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

1)局部齐次系统的标准结构：对于紧致三维空间上的局部齐次系统，给出了确定由初始数据集的所有微分同胚类组成的相空间的标准结构的一般算法，并且证明了其模自由度为运动常数。2)量子比安奇模型：在I)结果的基础上，用带头研究人员提出的新形式(Web形式)分析了紧致空间E3型纯引力系统的量子动力学，并与传统的Wheeler-DeWitt方程的处理方法进行了比较。最重要的…更多的结果是发现了这样一个事实：Web形式中的量子动力学不是用Wheeler-DeWitt方程的解来描述，而是用它们在分布意义上的推广来描述。3)扩展宇宙中标量场的微扰：详细研究了空间均匀和各向同性扩展宇宙背景上由标量场和引力组成的系统的规范不变微扰理论的结构。结果表明，规范不变微扰的长波极限与齐次背景解的齐次微扰之间存在着普遍的对应关系。在此基础上，对膨胀模型再热阶段超视界尺度微扰的演化进行了解析研究，建立了这一阶段的Bardeen参数守恒定律。4)黑洞热力学：基于纠缠的思想，建立了平坦时空和黑洞时空上量子标量场的热力学。结果表明，为黑洞时空构造的纠缠热力学与经典黑洞热力学具有相同的结构。较少

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