Symmetry Analysis and Quantization in Discrete Dynamical Systems

Introduction Discrete systems are widespread in applications. In particular, many nanostructures are symmetric discrete formations. From a fundamental point of view, there are many philosophical and physical arguments that discreteness is more suitable for describing physics at small distances than continuity which arises only as approximation or as a logical limit in considering large collections of discrete structures. We consider here deterministic and nondeterministic dynamical systems with non-trivial symmetries defined on discrete spaces and evolving in discrete time. As a tool for our study we are developing programs in C based on computer algebra and computational group theory methods. Basic constituents of discrete models The following constructions form the basis for all types of dynamical systems under study: 1. Space X is a k-valent graph with symmetry group G = Aut(X) — space symmetries. 2. Vertices x of X take values in a set Σ with symmetry group Γ ≤ Sym (Σ) — internal symmetries. 3. States of the whole system are functions σ(x) ∈ Σ . 4. We define whole system symmetry groupsW — unifying space G and internal Γ symmetries — as equivalence classes of split group extensions of the form 1 → Γ → W → G → 1 . (1)