Hermann Weyl: 1885-1985: Centenary Lectures [1986 ed.] 3540168435, 9783540168430

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Hermann Weyl © 1885-1985

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Maprrannr 9 November

1885

“eek

— 8 December

1955

HERMANN

WEYL

1885-1985 Centenary Lectures delivered by C.N. Yang, R. Penrose, A. Borel

at the ETH Ziirich Edited by K. Chandrasekharan

Published for the

Eidgendssische Technische Hochschule

Ziirich

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo

Komaravolu Chandrasekharan Professor of Mathematics Eidgendssische Technische Hochschule Ziirich

CH-8092 Ziirich

D&S 231 Mathematics

Subject Classification (1980): 00, 22, 70, 82

ISBN 3-540-16843-5 Springer-Verlag Berlin Heidelberg New York

ISBN 0-387-16843-5 Springer-Verlag New York Berlin Heidelberg

Library of Congress Cataloging in Publication Data Hermann Weyl, 1885-1985: centenary lectures. Bibliography: p. 1. Mathematics — 1961-. 2. Mathematical physics. 3. Weyl, Hermann, 1885-1955. 4. Mathematicians— Germany -Biography. I. Yang, Chen Ning, 1922-. Il. Penrose, Roger. III. Borel, Armand. IV. Chandrasekharan, K. (Komaravolu), 1920-. V. Eidgendssische Technische Hochschule Ziirich. QA7.H47 1986 51086-15573 ISBN 0-387-16843-5 (U.S.) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction

by photocopying machine or similar means, and storage in data banks. Under § 54 of the German

Copyright Law where copies are made for other than private use a fee is payable to “Verwertungsgesellschaft Wort”, Munich.

© ETH Ziirich and Springer-Verlag Berlin Heidelberg 1986 Printed in Germany Typesetting, printing and bookbinding: Briihlsche Universitatsdruckerei, Giessen 2141/3140-543210

Time present and time past

Are both present in time future,

And time future contained in time past. T.S. Eliot

Foreword This volume marks the celebration of the Centenary of Hermann Weyl by the Swiss Federal Institute of Technology (ETH), Ziirich. That celebration was given a special substance and meaning by the three Centenary Lectures delivered here by Professor Chen Ning Yang, by Professor Roger Penrose, and

by Professor Armand Borel. It is the purpose of this volume to make the text of those lectures available to a wider audience. The themes chosen represent only a fraction of Weyl’s mathematical interests. But they give us more than

a glimpse of the mighty effulgence of his mind. Among

all the mathematicians

who

began

their research

work

in the

Hilbert and

Henri

twentieth century, Weyl was the one who made major contributions in the greatest number of different fields. He alone could stand comparison with the last great universalists of the nineteenth century, David

Poincaré. No one else bounds about among the peaks of mathematics with

quite such dazzling aplomb.

To Weyl the world of ideas and concepts was as real as the world of human

beings. The force, the steadiness, the comprehensiveness,

and the versatility

of his intellect were matched by his generosity, sympathy, and support for striving young researchers spread around the globe.

He was peerless in the part he played in the international world of mathematics and mathematicians, as adviser, inspirer, referee, bringer on, and

godfather. And he was totally devoid of envy, rare even among the great.

Incelebrating his Centenary, we are not just savouring the past, or savaging

the present, but looking to the future with a sober idealism that transcends the

personality. While inscribing a copy of his book on Classical Groups for one of his admirers, Weyl used a quotation from Gauss which said: ‘you have achieved

nothing so long as there is something to be achieved’. There is much to be achieved. But the example of Weyl, as it flames and glows in our memory, will remain a source of inspiration for succeeding generations. The Centenary Lectures make this point in their own way. ETH

Ziirich, March

30, 1986

K. Chandrasekharan

Note The Hermann Weyl Centenary celebration took place, with the support and approval of the Schulleitung of ETH Ziirich, under the auspices of a Centennial Committee consisting of Prof. H. Ursprung (Chairman), Minister Dr. J. Burckhardt (Honorary Chairman), Prof. Hans Hauri, Mr. Hans J. Bar, Dr. Heinz Gétze, and

The

Centenary

International

K. Chandrasekharan.

Lectures

Mathematical

institut fiir Mathematik,

were

Union,

ETH

organized

with

in co-operation

Ziirich.

Mention

the

must

endorsement

with

of the

the Forschungs-

be made

of the help

received from Prof. E. Freitag, Prof. J. Frohlich, Prof. K. Hepp, Prof. J. Moser,

Prof. J. L. Olsen, and Prof. K. Osterwalder. A report on the celebration is given in an Appendix, which includes a pen portrait by Dr. Michael Weyl, as well as a list of Hermann Weyl’s publications. The pictures printed in this book are from private collections belonging to members of the Weyl family, to Mr. Hans J. Bar, to Professor Shizuo Kakutani, and to Professor J. L. Olsen.

Thanks are due to Springer-Verlag for the readiness and efficiency which they have executed the publication.

with

Table of Contents Opening Address by Prof. H. Ursprung (President, ETH 1

Hermann

Zirich)

Weyl Centenary Lectures

Hermann Weyl’s Contribution to Physics

by Prof. Chen Ning Yang i

Hermann Weyl, Space-Time and Conformal Geometry by Prof. Roger Penrose

23

Hermann

Weyl and Lie Groups

by Prof. Armand 33

Hermann

Weyl

83

Borel

Memorabilia

Appendix

Report on the Celebration

95

List of Publications by Hermann Weyl

109

Opening Address Heinrich

Ursprung

President, ETH

Was

Ziirich

hat derjenige, dem die Er6dffnung dieser Vorlesungsserie tibertragen

worden ist, noch zu sagen, wo doch die Vorlesungen von Yang, Penrose und Borel bevorstehen? Er k6nnte in Erinnerung rufen, dafs das erste Buch von Hermann Weyl (,,Die Idee der Riemannschen Flache*) erschien, als Weyl erst

28 war, oder daB schon fiinf Jahre darauf zwei weitere Bucher von ihm erschienen (,,Das Kontinuum* und ,,Raum, Zeit, Materie“, welches innert fiinf

Jahre fiinf Auflagen Vorgehen gewahlt.

und zwei Ubersetzungen

erfuhr). Ich habe ein anderes

When I got a chance to look through a selection of Weyl’s writings on general themes, I chose from among the rich material a few passages that seemed to shed some light on how

Weyl functioned in his environment, i.e.,

academia, and what he thought of it; what he thought of borders between the various sciences, between nations, and between professors and students. Lassen sie mich einige Eindriicke festhalten, die ich bei diesem Versuch

empfunden habe. Bei aller Stringenz seines wissenschaftlichen Werks mu

Weyl akademischem

Territorialdenken abhold gewesen sein. Das ergibt sich z. B. aus dem Vorwort zu

seinem

1928 erschienen Buch ,,Gruppentheorie und Quantenmechanik*.

