In the present note I wish to talk about formation of one line of works concerning theory of high energy electromagnetic processes in the substance which was initiated thanks to brilliant intuition of Alexander Illich Akhiezer who took active part in the development of these investigations till his last days.

In 1969 a short B.Ferretti's paper caught the eye of Alexander Illich, the work was devoted to quasiclassical theory of coherent radiation of relativistic electrons in a crystal. The work was written very unclearly and as it became apparent later, the considered variant of the problem was of little interest. But the statement of the problem in this article attracted Alexander Illich's attention. Reasons were the following.

In the sixties theoretical department of UPhTI directed by Alexander Illich conducted intensive studies of radiation corrections to quantum electrodynamical processes at high energies. Calculations of such corrections were not an easy matter, but they resulted only in small modifications to the probabilities of processes defined by the first order of quantum electrodynamical perturbation theory. The physical essence of such corrections was difficult enough to catch on.

By this time B.Ferretti, M.Ter-Mikaelyan and G.Juberall have developed the theory of coherent radiation of relativistic electrons in crystals. This theory was based on the first Born approximation of quantum electrodynamics. Its basic result was that coherent and interference radiation phenomena, thanks to which the probability of radiation process for electrons in a crystal could essentially surpass corresponding probability in the amorphous medium, manifested itself in the movement of high energy electrons in a crystal at a small angle to one of crystallographic axes or planes. It was this last result that has drawn Alexander Illich's attention. The intuition and a wide experience in the field of quantum electrodynamics suggested to him that if there was a considerable amplification of the first order process of perturbation theory then it was possible to hope, that contributions of higher orders of perturbation theory to this process also would increase. Owing to this consideration new trends in study of radiation corrections in substance at high energies might be opened.

One effect to which Alexander Illich paid attention was the dependence of radiation probability on a sign of particle charge. It was well-known that such dependence appears only after taking into account higher orders of perturbation theory. But for a number of processes known to that time, like elastic scattering and radiation in Coulomb field, dependence of scattering and radiation probabilities on a sign of particle charge at high energies was very weak. Therefore if for particle interacting with a crystal the relative contribution to the radiation from higher orders of perturbation theory increased, it should result in turn in strengthening the dependence of radiation probability on a sign of particle charge.

Alexander Illich offered Peter Ivanovich Fomin to be engaged in this problem. This offer was very natural as Peter Ivanovich had a wide experience of work with radiation corrections. Besides, he has mastered the existing by that time theory of coherent radiation of relativistic electrons in crystals and even gave lectures on this problem to experimenters. At this time, however, Peter Ivanovich, was very keen on a problem of electromagnetic mass of electron and could not devote much time to the offered problem. Several earlier I have addressed to Peter Ivanovich with the request to be a head of my predegree and degree practice as I wished very much to take up the problem of electromagnetic mass of electron. However, Peter Ivanovich already had a student, which dealt with this problem, therefore he deliberated for some time what to do with me, and after consulting with Alexander Illich he has connected me to a problem of radiation of relativistic electrons in a crystal.

Generally speaking the problem of taking into account higher corrections of the perturbation theory for electromagnetic processes at high energy in substance presented enough challenge. But dependence of radiation probability on a sign of a particle charge should appear already after introduction to radiation the contribution of the second Born approximation. We also have begun with this "elementary" problem.

Calculations moved ahead quickly enough. I would like to finish with this problem faster to be engaged afterwards with an electron mass problem. But the problem was growing continuously and it became clear that the work was only the beginning. Peter Ivanovich regularly reported to Alexander Illich on progress of the work. And, at last, when we succeeded to find an elementary case in which calculations became transparent, but at the same time the physical aspect of the problem was not diluted, my first meeting with Alexander Illich took place. It was felt that Alexander Illich and Peter Ivanovich were very happy with the result. First of all, the idea that contributions not only of the first, but also of the higher orders of the theory of perturbations could increase considerably at interaction of relativistic electrons with a crystal proved to be true. In particular, it has appeared that relative contribution of the second Born approximation to the probability of radiation increased quickly with reduction of particle angle of incidence on the crystal relative to one of its crystallographic planes. It, in turn, led to considerable dependence of probability of radiation on a sign of particle charge. It was essential also that the mentioned effect was quite available for experimental verification and that such researches could be performed at the accelerator in Kharkov where there was the possibility to carry out experiments both with relativistic beams of electrons and positrons. Alexander Illich very much appreciated works in which the theory incorporated with experiment so the last situation relished him especially.

After some time my second short meeting with Alexander Illich took place. It was at the Faculty of Physics and Technology of Kharkov State University in Pyatichatki. Alexander Illich together with George Anatolievich Milyutin and several other university fellows left a faculty building. Having seen nearby me, he has called me up and told only several phrases:: «So to hold! Only forward! Give Warsaw!» All have laughed. I was a little stunned. I was stunned not only with such unexpected way of support, but also with words he told. First two phrases were clear. But what did mean Warsaw here? Only much later I have learnt that else in the thirties when Alexander Illich worked with I.Ya.Pomeranchuk over his first problem concerning light scattering on light (this problem has been posed to them by L.D.Landau) they often recited this civil war L.D.Trotsky's slogan to encourage each other.

