Are nonlinear discrete cellular automata compatible with quantum mechanics?

TL;DR

This paper investigates whether nonlinear discrete cellular automata can be compatible with quantum mechanics, concluding that only linear CA align with local quantum dynamics due to nonlocal effects introduced by nonlinearities.

Contribution

It demonstrates that incorporating nonlinearities into Hamiltonian cellular automata leads to nonlocal effects, establishing linearity as essential for compatibility with quantum mechanics.

Findings

Nonlinear CA produce nonlocal effects incompatible with quantum mechanics.

Linear CA can be mapped to quantum models preserving locality.

Nonlinearities in CA lead to inconsistencies in continuous quantum descriptions.

Abstract

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.