture, Morgan, J, and Bass, H, editors, Academic Press, Inc., New York.
DeWitt, B (1967a), Quantum theory of gravity. I. The canonical theory, Physical Review 160, 1113–1148.
DeWitt, B (1967b), Quantum theory of gravity, II. The manifestly convariant theory, Physical Review 162, 1195–1239.
DeWitt, B (1967c), Quantum theory of gravity, III. Application of the covariant theory, Physical Review 162, 1239–1256.
DeWitt, B (1970), Spacetime as a sheaf of geodesics in superspace, in Relativity, M. Carmeli, S. Fickler, and L. Witten, editors, Plenum Press, New York.
Earle, C, and Eells, J (1969), A fibre bundle description of Teichmüller theory, J. Diff. Geom., 3, 19–43.
Ebin, D, (1970), The space of Riemannian metrics, Proc. Symp. Pure Math., Amer. Math. Soc. 15, 11–40.
Ebin, D, and Marsden, J (1970), Groups of diffeomorphisms and the motion of an incompressible fluid, Annals of Mathematics 92, 102–163.
Edmonds, A (1985), Transformation groups and lowdimensional manifolds, in Group Actions on Manifolds, Contemporary Mathematics, Volume 36, R. Schultz, editor, 339–366.
Ellis, G (1971), Topology and cosmology, General Relativity and Gravitation 2, 7–21.
Fischer, A (1970), The theory of superspace, in Relativity, M. Carmeli, S. Fickler, and L. Witten, editors, Plenum Press, New York.
Fischer, A, and Marsden, J (1975), Deformations of the scalar curvature, Duke Mathematical Journal 42, 519–547.
Fischer, A, and Marsden, J (1977), The manifold of conformally equivalent metrics, Canadian Journal of Mathematics, 1, 193–209.
Fischer, A, and Moncrief, V (1994a), Reducing Einstein's equations to an unconstrained Hamiltonian system on the cotangent bundle of Teichmüller space, in Physics on Manifolds, Proceedings on the International Colloquium in honour of Yvonne Choquet-Bruhat, M. Flato, R. Kerner, and A. Lichnerowicz, editors, Kluwer Academic Publishers, Boston, 111–151.
Fischer, A, and Moncrief, V (1994b), Classical and conformal superspace, linearization stability, and the reduction of Einstein's equations, in Proceedings of the Cornelius Lanczos International Centenary Conference, J. Brown, M. Chu, D. Ellison, and R. Plemmons, editors, Society for Industrial and Applied Mathematics, Philadelphia, 535–542.
Fischer, A, and Moncrief, V (1995a), A method of reduction for Einstein's equations of evolution and a natural symplectic structure on the space of quantum gravitational degrees of freedom, General Relativity and Gravitation.
Fischer, A, and Moncrief, V (1995b), Quantum conformal superspace, General Relativity and Gravitation.
Fischer, A, and Tromba, A (1984a), On a purely “Riemannian” proof of the structure and dimension of the unramified moduli space of a compact Riemann surface, Mathematische Annalen 267, 311–345.
Fischer, A, and Tromba, A (1984b), Almost complex principle fiber bundles and the complex structure on Teichmüller space, J. für die reine und angewandte Mathematik 352, 151–160.
Fischer, A, and Tromba, A (1984c), On the Weil-Petersson metric on Teichmüller space, Trans. Amer. Math. Soc. 284, 319–335.
Fischer, A, and Tromba, A (1987), A new proof that Teichmüller space is a cell, Trans. Amer. Math. Soc. 303, 257–262.
Freed, D, and Groisser, D (1989), The basic geometry of the manifold of Riemannian metrics and of its quotient by the diffeomorphism group, Michigan Math. J. 36, 323–344.
Friedman, J (1990), Space-time topology and quantum gravity, in Conceptual problems in quantum gravity, A. Ashtekar and J. Stachel, editors, Birkhäuser, Boston, 539–572.
Friedman, J, and Witt, D (1986), Homotopy is not isotopy for homeomorphisms of 3-manifolds, Topology 25, 35–44.
Freedman, M, and Yau, S-T (1983), Homotopically trivial symmetries of Haken manifolds are toral, Topology 22, 179–189.
Gil-Medrano, O, and Michor, P (1991), The Riemannian manifold of all Riemannian metrics, Quart. J. Math. Oxford (2) 42, 183–202.
Giulini, D (1994), 3-manifolds for relativists, International Journal for Theoretical Physics, 33, 913–930.
Giulini, D (1995), On the configuration-space topology in general relativity, Helvetica Physica Acta, to appear.
Gribov, V (1976), Quantisation of non-Abelian gauge theories, Nucl. Phys. B139, 1–19.
Hatcher, A (1976), Homeomorphisms of sufficiently large P 2-irreducible 3-manifolds, Topology 15, 343–347.
Hawking, S, and Ellis, G (1973), The Large Scale Structure of Space-Time, Cambridge University Press, Cambridge, England.
Hempel, J (1976), 3-manifolds, Annals of Mathematics Studies, Number 86, Princeton University Press, Princeton, New Jersey.
Hsiang, W (1967a), The natural metric on SO (n)/SO (n−1) is the most symmetric metric, Bull. Amer. Math. Soc. 73, 55–58.
Hsiang, W (1967b), On the bounds on the dimensions of the isometry groups of all possible riemannian metrics on an exotic sphere, Ann. of Math. 85, 351–358.
Hsiang, W (1971), On the degree of symmetry and the structure of highly symmetric manifolds, Tamkang Journal of Mathematics, Tamkang College of Arts and Sciences, Taipei, 73, 1–22.
