Preface
I consider in this book a formulation of Quantum Mechanics, which is often abbreviated as QM. Usually QM is formulated based on the notion of time and space, both of which are thought a priori given quantities or notions. However, when we try to define the notion of velocity or momentum, we encounter a difficulty as we will see in chapter 1. The problem is that if the notion of time is given a priori, the velocity is definitely determined when given a position, which contradicts the uncertainty principle of Heisenberg.
We then set the basis of QM on the notion of position and momentum operators as in chapter 2. Time of a local system then is defined approximately as a ratio |x|/|v| between the space coordinate x and the velocity v, where |x|, etc. denotes the absolute value or length of a vector x. In this formulation of QM, we can keep the uncertainty principle, and time is a quantity that does not have precise values unlike the usually supposed notion of time has.
The feature of local time is that it is a time proper to each local system, which is defined as a finite set of quantum mechanical particles. We now have an infinite number of local times that are unique and proper to each local system.
Based on the notion of local time, the motion inside a local system is described by the usual Schroedinger equation. We investigate such motion in a given local system in part II. This is a usual quantum mechanics.
After some excursion of the investigation of local motion, we consider in part III the relative relation or motion between plural local systems. We regard each local system's center of mass as a classical particle. Then as the relative coordinate inside a local system is independent of its center of mass, we can set an arbitrary rule on the relation among those centers of mass of local systems. We adopt the principles of general relativity as the rules that govern the relations of plural local systems. By the reason that the center of mass and the inner coordinate are independent, we can combine quantum mechanics and general relativity consistently.
We give an approximate Hamiltonian that explains partially the usual relativistic quantum mechanical phenomena in chapter 9.
In the final part IV, we consider some contradictory aspect of mathematics in chapter 10. Although this does not give directly that mathematics is inconsistent, this will give an introduction to the next chapter 11, where starting with the contradictory nature of the semantics of set theory in the sense that if we consider all sentences of set theory, they are contradictory, we regard that the Universe that is described by ourselves is of contradictory nature, and can be described as a superposition of all possible, infinite number of waves. As this is the state of the Universe, the Universe is described as a stationary state describing a superposition of all waves. We then give a formulation of the Universe and local systems inside it, in the form of a theory described by Axiom 1 to Axiom 5 in chapter 11. In the final chapter 12, we will prove that there is at least one Universe wave function phi in which all local systems have local motions and thus local times. This concludes our formulation of Quantum Mechanics.
Hitoshi Kitada
Dec. 15, 2003, Tokyo