  We introduce an action principle for a class of integer valued cellular
automata and obtain Hamiltonian equations of motion. Employing sampling theory,
these discrete deterministic equations are invertibly mapped on continuum
equations for a set of bandwidth limited harmonic oscillators, which encode the
Schr\"odinger equation. Thus, the linearity of quantum mechanics is related to
the action principle of such cellular automata and its conservation laws to
discrete ones.
