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arXiv:1705.04376

Cellular Automaton-Like Model of Arising Physical-Like Properties

M. Pietrow

Published May 11, 2017• 0 citations
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Abstract

A simple relation of the order of $n$ abstract objects generates an $n-2$ dimensional basis of three dimensional vectors. A cellular automaton-like model of evolution of this system is postulated. During this evolution, some quantities stabilize with time and form a discrete spectrum of values. The presented model may have some general aspects in common with a cellular automaton representation of a quantum system.

DOI
10.48550/arxiv.1705.04376

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A simple relation of the order of nnn abstract objects generates an n−2n-2n−2 dimensional basis of three dimensional vectors. A cellular automaton-like model of evolution of this system is postulated. During this evolution, some quantities stabilize with time and form a discrete spectrum of values. The presented model may have some general aspects in common with a cellular automaton representation of a quantum system.
Take a cellular automaton, consider that each configuration is a basis vector in some vector space, and linearize the global evolution function. If lucky, the r esult could actually make sense physically, as a valid quantum evolution; but do es it make sense as a quantum cellular automaton? That is the main question we a ddress in this paper. In every model with discrete time and space, two things ar e required in order to qualify as a cellular automaton: invariance by translatio n and locality. We prove that this locality condition is so restrictive in the q uantum case that every quantum cellular automaton constructed in this way - i. e., by linearization of a classical one - must be reversible. We also discuss some subtleties about the extent of nonlocality that can be encountered in the o ne-dimensional case; we show that, even when the quantized version is non local, still, under some conditions, we may be unable to use this nonlocality to trans mit information nonlocally.
The notion of ontological states is introduced here with reference to the Cellular Automaton Interpretation of Quantum Mechanics (QM) proposed by ’t Hooft. A class of discrete deterministic “Hamiltonian” Cellular Automata (CA) is defined that has been shown to bear many features in common with continuum quantum mechanical models, however, deformed by the presence of a finite discreteness scale [Formula: see text], such that for [Formula: see text], the usual properties result e.g. concerning linearity, dispersion relations, multipartite systems, and Superposition Principle. We argue that within this class of models, only very primitive realizations of ontological states and their dynamics can exist, since the equations of motion tend to produce superposition states that are not ontological. The most interesting, if not the only way out seems to involve interacting multipartite systems composed of two-state “Ising spins”, which evolve by a unitary transfer matrix. Thus, quantum like and ontological models appear side by side here, but distinguished by second-order and first-order dynamics, respectively.
Simple algebraic cellular automata and a diffusion confinement are used to model quantum behavior of a relativistic particle in a 1-D box. Comparisons and contrasts are drawn between such a quantum cellular automaton (QCA) and both orthodox and unorthodox interpretations of quantum mechanics. The persistent tension between deterministic pictures and probabilistic descriptions of quantum reality are clarified for the case of a 1-D quantum particle viewed as a cellular automaton surging for survival in space and time.
This study presents a unitary quantum cellular automaton (QCA) that, in the continuum limit, converges to the (1+1)-dimensional Generalized Dirac Equation (GDE). We outline the construction of the unitary, discrete-time evolution and derive the model's exact dispersion relation from its eigenvalue spectrum, showing it possesses the correct relativistic limit. We then explore its dynamics through numerical simulations. While the total probability density is shown to be insensitive to the chiral angle parameter (ρ)(ρ)(ρ) of the GDE, we demonstrate that this parameter has a profound and directly observable effect on the particle's local spin polarization. By measuring this quantity, we reveal the hidden internal dynamics of the spinor, providing a clear, visual confirmation of the physical consequences of the chiral mass term.