Er

schreibt in Ziirich, im August 1928: ,,Zum zweiten Male erkiihne ich mich, mit einem

Buche

auf den

Plan

zu treten, das

nur halb meinem

Fachgebiet

der

Mathematik, halb aber der Physik angehért.... Ich kann es nun einmal nicht

lassen, in diesem Drama

von Mathematik

und Physik — die sich im Dunkeln

befruchten, aber von Angesicht zu Angesicht so gerne einander verkennen und verleugnen — die Rolle des (wie ich gentigsam erfuhr, oft unerwiinschten) Boten

zu spielen.“ —(Hie Mathematik, hie Physik: das Problem hat uns bis heute nicht

verlassen. Aber zum Glick gibt es auch heute noch Boten). His vision of the need of unified thinking is infinitely more evident in the

lecture on the “Unity of Knowledge”, delivered at Columbia

University’s

Bicentennial Celebration, in New York, in 1954. “Doubts about the methodical

unity of the natural sciences have been raised. This seems unjustified to me.

2:

Heinrich Ursprung

Following Galileo, one may describe the method of science in general terms as a combination of passive observation refined by active experiment with that symbolic

construction

paragon.

methodical

Hans

to which

theories

ultimately

reduce.

Physics

is the

Driesch and the holistic school have claimed for biology a

approach

different

from,

and

transcending,

that

of

physics.

However, nobody doubts that the laws of physics hold for the body of an animal or myself as well as for a stone. Driesch’s attempts to prove that the organic processes are incapable of mechanical explanation rest on a much too narrow

notion of mechanical or physical explanation of nature. Here quantum physics

has opened up new possibilities. On the other side, wholeness is not a feature limited to the organic world. Every atom is already a whole of quite definite structure; its organization is the foundation of possible organization and structures of the utmost complexity. I do not suggest that we are safe against suprises in the future development of science. Not so long ago we had a pretty startling one in the transition from classical to quantum physics. Similar future breaks may greatly affect the epistemological interpretation, as this one did with the notion of causality; but there are no signs that the basic method itself, symbolic construction combined with experience, will change.” Was die Orte seines Wirkens betrifft, scheint Weyl ausgesprochene Loyalitat aufgebaut zu haben. ,,So kann ich nicht verhehlen, daB mein Herz es fast als

treulos empfand, daB ich Ziirich verlieB“, schreibt er in seinem ,,Riickblick auf

Zirich aus dem Jahre 1930*. ,,.Die schlimmste Plage wahrend meiner Ziircher Jahre waren fiir mich Berufungen nach auswarts. Einmal... geschah es, daB ich gleichzeitig nach Berlin und Gottingen berufen war. VerhaltnismaBig rasch entschloB ich mich zur Ablehnung von Berlin. Aber den Lehrstuhl von Felix Klein an der Géttinger Universitat ausschlagen — das war eine hartere NuB... . Als sich die Entscheidung nicht langer aufschieben lief, lief ich im Ringen

darum mit meiner Frau stundenlang um einen Hauserblock herum und sprang

schlieBlich aufein spates Tram... zum Telegraphenamt, ihr zurufend: ,,Es bleibt doch nichts anderes tibrig als annehmen.* Aber dann muf es mir das frdhliche Treiben... an diesem schOnen Sommerabend um und auf dem See... angetan haben: ,,ich... telegraphierte eine Ablehnung™.

Im selben Aufsatz schildert er eindriicklich, wie sich seine auf der Lektiire

von Gottfried

Keller

und

C. F. Meyer

beruhende

Vorstellung

der Schweiz

korrigierte, als er hier zu leben begann. ,,I[ch hatte zu lernen, daB die deutsche Kultur die Schweiz nicht so selbstverstandlich

geschienen

hatte;...

dafs die Schweiz

eben

umschlieBt, wie es bei Keller

nicht die deutsche

Schweiz

ist,

sondern das Dach Europas, unter dem sich germanische und romanische Kultur treffen.... Ich fiihle mich der Schweiz nicht viel weniger verbunden als Deutschland*.

Von noch einem seiner Wirkungsorte spricht er voll Affektion, und zwar in

seinem

Arnold

Lebensritickblick,

Reymond

1954, als die Universitit

verleiht,

in

Anerkennung

Lausanne

seiner

ihm den

Bemiihungen

um

Preis

die

Philosophie der Wissenschaften. Er sei an der schnsten Forschungsstiatte, die

Opening Address

3

es fir Mathematik in der Welt gibt“, festgehalten worden, ,,an dem Institute for Advanced Study in Princeton, New Jersey“. Wie erfrischend, schlielich, seine Ansicht zum Lehrer-Schiiler Verhdltnis und zum Verhdltnis zwischen Forschung und Lehre an der Hochschule. Im schon erwahnten ,,Riickblick auf Ztirich aus dem Jahre 1930“ ist eine Rede

wiedergegeben, die Weyl 1930 in Gottingen an die Mathematische Verbindung

richtete, der er als Student angeh6rt hatte. Er schildert dort u.a. tiber die Art und Weise, wie er an der ETH zu wirken versucht hatte.

Eine Hochschule ist meiner Uberzeugung nach nicht nur Schule, oberste

jener Institutionen, durch welche die Gesellschaft den Gehalt ihrer Kultur, insbesondere auch die gewonnenen wissenschaftlichen Erkenntnisse, techni-

schen

Erfahrungen

und

das

theoretische

Weltbild

der

heranwachsenden

Generation tradiert, sondern sie dient auBerdem der Forschung. Der Erkennende, der theoretisch Gestaltende ist so gut wie der Kiinstler ein Grundtypus

des Menschen, der in der gesellschaftlichen Organisation seinen Platz finden

mu, und er findet ihn heute nur an der Hochschule. Wer erkennt, den ,verlangt nach Rede’; so mégen denn die Jungen zu seinen FiBen sitzen und ihm zuhoren, wenn Rede aus ihm bricht. Dies betrachte ich als das Grundverhaltnis. Ich glaube nicht daran, daB das System der Erziehung von unten aufgebaut werden miisse; die Gegenbewegung darf nicht fehlen. Was der Natur und der Notwendigkeit angehért, wachst von unten her, der Geist und seine Freiheit aber brechen von oben herein. In dieser Weise, hoffe ich, werde ich Ihnen, liebe

Kommilitonen, Lehrer sein k6nnen; den Samen in den Wind streuend; es fassen kann.“

fasse, wer

This is what I wanted to say by way of introduction to these Hermann Weyl

Centenary

Lectures.

It is a particular pleasure for me to welcome to the ETH Ziirich Professor

Yang, who has many friends and followers here. It is a happy augury that such an eminent physicist, gifted as he is with a powerful mathematical intuition, has

agreed to speak about the work of such an eminent mathematician as Weyl with his profound perception of the structures of theoretical physics.