Subsequently my first scientific work together with Alexander Illich and Peter Ivanovich «Bremsstralung of electrons and positrons with ultrarelativistic energy in crystals» based on obtained results has been published in JETP Letters (A.I.Ahiezer, P.I.Fomin, N.F.Shulga, JETP Letters (1971) v.13. pp.713-715).

So a new field of research connected with electromagnetic processes at high energies in substance out of range of applicability of Born perturbation theory was becoming. Subsequently this research trend obtained considerable development in connection with problems of interaction of high energy particles with oriented crystals. Alexander Illich showed considerable interest to researches in this area, and actively worked in it. I will note here only some directions which appeared from developments of researches in the field.

It appeared that violation of one condition of Born theory applicability for high energy electron coherent radiation in crystals led to phenomenon of particles channeling. Consequently the problem of channeling phenomenon appeared in the developed theory naturally as a special case. It opened possibilities to look at the phenomenon of channeling from more general positions, and also to connect the phenomenon with a number of other phenomena at high energies in crystals. Earlier it seemed impossible.

Substantial growth of probabilities of radiation processes and formation of electron-positron pairs at high energies in the oriented crystals should lead to accelerated development of electromagnetic showers in crystals. It meant that the theory of electromagnetic showers offered by Landau and Rumer in the late thirties, should be modernized taking into account the specified effects. Alexander Illich remembered well, how Landau and Rumer’s theory was created and told a lot about it. Everybody felt that the problem of showers development in crystals was very dear to Alexander Illich because it was the development of Lev Davidovich Landau’s work, his teacher’s work.

The same could be said about another line of works. In the early 50’s Landau and Pomeranchuk have shown that multiple scattering can have a significant impact on process of bremsstrahlung of electrons with ultrarelativistic energy in amorphous environment. The analysis of process of particle scattering in crystal showed that such conditions could be created in the case of special crystal orientations when average values of particles scattering angles would surpass considerably corresponding values for amorphous environment. It opened new possibilities in the study of Landau and Pomeranchuk’s effect for particles interacting with crystals.

The following episode was connected with the last problem. In the late seventies I together with my then still pregraduate S.P.Fomin investigated influence of process of repeated scattering on radiation of high energy particles in thin layers of substance (Landau and Pomeranchuk considered a case of thick, actually infinite target). When working on the problem we could notice the following. To account for influence of repeated scattering on radiation it was necessary to average the function which defined the process of radiation over the stochastic process describing repeated scattering of particles in the environment. Landau and Pomeranchuk when carrying out such procedure of averaging could obtain only estimative results. Subsequently A.B.Migdal obtained quantitative formulas on the basis of kinetic equation method he proposed for the problem (Alexander Illich later told us that V.A.Fock liked Migdal’s work very much and presented it at 1954 for publication in DAN USSR, then he supported А.B. Migdal for his election as a corresponding member of the AS USSR).

We noticed that in the method developed in Landau and Pomeranchuk’s work the function subjected to averaging has a Gaussian form. A stochastic process describing repeated scattering of a particle in the environment was a Gaussian process. It meant that in the problem under consideration to carry out the procedure of averaging the method of functional integration or as it was still named — a method of path integrals could be used. The similar situation took place also for electron radiation in crystal. Thus, the given problem was reduced to mathematical problem of calculation of functional integral. For development of this method some time is necessary. I have been busy then with a number of other problems, therefore I asked for help several theorists who knew that was such functional integral. Besides some times I talked about the problem during my reports at theoretical seminars in KIPT.

Everyone agreed that the problem was interesting, but then it did not go. And only some years later then still very young staff of theoretical and one experimental departments of KIPT, N.V.Laskin and A.S.Mazmanishvili, took up this problem. They already had a wide experience of work with functional integrals. The result was impressive. A method of functional integration made it possible to reproduce completely the basic result of Migdal’s theory.

It opened new possibilities to describe processes of interaction of high energy particles with substance. All this was immediately reported to Alexander Illich. He was very glad of obtained results and suggested us to address A.B.Migdal and ask him to present our article on the work to Reports of Academy of Sciences of the USSR (Doklady). He also wrote a letter to Migdal with this request. Soon Migdal has sent the answer to Alexander Illich with the following text: «Dear Shura! When I solved this problem, I at first have thought up a functional integral, but it seemed to me that it is very difficult to work with it. I’m glad that it is not so. Your Arkady». The work has soon been published in Doklady (N.V.Laskin, A.S.Mazmanishvili, N.F.Shulga. The functional integral approach to account the influence of repeated scattering on radiation of high energy particles in crystals and amorphous environments. DAN of the USSR v.277, p.850, 1984). Subsequently on the base of this method were obtained also a number of other results.

I dwelled on the details of the episode, in particular to show how Alexander Illich rejoiced in the works of his pupils and how he supported them.

It is impossible in a short note to dwell on all I would like to tell about Alexander Illich. At seminars he used to speak: «Devils lurk in details». And there was plenty of "details"! I thank my lucky stars that they have brought me with Alexander Illich and have given the opportunity to work with him for many years. Now, years after Alexander Illich has gone, all of us especially felt , how it was easy to work with him and how much easier, easier and faster many questions would possibly be solved after having consulted with Alexander Illich. But life proceeds. We remember Alexander Illich. He continues to live in activity of his pupils.