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

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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.