Topolygy in Quantum Cosmology

在本研究项目中，我们主要关注紧致齐次宇宙的分类及其动力学。在我们发表在Journal of Mathematical Mathematics上的第一篇论文中，我们对除了三维双曲空间之外的几乎所有紧致齐次宇宙进行了分类，这仍然是数学中的一个重要课题。我们已经成功地将运动自由度识别为局部曲率自由度和Teichmuller参数。前者是普遍覆盖空间的自由度，后者涉及到构成商空间的等距群的离散子群的生成。在第二篇论文中，我们明确地证明了我们可以提取动力自由度的子集，特别是与时间演化一致的“可扩展等距群”。在我们准备发表的第三篇论文中，我们……More发现了一种系统的方法，其中整个动力学包含在空间度量中，而不是基本区域的参数中。这使我们能够用哈密顿描述紧致齐次宇宙的动力学。我们相信我们最初的目标已经实现了。我们的工作应该被视为紧致齐次宇宙的完整理论，并与Ryan和Shapley的教科书“齐次宇宙”相比较。主要合作者是谷本雅之博士和小池达彦博士。(2)我还研究了量子引力对时空奇点涂抹的可能性，希望所有经典奇点都被量子力学地抹去，时空最终将与闵可夫斯基时空因果相似。我提出了Smidt发明的广义仿射参数的量子力学版本，作为距离的度量，这也让人想起威尔逊环I规范理论。在（2+1）维黑洞中，由于时空的量子涨落，量子广义仿射参数是无限大的。这可以解释为到潜在奇点的距离是无限远的。实际上，这个时空看起来像闵可夫斯基时空。类似的事情也可以在史瓦西黑洞内部的超空间模型中显示出来。少

(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