The authors have declared that no competing interests exist.

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.

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

In a quantum case, the notion of a phase space trajectory loses its meaning and so does the notion of the Lyapunov exponent, which measures the separations between trajectories

We consider a possible existence of a quantum recurrence in a model of the following virtual reaction taking place in the physical vacuum:

Equation (1) describes a reversible electromagnetic formation of an electron-positron couple from two photons together with electron-positron annihilation giving a birth to two photons.

We discuss the usage of Anderson's model of a particle whose possible locations are the equidistant sites of a one-dimensional chain _{m} from neighboring reactions acts and the hopping of the particle from one site to its r-th neighbor is described by a hopping amplitude _{r}. The probability amplitude _{m} for finding the particle on the m-th site obeys the Shrodinger equation

Consider if _{m} are random numbers uncorrelated from site to site and distributed with a density _{m}_{r}, in contrast, will be taken to be nonrandom and to decrease fast for hops of increasing length r. As it was shown

(3)

Where _{ph}_{+}is the amplitude that photon will tunnel the barrier and _{ph}_{-} is the amplitude that a photon will be reflected from the barrier. K –characterizes the potential barrier.

The system (3) was used for a description of the Josephson junction dynamics

Summing up one can see that the two quantum problems (2) and (3) can be reduced to the standard mapping form. In the paper ^{n} through the following form of the Shrodinger equation:

If _{n}

Therefore a full wave function:

possesses a dissipation proportional to:

An impulse response of this system, which corresponds to an addition of the other eigenvalue states after the impulse, in a general case (due to such members as _{m}^{*}_{n}) has the members varying as

Accounting all mentioned the reaction (1) can be described by four coupled Shrodinger equations:

(5)

Where _{1 }- is the wave function of electron in (1), _{2 }- is the wave function of the first photon in (1), _{3 }- is the wave function of the second photon in (1), _{4} - is the wave function of a positron in (1). _{1},_{2},_{3},_{4} are random magnitudes of energy which correspond to the chaotically reversible exchange of energy between the vacuum cells. The multiplied wave function members in (1) reflect the non linear interaction processes. _{random}_{1},_{random}_{2},_{random}_{3},_{random}_{4} are random potentials.

For numerical analysis the system (3) was reduced to four coupled differential equations of the second order:

(6)

Where e(t),p(t) correspond to the electron and positron wave functions ψ_{1}, ψ_{4} in (5), and h(t),r(t) correspond to the wave functions of two photons ψ_{2}, ψ_{3} in (5);

_{1}, _{4} are random functions distributed within the interval (-1,1) corresponding to the positive and negative dispersion processes in the first and fourth equations in (5) having place due to energy interactions of electrons and positrons with other cells like (1); _{2}, _{3 }are random functions distributed within the interval (-1,1) corresponding to the dissipation processes in the second and third equations in (5) having place due to resonant interactions of photons with other cells like (1) _{1},_{2},_{3},_{4} are random functions distributed within the interval (0,1) corresponding to the random potentials _{random}_{1},_{random}_{2},_{random}_{3},_{random}_{4} in (5); _{1}, _{2}, _{3}, _{4} are positive constants. Spatial and temporal symmetries in (5) are realized in (6) through a symmetrical resulting Fourier transform image of the functions e, p, h, r in relation to the frequency axis (consequence of aliasing); ω_{1} corresponds to the spatial frequency in Eq. 1,4 in (5); ω_{2} corresponds to the temporal frequency in Eq. 2,3 in (5). To get the ratio between ω_{2} and ω_{1} we can equalize the energy for both spatial and temporal Shrodinger equations in (5).

Accounting that the electron and positron radius _{e,p}= 2.8*10^{-13}cm, temporal frequency/energy = 2.42*10^{14 }Hz/e.v., spatial frequency/energy= 8.06*10^{3}cm^{-1}/e.v.,and considering the length of spatial waves 4*_{e,p}if an electron and positron are next to each other we can get the ratio between the temporal frequency ω_{2 }in (5) and the spatial frequency ω_{1} in (5):

Accounting a quantum character of the dynamics in the cell (1) the initial conditions for the functions e, p, h, r were taken random.

A factual opposite direction of spatial (Eq.1,4) and temporal (Eq.2,3) coordinates in the system (5) brings the numerical analysis of the system (6) to the statistical observing of the Fourier image similarities appearing quasi periodically in multiple runs of the computer program during the solving process of the system (6). Since the potentials, dissipative members and initial conditions in (6) were random every run gave different Fourier images. A few hundred runs coverage allowed revealing the most frequent sequences of the Fourier images appearing quasi periodically in the results of computer analysis of the system (6) and to give them interpretations. (

One more numerical experiment was aimed at a possible application of the mathematical model. For that purpose all random functions _{1}, _{2}, _{3}, _{4} in (6) were substituted by positive constants. The result of this artificial situation is given in.

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. The sequence experiences random energy additions but no exponential separation occurs.

2. 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.

3. 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.

4. The quantum dynamics of every 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