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Quantization of diffeomorphism invariant theories of connections with local degrees of freedom
Quantization of diffeomorphism invariant theories of connections is studied and the quantum diffeomorphism constraint is solved. The space of solutions is equipped with an inner product that is shown ...

Linking topological quantum field theory and nonperturbative quantum gravity

J. Math. Phys. 36, 6417 (1995); doi:10.1063/1.531251

Issue Date: November 1995

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Lee Smolin
Center for Gravitational Physics and Geometry, Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802
School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey 08540
Quantum gravity is studied nonperturbatively in the case in which space has a boundary with finite area. A natural set of boundary conditions is studied in the Euclidean signature theory in which the pullback of the curvature to the boundary is self-dual (with a cosmological constant). A Hilbert space which describes all the information accessible by measuring the metric and connection induced in the boundary is constructed and is found to be the direct sum of the state spaces of all SU(2) Chern–Simon theories defined by all choices of punctures and representations on the spatial boundary [script S]. The integer level k of Chern–Simons theory is found to be given by k=6pi/G2Lambda+alpha, where Lambda is the cosmological constant and alpha is a CP breaking phase. Using these results, expectation values of observables which are functions of fields on the boundary may be evaluated in closed form. Given these results, it is natural to make the conjecture that the quantum states of the system are completely determined by measurements made on the boundary. One consequence of this is the Bekenstein bound, which says that once the two metric of the boundary has been measured, the subspace of the physical state space that describes the further information that may be obtained about the interior has finite dimension equal to the exponent of the area of the boundary, in Planck units, times a fixed constant. Finally, these results confirm both the categorical-theoretic ``ladder of dimensions'' picture of Crane and the holographic hypothesis of Susskind and 't Hooft. ©1995 American Institute of Physics.
History: Received 30 June 1995; accepted 11 July 1995
Permalink: http://link.aip.org/link/?JMAPAQ/36/6417/1
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KEYWORDS and PACS

Keywords
PACS
  • 04.60.-m
    General relativity and gravitation Quantum gravity
  • 12.10.-g
    Specific theories and interaction models; particle systematics Unified field theories and models
  • YEAR: 1995

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0022-2488 (print)   1089-7658 (online)
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