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arXiv:nlin/0404047 · Reviews in Mathematical Physics

A QUANTIZATION OF BOX-BALL SYSTEMS

R. INOUE, A. KUNIBA, M. OKADO

Published Nov 1, 2004• 5 citations
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Abstract

An L operator is presented related to an infinite dimensional limit of the fusion R matrices for [Formula: see text] and [Formula: see text]. It is factorized into the local propagation operators which quantize the deterministic dynamics of particles and antiparticles in the soliton cellular automata known as the box-ball systems and their generalizations. Some properties of the dynamical amplitudes are also investigated.

DOI
10.1142/s0129055x04002254

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An L operator is presented related to an infinite dimensional limit of the fusion R matrices for [Formula: see text] and [Formula: see text]. It is factorized into the local propagation operators which quantize the deterministic dynamics of particles and antiparticles in the soliton cellular automata known as the box-ball systems and their generalizations. Some properties of the dynamical amplitudes are also investigated.
A method of quantization of classical soliton cellular automata (QSCA) is put forward that provides a description of their time evolution operator by means of quantum circuits that involve quantum gates from which the associated Hamiltonian describing a quantum chain model is constructed. The intrinsic parallelism of QSCA, a phenomenon first known from quantum computers, is also emphasized. Introduction. Soliton cellular automata (SCA), is a class of cellular automata[1] operating on binary sequences with an updating rule function f for each cell, that depends on past and present time cells, the number of which determines the radius r of the automaton[2]. Contrary to usual CA ( that evolve their cells by means of past time cells only), SCA exhibit a variety of evolution patterns[4, 5, 6] that is mainly known to characterize the temporal behavior of
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.
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.
We study an integrable vertex model with a periodic boundary condition associated with U q A (1) n at the crystallizing point q = 0.It is an (n + 1)-state cellular automaton describing the factorized scattering of solitons.The dynamics originates in the commuting family of fusion transfer matrices and generalizes the ultradiscrete Toda/KP flow corresponding to the periodic box-ball system.Combining Bethe ansatz and crystal theory in quantum group, we develop an inverse scattering/spectral formalism and solve the initial value problem based on several conjectures.The action-angle variables are constructed representing the amplitudes and phases of solitons.By the direct and inverse scattering maps, separation of variables into solitons is achieved and nonlinear dynamics is transformed into a straight motion on a tropical analogue of the Jacobi variety.We decompose the level set into connected components under the commuting family of time evolutions and identify each of them with the set of integer points on a torus.The weight multiplicity formula derived from the q = 0 Bethe equation acquires an elegant interpretation as the volume of the phase space expressed by the size and multiplicity of these tori.The dynamical period is determined as an explicit arithmetical function of the n-tuple of Young diagrams specifying the level set.The inverse map, i.e., tropical Jacobi inversion is expressed in terms of a tropical Riemann theta function associated with the Bethe ansatz data.As an application, time average of some local variable is calculated.