Study of field theory and symmetries on continuum and latticized space-time

Fujikawa首先在路径积分公式中考察了非交换时空的量子化，并证明了一个修正的时空非交换理论的时间排序公式，它保持了统一性，但不可避免地打破了正能量条件。他还从扩展标准模型中中微子质量为零时增强对称性的角度研究了与小中微子质量有关的问题。Fujikawa和Deguchi提出了几何相位的第二个量化公式，它允许在路径积分和算子形式中同样地处理几何相位。Fujikawa进一步研究了该公式中出现的隐藏局部对称，并证明了所有几何相都被理解为与薛定谔方程中隐藏精确局部对称相关的完整相。这个新公式既符合量子力学的基本叠加原理，又符合所有物理观测值的规范不变性要求。几何相位由此产生于二次量化中隐藏的局部对称性。Deguchi用Atiyah-Singer指数定理研究了磁单极子的量子化。Deguchi还研究了球上规范场的量子化。Naka概括地阐明了q -变形和非局域场理论。

Fujikawa first examined the quantization of non-commutative space-time in the path integegral formulation, and showed that a modified time ordering prescription for space-time non-commutative theory, which preserves unitarity, inevitably breaks positive energy condition. He also studied the problem related to the small neutrino mass from a view point of the enhanced symmetry in the extended standard model when one sets the mass of the neutrino at zero.Fujikawa together with Deguchi presented a second quantized formulation of geometric phases, which allows the treatment of the geometric phases in path integral and operator formalism equally well. The hidden local symmetry appearing in this formulation has been further studied by Fujikawa, and it was shown that all the geometric phases are understood as holonomy associated with this hidden exact local symmetry in the Schroedinger equation. This new formulation is consistent with the basic superposition principle of quantum mechanics as well as the gauge invariance requirement of all the physical observables. The geometric phases thus arise from the hidden local symmetry appearing in second quantization.Deguchi studied the quantization of the magnetic monopole by using the Atiyah-Singer index theorem. Deguchi also studied the quantization of gauge filed on a sphere.Naka clarified the Q-deformation and non-local field theories in general.