P. A. M. DIRAC. Principles of Quantum Mechanics (Oxford University Press, 1930).

P. A. M. DIRAC. Die Prinzipien der Quantenmechanik, Deutsch von W. Bloch, Verlag S. Hirzel, Leipzig 1930.

P. A. M. DIRAC. Principles of Quantum Mechanics.

After several years of very rapid development, quantum mechanics has finally reached a state of relative completion. Its basic ideas, which at first seemed extremely abstract and paradoxical, have become familiar and customary; at the same time, the limits of applicability of the theory, as well as its fundamental ...

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difficulties that prevent it from crossing these boundaries; in other words, a period of crisis has set in. There is nothing surprising in the fact that theorists have felt the need to look back over the path already traveled, to sum up the results, and carefully to analyze the fundamental principles of the theory, in order to see in what direction it is necessary to move further. Dirac’s book on the principles of quantum mechanics is an expression of this need. It is known that the difficulties that have arisen before the theory are connected with the necessity of constructing a relativistic theory of quanta, i.e., of reconciling quantum mechanics with the principle of relativity. There can be no doubt that the fundamental propositions of quantum mechanics are correct for electrons and protons moving with velocities small in comparison with the velocity of light; in this region no further surprises are to be expected; phenomena not explained, or insufficiently explained, by quantum mechanics (for example, the phenomenon of superconductivity) present difficulties not because the basic principles of the theory prove to be incorrect, but because the conditions under which these phenomena occur are too complicated for analysis. It is another matter with phenomena in which the velocity of light plays a role (for example, the β-decay of radioactive elements): for the study of these phenomena a radical change in the principles of the theory is required, i.e., a new scientific revolution. The experimental material on the basis of which this expansion of the theory is to be carried out is extremely meager; therefore a very important role is played by the examination of quantum mechanics itself, with the aim of identifying its most vulnerable points, which become apparent in attempts to reconcile it with the principle of relativity. Even if such an investigation cannot clarify exactly what the new theory should look like, it can at least clarify what it must not look like.

Dirac set himself this task. Of course, his intention was not merely to write a good textbook of quantum mechanics; a book written by one of the most outstanding leaders of physical theory must be of interest not only to students but also to specialists. Dirac devoted the first five chapters of the book to the most general principles of the theory; these principles contain everything that distinguishes quantum theory from classical theory, namely the principle of superposition of states and the rules that make it possible to calculate the probabilities of various results of measurements performed on a physical system in a given state. The principles set forth in these five chapters are, in Dirac’s opinion, in agreement with the theory of relativity: they are, as he maintains, so general that they cannot depend on any special theory of space and time. In the following seven chapters Dirac expounds the application of these fundamental principles to special physical systems, whose mechanics can in the first approximation be treated as nonrelativistic; finally, the last (XIII) chapter of the book is devoted to the relativistic theory of one electron in a given electromagnetic field; the difficulty connected with the appearance of negative

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energy in this theory, leads Dirac to his well-known paradoxical ideas on the nature of protons. These ideas, in Dirac’s opinion, are supposed to remove the difficulties that have arisen before the relativistic quantum theory of the electron. Such is the general scheme of Dirac’s book; it is connected with Dirac’s notions of the possible paths of development of quantum theory. Recently it has become more and more clear that these notions are deeply mistaken. The analysis of the uncertainty principle in quantum mechanics, carried out by L. Landau and R. Peierls (Z. Physik. 69, 56, 1931), shows that in the relativistic theory of quanta such concepts as the momentum of the electron, its coordinates, energy, etc., must lose their meaning; in quantum mechanics for all these quantities there holds the so-called “principle of the possibility of predictions” (das Voraussagbarkeitsprinzip): this means that among the possible states of a system there necessarily are such states in which the energy (or momentum, or coordinate, etc.) has a perfectly definite value; in the relativistic theory of quanta, however, this principle ceases to be valid; in other words, under no conditions can, for example, the coordinate or the momentum of an electron, etc., be exactly measured, and this means that the very concepts of these observable quantities lose their exact meaning. As Pauli jokingly put it: “Die Observable ist eine Grösse, die man nicht messen kann”; the uncertainty principle of ordinary quantum mechanics is therefore defined too narrowly for the relativistic theory of quanta. From the point of view of these ideas, the principled doom to failure of all attempts to reform quantum mechanics without breaking with its fundamental principles, such as the principle of the possibility of predictions, becomes quite obvious (although this principle does not appear in Dirac’s book under this name, it is nevertheless a very essential part of Dirac’s concept of the “observable quantity”); “conciliatory” and doomed to failure are both those attempts to construct a new theory which rely on additional physical ideas such as Dirac’s “holes,” and those attempts in which the way out of the crisis is sought not in the direction of transforming the basic concepts, but in the direction of constructing a new special form of the wave equation (for example Schrödinger, Zur Quantendynamik des Electrons, Berl. Ber. 63, 1931). From this point of view, Dirac’s book, as an attempt at a logical analysis of the quantum theory with the aim of clarifying the possibilities of its further development, must be recognized as unsuccessful. This is connected with the insufficient attention that Dirac paid to Heisenberg’s uncertainty principle; at the same time he did not at all touch upon the question of the relation between the accuracy of measurement and the interval of time during which this measurement is made, and even admitted several erroneous assertions concerning the possibility of repeating measurements.

We have dwelt here in such detail on the shortcomings of Dirac’s book because they strike the reader’s eye much less than its completely

indisputable merits. Chief among these merits is simplicity. Comparing Dirac’s book with such expositions of quantum mechanics as, for example, the books by Weyl or by Born and Jordan, we begin to understand what a genuine physics book should be like, one in which a simple and clear physical result is not obscured by the pedagogical constructions of the mathematician. In this sense Dirac’s book is the best of the existing expositions of quantum mechanics. Dirac nowhere makes use of a cumbersome mathematical apparatus; for the purposes of exposition he introduces a special algebra, the symbols of which directly denote such physical concepts as the states of a system, observable quantities, etc., without any geometrical illustrations such as vectors in Hilbert space and linear transformations in it. It is true that no formal advantage is thereby achieved (Dirac’s assertion that the usual illustration of quantum-mechanical concepts by vectors and linear transformations communicates to quantum mechanics a greater number of properties than is necessary is incorrect), but the enormous advantage of a pedagogical character consists in the fact that Dirac’s symbolic algebra is far more vivid and less artificial than the geometrical scheme of the Göttingen school. The chapters containing concrete applications of the theory are written with unusual clarity, although somewhat concisely; such chapters, for example, as Chapter IX (perturbation theory) or Chapter XI (systems consisting of identical particles), are, in respect of their pedagogical merits, a model that cannot be surpassed. The exposition of quantum mechanics in Dirac’s book completely refutes the legend that modern theoretical physics is some dense thicket of mathematical formulas; all this proves to be “from the Evil One”: throughout the entire book the reader will not find the slightest traces of contrived erudition or pedantry.

M. Bronstein.