I give the floor now to Professor Yang. Den ZuhGrern rufe ich mit Hermann

Weyl zu: ,,Fasse, wer es fassen kann!“

Hermann Weyl Centenary Lectures

Hermann Weyl’s Contribution to Physics Chen Ning

Yang

I, In May 1954, at the age of 69, Hermann Weyl gave a lecture! in Lausanne,

as President Ursprung already mentioned. This lecture was largely autobiographical, centering upon various stages of his thinking, especially about philosophy. It touched upon Weyl’s first important work in physics:

The next epochal event for me was that I made an important mathematical discovery. It concerned the regularity in the distribution of the eigenfre-

quencies of a continuous medium, like a membrane, an elastic body, or the

electromagnetic ether. The idea was one of many, as they probably come to every young person preoccupied with science but while the others soon burst like soap bubbles, this one led, as a short inspection showed, to the

goal. I was myself rather taken aback by it as I had not believed myself capable of anything like it. Added to it was the fact that the result, although

conjectured by the physicists some time ago, appeared to most mathematicians as something whose proof was still in the far future. While I was feverishly working on the proof, my kerosene lamp had begun to smoke, and

I was no sooner finished than thick sooty flakes began to rain down from the

ceiling onto my paper, hands, and face. What Weyl was talking about was a very interesting piece of work which

had its origin in the Wolfskehl lecture by H. A. Lorentz in 1910 in Gottingen.

Lorentz had posed the problem?:

In conclusion there is a mathematical problem which perhaps will arouse

the interest of mathematicians who are present. It originates in the radiation theory of Jeans.

*A Hermann Weyl Centenary Lecture, ETH Ziirich, October 24, 1985 GA=H. Weyl: Gesammelte Abhandlungen, Vol. 1 to IV, (Springer-Verlag, Berlin-Heidelberg, 1968), Edited by K. Chandrasekharan. ' Weyl, 1954; GA IV, p. 636. English translation: The Spirit and the Uses of the Mathematical Sciences, edited by T. L. Saaty and F. J. Weyl, (McGraw Hill, New York, 1969), p. 286. ? See M. Kac, Am. Math. Monthly 73, 1 (1966).

8

Chen

Ning Yang

In an enclosure with a perfectly reflecting surface there can form standing electromagnetic waves analogous to tones of an organ pipe: We shall

confine our attention to very high overtones. Jeans asks for the energy in the frequency interval dv....

It is here that there arises the mathematical problem to prove that the

number of sufficiently high overtones which lies between vy and v+d\v is independent of the shape of the enclosure and is simply proportional to its volume.

Fig. 1.

A drum

Weyl was a young mathematician in the audience, and he took up Lorentz’s challenge. Using the powerful insight he always had about the right mathematical tools to use for a specific problem, he solved the question with the method of integral equations developed by his teacher David Hilbert. The problem concerns, for two dimensions,

a membrane which is like a drum with the edges

tied down to some solid material. One wants to study N(K?), the number of

eigenvalues less than K?, and is interested in N(K”) when K? is very large. The equation that has to be solved is the differential equation —V7u=Ku, where K=2nv/v,

and a physicist would recognize that v is the eigenfrequency and vis the velocity of propagation of waves on the membrane. What Weyl proved was that A

(1)

N(K?)> — K*,

as

4n

where A

K?>0o,

is the area of the membrane, thus verifying Lorentz’s conjecture that

the result is independent of the shape of the drum. By the way, the problem

was of importance in physics, because of its connection with the theory of black-body radiation, which had led Planck, about ten years earlier, to the

discovery of the quantum of action. This work

of Weyl started a new small field of mathematics, in which

there had been much activity. Instead of N(K?) it is more convenient to study (2)

nS

y

e

Kie

Hermann Weyl’s Contribution to Physics

Z)

where K? is the n-th eigenvalue. For small £, the summand 1

Kr>

B and ~1 for K?


——.,

as

f-30.

More accurate estimates of B are now known’.

B=

2

¥

n=1

e Kib—

Area 4np

L

1

——

7 4 V/4nB

1

+ — (1—r)+(term that 0 as 60). 6

B

The second term is proportional to L, the length of the circumference of the drum. Thus this term is not just dependent on

the area. The next order term is

even more interesting because it contains the parameter r, the number of holes in the drum. Thus the third term depends on the topology of the drum. In the years 1925-1927 Weyl was engaged in his deep work on the structure

of Lie groups and their representations, which he later regarded as the pinnacle of his mathematical achievements. As a physicist | am acquainted only with a

small part of this work, from studying his famous book The Classical Groups, but what I had learned was enough to allow me to have a glimpse of the grandeur, elegance and power of this achievement. It also allowed me to appreciate the meaning of the following passage in the preface to the first edition (1938) of the book:

The stringent precision attainable for mathematical thought has led many

authors to a mode of writing which must give the reader the impression of

being shut up in a brightly illuminated cell where every detail sticks out with

the same dazzling clarity, but without relief. I prefer the open landscape

under a clear sky with its depth of perspective, where the wealth of sharply

defined nearby details gradually fades away towards the horizon.

Indeed this passage states very clearly Weyl’s intellectual preference which had

a determining influence on the style of his work in mathematics and in physics.

In the midst of Weyl’s profound research on Lie groups there occurred a

great revolution in physics, namely, the development of quantum

mechanics.

We shall perhaps never know Weyl’s initial reaction to this development, but he soon got into the act and studied the mathematical structure of the new mechanics. There resulted a paper of 19273 and later a book*. (Recently David Speiser had written about both of these contributions®.) This book together 3 Weyl, 1927; GA III, p. 90.

+H. Weyl, Gruppen August

1928.

Theorie und Quantenmechanik,

5 David Speiser, Gruppen

Preface

to first German

edition dated

Theorie und Quantenmechanik, the book and its position in Weyl’s

work, preprint from the Institute of Theoretical Physics, Catholic University of Louvain, Belgium.

10

with

Chen Ning Yang

Wigner’s

articles and

Gruppen

Theorie

und

Ihre Anwendung

auf die

Quanten Mechanik der Atome were instrumental in introducing group theory

into the very language of quantum mechanics. Weyl

was

a mathematician

and

a philosopher.

He

liked

to deal

with

concepts and the connection between them. His book * was very famous, and was recognized as profound. Almost every theoretical physicist born before 1935 has a copy of it on his bookshelves. But very few read it: Most are not

accustomed to Weyl’s concentration on the structural aspects of physics and feel uncomfortable

with

his emphasis

abstract for most physicists.

on concepts.

The

book

was just

too

In 1930 a new German edition of the book was published and he wrote in the preface:

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.

This was a most remarkable passage in retrospect. The symmetry that he mentioned here, of physical laws with respect to the interchange of right and left, had been introduced by Weyl and Wigner independently into quantum physics. It was called parity conservation, denoted by the symbol P. The symmetry between the past and future was something that was not well understood in 1930. It was understood later by Wigner, was called time reversal invariance, and was denoted by the symbol T. The symmetry with respect to positive and negative electricity was later called charge conjugation invariance C. It is a symmetry of physical laws when you change positive and negative signs of electricity. Nobody, to my knowledge, absolutely nobody in the year 1930, was in any way suspecting that these symmetries were related in any

manner. It was only in the 1950's that the deep connection between them was

discovered. I will come back to this matter later. What had prompted Wey] in 1930 to write the above passage is a great mystery to me.

The “fundamental problem of the proton and the electron” that Weyl referred to was a crisis ushered in from 1928 to 1930 by Dirac who argued that

the electron must be described by a certain equation, now known as the Dirac

equation. He was immediately attacked because his equation has solutions which are evidently meaningless: They have negative energies which are unphysical. Dirac was bold enough to then suggest that the negative energy states are, usually, all occupied and thus not observed. It is only when a vacancy

appears in this “negative sea” of electrons that one can observe something and that something would have the opposite charge to the electron. He suggested

that such a vacancy should be a proton, the only particle known at that time

with a charge opposite to the electron.

