17.  Index of refraction for the unity of the four worlds

The above considerations have a weak point. The index of refraction (45) is taken from calculations with only retarded waves. Then if we want to know the index (or indexes) of refraction for the case when all four solutions are taken into account, we must repeat the usual deduction for this case. We must also include in consideration the dependence of waves and electrons on temperature through thermal fluctuations of metrics. These fluctuations and the usual viscosity are the sources of dissipation. But the formulas become too complex and to save simplicity I omit the dependence on temperature in all equations.

The index of refraction can be calculated from the usual equation that relates Green function G for waves in a medium to Green function  in vacuum. In the simplest case of rarefied gas of electrons we have:

                    .                   (58)

For the case of two worlds – retarded causal and advanced causal – we have:

                                                                                                                                (59)

                    .

According to Section 16, we have  for  and the equation (59) in momentum representation becomes the same as (58) – with the same index of refraction.

For  we can introduce new functions  and  that are defined as:

For  and we once more would have (58).

For causal  and anticausal  fields we have

                                      ÿ

                                                                                                                                (60)

                   ÿ .

Here the transition from  to  is accompanied by the change of the sign of  and of the sign of m.

Substituting the variables

we would have the pair of equations identical to (59).

There is another factor that must be taken into account: four fields  are radiated and scattered together. Let electron x radiate all . Electron y later on will perceive field  multiplied by factor (1 + v) and field  with the same (1 + v) – here  is going from the future but is anticausal in the usual direction of time, then we have the same (1 + v).

Electron y will move and radiate in the future the same two fields  and  with factor  and in the past it radiates fields  and  with the factor .

As a result, we have equations (signs before  and  are changed because , see (7-10)) that are identical to equations (59) after the change:

                     ;   .

That is the end of exploration of all consequences that we have with the simplest suppositions about the interaction of four worlds: four electrons from different worlds are sticking together. In the next section I shall sum up these consequences.

18.  What we have done and what we must do

All the above discussion is about the consequences of the simplest assumption (26) or the same assumption in the case of four worlds: particles of different worlds are sticking in tetrads

                    ,                                                         (61)

here   k is the number of a particle;

          indexes are the same as in Table 1.

It is interesting that this simple and, actually, erroneous assumption can give us some elements of the theory of mass, quantum mechanics and gravitation (see Sections 9‑11).

But it is not yet a theory because:

1.  Equalities (61) do not follow from the basic equation (20).

2.  In the absence of external forces we have for autooscillations of an electron the equation  – hence  and we do not have any oscillations.

3.  These oscillations have a unique undumping frequency  and because of that the electron can not move: any velocity generates Doppler shift and dumping (in the absence of external forces).

Hence something important is missing.

Those factors turn out to be:

1.  Gravitational waves.

2.  Transitions of particles from one world to another.

Inclusion of these factors solve the above three problems, but (61) is to be changed to more general relations. Equations (20) will be satisfied.

You could object: “What may be the role of gravitational waves that are  times weaker than electromagnetic waves?” Some hint to the answer you can take from Section 11: at undumping frequency  the magnitude of the gravitational constant is in  times greater than at any other frequency. It is precisely these waves at  that generate all quantum mechanics.

Transitions from one world to another solve the problem with Doppler shift because they generate distribution of particles even for one trajectory. Distributions can move without change of overall frequency. This fact explains many peculiarities of quantum behavior.