Abstract
A model of a Quantum recurrence in the dynamics of an elementary physical vacuum cell within the framework of four coupled Shrodinger equations has been suggested. The model of an elementary vacuum cell shows that a Quantum recurrence which represents the dynamics of virtual transformations in the cell, qualitatively differs from that of Poincare and the Fermi-Pasta-Ulam. Whereas these recurrences develop in time or space, the Quantum recurrence develops in a sequence of Fourier images represented by non exponentially separating functions. The sequence experiences random energy additions but no exponential separation occurs. The Quantum recurrence can be defined as the most frequent array of Fourier images that appear in a certain quantum system during a period of its observation. Different scenarios of the Fourier images sequences interpreted as bosons (electron and positron) and fermions (photons) apearing in the solutions of the model demonstrate that during some periods of its observation they become indistinguishable. The quantum dynamics of every physical vacuum cell depends on the dynamics of many other vacuum cells interacting with it, thus the quasi periodicity (during the period of observation) of the Fourier images recurrence can have infinite periods of time and space and the amplitudes of the Fourier images can vary many orders in their magnitudes. Such recurrence times does not correspond even roughly to the Poincare recurrence time of an isolated macroscopic system. It reminds the behavior of spatially coupled standard mappings with different parameters. The amount of energy in the physical vacuum is infinite but extracting a part of it and converting, it into a time-space form requires a process of periodical transfer of the reversible microscopic system dynamics into that of a macroscopic system. This process can be realized through a resonant interaction between the classical and quantum recurrences developing in these two systems. However, a technical realization of this problem is problematic.
Highlights
After Poincare had stated his classical form of a recurrence when both an amplitude and a phase of the system must recur after a certain period of time to their initial states [1] Fermi, Pasta and Ulam discovered in a system of non linearly coupled oscillators [2] a much more sophisticated form of recurrence when the recurrences to the initial states were not identical but manifested the appearance of stochasticity in the process of energy regrouping in a sequence of different recurrences
The obtained numerical results allowed making the following conclusions: 1. The model of an elementary vacuum cell shows that a Quantum recurrence which represents the dynamics of virtual transformations in the cell, qualitatively differs from that of Poincare and the Fermi-Pasta-Ulam
Whereas these recurrences develop in time or space, the Quantum recurrence develops in a sequence of Fourier images represented by non exponentially separating functions
Summary
After Poincare had stated his classical form of a recurrence when both an amplitude and a phase of the system must recur after a certain period of time to their initial states [1] Fermi, Pasta and Ulam discovered in a system of non linearly coupled oscillators [2] a much more sophisticated form of recurrence when the recurrences to the initial states were not identical but manifested the appearance of stochasticity in the process of energy regrouping in a sequence of different recurrences. In the paper [10] there was suggested a model of the Fermi-Pasta-Ulam recurrence within the framework of two coupled high and low frequency standard mappings This approach looks perspective for describing recurrences in quantum systems just like (1). Since the vacuum cell (1) is under the influence of similar cells the processes of the energy addition to and subtraction from the cell have to be included into the model For this purpose, the Pippard's idea about the Shrodinger equation with a complex potential has been used [11]. E−V ''t / If to take the dimensions of electron and positron in a coupled state (1) as symmetrical ones (-a+a), their wave functions can be described within the framework of coupled spatial Shrodinger equations having different directions of a spatial variable and having random potentials reflecting the influence of other cells like (1). Accounting a quantum character of the dynamics in the cell (1) the initial conditions for the functions e, p, h, r were taken random
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