Cayley parametrization of semisimple lie groups and its application to physical laws in a (3+1)-dimensional cubic lattice

Abstract

The assumption of a discrete space-time is expressed mathematically by restricting the space-time variables to the field of integer numbers, and by restricting to the field of rational numbers the functions describing the laws of motion. This rational character must be preserved under the transformations connecting different systems of reference. The Cayley parametrization of semisimple Lie groups, and in particular of the Lorentz group, satisfies this condition if we require these parameters to take only integer values. The rational points of the most frequently used transcendental functions are obtained with the help of the integer complex and hypercomplex numbers. Some applications are made concerning the laws of motion in special relativity defined over a (3+1)-dimensional cubic lattice.