On geometric ideas which lie at the foundation of quantum theory

Abstract

Part I advances the hypothesis that between any two events on the world-line of a particle there can be at most finitely many intermediate events. From this it draws the conclusion that the space-time coordinates of events do not have precise meanings. Part II modifies the Feynman integral in a way which is analogous to the replacement of white noise by Gaussian noise. It thereby obtains a version of the quantum mechanics for a nonrelativistic particle in a potential. In the new discrete quantum theory, space coordinates have probabilistic rather than precise meanings; the energy of a free particle is a bounded operator; and the potential energy is a compact operator. It is argued that the differential scattering cross section for elastic scattering is unchanged when taken in the Born approximation.