Bibliography

Gregor Wentzel. Einführung in die Quantentheorie der Wellenfelder.

Gregor Wentzel: “Introduction to the Quantum Theory of Fields,” Deuticke, Vienna, 1943.

If public opinion among all cultured humanity has called our time in science the atomic age, professional scientists, first of all, continue to speak more precisely of the epoch of nuclear physics; then, from the theoretical standpoint, one should above all emphasize the role of research into the entire system of elementary particles—this cornerstone of modern physics. It is known that both the fundamental and the concrete laboratory—so to speak, personal—boundaries between nuclear and cosmic physics are highly conditional. Obviously, in the future the everyday colloquial designations “nuclear physicists” (or rather “nucleotechnicians”) and “cosmic-ray physicists” will have to be generalized in some way.

An outward, convincing sign of the formation of the doctrine of elementary particles is the appearance of surveys and books such as the empirical monograph Particles of Modern Physics by Stranathan, Pauli’s Relativistic Theory of Elementary Particles (a review in Reviews of Modern Physics, vol. 13, No. 3, p. 203, 1941), and finally Wentzel’s book Introduction to the Quantum Theory of Fields, which is the subject of the present review. From the moment the first books appeared, one might apply to the theory of elementary particles a peculiar definition which the well-known Odessa mathematician-logician Shatunovsky gave in his lecture: “Higher algebra,” he said, “is called an object set forth in these two books,” while the professor demonstrated two volumes of Weber.

Of course, we are still far from a fully completed theory of particles arranged into some “natural system,” and we are compelled to be satisfied with a somewhat disordered “table,” in which a very great deal is lacking. Who knows, perhaps we are placing the question of Mendeleev’s natural system of elements in the first decades of the nineteenth century, when the first relations among metals, halogens, and so forth were being outlined.

Wentzel’s book consists of 6 chapters. Chapter 1 (pp. 1–28) contains the general foundations of the theory of arbitrary relativistic quantum field equations. Here the canonical formalism is presented, conservation laws and questions of relativistic invariance are discussed, and the transition to momentum space is made. The variational principle is naturally placed at the head of the entire theory. The corresponding Lagrangian function makes it possible to construct the energy-momentum tensor of the field, as well as the current vector—the charge density generating the electromagnetic field. Starting from, although by the usual and somewhat cumbersome route, Wentzel introduces the fundamental Pauli function \(D\), which determines the four-dimensional commutation rules.

Despite, generally speaking, the clear and in places very elegant exposition of this chapter, as of the entire book, it seems to us necessary to supplement the discussion of the general foundations of the theory with the following points:

a) The connection of conservation laws with invariance under relatively different groups of transformations in the spirit of Noether’s theorems.

b) The connection of the usual canonical energy–momentum density tensor with the Hilbert metric one.

c) The theory of spin moment, according to Belinfante.

d) More detailed investigations of both invariant solutions of the de Broglie equation* of the type of Pauli functions \(D\) and \(D_1\)** and an indication of their connection with the Green function. Unfortunately, in the book there is no complete exposition of the theory of the Dirac delta-function, the four-dimensional Green function, and other analogous questions (in a monograph by A. A. Sokolov, now being prepared for print, and the present author, an attempt is made to fill this gap. See our note in the “Bulletin of Moscow University,” No. 1—in print).

e) It is necessary to give a detailed exposition of Pauli’s theorem on the connection of spin with the type of statistics and the definiteness of energy (this point is set forth quite briefly by Wentzel in an additional chapter).

The remaining chapters are devoted to fields of spin \(0,1,\dfrac{1}{2}\) and to particles of higher spin. The theory of scalar fields, i.e. spinless particles, is very successfully and thoroughly presented in Chapter II (pp. 29–70). Here the classical theory of the real field of neutral and the complex field of charged particles is considered, and their secondary quantization is analyzed in detail (according to Pauli and Weisskopf). Questions of the scattering of neutral and charged spinless particles and of nuclear interactions mediated by spinless (neutral and charged) mesotrons (Yukawa’s theory) are also examined. The formation of pairs of spinless mesotrons is briefly analyzed.

Chapter III is devoted to the general theory of a complex vector field, i.e. the Proca equation for charged particles of spin “one” (pp. 71–106). Two paragraphs—§ 14 and § 15—are devoted to the theory of mesotron vector nuclear forces. We would have liked, above all, to supplement Wentzel’s exposition with the general theory of neutral vector mesotrons. As is known, such “classical mesodynamics,” developed especially by Bhabha, Sokolov, and Ivanenko, not only is acquiring ever more real significance in view of the recent, apparently final discovery of neutral mesotrons***, but also serves as an excellent heuristic method for developing the entire theory, allowing, for example, a very simple derivation of the general formula of nuclear forces, taking account of the influence of damping in scattering, etc.

