On the Quantization of Physical Space-time Operators

Abstract

§ 1. IntrDduction In the conventional field theory, space-time coordinates are treated as c-number parameters, and field operators (consequently energy-momentum density, current density and such quantities are expressed as the products of field operators also) are represented as their functions. On the other hand, many authorsl) investigated the possibility to regard space-time coordinates as operators, or to quantize them, in order to get rid of the so-called divergence difficulties. However, no promising guiding principle to quantize space-time coordinates has been found. Among many attempts, a series of investigations 2 )-4) which was begun by the famous Dirac's work 5 ) has an exceptional attraction for the present author, although these works were developed regardless of the divergence difficulties. For their quantization of the physical coordinate operators seems to express the uncertainty principle in the measurements, as will be seen in the following discussions. The purpose of this article is to make the physical meaning of the quantization clear and to analyse the nature of the extended Hilbert space. The meaning of the uncertainty principle in quantum mechanics is as follows. As long as the particle, for which the uncertainty relation between its position and momentum holds, is used in the observation of the motion of some particle, the position and the momentum of the object also cannot be measured simultaneously and exactly. If any particle which is free from the uncertainty principle does exist, we would be able to use this particle to make the simultaneous measurement of the position and the momentum of the object, whether they are elementary or not. It may in fact be possible to take the standpoint that the uncertainty relation is derived from such a theory that introduces some hidden variable free from the uncertainty principle, as Bohm