Weyl studied this suggestion in detail in the second edition of his book and

concluded that

Hermann

Weyl’s Contribution to Physics.

11

(Dirac’s hypothesis) leads to the essential equivalence negative electricity under all circumstances.

of positive and

He also concluded that the vacancy must have the same mass as the electron.

These conclusions of Weyl led, together with the work of others, to the ideas that the vacancies are positrons

(which

were later experimentally

found

in

1932,) and that the proton is another particle which has its own negative energy

states the vacancies in which are antiprotons, (which were experimentally found

in 1955.)°

Weyl’s conclusion about the “essential equivalence” of positive and negative electricity was the forerunner of the important concept of charge

conjugation invariance C which was fully formulated only in 1937°.

Thad earlier mentioned the discrete symmetries P, C, and T. Of the three the

one that Weyl was most concerned with was obviously the symmetry between the left and right. In his book Philosophy of Mathematics and Natural Science, first published in German in 1926, with an English translation published in

1949, Weyl had this to say about left and right: Left and right. Were I to name the most fundamental mathematical facts I

should probably begin with the fact (F,) that the counting of a set of elements leads to the same number in whatever order one picks up its elements, and mention as a second the fact (F,) that among the permutations of n(= 2) things one can distinguish the even and the odd ones. The

even permutations

form

a subgroup

of index 2 within the group of all

permutations. The first fact lies at the bottom of the geometric notion of

dimensionality, the second of that of ‘sense’.°* Indeed left-right symmetry, or parity conservation, was such a “natural” and

useful concept for physicists, it had always been taken for granted as a sacred

law of nature. In 1957, less than two years after Weyl’s death, to everyone’s

surprise, Wu, Ambler, Hayward, Hoppes, and Hudson found that left-right symmetry, was after all, not exactly observed ° by the laws of physics. The

violation was slight, but observable if one knows where to look for it. Interestingly in 1952 Weyl had written a beautiful little book Symmetry which

according to Speiser > Weyl had called his swan-song death. In this book we read the following passage; For in contrast

to the orient, occidental

shortly

before his

art, like life itself, is inclined

to

mitigate, to loosen, to modify even to break strict symmetry. But seldom is

asymmetry merely the absence of symmetry. Even in asymmetric designs

one feels symmetry as the norm from which one deviates under the influence of forces

of non-formal

character.

I think

the

riders from

the famous

Etruscan Tomb of the Triclinium at Corneto provide a good example. I have often wondered what Weyl would have said about the slight but very

important violation of the left-right symmetry of laws of nature if he had lived

just two years longer.

° For this history see Yang'® p. 236; also see C. N. Yang The Discrete Symmetries P, T, and C,

J. de Physique, Colloque C8, C8-439 (1982). © ‘Sense’ here means orientation.

12

Chen

Fig. 2.

Ning Yang

Reproduction of illustration in Weyl’s Symmetry

Now I come to another piece of work of Weyl’s which dates back to 1929,

and is called Weyl’s two-component neutrino theory. He invented this theory in 1929 in one of his very important articles’ which I will come to again later, as a mathematical possibility satisfying most of the requirements of physics. But it was rejected by him and by subsequent physicists because it did not satisfy leftright symmetry. With the realisation that left-right symmetry was not exactly

right in 1957 it became clear that this theory of Weyl’s should immediately be

re-looked at. So it was and later it was verified theoretically and experimentally that this theory gave, in fact, the correct description of the neutrino. Roughly speaking, it is something like this: In an ordinary neutrino theory there are four

components

of the neutrino:

the left-handed

neutrino

v,, the right-handed

neutrino vp, the left-handed anti-neutrino #, and the right-handed antineutrino Vp. In Weyl’s theory there are only two: the left-handed neutrino v, and the right-handed anti-neutrino Vz. This theory is not left-right symmetrical,

because if you make a reflection of a left-handed neutrino it becomes a righthanded neutrino, and the right-handed neutrino vp does not exist in the two-

component

neutrino

theory.

Thus

the very

existence

of the Weyl

neutrino

violates left-right symmetry. We call that violation weak because the neutrino is

very weakly coupled with all other particles in the universe. Another very important development in this area was the CPT theorem, a fundamental theorem in field theory, which was proved in 1953-1955. With

7H. Weyl, Z. Phyk 56, 330 (1929), Reprinted in GA. See 2+.

8 See ° above for this history. The usually quoted sources of the CPT theorem are Schwinger, Liiders and

Pauli.

contribution by J.S.

V.

Telegdi

Bell, Proc.

has kindly Roy.

Soc.

informed

(London),

me

that there was

423/, 479 (1955).

also an important

Hermann Weyl’s Contribution to Physics

13

the discovery of the violation of P invariance, the CPT theorem came to play a very important part in discussions of the mode of the violation. Again I cannot help wondering what Weyl would have said about the CPT theorem had he lived two years longer into 1957, not only because he had, quite mysteriously in 1930, written about these three symmetries C, P, and T

all in one sentence, as I

quoted before, but also because the theorem was given a more profound foundation by Jost° and that foundation involved the Lorentz group and the concept of analytic continuation, subjects that were dear to Weyl’s heart.

iE The next contribution of Hermann Weyl to physics that I shall discuss is

gauge theory. There were three periods during which Weyl] wrote about gauge theory which we shall now discuss separately.

During

the

first

we

find

three

papers,

all

written

in

the

years

1918-19191 11.12 The most important of these is the middle one and indeed

throughout his life, when he referred to gauge theory, Weyl always referred to this paper. The background of his thinking at that time can be traced through the preface of the various editions of his book Space, Time, Matter and through

his articles of 1917-1919. It seemed that Weyl, evidently inspired by the work of Einstein

on gravity (1916), and

also by the work

of Hilbert,

Lorentz,

and

that does

not

F. Klein, was searching for a geometrical theory that would embrace electromagnetism as well as gravity. He was also influenced by Mie who had, in 1912-1913,

attempted

to formulate

a theory of the electron

involve divergent field quantities inside of the electron.

In the beginning paragraphs of '' Weyl said that while Einstein’s gravity theory depended on a quadratic differential form, electromagnetism depended

ona

linear differential form 2¢,,dx,, (which in today’s notations is ¥A,dx"). The

next crucial sentences are,'?

The later work of Levi-Civita, Hessenberg and the author shows quite plainly that the fundamental conception on which the development of

Riemann’s geometry must be based if it is to be in agreement with nature, is

that of the infinitesimal parallel displacement of a vector.... But a truly infinitesimal geometry must recognize only the principle of the transference of

alength from one point to another point infinitely near the first. This forbids us to assume that the problem of the transference of length from one point to another at a finite distance is integrable, more particularly as the problem of °R. Jost, Helv. Phys. Acta 30, 409 (1957).

10 Weyl, 1918; GA II, p. 1.

1! Weyl, 1918; GA II, p. 29. 12 Weyl, 1919; GA II, p. 55

13 T quote here from a 1923 translation of '! in The principle of Relativity by H. A. Lorentz,

A. Einstein, H. Minkowski, and

H. Weyl, translated by W. Perrett and G. B. Jeffery, (first

published by Methuen and Co. 1923, reproduced by Dover Publications, 1952).

