Topolygy in Quantum Cosmology

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(1)In this research project we have been mainly concerned with the classification of compact homogeneous universes and their dynamics.In our first paper in Journal of Mathematical Mathematics, we have classified almost all the compact homogeneous universes except the three dimensional hyperbolic space, which is still a major topics in mathematics. We have succeeded in identifying the kinematical degrees of freedom as the local curvature degrees of freedom and the Teichmuller parameters. The former is the degrees of freedom of the universal covering space, while the latter are involved in the generators of the discrete sub group of the isometry group which make the quotient space.In the second paper, we have explicitly shown we can extract the dynamical degrees of freedom which are subset of the dynamical degrees of freedom paying special attention to the "extendible isometry group", which are consistent with the time evolution. In the third paper, which we are preparing to publish, we … More have found a systematic way in which the whole dynamics is contained in the spatial metric rather than the parameters of the fundamental region. This enables us the Hamiltonian description of the dynamics of compact homogeneous universes.We believe that our original target is fulfillled. Our work should be regarded as a complete theory of compact homogeneous universe and compared to be the textbook "Homogeneous Universe" by Ryan and Shapley. Main collaborators are Dr.Masayuki Tanimoto and Dr.Tatsuhiko Koike.(2)I have also studied possible smearing of the spacetime singularity by quantum gravity in hope that all classical singularities are quantum mechanically wiped out and the spacetime will end up with causally similar to the Minkowsky spacetime. I proposed a quantum mechanical version of the generalized affine parameter invented by Smidt as a measure of the distance, which is also a reminiscent of the Wilson loop i gauge theories.In (2+1) dimensional black hole case, the quantum generalized affine parameter turns out to be infinite due to quantum fluctuation of spacetime. This can be interpreted as that the distance to the would-be singularity is infinitely far away. In effect, the spacetime looks like the Minkowsky spacetime. Similar thing can be shown in the minisuperspace model of the interior of the Schwartzschild black hole. Less