BIBLIOGRAPHY

A. I. Akhiezer, V. B. Berestetskii. Quantum Electrodynamics, Gostekhizdat, 1953, 428 pp., price 18 rubles 30 kopecks.

The monograph by A. I. Akhiezer and V. B. Berestetskii, Quantum Electrodynamics, is not only the first exposition of modern quantum electrodynamics in Russian, but indeed the only systematic treatment of the subject in the world literature.

It is well known that until quite recently quantum electrodynamics encountered a number of substantial difficulties connected with the occurrence of infinities in higher approximations. In recent years (1947–1953), major advances have been achieved in this most important field of theoretical physics. The ideas of mass and charge renormalization indicated the way to excluding infinities from the theory. The development of a new, more perfect mathematical apparatus—the invariant theory of perturbations—made it possible actually to carry out the renormalization program and gave physicists a powerful tool for further theoretical investigations.

The authors of the monograph do not confine themselves to a sequential exposition of the foundations of the theory from contemporary positions, but apply it to the consideration of many physical phenomena.

The book begins with an exposition of the theory of unquantized electromagnetic and electron-positron fields (Chapters I and II). Chapter I is devoted to the quantum mechanics of the photon. Here questions are presented concerning the description of the photon in the coordinate representation and the classification of the states of a system of two photons. Of special interest is the following discussion of states with definite angular momentum and parity (see also § 10 of Chapter II) by means of spherical vectors, based to a significant degree on the work of one of the authors (V. B. Berestetskii). The mathematical apparatus developed here finds numerous applications. The second chapter sets forth the relativistic quantum mechanics of the electron. Particularly interesting here are the sections devoted to charge conjugation of wave functions, the parity of the positron, and the possibility of separating the solutions of the Dirac equation into electron and positron states in the presence of an external field. Some of these results also belong to V. B. Berestetskii. In addition, §§ 12 and 13, on the motion of an electron in the Coulomb field of a nucleus and on electron scattering, are very useful for the reader.

The authors then present the theory of quantization of the photon and electron fields, the basic equations of quantum electrodynamics, and the invariant perturbation theory (Chapters III, IV). In Chapter III, besides general questions of quantization of the electromagnetic and electron fields, the authors give a detailed theory of the various invariant \(\Delta\)-, \(D\)-, and \(S\)-functions, which is very important for practical applications. In Chapter IV the fundamental equations of quantum electrodynamics are formulated and their general properties are investigated.

The central part of this chapter is the invariant theory of perturbations. Questions to which Chapters III and IV are devoted are very complex. One should note the high modern level of the exposition and at the same time its accessibility to the reader, owing to its great coherence and consistency.

Chapter V sets out methods for calculating elements of the scattering matrix in the impulse representation and methods for eliminating the divergences from them. In this chapter the reader becomes acquainted with the basic practical methods of quantum electrodynamics and is given the possibility of using them for concrete calculations.

In Chapters VI and VII detailed calculations of the probabilities of various effects are carried out. The material presented here is valuable in two respects: first, the methods of invariant perturbation theory are applied here to concrete processes and therefore give the reader the opportunity to master the technique of quantum-electrodynamic calculations. Second, the calculations are carried through to final results; practically important formulas, tables, and graphs are given.

It should be emphasized that the authors dwell in detail on a number of questions that are of great physical interest and that have hitherto been insufficiently, or not at all, presented in monographic literature. These include the theory of the positronium, internal conversion (including pair formation), the theory of \(0-0\) transitions, the infrared catastrophe, and others.

In Chapter VIII the authors combine all the results obtained in recent times on the higher approximations of quantum electrodynamics. Here one finds the radiation corrections to electron scattering and to the Compton effect, the scattering of light by light (some of the results here belong to A. I. Akhiezer), the radiative shift of atomic levels and the anomalous magnetic moment of the electron, and a number of other questions. Such a complete exposition of the practical achievements of the contemporary theory, inseparably connected with the new mathematical apparatus, is very useful for theorists interested in applications of quantum electrodynamics.

The supplements to the book contain the general theory of wave fields, including such questions as the relation of statistics to spin. Also considered here are works devoted to the bound states of electrons, in particular the Bethe–Salpeter equation for a system of two electrons. The final paragraphs of the book contain an extremely useful mathematical supplement. In it, in particular, methods are presented for calculating integrals over a finite relativistically invariant region, and a summary is given of the most frequently encountered special functions in applications.

The book is not free of certain shortcomings. Thus, for example, the question of bound states (§§ 54, 55) is presented with insufficient clarity. One would also like the content of § 27 not to be limited only to a formulation of the basic ideas underlying the renormalization of the mass and charge of the electron, but to be supplemented by a more detailed analysis proving the possibility of eliminating all infinities that appear in any approximation of perturbation theory.

Finally, it would have been desirable for the book to reflect the results of recent works devoted to deriving exact equations for Green’s functions encountered in quantum electrodynamics. The development of quantum electrodynamics in recent years has proceeded, however, so rapidly that some of the works mentioned appeared only after the monograph had been written. Therefore this last remark should be regarded not as an indication of a shortcoming, but as a wish that these questions be included in the book in a third edition.

The entire monograph is written on the basis of the latest achievements of the quantum theory of wave fields, from a unified point of view and with great completeness.

Its content is a creative reworking of an enormous number of original papers (including papers by the authors of the monograph), carried out recently and published in various journals. The appearance of this monograph is an event for Soviet theoretical physicists. It will make an important and major contribution to the creation and training of new cadres of theorists in the USSR.

A. A. Abrikosov
I. Ya. Pomeranchuk
I. M. Shmushkevich