Quantum Theory of Totally Constrained Systems and its Application to Gravity Theories

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The main results obtained by the research from 1996 through 1998 are summarized as follows :1) Canonical structure of locally homogeneous systems : For a locally homogeneous system on a compact 3-dimensional space, a general algorithm to determine the canonical structure of the phase space consisting of all diffeomorphism classes of initial data sets was developed, and the moduli degrees of freedom were shown to be constants of motion. Further, with the help of the algorithm, the canonical structures of pure gravity systems of the Thurston type E3, Nil and Sol were completely classified to show that the canonical structure often becomes degenerate in the moduli sector.2) Quantum Bianchi model : On the basis of the result in I), the quantum dynamics of the type E3 systems with compact space was analyzed in terms of the new formalism (Web formalism) proposed by the head investigator and was compared with the treatment in terms of the conventional Wheeler-DeWitt equation. The most importa … More nt result is the discovery of the fact that the quantum dynamics in the Web formalism is described not by solutions to the Wheeler-DeWitt equation but rather by their extensions in the distribution sense in general.3) Perturbation of scalar fields on an expanding universe : the structure of the gauge-invariant perturbation theory was investigated in detail for the system consisting of scalar fields and gravity on a spatially homogeneous and isotropic expanding universe background. It was found that there is a general correspondence between the long-wavelength limit of gauge-invariant perturbations and the homogeneous perturbations of the homogeneous background solution. Further, with the help of this correspondence, the evolutions of superhorizon scale perturbations in the reheating stage of inflationary models were investigated analytically to establish the conservation law of the Bardeen parameter during this stage.4) Black hole thermodynamics : On the basis of the idea of entanglement, thermodynamics was constructed for a quantum scalar field on the flat spacetime and on black hole spacetimes. It was shown that the entanglement thermodynamics thus constructed for the black hole spacetimes has the same structure as that of the classical black hole thermodynamics. Less