Current Research

Quantum field theory

OUTLINE
I have been interested in various aspects of quantum field theory. It is one of fundamental theories of physics, which is advocated by Heisenberg and Pauli in 1929 and nearly completed in 1970s. It is, however, known that the theory is not mathematically established and that questions about its foundations are raised by recent quantum information theory. In order to resolve them, I am trying to develop a theory of quantum systems of infinite degrees of freedom including quantum field theory in terms of mathematics, physics and statistics.

Quantum theory is a theory treating genuinely probabilistic systems and is mathematically formulated by algebraic probability theory. Thus, for the purpose of understanding quantum theory, it is natural to deepen probabilistic and statistical aspects of quantum theory. Quantum systems of infinite degrees of freedom including quantum field theory are not exceptions. However, statistical inference for quantum systems of infinite degrees of freedom including quantum field theory is not sufficiently developed. Also, measurement theory in quantum systems of finite degrees of freedom is mathematically characterized for the last thirty years or so, but that in quantum systems of infinite degrees of freedom, in contrast, remains unclear. The aim of my study is to establish a mathematical theory of quantum systems of infinite degrees of freedom, in view of physical situations and experimental settings, from the viewpoint of mathematics, measurement theory and statistics. The followings are my investigations:
・A proposal for the use of the concept of local state in the framework of algebraic quantum field theory and a generalization of the concept of sector to local states (with Izumi Ojima and Hayato Saigo)
・A formulation of measurement theory in quantum systems of infinite degrees of freedom and its application to algebraic quantum field theory (with Masanao Ozawa)
・A proof of Born statistical formula based on sector theory and measuring processes (with Izumi Ojima and Hayato Saigo)
・Statistical inference for quantum systems of infinite degrees of freedom based on sector theory and classical (measure-theoretical) statistics and learning theory (with Izumi Ojima)