Mathematical Problems and Infinite Dimensional Analysis in Quantum Field Theory

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(1) A symmetric operator satisfying the weak Weyl relation with a Hamiltonian H (a self-adjoint operator H on a Hilbert space) is called a strong time operator of H. In this research, spectral properties of strong time operators are analyzed and important results have been obtained A. Arai, Spectrum of time operators, Lett. Math. Phys. 80 (2007), 11-17).(2) We develop a general theory of Heisenberg operators. A mathematically rigorous formulation was made on the Heisenberg equations of motion and a sufficient condition for the equation to have a unique solution was given. Moreover invariant domains are found with a discovery of new structures. This Theory is applicable not only to quantum mechanics with finite degrees of freedom, but also to quantum mechanics with infinite degrees of freedom, in particular, quantum field theory. Its range of applicability is very wide. For details, see: A. Arai, Heisenberg operators, invariant domains and Heisenberg equations of motion, Rev. Math. Phys. 19 (2007), 1045-1069.(3) Analysis was made for ground states of a quantum system interacting with a quantum field. For details, see: A. Arai, M. Hirokawa and F. Hiroshima, Regularities of ground states of quantum field models, Kyushu J. Math. 61 (2007), 321-372.