Classical and quantum discrete dynamical systems
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assical and Quantum Discrete Dynamical Systems V. V. Kornyak Joint Institute for Nuclear Research, Dubna, 141980 Russia Abstract—We study deterministic and quantum dynamics from a constructive “finite” point of view, since the introduction of the continuum or other actual infinities in physics poses severe conceptual and technical dif ficulties, and while all of these concepts are not really needed in physics, which is in fact an empirical science. Particular attention is paid to the symmetry properties of discrete systems. For a consistent description of the symmetries of dynamical systems at different time instants and the symmetries of various parts of such sys tems, we introduce discrete analogs of gauge connections. These gauge structures are particularly important to describe the quantum behavior. The symmetries govern the fundamental properties of the behavior of dynamical systems. In particular, we can show that the moving solitonlike structures are inevitable in a deter ministic (classical) dynamical system, whose symmetry group breaks the set of states into a finite number of orbits of the group. We demonstrate that the quantum behavior is a natural consequence of symmetries of dynamical systems. This behavior is a result of the fundamental inability to trace the identity of indistinguish able objects during their evolution. Information is only available on invariant statements and values related with such objects. Using general mathematical arguments, any quantum dynamics can be shown to reduce to a sequence of permutations. The quantum interferences occur in the invariant subspaces of permutation rep resentations of the symmetry groups of dynamical systems. The observables can be expressed in terms of per mutation invariants. We also show that in order to describe quantum phenomena it is sufficient to use cyclo tomic fields—the minimal extensions of natural numbers suitable for quantum mechanics, instead of a non constructive number system—the field of complex numbers. The finite groups of symmetries play the central role in this review. In physics there is an additional reason for such groups to be of interest. Numerous exper iments and observations in particle physics point to the importance of finite groups of relatively low orders in a number of fundamental processes. The origin of these groups has no explanation within presently recog nized theories, such as the Standard Model. DOI: 10.1134/S106377961301005X
CONTENTS 1. INTRODUCTION 2. CLASSICAL AND QUANTUM DYNAMICS 2.1. Basic Concepts and Structures 2.2. Dynamical Systems with a Space 2.2.1. Electrodynamics. Abelian prototype of all gauge theories 2.2.2. NonAbelian gauge theories in continuous spacetime 2.2.3. Emergence of space in discrete dynamics 2.3. Peculiarities of Deterministic Dynamics 2.3.1. Solitonlike structures in deterministic dynamics 2.3.2. About reversibility in discrete deterministic systems 3. CONSTRUCTIVE DESCRIPTION OF QUANTUM BEHAVIOR 3.1. Quantization of Discrete Systems 3.1.1. Standard and “finite” versions of q
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