Time: Hitoshi Kitada's Home Page

A possible solution to this contradictory state is therefore to define time as a notion associated with a system which can accommodate clocks which have actual sizes. Such systems are called "local systems," and their centers of mass are identified with the classical general relativistic points, with their inner structures being equipped with the structures of local systems. These local structures are assumed Euclidean ones tangent to the total 4-manifold, but no connections are imposed among their Euclidean structures. Instead we pose a relation between the inside and outside of a local system, which I will describe below in "A Solution of Unification of Quantum Mechanics and Special Theory of Relativity." With our identification of the centers of mass of local systems with classical particles, the general theory of relativity remains valid as a theory describing the "classical" world among the local systems. In this setting, time recaptures its meaning as a local time inside each local system, defined as a measure of "quantum mechanical" motion inside the local system. As a by-product, quantum mechanical and classical views of the world are reconciled by this formulation at the expense of the field equation.

From other viewpoints, sciences are founded on the notion of time. Sciences cannot be called "sciences" if they lose their descriptions of nature according to some time-coordinates. Physics, Chemistry, Biology, History, ..., these are the descriptions of nature or human beings along some time-coordinates. Without the notion of time, these academic areas cannot exist as academic activities.

Time has been, however, a notion whose existence nobody doubts. These academic activities have been assuming the existence of some time-coordinates, and people speak about things as if they move or change following the order prescribed by time. But what is time? Is it an existence in the same sense as the existence of other objective things? Time is not such an existence: Time does not appear until we measure it by some equipments, i.e., by clocks. Time is just a movement of the hands of clocks, and time is not an a priori existence which measures motion. Quite contrarily, just the motion of hands of analogue clocks, or just the change of figures of digital clocks measures time.

The definition of local time is based on this observation in addition to the afore-mentioned physical considerations. It is defined as a local notion effective only in each local system and is equivalent to the quantum mechanical evolution of that local system. By the locality of this notion of time, it is compatible with two basic principles of general theory of relativity in the first place.

The considerations on these problems are described in my recent papers and lectures in the following list. The lecture in the winter semester of 1996-1997 delivered to graduate and undergraduate students of liberal arts is not written in a form available here. However, the main content of the lecture is presented in my paper "Quantum Mechanics and Relativity --Their Unification by Local Time--" cited in the list as "Time IV." These are written in English except for one paper in Japanese, which is mainly concerned with the question "What should be done in Mathematical Sciences?" but on the deeper level, is related with my notion of time as suggested in the following introduction.