L. de Broglie: Théorie générale des particules à spin (méthode de fusion). Gautier Villars, Paris, 1943, p. 197.

Louis de Broglie. General Theory of Spin Particles (Fusion Method).

The central problem of contemporary physics, as is known, is the study of the general theory of elementary particles and fields, of which, by all indications, matter consists in the part of the universe accessible to investigation during the period of time studied. In view of the relative scarcity, or even absence,

the behavior of elementary particles and fields is in most cases governed by quantum mechanics, since the products of the “action” over the periods of the processes do not greatly exceed Planck’s quantum of action. In addition, the diverse transformations of particles and fields into one another, which occur predominantly at low velocities—for example, the transmutation of an electron-positron pair into photons—and the very frequent cases of motion with velocities close to the limiting one, that of light, require a description of particles on the basis of the theory of relativity. Thus the theory of particles must be relativistic quantum (r. q.) mechanics. However, at the same time, both nonrelativistic quantum mechanics (Schrödinger’s equation) by itself and the nonquantum relativistic theory by itself are sound theories, guaranteed within known limits, excellently describing the most subtle regularities; the construction of a unified r. q. theory is far from complete, despite the enormous successes achieved in this direction over the last 20 years. These successes relate above all to the construction of systems of equations describing particles and fields with various wave and spin properties, to the interpretation of kinematic magnetic moments, the type of statistics, to the theory of secondary quantization and the general theory of the interaction of particles through fields and particles. In developing r. q. theory, it has been possible, as is known, to predict the positron and the mutual transformations of electron-positron pairs into photons and to predict the meson. The general, so to speak, formal system of relativistic quantum theory, proceeding usually from a variational principle with a corresponding Lagrangian function and successively yielding the energy tensor, current vector, and all other fundamental quantities characterizing a field, has achieved great perfection. An exposition of this, so to speak, “official” theory may be found in the books by Wentzel (Introduction to the Quantum Theory of Wave Fields, Moscow–Leningrad, GTTI, 1948) and Pauli (Relativistic Theory of Elementary Particles, Moscow, GIIL, 1947).

Despite all these achievements, the impossibility of getting rid of the difficulties with the infinite energy of fields (electromagnetic, gravitational, mesonic, etc.) produced by any point particles, and a number of other difficulties close to this, do not allow r. q. mechanics to be regarded as a completed theory. Most painfully, perhaps, one feels the actual impossibility of constructing a theory of nuclear forces between nucleons on the basis of the conception of the realization of these forces by means of meson fields of any reasonable kind. It is not excluded, however, that only some part of this difficulty of interaction is due to the general shortcomings of r. q. theory; the principal share is connected with still insufficient empirical information about the mesons themselves, which carry the forces. In any case it is clear that the basic problem of the nature of the intrinsic mass of particles (connected, from the standpoint of field hypothesis, with the question of the energy of the fields produced by the particles), as well as the problem of the existence of various kinds of particles, can be solved only after further careful study of mesons. The discovery by the Alikhanov brothers of an entire spectrum of meson masses (varitrons) and the discovery by Powell’s group of the transition of some mesons into others are important new pieces of information about elementary particles, clarified in the last two years and graphically illustrating how premature were the attempts to understand the nuclear field on the basis of previously insufficient data.

In such a situation it is not surprising that some theorists continue to develop the general system of relativistic quantum mechanics and to apply it to various concrete problems. Other theorists (pessimists regarding the existing theory) have concentrated attention on the search for new radical means of eliminating the difficulties (nonlinear theories, equations with higher derivatives, quantized spaces, etc.). The author of the book under review, devoted to rela-

