## H Weyl: *Theory of groups and quantum mechanics* Introduction

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

**Introduction**

The quantum theory of atomic processes was proposed by Niels Bohr in the year 1913, and was based on the atomic model proposed earlier by Rutherford. The deduction of the Balmer series for the line spectrum of hydrogen and of the Rydberg number from universal atomic constants constituted its first convincing confirmation. This theory gave us the key to the understanding of the regularities observed in optical and X-ray spectra, and led to a deeper insight into the structure of the periodic system of chemical elements. The issue of *Naturwissenschaften,* dedicated to Bohr and entitled *Die ersten zehn Jahre der Theorie von Niels Bohr über den Bau der Atome* (Volume 11, p. 535 (1923)), gives a short account of the successes of the theory at its peak. But about this time it began to become more and more apparent that the Bohr theory was a compromise between the old "classical" physics and a new quantum physics which has been in the process of development since Planck's introduction of energy quanta in 1900. Bohr described the situation in an address on *Atomic Theory and Mechanics* (appearing in *Nature,* 116, p. 845 (1925)) in the words:-

From these results it seems to follow that, in the general problem of the quantum theory, one is faced not with a modification of the mechanical and electrodynamical theories describable in terms of the usual physical concepts, but with an essential failure of the pictures in space and time on which the description of natural phenomena has hitherto been based.

The rupture which led to a new stage of the theory was made by Heisenberg, who replaced Bohr's negative prophecy by a positive guiding principle.

The foundations of the new quantum physics, or at least its more important theoretical aspects, are to be treated in this book. For supplementary references on the physical side, which are urgently required, I name above all the fourth edition of Sommerfeld's well-known *Atombau und Spektrallinien* (Braunschweig, 1924), or the English translation *Atomic Structure and Spectral Lines* (London, 1923) of the third edition, together with the recent (1929) *Wellenmechanischer Ergänzungsband* or its English translation *Wave Mechanics* (1930). An equivalent original English book is that of Ruark and Urey, *Atoms, Molecules and Quanta* (New York, 1930), which appears in the *International Series in Physics,* edited by Richtmeyer. I should also recommend Gerlach's short but valuable survey *Experimentelle Grundlagen der Quantentheorie* (Braunschweig, 1921). The spectroscopic data, presented in accordance with the new quantum theory, together with complete references to the literature, are given in the following three volumes of the series *Struktur der Materie,* edited by Born and Franck:-

F Hund, *Linienspektren und periodisches System der Elemente* (1927);

E Back and A Lande, *Zeemaneffekt und Multiplettstruktur der Spektrallinien* (1925);

W Grotrian, *Graphische Darstellung der Spektren von Atomen und Ionen mit ein, zwei und drei Valenzelektronen* (1928).

The spectroscopic aspects of the subject are also discussed in Pauling and Goudsmit's recent *The Structure of Line Spectra* (1930), which also appears in the *International Series in Physics.*

The development of quantum theory has only been made possible by the enormous refinement of experimental technique, which has given us an almost direct insight into atomic processes. If in the following little is said concerning the experimental facts, it should not be attributed to the mathematical haughtiness of the author; to report on these things lies outside his field. Allow me to express now, once and for all, my deep respect for the work of the experimenter and for his fight to wring significant facts from an inflexible Nature, who says so distinctly "No" and so indistinctly "Yes" to our theories.

Our generation is witness to a development of physical knowledge such as has not been seen since the days of Kepler, Galileo and Newton, and mathematics has scarcely ever experienced such a stormy epoch. Mathematical thought removes the spirit from its worldly haunts to solitude and renounces the unveiling of the secrets of Nature. But as recompense, mathematics is less bound to the course of worldly events than physics. While the quantum theory can be traced back only as far as 1900, the origin of the theory of groups is lost in a past scarcely accessible to history; the earliest works of art show that the symmetry groups of plane figures were even then already known, although the theory of these was only given definite form in the latter part of the eighteenth and in the nineteenth centuries. F Klein considered the group concept as most characteristic of nineteenth century mathematics. Until the present, its most important application to natural science lay in the description of the symmetry of crystals, but it has recently been recognized that group theory is of fundamental importance for quantum physics; it here reveals the essential features which are not contingent on a special form of the dynamical laws nor on special assumptions concerning the forces involved. We may well expect that it is just this part of quantum physics which is most certain of a lasting place. Two groups, the group of rotations in 3-dimensional space and the permutation group, play here the principal role, for the laws governing the possible electronic configurations grouped about the stationary nucleus of an atom or an ion are spherically symmetric with respect to the nucleus, and since the various electrons of which the atom or ion is composed are identical, these possible configurations are invariant under a permutation of the individual electrons. The investigation of groups first becomes a connected and complete theory in the theory of the representation of groups by linear transformations and it is exactly this mathematically most important part which is necessary for an adequate description of the quantum mechanical relations. All quantum numbers, with the exception of the so-called principal quantum number, are indices characterising representations of groups.

This book, which is to set forth the connection between groups and quanta, consists of five chapters. The first of these is concerned with unitary geometry. It is somewhat distressing that the theory of linear algebras must again and again be developed from the beginning, for the fundamental concepts of this branch of mathematics crop up everywhere in mathematics and physics, and a knowledge of them should be as widely disseminated as the elements of differential calculus. In this chapter many details will be introduced with an eye to future use in the applications; it is to be hoped that in spite of this the simple thread of the argument has remained plainly visible. Chapter II is devoted to preparation on the physical side; only that has been given which seemed to me indispensable for an understanding of the meaning and methods of quantum theory. A multitude of physical phenomena, which have already been dealt with by quantum theory, have been omitted. Chapter III develops the elementary portions of the theory of representations of groups and Chapter IV applies them to quantum physics. Thus mathematics and physics alternate in the first four chapters, but in Chapter V the two are fused together, showing how completely the mathematical theory is adapted to the requirements of quantum physics. In this last chapter the permutation group and its representations, together with the groups of linear transformations in an affine or unitary space of an arbitrary number of dimensions, will be subjected to a thorough going study.

JOC/EFR August 2006

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