Questions of the theory of nuclear forces are set forth too briefly. The difficulties of the theory and the ways of eliminating them are analyzed far from sufficiently. Here the chief shortcoming of Wentzel’s book is evident—its remoteness from the empirics of the nucleus and of cosmic rays. In this sense the well-known book by Heitler on the quantum theory of radiation stands much higher. Chapters II and III should be supplemented by the theory of pseudoscalar and pseudovector particles, with an indication of the possible role of pseudoscalar mesotrons. It is desirable to give an exposition of the matrix theory of all spin \(0,1\) equations according to Kemmer–Duffin.

Chapter IV is devoted to quantum electrodynamics, or Maxwell’s equations, which are a special case of the Proca equations when equality

*) In accordance with a strange, rather widespread tradition, Wentzel calls the fundamental equation, discovered by O. Klein and constituting the relativistic generalization of the Schrödinger equation, the Schrödinger–Gordon equation (?). In accordance with a recent proposal by Schrödinger himself, we denote it as the de Broglie equation, although the equation itself is not found in de Broglie.

**) See, for example, the recent works of D. I. Blokhintsev.

***) See G. Frechtinger, Lloyd-Smith, and Krieger, a series of notes, Phys. Rev., 1945.

to zero the particle’s proper mass (pp. 107–175). This chapter contains the highlight of the entire book in the form of a detailed exposition of the many-time formalism (recently generalized (sic!) from quantum theory to the classical one by M. A. Markov) and of the theory of Wentzel’s so-called “lambda limiting process.” This ingenious (and rather enthusiastically taken up, for example by Pauli) formalism has so far proved capable only of eliminating a certain class of infinities in the problem of the proper zero mass, and has stimulated Dirac’s introduction of negative probabilities in order to eliminate another class of infinities: the idea of the “lambda process” consists in replacing the electromagnetic forces acting, for example, on the electron, etc., at the time \(t\), by a half-sum of the values at a somewhat earlier moment \(t-\lambda\) and a somewhat later moment \(t+\lambda\). In the final result \(\lambda\) is set equal to zero and may quietly leave the stage, like the Moor who has done his duty. Then, in particular, it turns out that the zero statistical mass of a point electron not only does not, as usual, go to infinity, but turns into zero. This theory has not yet brought any changes to the difficult situation with nuclear forces, or to other more or less concrete questions connected with empiricism.

Of course, one must treat with attention any formally irreproachable method that removes even part of the infinite difficulties. However, we believe that a sound solution of these difficulties will most likely come from the theory of the nucleus and cosmic rays, fertilized by rich empirical material.

In Chapter V (pp. 158–191) Wentzel considers the Lagrangian principle of the Dirac equation for particles of spin \(\frac{1}{2}\) (electrons, nucleons) and its second quantization. § 21 contains a good concise exposition of the foundations of the theory of the vacuum and of the nonlinearities in electrodynamics induced by the creation and annihilation of particle pairs.

The supplementary Chapter VI (pp. 192–204) contains a brief exposition of the theory of particles of higher spin (specifically spin 2). As Pauli and Fierz have shown, particles comparable with a weak gravitational field—gravitons—possess spin 2. Too briefly set forth here are Pauli’s theorems, stating that particles of half-integer spin obey Fermi–Dirac statistics, while particles of integer spin must obey Bose–Einstein statistics, and that in this case the energy of the particles and the density of their charge satisfy certain positivity conditions.

In view of their profundity, generality, and empirical significance, these Pauli theorems, which may be counted among the greatest successes of the general theory of elementary particles, should be presented in greater detail and assigned to Chapter I.

Thus, on the whole, Wentzel’s book gives a good account of the general foundations of the modern theory of elementary particles and fields, emphasizing sufficiently its greatest successes: the prediction and theory of the positron, the theory of nuclear forces, the prediction and theory of the mesotron, second quantization and the theory of the emission and absorption of particles, and Pauli’s theorems.

Of course, a speedy translation of Wentzel’s book into Russian is highly desirable. It is necessary only, if the book is not supplemented at the points listed above, at least to provide the translation in these places with exhaustive guiding explanations and to enlarge considerably the list of cited literature, which, as always up to now, is too meager, especially in the section relating to Soviet authors*).

D. Ivanenko

*) Of Wentzel’s recent works, he cites only the quantum theory of the many-time formalism and the foundations of the theory of paired nuclear forces.