On the relationship between the intensity of zero-point fluctuations of an electromagnetic field (ZPFs) and the magnitude of an elementary electric charge

The consequences of the idea of stimulation of spontaneous emission of photons and scattering of particles in quantum electrodynamics by zero-point fluctuations of an electromagnetic field (ZPFs) are considered. It is shown that this idea leads to a connection between the magnitude of the elementary electric charge and the intensity of the ZPFs, leads to the idea of the stochastic nature of the electromagnetic interaction in time, and due to this, to the possibility of “classical” passage of particles through a potential barrier.


Introduction
Almost immediately after the creation of quantum electrodynamics, Weiskopf and Wigner proposed that the spontaneous emission of photons by atoms or atomic nuclei is stimulated by zero-point fluctuations of the electromagnetic field (ZPF) [1].This idea was based on the description of the interaction of charged particles according to perturbation theory, when the Hamiltonian of a system of particles and fields is divided into the Hamiltonian of non-interacting particles and a free electromagnetic (EM) field and the Hamiltonian of the interaction between them.In this case, an elementary electric charge is introduced into the interaction Hamiltonian as a coupling constant between particles and the free EM field, consisting of ZPFs and photons.If we do not consider processes induced by photons, then it turns out that the spontaneous emission of a photon is induced by ZPFs.Similarly, the scattering of particles occurs due to their interaction with the ZPFs (see, e.g.[2,3] and references there).The probability of spontaneous photon emission and particles scattering depends on the intensity of ZPFs quanta.It turns out that the interaction constant itself depends on the intensity of ZPFs quanta.The random nature of the time distribution of the action of ZPFs quanta on particles leads to the stochastic nature of the EM interaction.These issues are discussed in this paper.

Some properties of zero-point fluctuations (ZPFs) of the EM field
As quantum mechanics shows, zero-point fluctuations (ZPFs) of physical objects are their inherent property.The reality of the existence of EM field ZPFs is confirmed, for example, by the Casimir effect -inside a cavity with metal walls there are no low-frequency EM waves (wavelengths are less than the size of the cavity) and the ZPFs pressure on the walls of the cavity from the outside is greater than their pressure from the inside.For other confirmations of the reality of ZPFs, see, e.g.[4].In free space, the energy density of ZPFs in the frequency interval dω is given by the formula Here c is the speed of light,  is Planck's constant divided by 2π.When ZPFs interact with a substance, their intensity can change (see review [3] and references there).Thus, in a metal, the intensity of lowfrequency ZPFs (ω up to ~ 1 keV) can decrease due to their Doppler displacement into a higher frequency region when EM waves are reflected from approaching atoms during the formation of the metal.In addition, ZPFs interact with atomic electrons, leading to their virtual transitions to the region of the continuous energy spectrum and back.During the formation of a metal, new permissible free states of electrons arise with the appearance of a conduction band instead of discrete atomic levels.In this case, the energy of interaction of electrons with ZPFs increases, and the energy of ZPFs themselves decreases near values of energy equal to the binding energies of electrons in atoms, which in the case of heavy elements can reach tens of kiloelectronvolts.After the formation of the metal, the intensity of ZPFs is restored due to the diffusion of ZPFs into the metal from free space.A change in the intensity of ZPFs leads to a change in the probability of spontaneous electromagnetic transitions.Nuclear isomers turned out to be convenient for experiments; for them, a slowdown of spontaneous EM transitions of energy up to 2 keV was really observed in metals [3], which qualitatively corresponds to the suppression of ZPFs in metals at transitions frequencies.A recent analysis [5] of Mössbauer experiments on changing the width of γ-lines depending on restrictions on the measurement time showed that the width of the γ-line is determined by the time that the absorber nucleus sees the emitter nucleus.Based on this, the structure of a photon and a ZPF was proposed -when an excited state of an atomic nucleus or atom is formed, an EM wave begins to be emitted that does not carry energy (0-wave).At the moment of the radiative quantum transition "on the tail" of this 0-wave a quantum of photon energy is appeared, localized in a region of space with a diameter of the order of the reduced photon wavelength .The wave properties of a photon are imparted by the 0-wave, which by its duration determines the width of the Mössbauer γ-line; a quantum of energy localized in space affects matter as a particle.The localization region of the energy quantum corresponds to the fact that in a smaller spatial region it is impossible to determine the parameters of the EM field of a photon [6] (see also [7]), and also corresponds to the cross section of 2 2 for the resonant stimulation of photon radiation by photons.
Based on this structure of ZPFs, for any point in free space it is possible to determine the timeaverage number nZPF of appearances of ZPFs quanta per unit time in a unit frequency interval -the dimensionless value of the spectral frequency of ZPFs quanta appearance.The ZPFs energy density in the frequency interval dω is given by Formula (1).Then the number of quanta of frequency ω passing through the area of natural radius  = c/ω during time dt in the frequency interval dω is equal to It is important that nZPF does not depend on the frequency of ZPFs.The moments when ZPFs quanta of different frequencies act on particles and stimulate EM interactions are randomly distributed in time and do not correlate with each other, which corresponds to complete uncertainty in the phase of a single ZPF according to the uncertainty relation between the phase of photons and their number (see e.g.[8]).This explains the Poisson statistics for spontaneous EM transitions that are stimulated by ZPFs, and, accordingly, the exponential law of decay of excited states in time [5].

