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arXiv:1807.08695 · Physical Review A

Scalar fermionic cellular automata on finite Cayley graphs

Paolo Perinotti, Leopoldo Poggiali

Published Nov 26, 2018• 5 citations
arXivHTMLPDF

Abstract

A map on finitely many fermionic modes represents a unitary evolution if and only if it preserves canonical anticommutation relations (CARs). Preliminarily general features of nonlinear fermionic cellular automata (FCAs) that can be extended to the case of cellular automata on infinite graphs are analyzed. We use the preservation of CARs for the classification of FCAs on Cayley graphs of finite groups in two simple but paradigmatic case studies. The physical properties of the solutions are finally discussed. One of the examples allows us to classify all scalar FCAs on the $\mathbb{Z}$ integers.

DOI
10.1103/physreva.98.052337

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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 Schrödinger 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.