14

Chen

Ning Yang

the transference of direction has proved being

assumption

recognized

false,

as

a

to be non-integrable. geometry

comes

Such

into

which...explains...also...the electromagnetic field. (Italics original)

an

being,

Thus was born'* the idea of a nonintegrable scale factor which appeared in one

Weyl paper, i.e. paper’®, explicitly as

Q

jae

(3)

e?

%

Weyl then argued that the addition of a gradient d(log/) to dg = X$,dx, should not change the physical content of the theory, thus concluding that

_ 0b, Op, 2 =3

F

(4)

has “invariant significance”. He naturally then identified F,, with the electro-

magnetic field and put (5)

¢, =(constant)A,

where A, is conceptually nonintegrable Invariance

Weyl

the electromagnetic potential. Thus electromagnetism was incorporated in this theory into the geometrical idea (3) of a scale factor. of the theory with respect to the addition dé—d¢ + d(log/) led

to the name’!

“MaBstab-Invarianz”

which

was

translated'*:'>

“measure-invariance” and “calibration invariance”. Later the German became

“Eich

invariance”,!>.

Invarianz”

and

the

English

term

became

as

term

“gauge

When Weyl’s paper'! was published in the Sitzber. Preuss. Akad. Wiss. in

1918 there were appended to the end of the paper a postscript by Einstein and a reply by Weyl. This unusual development came about, according to Hendry’’, because while at first Einstein was impressed with Weyl’s preprint, he later had

a strong objection. Nernst and Planck apparently shared Einstein’s objection and they demanded on behalf of the Berlin Academy that Einstein’s opinion be

appended to Weyl’s paper as a postscript.

What was the essence of Einstein’s objection? Einstein argued that if Weyl’s

idea of a nonintegrable scale factor is right, then if one takes two clocks and starts them from one point 0 and brings them along different paths back to the

same point 0, their scales would have continuously changed. Thus by the time

** In 1950 when Weyl reviewed 50 years of relativity (GA IV, p. 421)he referred to the origin

of this 1918 idea by saying that if a vector transported around a closed loop back to its original position could change its direction, “Warum nicht auch seine Lange?”

15 See p. 528 reference 1°. ‘© Chen Ning Yang, Selected Papers 1945-1980 with Commentary, (Freeman and Co. 1983). ‘7J. Hendry: The Creation of Quantum Mechanics and the Bohr-Pauli Dialogue, (Reidel

Publishing Co. 1984),

Hermann Weyl’s Contribution to Physics

1S

—)

Fig. 3. Illustration for Einstein’s Gedankenexperiment

they reached back to 0, since they have traced different histories, they would

have, in general, different sizes. They would thus keep time at different rates. Therefore, a clock’s measure of time depends on its history. If that is the case, Einstein argued, there cannot be physics, because everybody would have his

own laws, and there would be chaos. Weyl’s reply, also appended to this paper, did not really explain away the difficulty. In the years 1918-1921 he

came back'®:'° to this subject several times. He did not resolve the problem,

but his attempts clearly indicated a strong devotion to the original idea. His feelings can perhaps

be gleaned

from

a sentence

he wrote*?

in 1949 when

discussing the events after Einstein’s discovery of general relativity: A lone wolf in Ztirich, Hermann

Weyl, also busied

himself in this field;

unfortunately he was all too prone to mix up his mathematics with physical and philosophical speculations.

Pauli also objected to Weyl’s theory, but more on philosophical grounds.

According to Mehra and Rechenberg?', and to Hendry’’, Pauli’s objections

were of importance to the subsequent emphasis on the “observable” that was to

play a key role in the 1925 Heisenberg discovery of quantum mechanics. Now we come to the second period of Weyl and gauge theory. In 1925 to

1927, quite unrelated to Weyl’s gauge theory, a revolution took place in physics, namely quantum mechanics. One of the important points in quantum mechanics was that the momentum p, becomes a differential operator —ihd,.

In 1927, Fock and London independently pointed out that if p, is to be replaced

by —ihd,, then the quantity

We

e

he

should be replaced similarly by

(6)

—ihd,— < A,=—ih (2,- = A).

18 See the record ofa discussion between Weyl, Pauli, and Einstein at Bad Nauheim, Phys. Z. 21, 649-651 (1920) 19 Weyl 1921; GA II, p. 260. 20 Weyl 1949; GA IV, p. 394. 21 J. Mehra

and

H. Rechenberg, The

Chapter 5, (Springer-Verlag, 1982).

Historical

Development

of Quantum

Theory, Vol. 2,

Chen Ning Yang

16

(The

quantity p,— £e A, had already been of a charged

dynamics

“Quantum

particle.)

to be important

known

In London’s

article,

which

had

in the the

title

Mechanical Meaning of the Theory of Weyl”, it was pointed out

ie — A, in (6) is similar to the expression (0, +¢,) in he Weyl’s theory. Thus instead of (5) the identification should be

that the expression 0,—

§=—— 4,.

(7) Now =

ic

is a numerical constant. Therefore, (7) is really the same as Weyl’s

original identification (5) except for the insertion of —i (i=/ a

):

But this insertion, although trivial formally, has profound physical consequences, because it changes the meaning of the nonintegrable scale factor (3) into

2

(8)

;

exp (-! he i Als)

which is a nonintegrable phase factor. Thus Weyl’s theory is the theory of electromagnetism in quantum mechanics, provided one changes the idea of a

scale factor into a phase factor, with the insertion of a —i??. Fock and London

in 1927 did not explicitly have the concept of gauge

transformation (i.e., phase transformation.) That concept was for the first time

formulated in a decisive paper’:**+ of Weyl’s in 1929. I now quote froma

paper? of his, also published in 1929:

related

By this new situation, which introduces an atomic radius into the field equations themselves — but not until this step — my principle of gaugeinvariance, with which I had hoped to relate gravitation and electricity, is robbed of its support. But it is now very agreeable to see that this principle

has an equivalent in the quantum-theoretical

field equations which

is

>? For this history see '°, p. 525. For an analysis of the physical meaning of a nonintegrable

phase factor see **. The concept of the “nonintegrable” phase factor occurred to me only in 1967-1968 (see '®, p. 73). [I was not aware until ~1983 that Weyl had, in 1918, started

conceptually from the nonintegrable scale factor (3) and proceeded to the differential form 0, + ¢,-] Epistemologically this story is interesting and is representative of the style of Weyl’s ideas in physics, in contrast to that of the physicists: Weyl started from the integral approach and proceeded to the differential. Mills and I, physicists, learned the differential approach from

Pauli?* and only much later realized that one could also start from the integral approach. 23 W. Pauli in Handbuch der Physik, 2. Aufl. 24 part 1 (1933). 24 Weyl 1929, GA III, p. 245. 25 Weyl 1929, GA III, p. 229.

Hermann

exactly

like

simultaneous

arbitrary

Weyl’s Contribution to Physics

it in formal replacement

real function

respects;

the

of y by ep,

of position

and

laws

are

17

invariant oA

¢, by ¢,— >=—,

time.