Interaction of zero-point field fluctuations (ZPFs) with particles
In quantum electrodynamics, the interaction of two elementary charges e with radius vectors r1 and r2 in the lowest order of perturbation theory is described by the scattering matrix where J(r1, r2) is a smooth function of particles transitions currents, | R | = | r1 -r2 | is the distance between particles (see e.g.[8]).Formula (3) was obtained taking into account the interaction of particles with the ZPFs of all possible wavelengths  from 0 to ∞.Each of the ZPFs quanta with a wave vector k = 1/ causes an exchange of momentum k between particles.Because of the presence of an oscillating exponential in Formula (3) under the integral, it follows that the contribution to the interaction between particles is mainly made by ZPFs with wavelengths  > R. Considering that the size of a ZPF quantum is ~ , it turns out that interaction between particles is caused only by those ZPFs quanta that capture both particles at once, and it is the capture of particles into the field of one ZPF quantum that leads to the exchange of momentum between them (Figure 1).This picture is an alternative to the generally accepted interpretation of interaction today as the emission of a virtual quantum by one particle and its absorption by another particle.
Figure 1.Scheme of interaction of an ZPF quantum with two electrons for a rough estimation of the magnitude of the Coulomb force.Interaction between particles with momentum transfer k occurs if the ZPF quantum captures both particles at once.The case is shown when the ZPF quantum captures both particles only with its edge.Formula (3) for the scattering matrix in the case of interaction of two slow electrons leads to the Coulomb's law for the interaction force F = e 2 /R 2 .According to the above, F should be proportional to the average intensity nZPF of ZPFs.The only way to satisfy this requirement is to assume that the square of the elementary electric charge is proportional to nZPF.This conclusion was first made in [9].Based on this representation, it is possible to obtain an expression for the force F of interaction between elementary electric charges without using their magnitude based on the change in their momentum Δp over time Δt.For a rough estimation, we assume that an ZPF quantum with wave vector k has a spherical shape with a diameter  = 1/k; interaction between particles with momentum transfer k occurs if this ZPF quantum captures both particles at once (Figure 1).Then F is equal to Here it is taken into account that per unit solid angle Ω there are nZPF/4π quanta of ZPFs per unit time in a unit frequency interval.Using the value nZPF ≈ 1/π from Formula (2) and comparing Formula (4) with the Coulomb's law, we obtain an expression for the squared value of the elementary electric charge and for the fine structure constant α: Thus, the values of α and e 2 are determined by the intensity of the ZPFs quanta.The difference between the obtained value of α and the correct value 1/137 is perhaps insignificant, since in deriving Formula (4) only the rough estimated size of the ZPF quantum ~  was used.
It has long been known that in a closed theory of particles interaction, the dimensionless fine structure constant α = e 2 /c must be calculated based on the basic parameters of the theory, and not just measured experimentally (see, e.g.[10]).But numerous attempts to explain the value of α in this way were unsuccessful.Therefore, the urgency of the question of the need to calculate α gradually disappeared, but the problem itself remained.The presented approach solves this problem.Only the square of the elementary electric charge, which is an observable quantity, receives a physical meaning; the magnitude of the electric charge itself is not an observable quantity and has not a physical meaning.It would be interesting to apply this approach to calculating the constants of other interactions.Attempts have been made in this direction for a long time.It was believed that the laws of particles interaction could be obtained not only from experiment, but also theoretically derived from more fundamental principles.For example, back in the 18th century, Georges-Louis Lesage developed a model of gravitational attraction of bodies, previously considered by René Descartes, according to which attraction arises due to the mutual shading of bodies from the flow of ethereal particles, the concentration and momentum of which determine interaction constant [11].

