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

(1)

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 6. In our case, these are electron or positron in (1). At each site a random potential V_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 W_r. The probability amplitude p_m for finding the particle on the m-th site obeys the Shrodinger equation 6:

(2)

Consider if V_m are random numbers uncorrelated from site to site and distributed with a density ρ(V_m). The hopping amplitudes W_r, in contrast, will be taken to be nonrandom and to decrease fast for hops of increasing length r. As it was shown 57 this case can be reduced to the standard mapping form. On the other hand, the process of interaction between a photon and electron or positron can be interpreted as a random one, which represents either penetrating of a photon through or reflecting from a potential barrier of the electron or positron field. That brings to the temporal form of coupled Shrodinger equations:

(3)

Where ψ_ph+is the amplitude that photon will tunnel the barrier and ψ_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 8.The process described by (3) also can be interpreted as a kicked rotator and can be reduced to the standard mapping form. Together with this some popular examples of chaos are of this type 9

Summing up one can see that the two quantum problems (2) and (3) can be reduced to the standard mapping form. 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. He suggested the description of a particle moving in a potential field: -iVⁿ through the following form of the Shrodinger equation:

(4)

If ψ = 0 when x=0 and x=a, for an eigenvalue with a number n for which ψ_n = sin(nπ/a), a following equality must be fulfilled:

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 ψ_m^*ψ_n) has the members varying as and describing the oscillations at the frequency of beatings and decaying as 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). Two coupled photons in (1) are described within the framework of coupled temporal Shrodinger equations (3) having different directions of a time variable and also having random potentials. The link between the two couples is realized through a mutual parametric excitation by the differences between the wave functions of electron and positron and that of between the two photons. The differences have opposite signs in the coupled couples of the Shrodinger equations because of mentioned opposite directions of a spatial and temporal coordinates there. Since the number of the virtual reactions similar to (1) is infinite and they are interconnected through random potentials, we have to insert them into the coupled Shrodinger equations. Generally, the energy damping-increasing process is interpreted as a resonant chaotically reversible exchange of energy between the vacuum cells like (1).

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

(5)

Where ψ₁ - is the wave function of electron in (1), ψ₂ - is the wave function of the first photon in (1), ψ₃ - is the wave function of the second photon in (1), ψ₄ - is the wave function of a positron in (1). E₁,E₂,E₃,E₄ 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. V_random1,V_random2,V_random3,V_random4 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 ψ₁, ψ₄ in (5), and h(t),r(t) correspond to the wave functions of two photons ψ₂, ψ₃ in (5);

R₁, R₄ 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); R₂, R₃ 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) 11; A₁,A₂,A₃,A₄ are random functions distributed within the interval (0,1) corresponding to the random potentials V_random1,V_random2,V_random3,V_random4 in (5); C₁, C₂, C₃, C₄ 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); ω₁ corresponds to the spatial frequency in Eq. 1,4 in (5); ω₂ corresponds to the temporal frequency in Eq. 2,3 in (5). To get the ratio between ω₂ and ω₁ we can equalize the energy for both spatial and temporal Shrodinger equations in (5).

Accounting that the electron and positron radius r_e,p= 2.8*10^-13cm, temporal frequency/energy = 2.42*10¹⁴ Hz/e.v., spatial frequency/energy= 8.06*10³cm^-1/e.v.,and considering the length of spatial waves 4*r_e,pif an electron and positron are next to each other we can get the ratio between the temporal frequency ω₂ in (5) and the spatial frequency ω₁ 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. (Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23, Figure 24)

One more numerical experiment was aimed at a possible application of the mathematical model. For that purpose all random functions R₁, R₂, R₃, R₄ in (6) were substituted by positive constants. The result of this artificial situation is given in. Figure 25, Figure 26, Figure 27, Figure 28.