A New formulation of Quantum Gravity

在上述项目中，我一直致力于以下四个课题：1)我们在大约10年前提出了（Kawamoto, Watabiki）可以在任何维度上表述的新型广义规范理论。我们对广义的chen - simons作用进行了量子化，发现该系统是无限可约的，并且可以通过引入无穷多个鬼来量子化，但与原始经典作用具有相同的chen - simons形式。然后，我们对广义拓扑Yang-Mills作用进行了量子化，并发现N=2超级Yang-Mills作用是通过扭转过程量子化的自然结果。我们发现缠绕本质上与狄拉克-卡勒费米子公式有关。2)我们成功地在随机晶格上建立了三维chen - simons引力。结果表明，该模型与Ponzano-Regge模型是等价的。然后，我们成功地将该公式应用于随机晶格上的四维BF重力。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。我们了解到二维量子引力的本质是分形的性质，数值分形维数与理论预测是一致的。对该模型的三维扩展进行了数值研究。我们测量了弦的磁化率，发现了三维晶格引力存在连续极限的证据。4)利用SU（2|1）超群，将广义规范理论推广到广义Yang-Mills作用，并成功地应用于Weinberg-Salam模型的构造。

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.