George Ryazanov

15. The undumping frequency – the main result of the unity of worlds

The frequency which makes attenuation vanish is determined from the expression (52) for mass and from the condition for zero imaginary part of the mass

. (53)

Here the integral spread up to a maximum velocity is determined by the condition (51). As a result, condition (53) takes the form:

. (54)

The behavior of functions under integral is shown in Fig. 9.

Fig. 9. The function under the integral (54).

In order to satisfy the equality (53), the positive values along the whole interval should be counterbalanced by a negative peak near v = 1. Therefore the argument of the sine in (53) for large N is close to , then – the addition to in

(55)

is small. The positive part of the integral (54)

and the negative part

must give zero sum, then

From (55) we have for small

Substituting this expression in (52) we obtain

Then

. (56)

At the point we have balance of dumping from retarded waves and pumping from advanced waves. Near this point we didn’t have linear terms in because of the symmetry of and . Then for we have extremum of energy.

It is interesting that results of this and the above sections are opposite to the calculations of Wheeler and Feynman.

Their results for :

My results on :

On there is no radiation dumping for electrons. A simple estimation shows that we can do this only in a very narrow interval

, .

Outside this interval we have the results of Wheeler and Feynman [3]. Then we have very specific dependence of Re {m} on , depicted on Fig. 10.

Fig. 10. Dependence of the electron mass on the frequency .

The absence of radiation reaction on does not mean the absence of radiation: here retarded and advanced radiation reactions are counterbalanced. Then we have advanced waves on , but we can not have any special signal from them because all possible sources or receptors are also counterbalanced.

But if m = 0 for , from where have we usual mass of an electron? The answer will be given later on. It turns out that the usual mass is equal to the mass calculated above.

16. Genesis of Coulomb force is not a simple thing

In the usual expression for Coulomb force

and are the time-components of velocities of two electrons. Two directions of time imply two signs of u. Hence for symmetric combination of retarded and advanced electrons (see Fig. 8) we have

The main fact of electrodynamics disappears!

But don’t let it bother you. As had been shown before, the energy of electron has the form of the sum on all loops in time that connect this electron with other electrons in the universe. These other electrons can be divided in two parts. The first part is the distant electrons near the horizon. They are distant but their number is great.

The second part is a much smaller number of electrons (may be one electron), but they are near.

For this second kind of electrons we have for the energy of causal and anticausal loops in time (signs of mass are different):

Here is very small coefficient of dumping. But we have the same coefficient in the expression (52) for the mass of electron (I – integral in (52)):

(57)

Hence after introducing a renormalized change

we have for m in (57) the expression

The smallness of here did not play any role.

Now we can remember that we have four types of loops depicted on Fig. 8. They give us four types of interactions (7-10). At the undumping frequency all four interactions (7-10) appear on equal footing.

But in spite of this fact I want to demonstrate that for we have our usual field (7).

We have this interesting result because for we have expansion (or contraction) and dissipation (or pumping) along any ray. It was demonstrated by Wheeler and Feynman [3] that we can start from symmetric combination (in means initial)

but in the dissipating universe we would have the response

Then the observed field will be

To explain this result let us take a radiating particle 1 and a receiving particle 2. Without any scattering we have at the place of particle 2 two fields from particle 1: and . After the scattering on all particles of the universe we have an advanced spherical wave that is contracted to the particle 1, is passing through the focus at the place of particle 1 and then is going to the particle 2. But by passing through the focus the amplitude of the wave changes its sign – this fact is well known from optics. Hence we have at the point of particle 2 for the past cone:

After the passing through the focus this advanced wave can be scattered, will return to the particle 1, and after the second passing through the focus and the second change of sign (at the place of particle 1) it will be going to particle 2 as usual retarded wave. Then at the place of particle 2 we have for the future cone