Shift of Atomic Electron Levels and the Additional Magnetic Moment of the Electron According to the Latest Quantum Electrodynamics. Collection of articles. Translations and abstracts by V. I. Grigor'ev and N. P. Klepikov. Edited and with an introductory article by D. D. Ivanenko. Foreign Literature Publishing House. Moscow, 1950. Pp. 222. Price 14 rubles 50 kopecks.

The book under review is a collection of translated articles and abstracts devoted to the latest achievements in the field of relativistic quantum mechanics (chiefly those of its sections that concern quantum electrodynamics and the theory of the electron).

In order to form a clear idea of the meaning and significance of the works contained in the collection, one should recall the historical course of the development of relativistic quantum mechanics. Ordinary—nonrelativistic—quantum mechanics, mathematically based on the Schrödinger equation, despite a number of brilliant successes, proved unable to explain certain experimental facts, such as, for example, the fine structure of the spectrum of the hydrogen atom. Moreover, the very fact of spontaneous

the radiation of light by an excited atom could not be consistently explained within the framework of nonrelativistic theory, for it regards the electromagnetic field as something given externally; spontaneous emission, as we now know, is connected with the reaction of the field of the electron upon the electron itself. Further, the experimentally established existence of the electron’s own mechanical and magnetic moment received no explanation, and the idea of spin had to be introduced into the theory ad hoc, as a certain additional assumption.

These failures of nonrelativistic quantum mechanics were entirely natural and inevitable, since in all the cases listed the question concerned relativistic effects which, in essence, could not be encompassed by Schrödinger’s quantum mechanics. The need arose to create a quantum theory that would take account of the requirements of the special principle of relativity—the need to create relativistic quantum mechanics. The corresponding works appeared at the end of the twenties and the beginning of the thirties of our century; moreover, the very first steps of the emerging relativistic quantum theory were marked by considerable successes: a solution was obtained to the problem of spontaneous emission of light by atoms; the fine structure of the hydrogen spectrum was explained by the theory in complete agreement with the experimental data of that time; finally, the existence of spin proved to be a direct consequence of the relativistic quantum properties of the electron, described by Dirac’s equation. At the same time the mechanical moment of the electron’s magnetic moment, calculated theoretically, agreed excellently with the available experimental data.

Further investigations led to new successes of relativistic quantum mechanics; its seemingly most improbable consequence—the existence of the positron—was justified by experiment. Theoretical calculations of all the effects observed without exception in the interaction of electrons with the electromagnetic field, such as the photoelectric effect, scattering of light by electrons, bremsstrahlung radiation of light, the creation and annihilation of electron-positron pairs, and others, proved to be in excellent agreement with experiment. The cascade theory of showers in cosmic rays, created on this basis (mainly by Soviet scientists—Landau and Rumer, Ivanenko and Sokolov, Tamm and Belenky), succeeded in explaining a number of experimental facts and is now a reliable foundation for further investigations concerning more complex, non-electromagnetic processes in cosmic rays.

It would seem that the position of relativistic quantum mechanics is unshakable. However, this is not so. It should be noted above all that a number of fundamental questions, long posed before theoretical physics, did not find (and do not find) their solution in relativistic quantum mechanics. Why does the charge of the electron \(e\) (more precisely, the dimensionless quantity

\[ \frac{e^2}{hc}, \]

where \(h\) is Planck’s constant and \(c\) is the speed of light) have this value and not another? Why are elementary particles with a charge that is a multiple of \(e\) not observed? Why do the mass ratios of elementary particles have this value and not another? Relativistic quantum mechanics (just like the earlier theories) gives no answer whatever to all these questions. True, this circumstance by itself can hardly be regarded as a serious defect of the theory. It is bad, of course, that all the facts mentioned are in no way connected with the apparatus of the theory, but nevertheless one could reconcile oneself to this—considering that the solution of such questions lies beyond the limits of relativistic quantum mechanics and belongs to the competence of some future theory—if relativistic quantum mechanics itself were an internally consistent science. But in fact, apparently, this is not the case. Along with

