A New formulation of Quantum Gravity

在上述项目下，我一直致力于以下四个主题：1）我们在大约10年前提出了（Kawamoto，Watabiki）可以在任何维度上制定的新型广义规范理论。我们对广义Chern-Simons作用量进行了量子化，发现该系统是无限可约的，可以通过引入无穷多个鬼来量子化，但仍具有与原经典作用量相同的Chern-Simons形式.然后，我们将广义拓扑杨-米尔斯作用量量子化，发现N=2的超杨-米尔斯作用量是通过扭转过程量子化的自然结果。我们发现这种扭曲与狄拉克-卡勒费米子公式有本质的联系。2)我们成功地建立了随机格点上的三维Chern-Simons引力。结果表明，该模型与Ponzano-Regge模型是等价的。然后，我们成功地将这个公式应用到四维BF引力的随机格子。3)我们对二维量子引力耦合c=-2物质进行了高精度的数值研究。结果表明，二维量子引力的本质是分形性质，其数值分形维数与理论预言是一致的。数值研究了该模型的三维扩展。我们测量了弦的磁化率，发现了三维格点引力存在连续极限的证据。4)我们将广义规范理论推广到广义Yang-Mills作用量，并应用SU（2）成功地建立了Weinberg-Salam模型|1）超级集团。

I have been working on the following four subjects under the above project : 1) We proposed (Kawamoto, Watabiki) new type of generalized gauge theories which could be formulated in any dimensions about 10 years ago. We quantized the generalized Chern-Simons action and found out that the system is infinitely reducible and can be quantized by introducing infinitely many ghosts yet has the same Chern-Simons form as the original classical action. We have then quantized the generalized topological Yang-Mills action with instanton gauge fixing and found that N=2 super-Yang-Mills action comes out as a natural consequence of the the quantization via twisting procedure. We found the twinsting is essentitally related with Dirac-Kahler fermion formulation. 2) We have successfully formulated the three-dimensional Chern-Simons gravity on the random lattice. It turned out that the model is equivalent with the Ponzano-Regge model. We have then successfully applied this formulation into four dimensional BF gravity on the random lattice. 3) We have investigated two dimensional quantum gravity coupled c=-2 matter numerically with great accuracy. We have learned that the essense of the two dimensional quantum gravity is the fractal nature and the numerical fractal dimensions are consistent with the theoretical prediction. The three dimensional extension of this model is investigated numerically. We measured the string susceptibility and found an evidence that there exists a continuum limit of the three dimensional lattice gravity. 4) We have extended the generalized gauge theory to formulate the generalized Yang-Mills action and applied to formulate Weinberg-Salam model successfully by using SU(2|1) super group.