Symmetries, gauge invariance and quantization in discrete models

V. V. Kornyak

Numerical methods and programming, Tome 10 (2009) no. 4, pp. 415-421.

Résumé

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 - analogs of “spaceships” in cellular automata or “generalized coherent states” in quantum physics - 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 groups. 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.

Détail

Keywords: symmetries of discrete systems; gauge principle; quantization.