Quantum symmetry in field theory and its applications

藤川继续他的研究量子力学方面的二维引力，并试图澄清作用的量子异常量子引力。具体地说，他试图理解基于动力学三角剖分的离散引力与基于刘维尔作用量的更传统的二维引力之间的直接关系。特别是，他试图了解的表现形式外尔异常和相关的刘维尔行动的动态三角测量。在这个过程中，他得到了一个相当清晰的物理图像外尔对称性破缺和它的简化在动力学三角计划与概念的坐标。他还提出了QCD中单圈β函数的简单非线性计算。藤川还继续了过去在规范理论中与重子数破坏有关的量子隧穿的研究。他研究了如何克服缺点的瞬子演算，虽然大部分的主题是留给未来的调查。久保给了一般制定的量子异常和相关的异常的approximators的基础上，扩大相空间。这种方法可以适用于各种各样的主题，他获得了相当有趣的结果规范独立制定的二维重力和蒸发的黑洞。

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.