  A synopsis is offered of the properties of discrete and integer-valued, hence
"natural", cellular automata (CA). A particular class comprises the
"Hamiltonian CA" with discrete updating rules that resemble Hamilton's
equations. The resulting dynamics is linear like the unitary evolution
described by the Schr\"odinger equation. Employing Shannon's Sampling Theorem,
we construct an invertible map between such CA and continuous quantum
mechanical models which incorporate a fundamental discreteness scale $l$.
Consequently, there is a one-to-one correspondence of quantum mechanical and CA
conservation laws. We discuss the important issue of linearity, recalling that
nonlinearities imply nonlocal effects in the continuous quantum mechanical
description of intrinsically local discrete CA - requiring locality entails
linearity. The admissible CA observables and the existence of solutions of the
$l$-dependent dispersion relation for stationary states are mentioned, besides
the construction of multipartite CA obeying the Superposition Principle. We
point out problems when trying to match the deterministic CA here to those
envisioned in 't Hooft's CA Interpretation of Quantum Mechanics.
