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