Quantum theory of gauge fields, and fermions and solitons

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Fujikawa investigated field theory on the lattice, in particular, the possibility of new Dirac operators for fermions. He also studied the definition of Majorana fermions on the lattice and CP symmetry among others and studied the definition of supersymmetry on the lattice. The unitarity issue of space-time non-commutative theory was analyzed in path integral framework. Gave a new formulation of spin-statistics theorem in path integral, and investigated a kind of anomaly for a 2-dimensional supersymmetric theory with soliton solutions.Fujikawa also studied the origin of gauge symmetry in the geometric phases in Schroedinger problem, and clarified the basic difference detween the chiral anomaly and the geometric phase. It was shown that all the geometric phases are understood as holonomy associated with the exact hidden local symmetry in Schroedinger equation.Tsutsui investigated the mathematical property of quantum theory with singular potentials, and found interesting phenomena such as supersymmetry, duality and Berry's phase. He also extended the formulation to include the potential which contains divergence and examined the effects of the singularity on spectra. In particular, it was shown that a hind of copying states is possible when the potential induces caustics. By investigating the temperature dependence of the pressure for a quantum well, he showed that it is sensitive to statistics of particles in the well. It was alson shown that the quantum singularity is used as a qubit in quantum computation. A solution more general than previously known was obtained for the case of N=3 in the soluble Calogero model with singular potential.Tsutui also quantized the game theory as an application of entanglement and studied in detail the effects of entanglement on the stable solutions in game theory. By using the representation of mixed states called Schmidt decomposition, he succeeded in constructing a more general theoretical framework than previously known ones.