As for the history of the philosophical foundation of time and the recent treatment of its problems in mathematical science, I have focused on a few themes and discussed them [Fujimoto 2017]. However, although the issue of time generation was addressed through a few reference to some philosophers’ ideas (e.g., G.W.Leibniz and B.Bolzano), I could not explicitly address in the context of mathematical sciences.

It is anticipated that time operators will play an important role in physics, especially quantum physics, and are being discussed in many articles and textbooks today. Unlike W. Pauli who once argued it negatively, it is sometimes extended to the discussion of non-commutative space-time theory. However, the time operator has a complicated appearance, not only in its mathematical representation theory, but also depending on the topology and the domain in which it operates. In addition, this operator was originally revisited as a problem of “survival probability” of particles (related to the scattering problem) based on uncertainty relations, and directly linked to the discussion of space-time non-commutativity would have to be careful.

The most important problem is that a time operator is linked to a Hamiltonian of the system. The Hamiltonian, as is well known, controls the translation of time by the Noether’s theorem. When a“time”range is fixed, the Hamiltonian determines the symmetry of the system with respect to time through showing energy as the conserved quantity of the physical system. Therefore, a particular time operator can only be applied to the behavior of particles and fields in the local world of one of “reference systems”. In other words, the theory of time operator is not generally applicable to non-inertial systems where the Hamiltonian form cannot be used. In addition, the theory of time operators is, by itself, somewhat narrow in terms of how time “comes”.

In this paper, taking consideration of the above problems, I present another view on the generation of time and attempt to combine it with the discussion of time operators. At that time, I would like to deal with two specific issues (entanglement and cluster decomposition,) related to the quantum field theory.