場の理論における低エネルギー定理とその応用についての研究

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A low energy theorem in three dimensional fermion theory, which is connected with even dimensional axial anomaly, is derived and extended. An extension of the theorem to systems without translational invariance is made. The theorem is also applied to the system with magnetic field. The exact quantization of the Hall conductance in the localized state region regardless of interactions and disorders in shown. Ward-Takahashi identity in the magnetic field leads that the Hall conductance is a topological winding number and satisfies the exact low energy theorm. The material effect and four dimensional quantum electrodynamics effect do not modify the quantized Hall conductance.