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The Dirac Equation, Mass and Arithmetic by Permutations of Automaton States

Hans-Thomas Elze

TL;DR

The work develops a deterministic cellular automaton framework in which Dirac dynamics in $1+1$ dimensions arise from permutations of ontological Ising-spin states arranged in a torus-like 'Necklace of Necklaces'. A mass term is incorporated via a scattering operator that governs arithmetic updates, complementing the left/right mover kinematics derived from Weyl-type equations in the continuum limit. By reverse-engineering from the Dirac equation, the authors show how mass contributions can be encoded deterministically through permutations, and they introduce a sophisticated encoding scheme using shifted up-spins and block variables. Although the automaton remains classical and non-superposable, the construction illuminates how quantum-like structure can emerge in coarse-grained descriptions and points to extensions to higher dimensions, Hamiltonian derivations, and gauge interactions. Overall, the paper advances a concrete, permutation-based path to Dirac dynamics within a torus-compactified state space, bridging deterministic ontologies with relativistic fermion equations.

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.

The Dirac Equation, Mass and Arithmetic by Permutations of Automaton States

TL;DR

The work develops a deterministic cellular automaton framework in which Dirac dynamics in dimensions arise from permutations of ontological Ising-spin states arranged in a torus-like 'Necklace of Necklaces'. A mass term is incorporated via a scattering operator that governs arithmetic updates, complementing the left/right mover kinematics derived from Weyl-type equations in the continuum limit. By reverse-engineering from the Dirac equation, the authors show how mass contributions can be encoded deterministically through permutations, and they introduce a sophisticated encoding scheme using shifted up-spins and block variables. Although the automaton remains classical and non-superposable, the construction illuminates how quantum-like structure can emerge in coarse-grained descriptions and points to extensions to higher dimensions, Hamiltonian derivations, and gauge interactions. Overall, the paper advances a concrete, permutation-based path to Dirac dynamics within a torus-compactified state space, bridging deterministic ontologies with relativistic fermion equations.

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.

Paper Structure

This paper contains 10 sections, 28 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Results: An Automaton Beneath the Dirac Equation
  3. The Weyl Equation
  4. Reverse Engineering an Automaton Based on Permutations from the Dirac Equation
  5. Doing Arithmetic by Permutations
  6. Discussion
  7. Conclusions
  8. Permutations and the Cogwheel Model
  9. A Chain of Ising Spins with Exchange Interactions
  10. Illustration of Scattering Operator $\hat{\cal M}$ and Automaton Update

Figures (3)

  • Figure A1: A Cogwheel jumps periodically through $N$ states, taking time $T$ for each jump. FromElzehybridCA.
  • Figure A2: ( Top): Spin variables on odd/even sites are transported to the next odd/even sites to the left/right, respectively, in one update by $\hat{U}$, Equation (\ref{['A7']}), of the spin chain automaton. ( Bottom): Occupation numbers on neighboring even sites are added to form right-moving block variables, $n_{2k}+n_{2k+2}=:\tilde{n}_{2k}$; likewise, left-moving ones, $n_{2k+1}+n_{2k+3}=:\tilde{n}_{2k+1}$. They are illustrated by arrows displaced by one unit to the right for better visibility.
  • Figure A3: ( Left): Two initial conditions (heavy dots, crosses) for $|\Psi\rangle_n$ on two neighboring transverse necklaces attached to the even and odd longitudinal sites $2k$ (blue) and $2k+1$ (red), respectively. Only the transverse positions (drawn vertically, ranging in $[-3,+3]$) of the single up spins are marked; all others are down. ( Middle): Update of the initial state by the scattering operator $\hat{\cal M}(k)$, Equation (\ref{['scatteringopk']}), gives the intermediate result $|\Psi\rangle_n'$. Note that the blue cross has felt the periodic boundary condition in the vertical direction, according to $-2+(-3)\equiv +2$. ( Right): To obtain the final result of the one-step update, $|\Psi\rangle_{n+1}$, the up spins at even and odd longitudinal positions have to move two positions to the right and left, respectively.