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quantum mechanics, mainly from the point of view of spin properties, stand on an essentially “optimistic” position and suppose, more or less explicitly, that the theory of elementary particles can in the main be developed on the basis of the existing formalism. Therefore in his book de Broglie touches only briefly on particular difficulties of the theory, mentioning, for example, in passing on p. 194 nonlinear generalizations and Born’s electrodynamics, etc. It must be noted that in recent years the “optimists” have managed to score a major success by explaining the additional non-kinematic magnetic moment of the electron by taking account of its interaction with the zero-point oscillations of the electromagnetic field, necessarily introduced by relativistic quantum mechanics and previously considered subject to discarding. De Broglie in his book does not merely set forth and analyze Dirac’s equation, Maxwell’s equations, and others, but gives an original method for treating fields of any spin, starting from the Dirac equation for particles of spin \(1/2\). In this lies the chief interest of his book, which continues the author’s previous investigations in the monograph The Wave Mechanics of the Photon (in French). At the same time the present book is, to a certain extent, a continuation of his course on Dirac’s theory, The Magnetic Electron, which has also appeared in Russian translation.

The first part of the book (pp. 1–91) is devoted to an exposition of quantum mechanics and Dirac’s theory (ch. 1, “Fundamental ideas and general equations of quantum mechanics”; ch. 2, “Physical interpretation of wave mechanics”; ch. 3, “Quantum mechanics of kinetic momenta (i.e., angular momenta) and spin”; ch. 4, “Proper kinetic moments from the relativistic point of view”; ch. 5, “Dirac’s theory of the spinning electron”; ch. 6, “The general formalism and physical interpretation of Dirac’s theory”).

Setting forth, in his usual clear form, the well-known foundations of the theory, de Broglie gives a number of original remarks on the properties of angular momenta. The exposition of Dirac’s theory cannot replace its complete theory, including the variational principle and the analysis of the algebra of Dirac matrices. Moreover, here and in what follows de Broglie does not touch on second quantization and the theory of the interaction of particles and fields. The impossibility of dispensing without the variational Lagrangian principle is especially clear in de Broglie’s treatment of the energy tensor (more precisely, the density of the energy–momentum stress). Indeed, the question of the symmetry of this quantity (p. 127 and others) can evidently be satisfactorily analyzed only when both the canonical and the metric energy tensors are taken into account and when the proper spin moment is analyzed. Otherwise one has to resort to little-satisfactory additional symmetrizations. Unfortunately, this circumstance has not yet found due reflection in any exposition of classical or quantum field theory. De Broglie also does not emphasize sufficiently clearly in his book the presence of pseudoscalar and pseudovector quantities in the theory, although he himself analyzes a number of subtle questions. A shortcoming of de Broglie’s exposition is also the absence of a clear indication that the spin of a field is characterized by a tensor of rank 3 (see, for example, Pauli’s book and our work \(^1\)).

The second part of de Broglie’s book, in which its chief interest is contained, is devoted to the theory of particles of spin 1, 2, and higher spin, treated by the method of “fusion” of particles of spin \(1/2\) (ch. 7, “Theories of particles of spin 1—photons, mesotrons...”; ch. 8, “Wave mechanics of particles of maximum spin,” continuation; ch. 9, “General theory of spin particles obtained by ‘fusion’”; ch. 10, “Study and nomenclature of spin states; tensor quantities and ch. 11, “Theory of particles with maximum spin 2.” In the very brief form given here, it is a question of taking the wave function of a “complex” particle of higher spin in the form of a product

wave functions of the “component” particles of lower spin: \(\psi=\psi_1\psi_2\). Then in fact from two Dirac equations for the components de Broglie obtains, taking into account that \(\psi_1\) and \(\psi_2\) must correspond to the same value of the energy and momentum \(\left(\psi_1\,\dfrac{\partial\psi_2}{\partial t}=\psi_2\,\dfrac{\partial\psi_1}{\partial t}\ \text{etc.}\right)\), equations for particles of higher spin (see pp. 97–99 and others).