brilliant successes, relativistic quantum mechanics, from the very moment of its birth, led and still leads to a number of absurd, physically meaningless results that testify to its imperfection (and perhaps to its internal inconsistency). One of such meaningless results is encountered in the very first problem with which quantum electrodynamics had to deal—in the theory of the spontaneous emission of light by atoms. Namely, an attempt to calculate the so-called “natural” width of a spectral line, caused by the finite “lifetime” of an excited state, led to the absurd conclusion of an infinitely large shift of the frequency of the emitted light (which in reality, of course, is not observed at all). It is curious, however, that alongside this the very width of the spectral line turned out to be quite correct, so that if it were possible to “forget” about the infinite frequency “shift,” everything would have been in order. Later it turned out that this state of affairs is characteristic of all cases, without exception, considered in quantum electrodynamics: when attempting to take into account more accurately the interaction of the electron with the radiation field, we arrive at infinite (i.e. physically meaningless) expressions for all those effects which, in the first approximation to the perturbation theory, were explained by the theory. Some of these infinities had already long been well known in physics. They occurred even in classical electrodynamics, in which, as is well known, the electromagnetic mass of the electron turned out to be infinite if the electron was regarded as a point charge. In quantum theory this difficulty with the infinite self-mass has remained in full measure. Moreover, quantization has led to new complications that have no analogue in the classical theory and are due precisely to the quantum character of the field. The point is that, as quantum electrodynamics shows, even in a vacuum, when there are no photons, the strength of the electromagnetic field is equal to zero only on the average; its exact value is not definite and fluctuates around zero. Owing to interaction with these so-called zero oscillations, the electron acquires additional energy, which formally turns out to be infinite (the so-called “transverse” self-mass; it is due to interaction with the transverse oscillations of the field). This last result—the infinite value of the self-energy—is, of course, physically meaningless and testifies only to certain defects of the theory. It is essential, however, that one cannot simply “wave it away,” for the very same zero oscillations of the field in a vacuum determine, in the final analysis, also the spontaneous emission of light by an excited atom.

Similar difficulties (the appearance of a formally infinite additional charge of the electron) also arise in the consistent quantum-mechanical treatment of the electron-positron vacuum.

A paradoxical situation was created, probably unique in the history of science. On the one hand, calculations of various observed effects, carried out in the first nonvanishing approximation of perturbation theory, lead to excellent agreement with experiment in all cases without exception. On the other hand, attempts to improve the calculation, to take into account more accurately and consistently the interaction of the electron with the radiation field, lead to the appearance of physically meaningless divergent expressions—again in all cases without exception. At the same time, the physical cause of the appearance of a number of infinities—the interaction with the “quantum vacuum” (i.e. with the zero oscillations of the field)—is the same as the cause determining a number of observed effects (for example, the spontaneous emission of light by excited atoms).

Seeing no ways to solve the problem that had arisen, theorists were compelled simply to ignore the infinite expressions that appeared, while огра-

limiting themselves only to the results of the first nonvanishing approximation of perturbation theory. It was hoped that the future theory would be able to deal with the processes occurring at high energies, and that the several anomalous formulas of electrodynamics which had now diverged would turn out to be finite and would entail only a small correction—lying beyond the limits of accuracy of contemporary experiment—to the results of the first approximation. This approach to the matter, justified to a certain extent by the good agreement of the theory’s predictions with experiment, proved rather fruitful. Thus, for example, A. A. Sokolov, Gaitler, and Wilson succeeded in constructing a consistent quantum-mechanical theory of the reaction of radiation in the scattering of mesons by nucleons (or of light by electrons).

From a fundamental point of view, however, this state of affairs was quite unsatisfactory, and even in the prewar and war years a considerable number of works appeared devoted to attempts to analyze and remove the difficulties that had arisen. Yet all of them were to a large degree “academic” in character, being in no way connected with experiment, whose accuracy at that time was insufficient for these purposes. It is therefore not surprising that these investigations did not lead to any positive results. As has always happened in such cases, the new impetus to the development of the theory was given by new facts, whose discovery became possible thanks to the appearance of new experimental techniques.

