Gauge Invariance in Discrete Dynamical Systems

The fundamental laws of physics are described from the modern point of view by gauge theories with continuous gauge groups. With some modification the gauge principle can (and should) be applied to discrete dynamical systems with nontrivial symmetries also. We consider specifics of introduction of gauge invariance into W -equivariant discrete dynamical systems with finite sets of states Σ, where W is the group of symmetries of Σ. Typical examples of such systems are cellular automata on finite lattices. In this case the set of states Σ is the set of functions on a lattice X with values in a set of local states Φ: Σ = ΦX . If the groups of symmetries of local states Φ and lattice X (space group) are Γ and G, respectively, the whole symmetry group of the system can be represented either as the direct product