The Structure of Space and the Formalism of Relativistic Quantum Theory. I

Abstract

The structure of a Euclidean space can be approached, with an unlimited accuracy, by a part of a maximally ordered finite linear space. Accordingly, all the physical theories based on the space‐time continuum can also be considered in such a finite space‐time. The finiteness of the underlying space makes also some new kinds of theories possible. Among them is a purely group theoretical formalism of relativistic quantum theory, including a free‐particle theory as well as a group formalism of interaction of particles. The free‐particle formalism of a finite space‐time is considered here (Part I). An essential difference in comparison with the formalism of continuous space‐time is that there is, as a consequence of the relations of Euclidicity to be imposed on observable 4‐vectors, a nontrivial spectrum of momentum, mass, and energy in a finite geometry.