BIBLIOGRAPHY

L. I. Mandelstam, Collected Works, Vol. V, edited by Acad. M. A. Leontovich. Publishing House of the Academy of Sciences of the USSR, 1950. Pp. 467. Price 26 rubles.

The fifth volume of the collected works of Academician Leonid Isaakovich Mandelstam contains lectures and seminars relating to the period 1932–1944. Although the author did not manage during his lifetime to review and edit the notes of his lectures, one may nevertheless think that, thanks to the careful recording made by his pupils, the printed text conveys the content of the lectures with sufficient fullness and accuracy.

In these lectures and seminars those general questions are touched upon which most interested L. I. himself and to the correct understanding of which he attached great importance.

The main part of the fifth volume consists of lectures on selected questions of optics (1932–1933), on the physical foundations of the theory of relativity (1933–1934), and on the foundations of quantum mechanics (1939).

These lectures were not intended for an initial acquaintance with the subject, and in them L. I. did not set out to give a systematic exposition of it. L. I.’s aim was different. Having in mind listeners already acquainted with the subject to the extent of an ordinary university course, L. I. wished to illuminate individual selected questions that have fundamental significance and present, despite their simplicity, the greatest difficulty for understanding. The very choice of these questions, the brilliant form of exposition, and the subtle analysis of physical concepts are characteristic of the great scholar that L. I. Mandelstam was. Questions are often posed in the form of paradoxes.

Thus, considering in his lectures on optics the question of the modulation of light and of the reality of monochromatic components of a modulated oscillation, L. I. draws attention to the following circumstance. A discontinuous oscillation can, as is known, be represented in the form of a superposition of monochromatic continuous oscillations. An instrument capable of perceiving a separate monochromatic oscillation will oscillate continuously, i.e., even when no oscillations are arriving from outside. From where, then, does it draw its energy? L. I.’s analysis shows that such an instrument must necessarily possess inertia (a delay time), which fills the gaps between the incoming impulses. This example serves L. I. Mandelstam to illustrate the general proposition that the behavior of a physical object (in the given case, light) should be studied not abstractly, but in relation to the instrument investigating this object.

In the lectures on optics, diffraction by a slit is analyzed in detail, and on the basis of the approximate theory obtained, a number of questions of wave optics and quantum mechanics are considered. In particular, the important role of waves attenuated in space and coming from the slit is elucidated, and it is shown that neglect of these waves can also lead to paradoxes.

Very interesting are the author's discussions on the question of the uniqueness of the Umov–Poynting vector. If one replaces the usual expression

\[ \mathbf{S}=\frac{c}{4\pi}[\mathbf{E}\times\mathbf{H}] \]

for this vector by \(\mathbf{S}'=\mathbf{S}+\operatorname{rot}\mathbf{A}\) (where \(\mathbf{A}\) is an arbitrary vector), then the flux of the vector \(\mathbf{S}\) through a closed surface will not change. On this basis the author concludes (p. 22) that, so long as the vector \(\mathbf{S}\) is interpreted only as an energy flux, it cannot be determined uniquely. It seems to us, however, that this reasoning is incomplete, since it does not take into account the connection, following from the theory of relativity, between the energy flux and the density of momentum (this connection is mentioned only on p. 148, in the lectures on the theory of relativity). Moreover, the author assumes that the energy density is in any case given; but if it is given in all reference systems, then from this not only the energy flux (and the proportional density of the amount of motion) is uniquely determined, but also all the other components of the energy tensor. The author's interesting discussions are not carried through to the end in this way, and it remains unclear to the reader whether, in the final analysis, the vector \(\mathbf{S}\) is determined uniquely or not.

The next large section of the book consists of lectures on the theory of relativity. Of the fourteen lectures, the first six are devoted to a historical survey of the electrodynamics of moving media, the development of which led to the theory of relativity. This survey can truly be called brilliant. It is evident that the author not only has a deep knowledge of the history of the question, but also himself lived through all the stages in the creation of the theory of relativity. The survey gives a clear idea of what questions confronted physicists during the period when the theory of relativity was being created, and how great was the role of this theory in resolving the accumulated contradictions.

Beginning the exposition of the theory of relativity, the author focuses his attention on two facts representing a generalization of an enormous accumulation of experimental material: on the independence of phenomena from the unaccelerated motion of a closed system and on the independence of the speed of light from the speed of the source. The author shows that within the framework of the classical (pre-relativistic) theory these two facts lead to contradictions. Only the theory of relativity provides a way out of these contradictions. For its construction a deeper analysis of temporal and spatial relations is necessary, leading to new definitions of the concepts of space and time. The subsequent lectures are devoted to this analysis.

On the question of defining physical quantitative concepts, L. I. Mandelstam says: “The definition of fundamental concepts consists in the fact that I present a definite object, give a definite process, and by this object and process define the concept...” It seems to us that the presentation of an object and a process is not sufficient; in addition to them, and above all, it is necessary to give a theory that would encompass not only the behavior of the given object in the given process, but also reflect the general laws. Only under this condition will the definition not be artificial, but will correspond to nature with the accuracy with which the given physical theory is valid. In this connection it should be emphasized that definitions are just as little arbitrary as are the physical theories on which they are based.

