Discrete Reflection Groups and Induced Representations of Poincaré Group on the Lattice

Abstract

The assumption of a physical model on discrete space and time leads to some changes of the mathematical equations. In particular, the symmetries of the model has to be restricted to integral transformations acting on some vector space defined over the integers. The space-time groups are subgroups of the set of non-singular integral matrices, and the classical and quantum laws are written with the help of difference operators. In [1] we have developed some properties of non-compact groups acting on lattice with non-Euclidean metric. In [2] we have modified some of the postulates of quantum mechanics in order to incorporate the hypothesis of discrete space and time. In [3] we have proposed a new scheme for the Klein-Gordon and Dirac wave equation. The Quantum fields on the lattice is the subject of very extensive literature [4] and the theoretical difficulties of it have not yet been solved.