Methods in quantum field theory and their applications

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We first investigated the quantum theory of membranes. The membrane is an extension of string theory and it is expected to play a fundamental role in the formulation of the so-called M theory. In this connection, the matrix formulation of the membrane is important. In the past formulation of the matrix regularization of the membrane, the lightcone gauge has been used. In our approach, we studied to what entent a Lorentz covariant matrix regularization of the membrane is possible. We have shown that the Bosonic membrane can be formulated as a matrix theory except for a subtle property related to the Faddeev-Popov ghost. As for a supersymmetric membrane, we encountered a more fundamental complication, which may be solved only when we formulate it in a way completely different from the present formulation of membrane.We have recently witnessed a remarkable progress in the treatment of lattice fermion operators. We clarified the meaning of index theorem on the lattice and the physical meaning of the new fermionic operator. More recently, we have extended the so-called Ginsparg-Wilson relation to a form characterized by non-negative integers. It was shown that these new lattice Dirac operators are free of species doubling and satisfy the correct form of index theorem in the smooth continuum limit.