Quantum models as classical cellular automata

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 Schrödinger 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.

11 pages; based on invited talks at Testing Quantum Gravity (Torino, May 2016) and IARD 2016 (Ljubljana, June 2016), to appear in JPCS. arXiv admin note: text overlap with arXiv:1604.06652

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Cited by in corpus (4)

• Ontological states and dynamics of discrete (pre-)quantum systems
• Qubit exchange interactions from permutations of classical bits
• Non-local imprints of gravity on quantum theory
• Entropy, topological theories and emergent quantum mechanics