## H Weyl: *Theory of groups and quantum mechanics*Preface to Second Edition

Here is Weyl's Preface to the First German Edition We give below an extract from Weyl's Preface to the Second German Edition:- |

### From the Author's Preface to the Second German Edition

During the academic year 1928-20 I held a professorship in mathematical physics in Princeton University. The lectures which I gave there, and in other American institutions, afforded me a much desired opportunity to present anew, and from an improved pedagogical standpoint, the connection between groups and quanta. The experience thus obtained has found its expression in this new edition, in which the subject has been treated from a more thoroughly elementary standpoint. Transcendental methods, which are in group theory based on the calculus of group characteristics, have the advantage of offering a rapid view of the subject as a whole, but true understanding, of the relationships is to be obtained only by following an explicit elementary development. I may mention in this connection the derivation of the Clebsch-Gordan series, which is of fundamental importance for the whole of spectroscopy and for the applications of quantum theory to chemistry, the section on the Jordan-Hölder theorem and its analogues, and above all the careful investigation of the connection between the algebra of symmetric transformations and the symmetric permutation group. The reciprocity laws expressing this connection, which were proved by transcendental methods in the first edition, as well as the group-theoretic problem arising from the existence of spin have also been treated from the elementary standpoint. Indeed, the whole of Chapter V - which was, in the opinion of many readers, much too condensed and more difficult to understand than the rest of the book-has been entirely re-written. The algebraic standpoint has been emphasized, in harmony with the recent development of "abstract algebra," which has proved so useful in simplifying and unifying general concepts. It seemed impossible to avoid presenting the principal part of the theory of representations twice; first in Chapter III, where the representations are taken as given and their properties examined, and again in Chapter V, where the method of constructing the representations of a given group and of deducing their properties is developed. But I believe the reader will find this two-fold treatment an advantage rather than a hindrance.

To come to the changes in the more physical portions, in Chapter IV the role of the group of virtual rotations of space is more clearly presented. But above all several sections have been added which deal with the energy-momentum theorem of quantum physics and with the quantization of the wave equation in accordance with the recent work of Heisenberg and Pauli. This extension already leads so far away from the fundamental purpose of the book that I felt forced to omit the formulation of the quantum laws in accordance with the general theory of relativity, as developed by V Fock and myself, in spite of its desirability for the deduction of the energy-momentum tensor. The fundamental problem of the proton and the electron has been discussed in its relation to the symmetry properties of the quantum laws with respect to the interchange of right and left, past and future, and positive and negative electricity. At present no solution of the problem seems in sight; I fear that the clouds hanging over this part of the subject will roll together to form a new crisis in quantum physics. I have intentionally presented the more difficult portions of these problems of spin and second quantization in considerable detail, as they have been for the most part either entirely ignored or but hastily indicated in the large number of texts which have now appeared on quantum mechanics.

It has been rumoured that the "group pest" is gradually being cut out of quantum physics. This is certainly not true in so far as the rotation and Lorentz groups are concerned; as for the permutation group, it does indeed seem possible to avoid it with the aid of the Pauli exclusion principle. Nevertheless the theory must retain the representations of the permutation group as a natural tool in obtaining an understanding of the relationships due to the introduction of spin, so long as its specific dynamic effect is neglected. I have here followed the trend of the times, as far as justifiable, in presenting the group theoretic portions in as elementary a form as possible. The calculations of perturbation theory are widely separated from these general considerations; I have therefore restricted myself to indicating the method of attack without either going into details or mentioning the many applications which have been based on the ingenious papers of Hartree, Slater, Dirac and others.

The constants *c* and *h*, the velocity of light and the quantum of action, have caused some trouble. The insight into the significance of these constants, obtained by the theory of relativity on the one hand and quantum theory on the other, is most forcibly expressed by the fact that they do not occur in the laws of Nature in a thoroughly systematic development of these theories. But physicists prefer to retain the usual e.g.s. units - principally because they are of the order of magnitude of the physical quantities with which we deal in everyday life. Only a wavering compromise is possible between these practical considerations and the ideal of the systematic theorist; I initially adopt, with some regret, the current physical usage, but in the course of Chapter IV the theorist gains the upper hand.

An attempt has been made to increase the clarity of the exposition by numbering the formulae in accordance with the sections to which they belong, by emphasizing the more important concepts by the use of boldface type on introducing them, and by lists of operational symbols and of letters having a fixed significance.

H Weyl

Göttingen,

November, 1930

Here is Weyl's Introduction to the Second German Edition

JOC/EFR August 2006

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