Relativistic Theory of Discrete Momentum Space and Discrete Space-Time

An examination is made of the result of restricting translations and Lorentz transformations in space-time to those with rational coefficients. This removes the major defect of Schild's model of discrete space-time by elimination of the lower bound on the relative velocities of reference systems. The theory is formulated in two stages. In the first the energy and momentum components of a particle are restricted to a countable set satisfying the relativistic energy-momentum relation while the space-time variables are continuous. This gives a theory of discrete energy-momentum space in which wave functions are almost periodic functions. In the second stage the space-time variables are restricted to rational values. This leads to the theory of discrete space-time.