23.  The unusual rigidity of the ground state of electron

Circles depicted on Fig. 11 seem to be somehow invented, somehow contradicting to the spirit of quantum mechanics. But actually these circles are an inevitable and unique consequence of the first principles. These new (in comparison with the usual quantum mechanics) features seem to be unnecessary, but in Section 25 I shall point out the experiments where all these features can be observed.

The most interesting feature of the circle is its unusual hardness. Its parameters are determined with precision . That is the maximal precision in our universe, because local dynamics is generated by the response from our N-particle universe.

As an example, let us take  – the frequency of rotation of the circle that I have calculated before. This  was calculated from condition

                    .

If  differs from

                    ,

we would have the dumping

                    .

Hence we must have

and for

                    .                                                                                         (96)

Now I shall show that for velocities  we have conditions

                                                .           (97)

These estimations follow from the fact that  and  enter the expressions of the current and the lagrangian of selfaction in combinations

                    ,       ,      and so on.                  (98)

For the simplest state we have

If we put these  into the combinations (98), we would see that terms proportional to A and  give higher harmonics of  in radiation. Here A is the number of  pairs for an electron moving around a circle. This number is connected to the d – density of turning points through the evident relation

where  is the average length of free path.

This length can be estimated from the change of  along the circle. We see that  is of the same order as a (the radius of the circle). Hence

The turning points are generated by the waves (or short impulses) on frequencies near . The condition (53) can be rewritten as

where  is coefficient of dissipation generated by the imaginary part of mass. You can see that this condition takes place not only for , but also for frequencies near .

Linear density d can not be greater than , where  is the length of waves with the frequencies near , because for greater density turning points cease to be independent. Hence

And from the above expression for A we have

           .

Actually, any electron is smearing along the circle and the correct image is not Fig. 11, but Fig. 16. Hence the condition, say

                    ,

must be changed to

                    ,

where the line denotes averaging.

We must rewrite the condition (97) for average values of velocities.

Fig. 16. Distributions of four electrons along the circle.

24.  Why quantum mechanics?

The most immediate consequence of the worlds different in two signs of time and two signs of causality is the response from the universe to any move of a particle. The ground state of a particle is supported by this response but we have the needed response only for some distributions of these ground states. Quantum mechanics describes the dynamics of these distributions.

In some circumstances the dynamics of these distribution can be near classical dynamics, but conditions for stability of distributions generate different quantum effects.

This supporting response is the response to waves generated by ground state that is supported by this response. The only stable state here is the oscillating state – then de Broglie waves. These waves are collective phenomena for great number of particles. But there are another phenomena where small difference between the great number of retarded and the great number of advanced particles is important – here we see a spot on photographic plate.

Then quantum mechanics is a particular case of classical statistics. This statistics has two peculiar features:

1.  Trajectories move through four worlds that are simple and slightly different.

2.  Trajectories radiate electromagnetic waves that generate the response from the universe. Particles move in the field of this response.

I am speaking about trajectories though absence of trajectories behind quantum dynamics is generally accepted. In a paper by Dirac there is a proof that no statistics can explain quantum behavior. It is true, but not with our four worlds and not on the background of the response from the universe.

With our four worlds we have as the ground state of electron not “a point particles at rest” but a complex dynamical picture described in the above sections. And now I want to draw your attention to some features of ground state that were not discussed.

The first evident objection to the above picture: the ground state of an electron is supported by waves that oscillate with a unique frequency  (it is the only frequency that ensures the undumping of waves in the universe). The frequency of a moving electron has a Doppler shift, then we have here a dumping. The response from the universe disappears and the circle Fig. 16 disappears.

And any external field will change the frequency of de Broglie oscillation. Here we will also have dumping and disappearance of electron.

This objection has a simple answer. Let the circle move with the velocity . From (84) we have

                    .

The solution:

                     .

For electron at rest we have

                    .

For electron moving with velocity  we have

                     .

Hence

                     .

At any point we have waves from all space, i.e. we have

                          .

Hence Doppler shift is canceled by the factor

                          .

For

the Doppler shift will be canceled after averaging over  and  if

                    .                                                                                          (99)

Then the principle of uncertainty is a simple consequence of the response from the universe.

Hence the object of quantum mechanics is not a trajectory but an ensemble of trajectories.

But in spite of all that the trajectories can exist and can be discovered in some experiments of the new kind (see Section 26).

In this connection I shall dwell on my attempt to rewrite quantum mechanics as an universal distribution analogous to the Gibbs distribution in statistical mechanics.

There is an interesting expression that I proposed [8]:

                                                                                                 (100)

here   W(a) – probability of an event a;

          – the integral over all trajectories ...... that have the property a;

;   (we have two signs of time – this ± is the main point!);

          S – action along trajectory ......

The expression (100) is similar to the Gibbs distribution, because (100) contains all quantum mechanics.

The expression (100) has sense only for the event if a  has the form of a number of independent variable. But now we can improve (100) rewriting it as

                    .                                                                   (101)

Here R(a) = 1 if the state radiates only on the frequency  and R(a) = 0 for all other cases. With this addition we have a full analogue of the Gibbs distribution – no restriction on a.

The trajectories in (101) have sense only as an element of a distribution that satisfies the condition (99).

But  is only a part of the correct probability . This correct probability is positive (in (101) the constant A is omitted) and it can be measured, for example, through the effects that will be discussed in the Section 26. The correct probability can be constructed as multiplication of probabilities of transitions along the path. The probability of a small section along a trajectory is , where  is the change of action along this section. Hence the correct form of (101) is

            .

After decomposition in  and after integration all terms of this decompositions turn out to be zero beside 1 and

            .

Hence

Here the first term is unobservable, the second term is identical to (101).

That is the source of the mystery of “negative probabilities” in quantum mechanics.

            It follows from (101) and from lagrangian of self-action through the response from the universe that probability for a particle to pass from one point to another has the form of the sum over loops in time. This fact explain the interference in two slit experiment. We have here three kinds of loops: loops that pass through slit1, loops that pass through slit2, and loops that pass through both slits. Interference is due to last kind of loops.

             There is also a simple explanation of the fact that continuous wave function generate discrete spots on a photographic plate. In above discussion I do not take into account that for any particle wandering with different signs of time and with different signs of causality we have at any time a huge numbers of particles with those different signs but the difference of those numbers must be 1 or 0. We can write down the equation for probability of this number to be 1.

             This equation absent in usual quantum mechanics and because of that this quantum mechanics can not describe act of measurement!

              The probability of ionization of an atom of photographic plate is proportional to above probability.

               The second question: if we have a huge amount of particles why we do not have multiple ionization?

               The answer is that in the first approximation we have for those huge amounts the linear equations. If we take into account the above numbers 1 and 0 we would have nonlinear terms for those huge amounts. These amounts turn out to have maximum values for the above number to be 1 and turn out to be zero for this number to be zero. It is the collapse of wave function.

                We have now a full theory of this collapse. This collapse turn out to be dependent on the state of near environment. This dependence generate many new physical effects.

                 In near future I hope to describe all that in details.