The matter still concerns the same phenomena with the explanation of which the development of relativistic quantum mechanics began—the fine structure of the hydrogen spectrum and the magnetic moment of the electron. With the aid of the very precise radio-spectroscopic methods of investigation developed in recent years, it was established that: a) the fine structure of the hydrogen spectrum is not fully described by Dirac’s theory (the levels \(2S_{1/2}\) and \(2P_{1/2}\), which theoretically should coincide, are in fact separated by an interval of \(1062\) Mc/s); b) the magnetic moment of the electron \(\mu\) is not equal to one Bohr magneton \(\mu_0\) (as is required by Dirac’s theory), but exceeds this value somewhat:

\[ \mu = \mu_0 + \mu_0 \delta, \]

where \(\delta = 0.0012 + 0.0002\).

Thus, for the first time experiment diverged from the predictions of the first approximation of relativistic quantum mechanics. The idea of an explanation of the level shift (advanced as early as 1938 by D. I. Blokhintsev) is very simple. As we have already said above, owing to the interaction of the electron with the zero-point oscillations of the radiation field, the energy levels of the electron in the atom are displaced (the frequency shift of the spectral line). Formally, in the contemporary theory this shift turns out to be infinite (which is connected with the formally infinite self-mass of the electron), and therefore until now it has simply been ignored. If, however, in a future theory still unknown to us, the displacement of the levels proves finite, then for different levels it will, generally speaking, be different, and the distance between the levels will change in comparison with that calculated from the ordinary Dirac equation. The question arises: is it not possible, still within the framework of the contemporary theory, to find the difference of the displacements of two different levels and, if it proves finite, to regard it as a real effect, while attributing the infinite absolute values of the displacements to the imperfection of our theory, which supposedly will be eliminated in the future? This device received the name of mass renormalization, for in practice it amounts to the fact that in the displacement of a level there is singled out a divergent part formally coinciding with the infinite self-mass of a free electron; it is then combined with the “mechanical” (“original,” appearing in Dirac’s equation) mass into the experimentally observed, finite mass of the elec-

tron \(m\). The final remainder from this subtraction operation is then interpreted as the observed effect.

In this way Bethe was indeed able to give a quantitative explanation of the shift of levels in the hydrogen atom. Similar ideas were also used in calculating the additional magnetic moment of the electron (Luttinger), and the theoretical value, equal to \(\mu_0 \dfrac{1}{2\pi}\dfrac{e^2}{\hbar c}\), proved to be in excellent agreement with the experimental data.

Thus, the interaction of the electron with the “quantum vacuum” turned out to be an essential and quite real effect. The problem of investigating this interaction came to occupy a central place in contemporary relativistic quantum mechanics.

It should be noted, however, that despite its outward success, Bethe’s work could hardly be recognized as convincing. The point is that the subtraction of divergent expressions is, mathematically, a completely ambiguous operation, capable of yielding any result; therefore the agreement with experiment is, strictly speaking, not something at which one should be surprised. Connected with this ambiguity there also arises a possible violation of the relativistic invariance of the calculation (despite the formally invariant form of the theory). There arose the necessity for a formulation of quantum electrodynamics, clearer than before, that would make it possible to control invariance at every stage of the calculations. Such a formulation was given in a number of works; many works were also devoted to the application of new methods to the solution of a number of concrete problems which until then had not yielded to satisfactory theoretical treatment. As a result there was created, in essence, a new branch of relativistic quantum mechanics, whose task is the study of the interaction of elementary particles with the zero-point oscillations of the electromagnetic and mesonic fields.

In the works of the new “vacuum” direction a number of successes have been achieved; however, it should nevertheless be noted that the basic fundamental difficulties of relativistic quantum mechanics still remain unresolved. These problems are handed down as a legacy to the “future theory”; the authors of the new direction see their own principal task only in developing unambiguous and at least formally invariant computational methods, which would make it possible to investigate radiative corrections while pushing into the background the difficulties connected with divergences.

