  We summarize a recent study of discrete (integer-valued) Hamiltonian cellular
automata (CA) showing that their dynamics can only be consistently defined, if
it is linear in the same sense as unitary evolution described by the
Schr\"odinger equation. This allows to construct an invertible map between such
CA and continuous quantum mechanical models, which incorporate a fundamental
scale. Presently, we emphasize general aspects of these findings, the
construction of admissible CA observables, and the existence of solutions of
the modified dispersion relation for stationary states.
