H. WEYL, Gruppentheorie und Quantenmechanik, 2nd ed.
Verlag S. Hirzel, Leipzig, 1931.

H. WEYL. Group Theory and Quantum Mechanics.

The new edition of Weyl’s well-known book has been very substantially revised in comparison with the old edition; suffice it to say that instead of the 288 pages of the first edition we now have 366 pages. All the changes and additions pursue the aim of increasing the elementary character and pedagogical merits of the book, whose first edition had acquired the reputation of being a very solid and difficult work, at times inaccessible even to theoretical physicists. Weyl’s work, as is clear from the title itself, is not a purely physical book; the content of this large work consists chiefly of an exposition of the applications of group theory in quantum mechanics; but in order that both mathematicians and physicists may be able to read the book, the exposition of the applications of group theory in the theory of quanta

provided a brief exposition of the foundations of quantum mechanics itself and of group theory itself and their representations. Then follow applications of group theory: first the group of spatial rotations is discussed, and, in connection with it, the spectral terms of atoms and ions, the intensities of spectral lines, multiplets, and the Zeeman effect; next comes the Lorentz transformation group and the application of the properties of this group to Dirac’s relativistic equation for a single electron in a given external field. The last, very extensive, chapter of the book is devoted to the theory of representations of the permutation group in connection with the quantum-mechanical theory of systems consisting of identical particles; the aim of this chapter is the study of the spectral terms of an atom or ion with any number of electrons, taking into account the Pauli principle and the presence of electron spin. The exposition everywhere is distinguished by Weyl’s characteristic elegance; however, it cannot avoid reproaches for excessive refinement, even if one makes the appropriate allowance for the circumstance that Weyl’s book is not a physics book, but only a mathematical book occasioned by physics. Anyone who wishes to understand quantum mechanics will make a great mistake if he studies it for the first time from Weyl’s book; the book has an aesthetic character and therefore can be recommended only to a reader who is a mathematician, but not to a physicist (even a theoretician). For a long time it was thought that the application of group theory was absolutely necessary in a whole series of problems relating to complex atoms and to molecules; however, a clever method proposed by Slater (Phys. Rev. 34, 1293, 1929) makes it possible to avoid the use of group theory and to get by with a very elementary mathematical apparatus. In an article by Born (Z. Physik 64, 729, 1930) there is an application of this method to the theory of chemical forces; consideration of the full permutation group is rendered superfluous because from the very beginning only antisymmetric wave functions are introduced; of the properties of the rotation group it is enough to know a few very simple theorems on the properties of the orbital angular momentum and the electron spin. The physical results to which Weyl’s book leads the reader could have been obtained by a much shorter route; but if one takes into account the fact that the book was written by a mathematician for mathematicians, this ceases to seem such a great shortcoming of the exposition, if only because “he who walks will never make a hook.”

Among the physical questions included in the second edition of Weyl’s book, one should especially note the quantum electrodynamics of Heisenberg and Pauli, as well as the problem of the proton and electron. The construction of a relativistic form of quantum mechanics, as is well known, encounters great difficulties. Heisenberg and Pauli proved that a quantum theory of wave fields can be constructed in agreement with the principle of relativity; in doing so, however, the difficulty arises that the interaction of the electron with itself turns out to be infinitely large. Thus the problem of the electron remains unresolved. Noting

the fact that the quantized field equations are not symmetric with respect to both directions of time, Weyl concludes that “the distinction between the two electricities presents an even deeper riddle of nature than the distinction between past and future.” The exposition of these difficult questions, which show, in Weyl’s words, that “clouds of a new serious crisis are gathering over quantum theory,” is given in Weyl’s usual manner: a combination of external mathematical brilliance with poverty of physical ideas.

From all that has been said it is easy to conclude that Weyl’s book, despite the erudition of its author, is not the best of the existing survey books on quantum mechanics; being the most difficult of all, it does not surpass the other books even in the abundance of physical material. A Russian translation of Weyl’s book could hardly be considered advisable.

M. Bronstein.