Quantum symmetry in field theory and its applications

藤川继续他对二维引力的量子力学方面的研究，并试图澄清量子异常在量子引力中的作用。具体来说，他一方面试图理解基于动态三角测量的离散重力与基于刘维尔作用的更传统的二维重力之间的直接关系。特别是，他试图理解在动态三角测量中Weyl异常和相关的Liouville作用的表现。在这个过程中，他得到了Weyl对称性破缺的一个相当清晰的物理图像，以及它在动态三角剖分方案中的还原，以及坐标的概念。他还提出了QCD中一环beta函数的简单非图解计算。Fujikawa还继续了过去在规范理论中与重子数违反有关的量子隧道效应的研究。他研究了如何克服瞬子演算的缺点，尽管这个问题的大部分有待将来的研究。Kubo在扩展相空间的基础上给出了量子异常和相关反常换向子的一般公式。这种方法可以应用于各种各样的主题；他在不依赖规范的二维引力公式和黑洞蒸发方面获得了相当有趣的结果。

Fujikawa continued his research of quantum mechanical aspects of two-dimensional gravity and tried to clarify the role of quantum anomaly in quantum gravity. To be specific, he tried to understand the direct relation between the discretized gravity on the basis of dynamical triangulation, on the one hand, and the more conventional two-dimensional gravity on the basis of Liouville action. In particular, he tried to understand the manifestation of the Weyl anomaly and the associated Liouville action in the dynamical triangulation. In the process, he obtained a rather clear physcal picture of the Weyl symmetry breaking and its retoration in the dynamical triangulation scheme together with the notion of coordinates. He also presented a simple nondiagramatic calculation of one-loop beta function in QCD.Fujikawa also continued the past study of quantum tunneling in connection with the baryon number violation in the gauge theory. He investigated how to overcome the shortcoming of the instanton calculus although much of the subject is left for future investigation.Kubo gave a general formulation of quantum anomaly and related anomalous commutators on the basis of the extended phase space. This method can be applied to a wide variety of subjects ; he obtained quite interesting results in gauge independent formulation of two dimensional gravity and the evaporation of black holes.