# Quantum models as classical cellular automata

## TL;DR

This paper explores how certain discrete cellular automata can be mapped to quantum mechanical models, highlighting their linearity, conservation laws, and the challenges in aligning them with 't Hooft's deterministic quantum interpretations.

## Contribution

It demonstrates a one-to-one correspondence between Hamiltonian cellular automata and quantum models using Shannon's Sampling Theorem, emphasizing linearity and conservation laws.

## Key findings

- Establishes invertible mapping between CA and quantum models
- Shows CA dynamics can mimic quantum linear evolution
- Discusses issues in matching CA with deterministic interpretations

## Abstract

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.