Symmetries, gauge invariance and quantization in discrete models

Abstract

Different aspects of discrete symmetry analysis in application to deterministic and non-deterministic lattice models are considered. One of the main tools for our study are programs written in C. In the case of {deterministic dynamical systems}, such as cellular automata, the non-trivial connections between the lattice symmetries and dynamics are discussed. In particular, we show that the formation of moving soliton-like structures mdash; analogs of "spaceshipdf" in cellular automata or "generalized coherent states" in quantum physics mdash; results from the existence of a non-trivial symmetry group. In the case of {mesoscopic lattice models}, we apply some algorithms exploiting the symmetries of the models to compute microcanonical partition functions and to search phase transitions. We also consider the {gauge invariance} in discrete dynamical systems and its connection with {quantization}. We propose a {constructive} approach to introduce {quantum structures in discrete systems} based on finite gauge groupdf. In this approach, quantization can be interpreted as the introduction of a gauge connection of a special kind. We illustrate our approach to quantization by a simple model and propose its generalization.