Also

under

the

where

/ is an

the relation

of this

OX,

property of invariance to the law of conservation of electricity remains exactly as before...the law of conservation of electricity

2x _9

OXg

follows from the material as well as from the electromagnetic equations. The

principle of gauge-invariance has the character of general relativity since it

contains an arbitrary function /, and can certainly only be understood in terms of it. Weyl’s emphasis in this passage on the current density @, and its divergencelessness as basic to the law of conservation of electricity echoes what

he had already said in 1918 1°.

For we shall show that as, according to investigations by Hilbert, Lorentz,

Einstein, Klein, and the author, the four laws of the conservation of matter (the energy-momentum tensor) are connected with the invariance of the action quantity (containing four arbitrary functions) with respect to transformations of coordinates, so in the same way the law of the conservation of electricity is connected with the “measure-invariance”. But in 1929 he developed further the idea and expressed it as the divergenceless-

ness of the current density g,. In the language of physics today, this is called

local current conservation. It was elaborated on by Pauli (7%, p.

111, and 7°) and

exerted a great influence on my own thinking, as we shall discuss later.

The quote above from Weyl’s 1929 paper also contains something which is

very revealing, namely, his strong association of gauge invariance with general relativity. That was, of course, natural since the idea had originated in the first

place with Weyl’s attempt in 1918 to unify electromagnetism with gravity. Twenty years later, when Mills and 177.78 worked on non-Abelian gauge fields, our motivation was completely divorced from general relativity and we did not appreciate that gauge fields and general relativity are somehow

related.

Only in the late 1960’s did I recognize the structural similarity mathematically of non-Abelian

gauge fields with general relativity and understand

both were connections mathematically??. Before

proceeding

further

let us ask

what

has

happened

that they

to Einstein’s

original objection after quantum mechanics inserted an —i into the scale factor (3) and made it into a phase factor (8)? Apparently no one had, after 1929, 26 W. Pauli, Rev. Mod. Phys. /3, 203 (1941). 27 C_N. Yang and R. Mills, Phys. Rev. 95, 631 (1954); reprinted in '°, p. 171. 28 C_N. Yang and R. L. Mills, Phys. Rev. 96, 191 (1954).

29.73 of !®. The detachment of gauge field concepts from general relativity, in retrospect, was

an advantage because it allowed us to concentrate on one problem at a time.

18

Chen

Ning Yang

*

Fig.4. Aharonov-Bohm experiment. S is a solenoid with magnetic flux perpendicular to plane of paper.

relooked at Einstein’s objection until I did in 1983%°. The result is interesting and deserves perhaps to be a footnote in the history of science: Let us take

Einstein’s Gedankenexperiment

in Fig. 3. When

back,

the two clocks come

because of the insertion of the factor —i, they would not have different scales

but different phases. That would

not influence their rates of time-keeping.

Therefore, Einstein’s original objection disappears. But you can ask a further question: Can one measure their phase difference? Well, to measure a phase

difference one must do an interference experiment. Nobody knows how to do

an interference experiment with big objects like clocks. However, one can do

interference experiments with electrons. So let us change Einstein’s Gedankenexperiment to one of bringing electrons back along two different paths and ask: Can one measure the phase difference? The answer is yes. That was in fact

a most important development in 1959 and 1960 when Aharonov and Bohm*! realized — completely independently of Weyl — that electromagnetism has some meaning which was not understood before. They proposed precisely this experiment, with a slight variation of inserting a solenoid which carries a

magnetic

flux

inside.

Changing

the

flux

one

can

manipulate

the

phase

difference between the two paths. The experiment was done by Chambers*? in 1960.

For

an

analysis

of its significance

and

its

relationship

with

the

identification of the nonintegrable phase factor (8) as the essence of electromagnetism, see reference**. For discussions of other experiments related to

the Aharonov-Bohm effect see*°.

The third period during which Weyl wrote about gauge theory covers the

years from 1930 to his death in 1955. One finds Weyl referring to gauge theory in many of his papers throughout this period. For example, he referred to it in

1931 in a paper called “Geometrie und Physik.” He referred to it again in 1944 in a paper called “How far can one get with a linear field theory of gravitation in

flat space-time?”. If additional evidence is needed to demonstrate Weyl’s deep attachment to the gauge idea, one can look at the postscript (to the 1918 gauge

theory paper'') which he wrote, for inclusion in his Selecta, six months before

*° Chen Ning Yang in Proc. Int. Sym. Foundations of Quantum Mechanics (Tokyo, 1983),

Edited by S. Kamefuchi, H. Ezawa, Y. Murayama, M. Namiki, S. Nomura, Y. Ohnuki, Yajima, p. 5 (Phys. Soc. of Japan, 1984).

and T.

*"'Y, Aharonov and D. Bohm 1/5, 485 (1959). See also W. Ehrenberg and R. E. Siday, Proc.

Phys. Soc. London

B62, 8 (1949).

*? R.G. Chambers, Phys. Rev. Lett. 5, 3 (1960). In this connection see the discussion in 3° of

other experiments that are related to the Aharonov-Bohm effect. *° Tai Tsun Wu and Chen Ning Yang, Phys. Rev. D/2, 3845 (1975).

Hermann Weyl’s Contribution to Physics

19

his death in 1955. In this postscript one finds, explicitly stated once more, the

reason for his devotion to the idea**:

Das starkste Argument fiir meine Theorie schien dies zu sein, daB die Eich-

invarianz dem Prinzip von der Erhaltung der elektrischen Ladung so entspricht wie die Koordinaten-Invarianz dem Erhaltungssatz von Energie-Impuls. IIL. Weyl’s reason, it turns out, was also one of the melodies of gauge theory that

had very much appealed to me when as a graduate student I studied field theory

by reading Pauli’s articles**:*°. I made a number of unsuccessful attempts to generalize gauge theory beyond electromagnetism*°, leading finally in 1954 toa

collaboration

with

Mills

in

which

we

developed

theory?”:?5. In 77 we stated our motivation as follows:

a non-Abelian

gauge

The conservation of isotopic spin points to the existence of a fundamental

invariance law similar to the conservation of electric charge. In the latter case, the electric charge serves as a source of electromagnetic field; an important concept in this case is gauge invariance which is closely connected with (1) the equation of motion of the electro-magnetic field, (2) the existence of a current density, and (3) the possible interactions between a charged field and the electromagnetic field. We have tried to generalize this concept of gauge invariance to apply to isotopic spin conservation. It turns out that a very natural generalization is possible. Item (2) is the melody referred to above. The other two melodies, (1) and (3), were what had become pressing in the early 1950’s when so many new particles

had been discovered and physicists had to understand how they interacted with

each other. Thad met Weyl in 1949 when I went to the Institute for Advanced Study in

Princeton as a young “member”. I saw him from time to time in the next years, 1949-1955. He was very approachable, but I don’t remember having discussed

physics or mathematics with him at any time. His continued interest in the idea

of gauge fields was not known among the physicists. Neither Oppenheimer nor Pauli ever mentioned it. I suspect they also did not tell Weyl of the 1954 papers of Mills’ and mine. Had they done that, or had Weyl somehow come across our paper, I imagine he would have been pleased and excited, for we had put 34 Weyl, 1955; GA II, p. 42. In sharp contrast, Pauli had a negative attitude about the idea of

gauge fields in the last years ofhis life °°. In 1956 he wrote a series of Supplementary Notes for

the English translation of his 1921 article Relativitdtstheorie. The note on the “Theory of Weyl” was no more positive than his 1921 original article in German. 35 See A. Pais’s forthcoming book on the history of elementary particle physics and an article

by C. P. Enz on Pauli (reported at the June 1985 Symposium on the Foundation of Physics in

Joensuu, Finland.) See also °°. 38/Sce [16], p. 19.