Meanwhile, an indifferent attitude toward the solution of the fundamental problems of the theory does not pass without consequences, for if there are fundamental physical defects in it, they will inevitably sooner or later manifest themselves, despite any purely mathematical “regularization” refinements. And indeed, even the mathematically most perfect of all the foreign works in this field—the article by Pauli and Villars—does not in essence provide an invariant method of calculation within the framework of the existing theory, for, as the authors themselves note, they do not satisfy Laue’s theorem. This difficulty, in our view, has not yet received an entirely consistent resolution. There is reason to think that it is not accidental, but is caused by very deep reasons connected with the purely formal character of the invariance of modern quantum electrodynamics.

All that has been set forth gives grounds to believe that the problem of the interaction of the electron with the “quantum vacuum” is still far from solved. It is not clear, moreover, whether it can be completely solved within the framework of the contemporary theory. Nevertheless, even the very formulation of this problem is a serious achievement and is of great interest; it is no accident that works in the field of the “theory of the vacuum” now enjoy great attention among theorists. It is precisely works of this new direction that the reviewed collection is devoted to. Opening with an introductory article by the editor, the collection contains 10 fully translated articles and 82 abstracts of experimental

in theoretical papers. Translated in full are: Weisskopf’s article “The Latest Development of the Theory of the Electron,” containing a survey of the development of electrodynamics from Lorentz up to the very latest time; three experimental papers devoted to the discovery and study of the shift of levels in the hydrogen atom and of the additional magnetic moment of the electron; the above-mentioned theoretical papers by Bethe and Luttinger; the articles by Dyson (a survey of new methods in quantum electrodynamics) and by French and Weisskopf (an investigation of the level shift by earlier methods); and, finally, the work of Källén and Villars, “On the Invariant Regularization in Relativistic Quantum Theory,” which, in mathematical terms, is perhaps the most perfect of all the foreign works in this field. The abstracts cover a very wide range of literature on the theory of the “quantum vacuum” and related questions, beginning with the old works of Dirac and Heisenberg and ending with 1950. The selection of articles for translation has been made, in our opinion, very successfully; it is a pity only that Dyson’s second article (“The \(S\)-Matrix in Quantum Electrodynamics”) has not been translated. Surprisingly, this very important work has not even been abstracted in the collection. In our view, it would also have been advisable to translate (and not merely abstract) Welton’s work “Some Observable Effects Due to Fluctuations of the Electromagnetic Field,” which compares favorably with most works of the new trend by its physical “transparency” and absence of formalistic aspirations. Finally, one would like to see translated Feynman’s works on the new formulation of quantum mechanics and quantum electrodynamics, which, in the reviewer’s opinion, are the most interesting and profound of all the foreign works in this field.

The quality of the translations and abstracts is, on the whole, satisfactory; however, in one place a substantial error has nevertheless crept in: in abstract No. 56 (devoted to Snyder’s work “On External Vacuum Polarization”) it is stated (p. 204) that \(a_k\) is the positron annihilation operator, whereas in fact this quantity is defined by Snyder as the positron creation operator.

The introductory article characterizes very clearly and in detail the conceptual side of the new trend and sets forth the history of its development. It gives due recognition (as do the editor’s notes in the text of the articles) to the role and significance of the works of Soviet scientists in this field. In the reviewer’s opinion, however, the article does not sufficiently emphasize the formal character of many achievements of the “vacuum” trend. The fact that the fundamental difficulties of the theory still remain unresolved is noted in the introductory article, but, in our view, is not sufficiently emphasized there.

The shortcomings noted, however, do not in any serious way diminish the value of the collection under review. It contains rich and well-systematized material and will prove very useful both to theorists working in this field and, in general, to physicists wishing to acquaint themselves with the latest development of relativistic quantum mechanics. One would wish publications of such collections to continue in the future.

V. Bonch-Bruevich