Neo Formulation of Quantum Gravity and its Dynamical Aspects.

最近我一直在研究两种不同的课题：(1)广义规范理论的各种应用，广义规范理论是由广义chen - simons理论形成的。(2)量子引力的分形结构，数值和分析研究。我们的调查已经有了一些结果，但它们还没有以发表论文的形式出现。这在一定程度上是因为我在过去的一两年里从东京搬到了札幌。(1)我们成功地在任意维度上推广了标准的chen - simons作用。在广义规范理论方法中，我们研究了广义chen - simons理论的量子化是如何工作的。我们成功地量化了gl （1， R）广义的二维chen - simons作用。由于模型的可约性，该模型非常简单，但对于量化来说非常重要。在这个方向上的最新成果是将广义chen - somons规范理论与Yang-Mills作用的非交换代数表述联系起来。我们证明了Weinberg-Salam模型可以用我们规范论的广义形式表述，并与规范论的非交换代数表述有直接联系。这个理论特别地只能由包括物质场在内的形式来表述。(2)在我们之前的工作中，我们已经用数值方法证明了分形结构是二维c=-2模型的量子引力的基本性质。在随后的分析研究中，我们成功地获得了二维量子表面的传递矩阵，从而使计算c=0模型下二维重力的分形维数成为可能。最近，我们在二维量子引力表面上开始了另一种类型的数值研究。这是从有限尺度的角度研究分形结构的一项研究。

Recently I have been working on two different kinds of subjects : (1)Various applications of generalized gauge theory, which was formulated from generalized Chern-Simons theory. (2)Fractal structures of quantum gravity, numerical and analytical investigations. There are already several results in our investigations but they are not yet in the form of published papers. This is partly because of my moving time from Tokyo to Sapporo for last one or two ysars.(1)We have succeeded to generalize the standard Chern-Simons action in arbitrary dimensions. In the generalized gauge thoery approach, we have investigated how the quantization of our generalized Chern-Simons theory works. We have succeeded to quantize gl (1, R) generalized Chern-Simons action in two dimensions. The model is simple but highly non-trivial for the quantization because of the reducibility of the model. The most recent result in this direction is the connection of the generalized Chern-Somons gauge theory to non-commutative algebra formulation of Yang-Mills action. We have shown that Weinberg-Salam model can be formulated in the generalized version of of ourgauge thory and have a direct connection with non-commmutative algebra formulation of gauge thorey. The theory is formulated in particular only by formsincluding matter fields.(2)In our previous work we have shown numerically that the fractal sructure is the fundamental nature of quantum gravity for c=-2 model in two dimensions. In the later analytic investigations we have scceeded to obtain transfer matrix of two dimensional quantum surface, which makes it possible to calculate the fractal dimension of two-demensional gravity for c=0 model.Recently we have started another type of munerical investigations on two-dimensional quantum gravity surface. This is an investigation of studying fractal structure from finite size scaling points of view.