Consequences of the stochastic nature of the Coulomb force
As was shown above, the Coulomb interaction of particles is induced by ZPFs quanta, the intensity of which at specific points in space has a random Poisson distribution in time.As a result, the magnitude of the Coulomb force also acquires a stochastic character, which leads to the possibility of considering the new mechanism of passage of a particle through a potential barrier (tunnel effect) when the particle energy is less than the height of the barrier.This mechanism is additional to the tunneling known from quantum mechanics due to the wave properties of particles, although it is worth keeping in mind that the particles wave properties themselves are possibly caused by particles interaction with ZPFs (see, e.g.[12]).
As particles approach each other, the number of ZPFs quanta causing particle repulsion may accidentally be so much less than the average value that the potential barrier will effectively decrease to allow the particle to classically pass over the barrier.This process can be modeled by dividing the classical trajectory of a particle into separate small times intervals dt, and taking into account the Poisson nature of the time distribution of the action of ZPFs quanta on particles, constructing a simple Markov chain, for which the probability of subsequent states of the particle depends only on its current state and does not depend on previous states (see, e.g.[13]).It is easy to estimate the probability of a particle passing a one-dimensional potential barrier at speed v0 (Figure 2), when the interaction between particles is completely turned off for the duration of the barrier passage.The average number of ZPFs quanta acting on a particle during time dt in the frequency interval dω is determined by Formula (2).The probability that no quanta will appear at all in the interval dt dω is equal to The probability PF = 0 of the absence of the action of ZPFs quanta on a particle in the total frequency range of ZPFs for the total time of passage of the barrier T = (R0 -R2) / v0 is obtained by the product of the probabilities according to Formula (6) for all time intervals and all frequency intervals satisfying the condition ω < c/R(t) .As a result we obtain:  Figure 3.The spectrum of γ-quanta from a polycrystalline Hg-matrix condensed together with molecules of hydrogen isotopes (1).The excess of this spectrum (1) over the background spectrum ( 2) is highlighted in dark gray.
It is interesting to compare the probability PF = 0 according to Formula (7) with the probability PQM of passing through the Coulomb barrier according to the semiclassical expression of quantum mechanics at a high potential barrier e 2 /R2 >> E (see, e.g.[14]) Here m is the reduced mass of the particle.It can be seen that the probability PF = 0 of the absence of interaction when passing the Coulomb barrier, taking into account the random nature of the Coulomb force according to Formula (7), and the probability PQM of tunneling according to the quantum mechanics Formula (8), depend on the same quantity.That is, in quantum mechanics, the transparency of the Coulomb barrier is apparently also determined by the intensity of ZPFs quanta acting on the particles.Of course, the probability PF = 0 of the complete absence of Coulomb interaction between particles when passing through a barrier is negligible.For example, let for the case of cold fusion of hydrogen and deuterium nuclei m = 10 -24 g, R0 = 10 -8 cm, R1 = 10 -9 cm (in this case E = 144 eV, initial velocity v0 = 2•10 7 cm/s), R2 = 10 -12 cm (Figure 2).Then PF = 0 ~ 10 -1909 .For comparison, under the same conditions, according to Formula (8), the transparency of the Coulomb barrier PQM ~ 10 -30 .
The idea of the tunneling mechanism due to the random distribution of the Coulomb force in time served as the basis for the assumption of the possibility of fusion of light atomic nuclei when they are introduced into the forming metal matrix.Since, as noted in Section 2, during the formation of a metal matrix the intensity of ZPFs decreases for some time [3], the transperency of the Coulomb barrier separating atomic nuclei increases accordingly.Indeed, when an equilibrium gas isotopic mixture of H2, HD, D2 molecules was captured into the forming polycrystalline mercury Hg-matrix, nuclear fusion reactions p + d → 3 He + γ5.5 MeV were detected by emission of γ-quanta of energy 5.5 MeV. Figure 3 shows the γ-spectrum obtained over 5000 s in one of these experiments from an Hg-matrix condensing with molecules of hydrogen isotopes, and the background γ-spectrum measured one day after the formation of the Hg-matrix [14].A photopeak with an energy of 5.5 MeV, an escape peaks near 4.7 MeV, and a contribution from Compton scattered γ-quanta of 5.5 MeV are visible.In these experiments, no neutrons were visible from the reaction d + d → 3 He + n, for which the barrier transparency is less due to the larger reduced mass of particles, although the sensitivity of the experimental setup was approximately the same for detecting 5.5 MeV γ-quanta and neutrons from the Hg-matrix.
Another consequence of the stochasticity of the Coulomb force is the fluctuation of the energy of particles during their scattering, although these energy fluctuations can be very small.For example, when particles of the same charge approach each other, the intensity of ZPFs quanta can randomly decrease, but with subsequent repulsion of particles, the intensity of ZPFs quanta mainly has an average value.After such scattering, the kinetic energy of the particles increases.On the contrary, if, as the particles approach each other, the intensity of the ZPFs quanta randomly increases, then after scattering the kinetic energy of the particles decreases.Thus, the law of conservation of energy in individual acts of particle elastic scattering may be violated.But the energy is conserved on average for a large number of particles scattering events.Perhaps it is these fluctuations in particles energy that prevent the so-called thermal death of the Universe.

Conclusions
The considered approach to the EM interaction of particles explains the squared value of the elementary electric charge via the intensity of the zero-point fluctuations (ZPFs) of EM field and allows us to understand the mechanism of tunneling of a particle through a potential barrier as a consequence of the stochastic nature of the EM interaction.This approach opens up other new research opportunities and appears to have practical prospects.However, many unclear questions remain here.Does only the square of the elementary electric charge really have a physical meaning, as a value proportional to the timeaverage intensity of ZPFs according to Formula (5)?And is it then justified in quantum electrodynamics (see, e.g.[8]) to use only the magnitude of the electric charge itself, which has no physical meaning, as a coupling constant between the field of free particles and the free EM field?And then what about the fractional electric charge of quarks?Also, for example, it remains unclear whether in individual acts of elastic interaction of particles it is really possible to violate the law of conservation of energy and whether this law is satisfied only on average over a large number of scattering acts?Can the tunnel effect really be explained in some cases by the stochastic nature of the interaction of ZPFs with particles?All these questions are the subject of future research.

Figure 2 .
Figure 2. Passage of a particle through the Coulomb potential barrier.