aqora

Mathematical Modeling of Finite Quantum Systems

Vladimir V. Kornyak

Lecture notes in computer scienceJan 1, 201211 citations

Description of quantum systems by the collective behavior method

Yu. I. Ozhigov

We develop the constructive viewpoint to quantum theory, which means the using of constructive mathematics as the basic formalism. It is shown how the heuristic of collective behavior leads to the description of complex quantum systems with the inbuilt decoherence on classical computers. Decoherence comes from the finite total number of samples for each quantum particles. When this number converges to infinity, its trajectory converges to the solution of Schrödinger equation. This model rests on the simple rule of the sort of momentum exchange and gives the linear growth of the complexity.We develop the constructive viewpoint to quantum theory, which means the using of constructive mathematics as the basic formalism. It is shown how the heuristic of collective behavior leads to the description of complex quantum systems with the inbuilt decoherence on classical computers. Decoherence comes from the finite total number of samples for each quantum particles. When this number converges to infinity, its trajectory converges to the solution of Schrödinger equation. This model rests on the simple rule of the sort of momentum exchange and gives the linear growth of the complexity.

Optics and SpectroscopyFeb 1, 20100 citations

Discrete dynamical models: combinatorics, statistics and continuum approximations

Vladimir V. Kornyak

This essay advocates the view that any problem that has a meaningful empirical content, can be formulated in constructive, more definitely, finite terms. We consider combinatorial models of dynamical systems and approaches to statistical description of such models. We demonstrate that many concepts of continuous physics --- such as continuous symmetries, the principle of least action, Lagrangians, deterministic evolution equations --- can be obtained from combinatorial structures as a result of the large number approximation. We propose a constructive description of quantum behavior that provides, in particular, a natural explanation of appearance of complex numbers in the formalism of quantum mechanics. Some approaches to construction of discrete models of quantum evolution that involve gauge connections are discussed.This essay advocates the view that any problem that has a meaningful empirical content, can be formulated in constructive, more definitely, finite terms. We consider combinatorial models of dynamical systems and approaches to statistical description of such models. We demonstrate that many concepts of continuous physics --- such as continuous symmetries, the principle of least action, Lagrangians, deterministic evolution equations --- can be obtained from combinatorial structures as a result of the large number approximation. We propose a constructive description of quantum behavior that provides, in particular, a natural explanation of appearance of complex numbers in the formalism of quantum mechanics. Some approaches to construction of discrete models of quantum evolution that involve gauge connections are discussed.

arXiv:1501.07356Mathematical Modelling and GeometryApr 1, 20154 citations

Applicability of quantum models

Miroslav Svítek

Elsevier eBooksJan 1, 20210 citations

Mathematical Modeling of Finite Quantum Systems

Vladimir V. Kornyak

We consider the problem of quantum behavior in the finite background. Introduction of continuum or other infinities into physics leads only to technical complications without any need for them in description of empirical observations. The finite approach makes the problem constructive and more tractable. We argue that quantum behavior is a natural consequence of symmetries of dynamical systems. It is a result of fundamental impossibility to trace identity of indistinguishable objects in their evolution - only information about invariant combinations of such objects is available. We demonstrate that any quantum dynamics can be embedded into a simple permutation dynamics. Quantum phenomena, such as interferences, arise in invariant subspaces of permutation representations of the symmetry group of a system. Observable quantities can be expressed in terms of the permutation invariants.We consider the problem of quantum behavior in the finite background. Introduction of continuum or other infinities into physics leads only to technical complications without any need for them in description of empirical observations. The finite approach makes the problem constructive and more tractable. We argue that quantum behavior is a natural consequence of symmetries of dynamical systems. It is a result of fundamental impossibility to trace identity of indistinguishable objects in their evolution - only information about invariant combinations of such objects is available. We demonstrate that any quantum dynamics can be embedded into a simple permutation dynamics. Quantum phenomena, such as interferences, arise in invariant subspaces of permutation representations of the symmetry group of a system. Observable quantities can be expressed in terms of the permutation invariants.

arXiv:1107.5675Jul 28, 20112 citations

Model Theory in Quantum Foundations

Nicholas Stipanovic

Apr 1, 20260 citations

Finite quantum models: constructive approach to description of quantum behavior

Related papers

Finite quantum models: constructive approach to description of quantum behavior

Related papers