Hu, S-T (1959), Homotopy Theory, Academic Press, New York.
Itoh, M (1991), Yamabe structures and the space of conformal structures, International Journal of Mathematics 2, 659–671.
Kneser, H (1929), Geschlossen Flächen in dreidimensionalen Mannigfaltigkeiten, Jber. Deutsch. Math.-Verein 38, 248–260.
Lelong-Ferrand, J (1969), Transformations conformes et quasiconformes des variétés riemanniennes; application à la démonstration d'une conjecture de A. Lichnerowicz, C. R. Acad. Sci. Paris 269, 583–586.
Lelong-Ferrand, J (1971), Transformations conformes et quasiconformes des variétés riemanniennes compacts (démonstration de la conjecture de A. Lichnerowicz), Acad. Roy. Belg. Cl. Sci. Mem. Coll. 8°(2) 39, no. 5.
Massey, W (1967), Algebraic Topology: An Introduction, Harcourt, Brace and World, New York.
Mess, G (1995), Homotopically trivial symmetries of 3-manifolds are toral, to appear.
Milnor, J (1956a), Construction of universal bundles: I, Ann. Math., (2)63, 272–284.
Milnor, J (1956b), Construction of universal bundles: II, Ann. Math., (2)63, 430–436.
Milnor, J (1961), A unique decomposition theorem for 3-manifolds, American Journal of Mathematics, 1–7.
Mitter, P, and Viallet, C, (1981), On the bundle of connections and the gauge orbit manifold in Yang-Mills theory, Commun. Math. Phys. 79, 455–472.
Moncrief, V (1976), Space-time symmetries and linearization stability of the Einstein equations. II, J. Math. Phys. 17, 1893–1902.
Moncrief, V (1989), Reduction of the Einstein equations in 2+1 dimensions to a Hamiltonian system over Teichmüller space, J. Math. Phys. 30 (12), 2907–2914.
Moncrief, V (1990), How solvable is (2+1)-dimensional Einstein gravity?, J. Math. Phys. 31 (12), 2978–2982.
Munkres, J R (1960), Obstructions to smoothing piecewise-differentiable homeomorphisms, Ann. of Math. 72, 521–554.
Obata, M (1971), The conjectures on conformal transformations of Riemannian mnifolds, J. Diff. Geom. 6, 247–258.
Omori, H (1970), On the group of diffeomorphisms of a compact manifold, Proc. Symp. Pure Math., Amer. Math. Soc. 15, 167–183.
Orlik, P (1972), Seifert manifolds, Lecture Notes in Mathematics 291, Springer-Verlag, New York.
Orlik, P, and Raymond, F (1968), Actions of SO (2) on 3-manifolds, in Proceedings of the Conference on Transformation Groups, New Orleans, 1967 Springer-Verlag, New York.
Palais (1966), Homotopy theory of infinite dimensional manifolds, Topology, 5, 1–16.
Raymond, F (1968), Classification of the action of the circle on 3-manifolds, Trans. Amer. Math. Soc., 131, 51–78.
Satake, I (1956), On a generalization of the notion of manifold, Proc. Nat. Acad. Sci. U.S.A. 42, 359–363.
Scott, P (1983), The geometries of 3-manifolds, Bull. London Math. Soc., 15, 401–487.
Singer, I (1978), Some remarks on the Gribov ambiguity, Commun. Math. Phys. 60, 7–12.
Sorkin, R (1989), Classical topology and quantum phases: quantum geons, in Geometrical and Algebraic Aspects of Nonlinear Field Theory, S. De Filippo, M. Marinaro, G. Marmo, and G. Vilasi, editors, Elsevier Science Publishers, 201–218.
Spanier, E (1966), Algebraic Topology, McGraw-Hill Book Company, New York.
Thurston, W (1978), The geometry and topology of 3-manifolds, preprint, Princeton University, Princeton, New Jersey.
Thurston, W (1982), Hyperbolic geometry and 3-manifolds, in Low-dimensional topology, London Mathematical Society Lecture Note Series 48, R. Brown and T. L. Thickstun, editors, Cambridge University Press, 9–25.
Thurston, W (1982), Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. 6, 357–381.
Tromba, A (1992), Teichmüller Theory in Riemannian Geometry, Birkhäuser Verlag, Basel.
Waldhausen, F (1967), Gruppen mit Zentrum und 3-dimensionale Mannigfaltigkeiten, Topology 6, 505–517.
Waldhausen, F (1968), On irreducible 3-manifolds which are sufficiently large, Ann. Math. 18 56–88.
Wheeler, J A (1962), Geometrodynamics, Academic Press, New York.
Wheeler, J A (1964), Geometrodynamics and the issue of the final state, in Relativity, Groups and Topology, C. DeWitt and B. DeWitt, editors, Gordon and Breach Science Publishers, New York.
Wheeler, J A (1968a), Superspace and the nature of quantum geometrodynamics, in Batelle Rencontres—1967 Lectures in Mathematics and Physics, C. DeWitt and J. A. Wheeler, editors, W. A. Benjamin, Inc., New York.
Wheeler, J A (1968b), Einstein's Vision, Springer-Verlag, New York.
Wheeler, J A (1970), Superspace, in Analytic Methods in Mathematical Physics, R. Gilbert and R. Newton, editors, Gordon and Breach Science Publishers, New York.
Whitehead, J H C (1961), Manifolds with transverse fields in euclidean space, Ann. of Math. 73, 154–212.
Witt, D (1986), Symmetry groups of state vectors in quantum gravity, J. Math. Phys. 27 (2), 573–592.
Wolf, J A (1972), Spaces of Constant Curvature, second edition, Publish or Perish Press, Berkeley, California.
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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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