The wave function \(\psi\) will describe not a particle of some definite spin 0 or 1, but a “particle of maximum spin 1,” which can be in two spin states, 0 or 1. In the end, for the state 1 de Broglie obtains the Proca equation (p. 108), i.e. Maxwell’s equations in vacuum, supplemented by a term with mass, and for the spin state 0—equations close to the pseudoscalar ones (p. 108). All these arguments can be generalized to higher spins and, in particular, used to construct a theory of particles of maximum spin 2. In the absence of rest mass, particles of spin 2 obey the same linear equations as the weak gravitational field, as Pauli and Fierz noted. Let us recall that this circumstance led further to a deciphering of the quadrupole character of the weak gravitational field, which makes it possible, with great simplicity, to develop its theory by directly obtaining important formulas for the magnitude of the radiated energy, for the force of gravitational radiation damping, etc.^1)

In de Broglie’s theory, particles of spin 2 naturally stand alongside particles of spin 1 and spin 0. In this circumstance de Broglie attempts to discern some unification of the weak field of gravitation (spin 2), electromagnetism (spin 1), and some other obscure “spinless” field (spin 0, a “non-Maxwellian” field in de Broglie’s terminology, p. 188). It is doubtful whether these latter considerations have any physical meaning. It is just as doubtful that an exposition of the theory should speak of actually “complex,” somehow not quite elementary “corpuscles” (analogous to “molecules,” in de Broglie’s words, p. 95), composed by a peculiar “fusion” (fusion) from what are in fact elementary “particles” (so to speak, “atoms”). Similar ideas, as is known, underlie the unsuccessful, or at least little-recommended, hypothesis of the neutrino theory of light, proposed by de Broglie himself (see also the cited book by Pauli, note on p. 62). Thus we are compelled to note a certain separation of de Broglie from the newest empirical material of elementary-particle physics, which considerably diminishes the chances of success of his attempts to construct a general theory.

Be that as it may, de Broglie’s method of fusion nevertheless makes it possible to arrive simply at a theory of particles of higher spin and then to separate states with individual spin values, for example 2, 1, 0, and from this point of view it deserves attention. The discovery of new elementary particles, the number of which is obviously far from exhausted, as well as the not excluded possibility of the existence of excited states of particles possessing higher spin, all these circumstances make the development of a theory of higher spin desirable. In recent years an increase has been observed in the number of works in this field by both Soviet and foreign authors. Therefore we would consider it desirable to acquaint Soviet physicists with the original exposition of de Broglie’s theory, perhaps preferably by translating, very incompletely, we repeat, chapters 2–3 of the book under consideration, if the question of its complete translation is not raised. Such a translation, supplied with some commentary, would supplement the picture of the present state of relativistic quantum theory. In conclusion it should be noted, unfortunately rarely, the absolute sil—

^1) D. Ivanenko and A. Sokolov, DAN. 58, 1633, 1947. “Classical Field Theory,” Gostekhizdat, Moscow–Leningrad, 1949.

…the suppression by de Broglie of Soviet authors, despite his use of their work. In this book there is no mention, for example, of Frenkel’s theory of the electric moment of the electron (along with the magnetic one) and of all Soviet works on Dirac’s theory and on gravitons. In this connection it must be mentioned that, in his recent two-volume monograph on nuclear theory, de Broglie took the worst path of directly transferring the priority and authorship of our Soviet works on the theory of the nucleus and nuclear forces to foreign physicists, including even persons who expressed their ideas in this field not in print but in private correspondence. Does not a certain deciphering of such behavior by de Broglie lie in the fact that some of his works were published even in the time of the Vichy government? In any case, de Broglie’s behavior serves as yet another good illustration exposing the myth of the nonpartisanship of science, of some single world science, etc. On the other hand, we know that such an attitude toward Soviet physics by no means characterizes all of French science, which has shone in the last decade with the names of advanced scholars, courageous fighters against fascist obscurantism, such as Langevin and the Joliot-Curies.

D. Ivanenko