The author then analyzes very clearly the basic and most difficult question of the simultaneity of distant events, and then proceeds to the derivation of the Lorentz transformations. In this derivation certain mathematical imprecisions are admitted, which, however, are not essential. On the basis of the transformations obtained, the author considers the change in the length of scales, the theorem of addition of velocities, and other questions, and then analyzes the concept of an invariant interval, quite correctly

emphasizing that the theory of relativity deals not only with relative quantities, but also with absolute quantities. In the last two lectures the author studies the group properties of Lorentz transformations.

Both the historical survey and the analysis of the basic physical concepts of the theory of relativity are written with extraordinary liveliness and appeal.

We turn to the review of the lectures on the foundations of quantum mechanics (the theory of indirect measurements). As in the preceding courses, the author here examines the principal and difficult fundamental physical questions. The author states from the very outset that in the field of quantum mechanics he does not undertake to make assertions with the same confidence as in the field of classical physics. Indeed, alongside extremely interesting and unquestionably correct ideas, the lectures also contain propositions that seem to us disputable; we shall note them below.

It should not, however, be forgotten that in 1939, when the lectures on quantum mechanics were delivered, certain questions were still so unclear that they were not even posed in explicit form. L. I. Mandelstam’s very formulation of these questions (for example, the question of the statistical collective to which the probabilities given by quantum mechanics refer) is a substantial step forward.

According to quantum mechanics, the maximally complete description of the state of a micro-object is effected by means of a wave function, which makes it possible to find the statistics of the results of measurements on an object that is (prior to the measurement) in the given state. Quantum mechanics is in this sense a statistical theory. L. I. Mandelstam draws attention to the fact that in any statistical theory it is first of all necessary to establish the concept of the collective to which the statistics refer. However, in the treatment of this question given by him there are, as it seems to us, obscurities. Suppose that over a micro-object in a definite state measurements are made of some quantity by means of one or another apparatus adapted for this purpose. Then we are indisputably dealing with a definite statistical collective. But on an object in the same state one can also make measurements of another quantity (which requires another apparatus), and then the statistical collective will already be different. Is it possible, in such a state of affairs, to connect the concept of a statistical collective with the concept of a state (with the wave function)? It seems to us that one cannot: for a given wave function the character of the statistical collective is not yet predetermined. The statistical collective is determined not only by the specification of the wave function, but also by the specification of the quantities measured (and of the corresponding measuring apparatus, which carries out the interaction with the object needed for the measurement). Meanwhile the author speaks (p. 356) of a micromechanical collective to which the \(\psi\)-function refers. With this obscurity another is also connected. In many places (for example, on p. 388) the author speaks of selecting subcollections from a certain aggregate of measurements. Meanwhile, in fact we are dealing with different collectives, obtained by applying different measuring apparatus, and one of the “subcollections” of which the author speaks belongs to one collective, while another belongs to an entirely different one. The indicated obscurities also tell in the discussion of the well-known dispute between Einstein and Bohr. The author’s conclusion (p. 389) that Einstein’s objection gives no reason for a revision of wave mechanics is indisputably correct, but the author’s words that the whole matter here lies in an incorrect application of probability theory remain not entirely clear, since it is unclear in what precisely this incorrectness consists. In our opinion, the essence of the matter lies in those specific features of quantum mechanics by virtue of which the specification of a state does not yet predetermine the collective to which probability theory must be applied.

In the theory of indirect measurements the author for the first time formulates the requirement that must be imposed on an apparatus suitable for the indirect measurement of a certain quantity pertaining to the system \((x)\). This requirement concerns the nature of the interaction between the given system \((x)\) and another system \((y)\), on which direct measurements are already possible. It is necessary that, by measuring the state of the system \((y)\) after the interaction, we should be able to infer the state of the system \((x)\) before the interaction. The character of the interaction must ensure this. Under the assumption of a one-to-one correspondence between the quantities pertaining to the systems \((x)\) and \((y)\), this requirement reduces to the following: after the interaction, the part of the wave function corresponding to the scattered wave must have the form of the product of a function of \(x\) by a function of \(y\). As indicated in the note on p. 367, and also at the end of the lectures (p. 401), the assumption of mutual uniqueness is not necessary and is introduced only for simplicity; if it is not introduced, then the simple product may be replaced by a sum of products of terms of a definite kind.

On the basis of the requirement thus formulated, a very interesting example of the indirect measurement of the momentum of a neutron is analyzed, as well as an example of a device for measuring the coordinate of a particle (a Heisenberg microscope). In the analysis of these examples use is made of results obtained in the lectures on selected questions of optics.

A further very interesting example is the measurement of the coordinate of a particle situated at the top of a potential barrier.

Despite the brevity of the course of lectures on the foundations of quantum mechanics (there are only five lectures), this course is exceedingly rich in content, and the ideas of L. I. Mandelstam set forth in it deserve further development, which might also remove the obscurities noted in the present review.

In addition to the courses of lectures mentioned, the fifth volume includes introductory lectures to a number of seminars on optics, the theory of oscillations, and other physical problems. In these introductory lectures, extraordinarily vivid and engaging, the problems that are to be examined at the seminar are enumerated, and each area of physics about which L. I. Mandelstam speaks appears before the reader as living, with all its current questions. Particular mention should be made of the lectures on questions of the theory of oscillations—an area in which L. I. was a profound expert.

The book as a whole is extraordinarily interesting and deserves the most attentive study.

V. Fock