Chen Ning Yang

20

together two things that were very close to his heart: gauge invariance and nonAbelian Lie groups.

Lie groups are mathematical objects that are deeply related to the concept

of symmetry in both everyday language and in the language of physics. One might say they are mathematical constructs representing (at least a major part of) the essence of the concept of symmetry. I venture to guess that Weyl’s fondness for symmetry, which I had mentioned before, had originated with his

deep penetration of the structure of non-Abelian Lie groups.

Symmetry, Lie groups and gauge invariance are now recognized, through

theoretical and experimental developments, to play essential roles in determining the basic forces of the physical universe. I have called this the principle that

symmetry dictates interaction>’. Furthermore, and this is very exciting, while great successes have been achieved in these developments, we are still far from a grand synthesis. I believe this is because the full meaning of the word symmetry is not yet understood and key additional concepts are still missing. In this

connection it is interesting to read what Maxwell had written a century ago when he discussed Faraday’s lines of force and the epistemological

mathematical ideas in physics**:

origin of

From the straight line of Euclid to the lines of force of Faraday this has been

the character of the ideas by which science has been advanced, and by the free use of dynamical as well as geometrical ideas we may hope for a further

advance. ...We are probably ignorant even of the name of the science which

will be developed out of the materials we are now collecting...

It is exciting that a century later we can be as hopeful as Maxwell was of great future developments in our evolving understanding of nature.

LV: I have outlined above a number of contributions that Weyl had made to physics and the impact they had on later developments. It is important to realize that together they represent only a small part of his thinking about

physics. As a physicist-philosopher, Weyl had written extensively about space, time, matter, energy, force, geometry, topology, etc., key concepts that provide the basis upon which modern physics is erected. When one reads Weyl’s papers

one is constantly amazed by how he had tried to puzzle out the structure of matter and of space with mathematical constructs. An interesting example is to be found in his 1924 paper called “Was ist Materie?” in which he raised the

question of topological structures in matter*°, a subject that has now become very popular.

37 See 19, 9.563.

58 J. C. Maxwell, Scientific Papers, Vol. 2, No. 61 (Cambridge Univ. Press, 1890).

39 Weyl, 1924: GA II, p. 510.

Hermann

Weyl’s Contribution to Physics

21

Weyl wrote beautifully. I do not know whether he also wrote poetry, but he

certainly enjoyed reading poems. In 1947 in the preface to his book Philosophy

of Mathematics and Natural Science he quoted T. S. Eliot: Home

is where one starts from. As we grow older

The world becomes stranger, the pattern more complicated Of dead and living. I venture to say that if Weyl were to come back today, he would find that amidst the very exciting, complicated and detailed developments in both physics and mathematics, there are fundamental things that he would feel very

much at home with. He had helped to create them.

Institute for Theoretical Physics

State University of New York at Stony Brook Stony

Brook, New

York

11794-3840,

(Received 6 January 1986)

U.S.A.

Hermann Weyl, Space-Time and Conformal Geometry Roger Penrose Itis a great pleasure for me to be in Zurich, with its many associations: Weyl,

Einstein,

Minkowski

— whose

work

has

been

so

close

to my

heart.

But

particularly, on this special occasion, I am honoured, privileged, and indeed

very flattered, to have this opportunity to pay my respects to Hermann Weyl,

whom I take to be the greatest mathematician of this century. In saying this, I am avoiding any arguments about Hilbert, Poincaré, or even Cartan, by insisting that when

I say “of this century”,

I mean

whose work

lies entirely

within the nineteen-hundreds. Since Weyl’s one-hundredth birthday is in two

day’s time, he must have been a mere fourteen at the turn of the century, so unless he was doing great things even at that time, I think that we can count him as being a mathematician entirely of this century! I should say, also, that as well

as being such a wonderful mathematician, Weyl made several very profound

contributions to theoretical physics. He was, I think, quite unusual, among pure mathematicians, in the extent and the great depth of his insights into the workings of the physical world. Professor Yang, in his lecture, has extensively covered several of the most

influential of Weyl’s contributions to physics; and likewise Professor Borel will be dealing with the part of Weyl’s work which has had the most impact on mathematics.

I am

not altogether

sure where

that leaves

me.

There

are, of

course, many other things in Weyl’s mathematical output which were of great

significance and profundity — and he had influence also in other areas such as logic, philosophy and perhaps even art. I could, I suppose, have chosen to base

my own lecture on any one of these. There will, indeed, be relevance to what I

have to say in Weyl’s contributions to some of these areas. But rather than

selecting one such topic, I shall use the fact that others will have dealt more

systematically with Weyl’s major contributions as an excuse for my being allowed to pay my respects while indulging in some flights of fancy of my own. I must apologize for this, but it would be hard for me to present my appreciation of Weyl in any other way. I would not be good at giving, say, a * A Hermann

Weyl Centenary

Lecture, ETH

Ziirich, November

7, 1985

24

Roger Penrose

survey of some particular topic or a historically accurate appreciation of the

impact of some of Weyl’s work. In fact I shall often simply take the fact that Weyl’s

has become

name

attached to a concept

as the indication of Weyl’s

seminal influence — despite the fact that I know that with some other names that would be a risky presumption! Indeed, I have many times been struck by the numerous and varied ways that Weyl’s name so often comes up in association

with ideas which have strongly influenced my own research. In much of what I shall say, I shall indeed refer to ideas which I have been involved with myself. No doubt this is a most inappropriate way to express my appreciation of another, and I ask that my audience forgive me for such indulgences. But perhaps Weyl would not be displeased that the influence of his work continues to be so strongly felt through research that some others are now actively pursuing.

Symmetry and Patterns I have mentioned that Weyl had an interest in art. The interplay between beauty and truth is something that held great fascination for Weyl. It is clear

from his writings that he believed that aesthetic qualities are of profound importance

in

mathematics,

and

that

truth

and

beauty

are

profoundly

intertwined. He was once asked where his sympathies would lie whenever truth and beauty might come into conflict — to which he replied: beauty, every time

(see Dyson

1979).

Weyl wrote a well-known book entitled “Symmetry” (Weyl 1952). This was

not really a book on art or aesthetics, but it brought together many ideas from the worlds of art, mathematics, physics and biology, and unified them in terms

of the concept of symmetry. There is perhaps also a hint of a subtle interplay

between the roles of symmetry and asymmetry in art in Weyl’s book. Figure 1, which is taken from the book — a photograph of a Greek statue from the 4"

century B.C. — shows a work of art with a magnificent symmetry, yet it is as much the slight deviations from exact bilateral symmetry which give this work

its particular

artistic

qualities.

