LINKING TOPOLOGICAL QUANTUM-FIELD THEORY AND NONPERTURBATIVE QUANTUM-GRAVITY

Quantum gravity is studied nonperturbatively in the case in which spac e has a boundary with finite area. A natural set of boundary condition s is studied in the Euclidean signature theory in which the pullback o f the curvature to the boundary is self-dual (with a cosmological cons tant). A Hilbert space which describes all the information accessible by measuring the metric and connection induced in the. boundary is con structed 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 rep resentations on the spatial boundary L. The integer level k of Chern-S imons theory is found to be given by k = 6 pi/G(2) Lambda + alpha, whe re Lambda is the cosmological constant and alpha is a C P breaking pha se: Using these results, expectation values of observables which are f unctions of fields on the boundary may be evaluated in closed form. Gi ven these results, it is natural to make the conjecture that the quant um states of the system are completely determined by measurements made on the boundary. One consequence of this is the Bekenstein bound, whi ch says that once the two metric of the boundary has been measured, th e subspace of the physical state space that describes the further info rmation that may be obtained about the interior has finite dimension e qual to the exponent of the area of the boundary, in Planck units, tim es a fixed constant. Finally, these results confirm both the categoric al-theoretic ''ladder of dimensions'' picture of Crane and the hologra phic hypothesis of Susskind and 't Hooft. (C) 1995 American Institute of Physics.