  We consider discrete and integer-valued cellular automata (CA). A particular
class of which comprises "Hamiltonian CA" with equations of motion that bear
similarities to Hamilton's equations, while they present discrete updating
rules. The dynamics is linear, quite similar to unitary evolution described by
the Schroedinger equation. This has been essential in our construction of an
invertible map between such CA and continuous quantum mechanical models, which
incorporate a fundamental discreteness scale. Based on Shannon's sampling
theory, it leads, for example, to a one-to-one relation between quantum
mechanical and CA conservation laws. The important issue of linearity of the
theory is examined here by incorporating higher-order nonlinearities into the
underlying action. These produce inconsistent nonlocal (in time) effects when
trying to describe continuously such nonlinear CA. Therefore, in the present
framework, only linear CA and local quantum mechanical dynamics are compatible.
