﻿ Introduction

## Introduction

As an introductory question, let us consider the light clock which Einstein used in his definition of an ideal clock in special theory of relativity. A light clock consists of two mirrors stood parallel to each other with light running mirrored to each other continuously. The time is then measured as the number of counts that the light hits the mirrors. This clock is placed stationary to an inertial frame of reference, and the time of the frame is defined by the number of the light-hits of this clock. Insofar as the light is considered as a classical wave and the frame is an inertial one which has no acceleration, this clock can measure the time of the frame. One feature of this clock is that the time is defined by utilizing the distance between the two mirrors and the velocity of light in vacuum which is assumed as an absolute constant in special theory of relativity. Thus time is measured only after the distance between the mirrors and the velocity of light are given, and it is not that time measures the motion or velocity of light wave.

This light clock occupies a certain volume in space, and if an acceleration exists, the mirrors in the clock should be distorted according to general theory of relativity. In this case the number of counts of light between two mirrors cannot be regarded as giving the time of a certain definite frame of reference. To any point in the clock, different metrical tensor is associated, and one cannot determine the time of which point the clock measures. Here appears a problem of the size of the actual clock which cannot be infinitesimally small. In this sense, the operational definition of the clock in General Theory of Relativity (NOT Special Theory of Relativity) has a problem. It seems that this problem may be avoided by an interpretation that the general theory of relativity is an approximation of reality, and the theory gives a sufficiently good approximation as experiments and astronomical observations show. This problem, however, will be a cause of difficulty when one tries to quantize the field equation of general theory of relativity, for the expected quantized theory should be covariant under the diffeomorphism which transforms a point of a space-time manifold to a point of the manifold, and no point can accommodate any clocks with actual sizes. This defect of the field equation constitutes a fundamental theoretical problem in quantizing general theory of relativity.

The problem here lies primarily in the assumption that any matter can be described by the notion of contiuous field, which allows the description of the universe as a solution of the Einstein's field equation. Any assumption of continuous field leads one to think that elementary ingredients of any matter should be sizeless points. We thus identify the problem to reside in the Einstein's field equation by the primary reason that it is based on the notion of matter field. We have also subsidiary reasons to abandon the field equation. One of them is that to solve the equation requires one to know all of the data about the matter distribution in the universe in advance of solving the equation, which is equivalent to getting a solution of the equation before "solving" the equation.

To have sound foundations to resolve these problems, we therefore have to find, firstly, a definition of time which should be given through length (or positions) and velocity (or motion) to accord with the spirit of Einstein's light clock, and secondly, our notion of time should have a certain "good" residence just as the inertial frame of reference in special theory of relativity accommodates the light clock.

As a residence, I prepare a Euclidean quantum space, and within that space I define a quantum-mechanical clock which measures the common parameter of quantum-mechanical motions of particles in a (local) system consisting of a finite number of particles. Since clocks thus defined are proper to each local system, and local systems are mutually independent as concerns the relation among the coordinates of these systems, we can impose relativistic change of coordinates among them. And the change of coordinates gives a relation among those local systems which yields relativistic quantum-mechanical Hamiltonians, explaining the actual observations.

These are technical explanations. Behind these, I have an image of the universe as a whole within which is all and which cannot be grasped. As such an existence we cannot impose any global time on the universe by nature. The universe inasmuch as it is the universe, it is nonsense to assume any global time-coordinates for the total universe. I thus take a universe without time.

These statements may sound unusual in the current physical context. However, these are not ridiculous, as Lancelot R. Fletcher (URL: http://www.freelance-academy.org/) pointed out. He is a philosopher and is a reader of Spinoza. As touched in our joint paper: Local Time and the Unification of Physics, Part I. Local Time, Spinoza stated (Spinoza, Ethics, Part I, Definition 8, from E. Curley 1994 p. 86, Princeton Univ. Press)

D8: By eternity I understand existence itself, insofar as it is conceived to follow necessarily from the definition alone of the eternal thing.

Explanation: For such existence, like the essence of a thing, is conceived as an eternal truth, and on that account cannot be explained by duration or time, even if the duration is conceived to be without beginning or end.

Lance noticed that Spinoza is one of the precedent observers of the non-existence of time, and my observation is not a new one.