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Updated: May 10, 2025

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The Dirac Equation, Mass and Arithmetic by Permutations of Automaton States
1Dipartimento di Fisica "Enrico Fermi", Universitá di Pisa, Largo Pontecorvo 3, 56127 Pisa, Italy.
Abstract:
The cornerstones of the Cellular Automaton Interpretation of Quantum Mechanics are its underlying ontological states that evolve by permutations. They do not create would-be quantum mechanical superposition states. We review this with a classical automaton consisting of an Ising spin chain which is then related to the Weyl equation in the continuum limit. Based on this and generalizing, we construct a new "Necklace of Necklaces" automaton with a torus-like topology that lends itself to represent the Dirac equation in 1 + 1 dimensions. Special attention has to be paid to its mass term, which necessitates this enlarged structure and a particular scattering operator contributing to the step-wise updates of the automaton. As discussed earlier, such deterministic models of discrete spins or bits unavoidably become quantum mechanical, when only slightly deformed.
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