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Optics and Spectroscopy

Description of quantum systems by the collective behavior method

Yu. I. Ozhigov

Published Feb 1, 2010• 0 citations
Publisher

Abstract

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.

DOI
10.1134/s0030400x10020219

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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.
With the help of a collective behavior method, quantum dynamics is represented as a result of mutual transformations of a swarm of classical samples of quantum particles. The method used is shown to generalize the Monte Carlo diffusion method to the case of many-body quantum dynamics. The method is illustrated by the example of the Schrödinger equation. It is shown that the diffusion representation of the Coulomb field linearly simulates a multiparticle system compared to the traditional method of quadratic complexity. A simple method of description of entangled states of any complexity with the help of association reactions of samples of quantum particles is given.
The simple algorithm for the simulation and visualization of non relativistic quantum dynamics is proposed that is based on a collective behavior of classical particles. Any quantum particle is represented as the swarm of its classical samples which interact by simple rules including emission and absorption of samples of tied photons. The quantum dynamics results from the collective behavior of such a swarm where the eigenstates are treated as the equilibrium states relatively to emission-absorption of photons. The entanglement is treated as a correlation between samples of the different swarms that is stored in the space-time part of the model inaccessible for a user. The amplitude is always grained. The Coulomb field between quantum particles is simulated, analogously to free flow of quantum package, by the point wise interaction between its samples and scalar photon samples which propagate by diffusion. It gives square root speedup in comparison to each-with-each method. This method obviously includes decoherence and admits the natural generalization on the QED of many particles with the linear computational cost.
In this course, I talk about the source of mathematical constructivism and its role in the future development of theoretical physics. I describe what physical constructivism is and why it is necessary for the penetration of exact methods of theoretical physics to the area of complex systems, which formally belong to the others natural disciplines. I describe the concrete heuristic for the creating of models of constructive quantum theory the method of collective behavior. I represent the constructive viewpoint on the quantum computer, which treats it as the model object of many particle quantum physics, and the practical recommendation concerning its building. Due to the known inertia of the educational system constructive methods in mathematics remains mostly unknown to the wide physical community, and I hope that this course will stimulate the interest to the new possibilities, which these methods open. These possibilities represent interest to natural scientists and for programmers as well.
A model of entangled quantum states of many bodies using tuples consisting of samples of individual real particles is proposed. This approach generalizes Bohm’s method for complex systems. The dynamics can be described both in the form of quantum hydrodynamics and in the form of dynamic diffusion which is examined in detail in this study. The latter method fundamentally differs from the choice of the grain of spatial resolution and cannot be reduced to differential equations; therefore, it requires direct simulation. The limitation on the number of samples is the fundamental factor causing decoherence, a deviation of the swarm dynamics from the exact solution to the Schrödinger equation. The model complexity is linear with respect to the number of particles.
\nUsing the fact that any linear representation of a group can be embedded into permutations, we propose a constructive description of quantum behavior that provides, in particular, a natural explanation of the appearance of complex numbers and unitarity in the formalism of the quantum mechanics. In our approach, the quantum behavior can be explained by the fundamental impossibility to trace the identity of the indistinguishable objects in their evolution. Any observation only provides information about the invariant relations between such objects.\nThe trajectory of a quantum system is a sequence of unitary evolutions interspersed with observations—non-unitary projections. We suggest a scheme to construct combinatorial models of quantum evolution. The principle of selection of the most likely trajectories in such models via the large numbers approximation leads in the continuum limit to the principle of least action with the appropriate Lagrangians and deterministic evolution equations\n
A straightforward algorithm is proposed for simulating and visualizing nonrelativistic quantum dynamics in terms of the collective behavior of classical particles. In this approach, a quantum particle is treated as a swarm (collection) of its classical samples interacting with one another by simple rules involving the emission and absorption of associated photons. The quantum dynamics is regarded as resulting from the collective behavior of the swarm, and the eigenstates are treated as equilibrium states of the process of photon emission and absorption. Entanglement is seen as classical correlation between samples from distinct swarms; the information on this correlation is stored in a spatiotemporal section of the model inaccessible to the user. Amplitudes are invariably treated as consisting of amplitude quanta. A Coulomb field experienced by quantum particles is represented in a similar manner to the free spreading of a wave packet, by point interaction between the particle samples and the samples of scalar photons propagating by diffusion. This provides a square-root gam in computational speed against the direct method involving item-by-item examination. The approach presented covers decoherence and allows natural generalization to multiparticle quantum electrodynamics, the amount of memory required growing linearly.