Transition from a continuous to a discrete space-time scheme

Abstract

This paper is concerned with a discrete space and time description of physical events. This involves transforming partial differentials with respect to space and time into finite differences evaluated between neighbouring space-time points. By considering a set of such finite-difference forms of the Euler equations for classical fields, a rule for such a transformation is obtained. It is found that each partial differential must be replaced by the corresponding central difference. Energy and momentum eigenvalues are found for particles moving in one dimension along which the separation between adjacent space points is ξ. It is found that these quantities have maximum values proportional to ξ−2 and ξ−1 respectively, and that there are finite numbers of distinct eigenvalues of each sort. The Schrodinger equation is transformed into a difference equation and is solved formally to give theS-matrix. It is found that the wave function does not tend to a limit at infinite times.