Representations of Classical Groups on the Lattice and Its Application to the Field Theory on Discrete Space-time

The idea of a discrete space-time has been introduced by physicists in the past in several different ways. Heisenberg advocated a fundamental length, Snyder has proposed a position operator with discrete spectrum, Ahmavaara has carried out a physical model on a cubic lattice, which is embedded in a finite linear space over a Galois field. Recently the use of discrete space-time variables in Lattice Gauge Theories have become very popular to construct models based on experimental date. An other succesful line of research deals with integrable models on discrete space-time. Many other hypothesis can be mentioned but they are reduced to two types: i) those using the space-time lattice as an artificial model, and ii) those proposing the discreteness of space-time as some fundamental structure of reality (Lorente 1986c). The first assumption is very old and of technical character and do not present conceptual problems. The second assumption is more recent, although is based on some old pressupositions, namely, the relational character of the structure of space and time in contradistinction to the absolute idea of space and time (Earman 1989). Both assumptions were defended in XVII century by Leibniz and Newton, respectively, and although the last one become more succesful the former remained as a consistent model (Jammer 1969). Acording to Leibniz “Time is the order of monads not existing simultaneously. Space is