Discrete Dynamics: Gauge Invariance and Quantization

TL;DR

This paper explores gauge invariance in discrete systems, proposing a novel approach to quantization via finite gauge groups and illustrating it with simple models, bridging discrete processes with continuous phenomena.

Contribution

It introduces a new method for quantizing discrete systems using finite gauge groups and constructs a unified framework for space and internal symmetries.

Findings

Finite gauge groups enable a new quantization approach.

Discrete models can approximate continuous phenomena.

A C program supports the analysis and modeling.

Abstract

Gauge invariance in discrete dynamical systems and its connection with quantization are considered. For a complete description of gauge symmetries of a system we construct explicitly a class of groups unifying in a natural way the space and internal symmetries. We describe the main features of the gauge principle relevant to the discrete and finite background. Assuming that continuous phenomena are approximations of more fundamental discrete processes, we discuss -- with the help of a simple illustration -- relations between such processes and their continuous approximations. We propose an approach to introduce quantum structures in discrete systems, based on finite gauge groups. In this approach quantization can be interpreted as introduction of gauge connection of a special kind. We illustrate our approach to quantization by a simple model and suggest generalization of this model. One of the main tools for our study is a program written in C.