20.  Trajectories with turning points in time and causality. The mystery of the wave function.

In the classical mechanics we have no turning points in time. But as was shown by Feynman [6] these turning points can be solutions for potentials greater than. I start from m = 0 and we can expect a great number of turning points.

Here trajectories inverted in time or in causality are moving not in our cosmology but in cosmological models that are also inverted.

A trajectory with turning points in time is depicted on Figs. 1-2. If we take into account both dynamical time t and statistical time s, we would have for any trajectory a path depicted on Fig. 15. Along any trajectory we have four types of transitions:

Fig. 15. The turning points in time and causality.

These four types of transitions belong to the four worlds discussed at the beginning of this chapter.

At any moment of dynamical time t (or statistical time s) we have a number of transitions of all four types, i.e. we have four numbers

                   , , ,  .

The kinetics of these  is unknown but it can be deduced from general principles of interaction of the four worlds – see Section 4. It turns out that in some situations four kinetic equations for these four  can be united in one equation for one function

                    .

That is the meaning of the wave function.

For a great number of turning points we have small fluctuations of  and the usual lagrangian of classical electrodynamics:

                   ÿ                                                   (74)

can be rewritten as

                   ÿ .                                               (75)

Later on (in a next chapter) I shall calculate . They actually turn out to be very large.

Fluctuations of  are important in two cases:

1.  When we observe the difference, say,  (as in the act of measurement).

2.  For very high frequencies .

Here we shall have two “other quantum mechanics.”

After introducing those turning points we have all the above difficulties to disappear and instead of the above dynamics of circles we would have full-fledged quantum mechanics and gravitation.

We must simply repeat for (75) the same calculation of selfaction through the response from the universe that was carried out above for (74).

21.  The lagrangian for selfaction throughout the universe

Now I have for N particles of the universe the lagrangian

                   ÿ .                                         (76)

In our dealing with nature we do not follow the trajectories of all particles in the universe. Then all these trajectories must be excluded from (76) besides those that are objects of measurements.

To carry out this elimination we must at first exclude the field  and introduce the interaction of particles. Then we must write down the equations of motion for all particles besides the chosen one, solve these equations and put the solutions in (76). Then the lagrangian (76) becomes the lagrangian of selfaction throughout the universe for the chosen particle.

I.e. we must repeat the calculations of preceding sections with this lagrangian. Some new features:

(1)     Charge e must be changed to .

(2)     In expressions (28) for E and (64-65) for  we have additional terms proportional to .

(3)     Amplitudes of scattering of electromagnetic and gravitational waves on a circle will get additional terms through the dependence of  on these fields.

I start from (1). Electric strength generated by a current  is proportional to

                    .                                                                                              (77)

This term does not change through transformation . Hence the forces acting on  and  (so as on  and ) are equal and the circle is moving here as a whole. In terms of  the response from the universe is proportional to

                    .

Hence we have for electromagnetic response the same expressions as above with e changed to .

But in gravitational field we have instead of (77) the expressions

                    .

This expression changes sign through transformation . Hence the forces acting on  and   have different signs, the diameter of circle of a particle in a field radiated by another particle is expanded or contracted. This change generates an additional  and at the place of initial (or chosen) particle

                    .                                     (78)

The lagrangian of electromagnetic selfaction for one particle has the form

                                                                                                              (79)

with m from (52).

Putting distributions  in configuration Fig. 11 we have instead of (79) (any  is multiplied by its own ):

                    .                            (80)

In the same manner we have from (72) and the first term in (78):

                    .                                                                               (81)

And from the second term in (78) (here I drop ,  because they will be integrated):

                      (82)

The whole lagrangian is

                                                            (83)

Here  and  are calculated as a result of scattering the waves at the frequency  on distant particles of the universe. But contribution of some near particles must be considered separately because their contribution may be compatible to the contribution of all distant particles.

Let there be near an electron that is described by (83), another electron at the distance r. The function  in (83) describes the internal motion around the circle. In the presence of another electron we must take into account the distribution of the circles. I denote the distribution of the centers of the circles by . Then in derivation of (83) we must change  to  and after putting  (the line denotes averaging) we would have:

                                                                                         (84)

For two electrons at the distance r we have the contribution ( – velocity of the center of the circle,  – velocity along the circle):

                                                     (85)

Here the signs + and – must be taken for causal and anticausal loops.

I leave in (85) the factor  .

Here s, p are indexes of four worlds. Numbers 1, 2 are related to the second particle. Velocities with different indexes belong to different worlds, but here they interact because four electrons are parts of one hard disc.

Here  have different directions according to Fig. 11, but  are velocities of the center and are the same for all four electrons of the circle. Difference in signs  can compensate the difference in the signs of loops for the future and for the past cones. Hence all terms quadratic in  are smaller than the linear in  terms by the factor . Hence quadratic terms can be omitted, we have only the term

Here  and  are stochastic velocities of the centers of circles. They are needed to have the constant response from the universe (see Section 24). Here the pair of particles is moving as a whole because the vector  between the particles is specified. Hence after averaging

                    .

This  is canceled by the same factor from m in the denominator – m is proportional to .

Hence we have from (85) after averaging on  (and for small ):

                    .

Or

                                                                                               (86)

                                        (actually, )                           (87)

                    .

Adding (86) to (84) we have

                    .                                           (88)

And

                                                                                                         (89)

Current is square in probability – one of the most enigmatic feature of quantum mechanics is explained!

22.  The mystery of spinors and two-valueness

The current  radiates at the distance  the field (– initial phase, )

                            (90)

Here to exclude dumping we must exclude all higher harmonics. From the above expression you can see that for  we have

                    for  .

Hence from (90)

                    .                                                                        (91)

Decomposing  into a representation of Lorentz group

                                                                (92)

we can see that to ensure the condition (90) we must have:

                    if  .

In a general case, the probability  of finding an electron at a point  of a circle with the center at x is

                    ,

where  is the probability to find a circle with the center at x. Actually, these two probabilities depend on , hence we have here

                    .                                                     (93)

After decomposition of  as in (92) and after averaging (93) over  we have

                    .

The equation for  (from variation of (84)) has the form:

                    .                                                (94)

After averaging this equation over  we would have the equation for

                                          (95)

here  is a Dirac bispinor.

Or in the usual notation

           .

It is clear that , .

The meaning of these spinors and bispinors is evident: any electron can move from one world to another, hence the space of electrons dynamics is not , but . All that we have in projection on a circle and in dynamics where any  after going a full circle around turn out to be in another world and will be changed to . Only after the second turn around the circle  returns to its initial value.

Hence in our usual space (only one world)  turns out to be two-valued. In our experiments we see the average of these two values. Hence we have in lagrangian for

                                                    .

That is the sticking together that I supposed at the beginning. Now I have proved the self-consistency of this assumption!

For the usual  with  we do not have the response from the universe (because of dumping), but we can have the response from the near environment. This response does not depend on expansion of the universe and is the same as in Wheeler and Feynman [3]. The role of this response in quantum mechanics was discussed by Hoyle and Narlikar [7]. They demonstrated that taking this response into account in usual quantum mechanics we would have all effects of spontaneous transitions and of second quantization. I have in my theory the same response from the near environment. Hence I have in my theory also the two above types of effects.

Then I have derived all quantum mechanics.

But above I assumed the ground state of an electron to be the hard circle depicted on Fig. 11. Then I must prove the hardness of this circle. This will be done in the next section.