(Later

on,

I shall

refer to one

of Weyl’s

contributions to physics for which significance lay in the fact that it introduced

a subtle asymmetry into the background of previously known symmetrical physics.) Figure 2 illustrates something taken from the world of biology. It illustrates star-fish and other creatures, each exhibiting a remarkable fivefold

symmetry. This is to be contrasted with the sixfold symmetry exhibited by the snowflakes of Fig. 3 — taken from the world of physics. Indeed, there is a standard argument, elegantly recounted in Weyl’s book, which shows that for any crystalline structure (as with the snowflakes just referred to) twofold,

threefold, fourfold, or sixfold symmetries can occur, but fivefold symmetries are forbidden. There is an incompatibility between the exact translational symme-

try, which a perfect crystal must possess, and a fivefold rotation axis. So it seems

Hermann

Weyl, Space-Time and Conformal

Geometry

Fig. 1. Greek sculpture, fourth century B.C

ps}

Roger Penrose

26

Fig. 2. Fivefold symmetry

that while fivefold symmetry

may

among lower animals

be encountered

in the world

of art or of

biology, we are not to find it in the crystalline structures of physics. Weyl also

exhibited crystal-like symmetries taken from the world of art — many islamic

designs, and, in particular, a beautiful fourteenth century window in a mosque

in Cairo (Fig. 4). Now have a look at Fig. 5. Though at first sight it seems to have the regularity of a crystalline structure, closer examination reveals that it is not

Hermann

Weyl, Space-Time and Conformal

Fig.

3.

Sixfold symmetry

Geometry

ei

in snowflakes

exactly repetitive — and, moreover, it seems to exhibit strong elements of the forbidden fivefold symmetry! This is not an illustration taken from Weyl’s book. The configuration was not known about at the time. It is something I came across somewhat over ten years ago, which, on a large scale, has many features in common with crystalline arrangements (Penrose 1974). Yet the completed pattern has infinitely many points about which it has approximate fivefold symmetry — to any preassigned degree of accuracy short of perfection.

The pattern is made up of four different shapes: regular pentagons, rhombuses,

pentagrams (i.e., five-pointed stars) and three-pointed “half-stars” — which I shall refer to as jester’s caps. Such an arrangement of shapes covering the plane without gaps and without overlapping is referred to as a tiling. There are, of

N O L O

tpG

peasererecsscntec!

oeaes

eee ee eG eKG de pee ess a cc COOH AREER

sioee

SO)

ch

ees eae

Sees

Fig. 5. Pattern with fivefo 1 id quasi-symmetry

Hermann

Weyl, Space-Time and Conformal

Geometry

29

Fig. 6. Six tiles which can be assembled only according to the pattern of Fig. 5

course, many other ways of tiling the plane with these particular shapes. In particular, we could simply dispense with all the shapes save the rhombus, and tile just with that one shape in the standard periodic way. However, we can

actually

force

the type of non-periodic

tiling shown

in Fig. 5 merely

by

modifying the shapes of the pieces slightly, according to the scheme shown in Fig. 6. There are now six different shapes, the pentagon occurring in three

different versions according to whether it is to be adjacent to five, three, or just

two other pentagons. The structure of the resulting assembly turns out to be

necessarily hierarchical in nature. The nature of this hierarchy is illustrated in

Fig. 7.

There are, however, many

regular and uniform features that the pattern

possesses which seem to go beyond its hierarchical construction. In particular, you will notice that there are many regular decagons in the pattern, each time the decagon being divided up in exactly the same way into three pentagons, two rhombusses and a jester’s cap. Sometimes two decagons overlap with one another, having a single rhombus in common. Moreover, wherever there is one such decagon, it is always surrounded with a complete ring of ten pentagons.

This happens even when two decagons overlap, the two rings of pentagons simply passing through one another. A further striking feature of the pattern is the way things line up in a very precise way. This is most easily seen by looking at the pattern along the page at an oblique angle. Each line segment is seen to

line up with a large number of others, and in fact this feature continues indefinitely. In a certain sense the pattern is built up from strips which are almost repetitive, somewhat resembling a crystalline structure. But the strips

occur in five different directions and are angled to one another at multiples of 36° (=7/5), which is crystalographically impossible! Although the pattern has translational motions that can be chosen to be as close as one likes to leaving

30

Roger Penrose

et tg gte Cg' s \/ mat oe

ee

er

eeegegt@:.

DOV C1 Shah

POF

DOVS-@. ett

eg

Ogee

wee

y~.

e'

ax) SAK fon% eae

Fig. 7. The hierarchical nature of the pattern of Fig. 5

the completed pattern unchanged, none of these motions is actually an exact

symmetry.

The set of six tiles depicted in Fig. 6 thus has a remarkable property: its

shapes will tile the plane only in a non-periodic arrangement. But it was by no

means the first set of tiles with this property. In fact, I should say something about the history of this problem. Suppose

that we are given a

finite set of

polygonal shapes and we are asked the question whether or not these shapes will tile the entire Euclidean plane. Is there a decision procedure (i.e.,a computer

program which will provide a definitive answer “yes” or “no” in each case) for

this problem? In around 1961, Hao Wang showed that there would indeed be a decision procedure, if it could be shown that any set of tiles, which can tile the

plane in some way, can actually tile it periodically (Wang

1961). I think that it

may have been generally felt at that time that the answer to this question would

surely be “yes”. (In fact Hao Wang’s original result applied to a slightly different problem concerning tiles with vertices on square lattice points, but this is not an essential distinction for us here.) However, in about

Wang’s leads, Robert Berger was able to show (Berger

1964, and following Hao

1966) that there is, in fact,

no decision procedure for this problem. (The argument involved showing how

one may, in effect, translate a universal Turing machine into a tiling problem.)

This established the existence of a set of tiles which will tile the plane only non-

periodically. The argument originally led to a set of something like 20,000 tiles

Hermann Weyl, Space-Time and Conformal Geometry

31

y— Fig.

8. Raphael

Robinson’s set of six non-periodic tiles

Fig. 9. Kites and darts — a non-periodic pair

with this property, though Berger was then able to reduce this Raphael Robinson (1971) was finally able to reduce the set down have depicted in Fig. 8. When I was told (or reminded) of this clear to me that my own set of six could be reduced down to at

number to 104. to the six that I fact, it became most five, since

the modified pentagon on the lower right can be eliminated by attaching it once to the bottom of the jester’s cap and twice to the rhombus. With some more

effort it proved possible to reduce the set to just two, as depicted in Fig. 9, where

32

Roger Penrose

Fig. 10. Assembled kites and darts and (on the right) their relation to Fig. 5

Fig. 11. A pair of non-periodic rhombuses the tiling arrangement and the relation to my original set of six are given in Fig. 10. These two shapes are referred to as kites and darts [as described by Martin Gardner (1977) in his column in the Scientific American]. An alternative

pair, consisting basically of two rhombusses, is shown in Fig. 11. An array tiled with these is depicted in Fig. 12. The pair is closely related to the original kitedart pair and, in an appropriate sense, lies between two stages of hierarchy for the kites and darts. There are other sets of tiles known which will tile the plane only in analogous quasi-periodic arrangements. Figure 13 exhibits a set of three tiles,

Hermann

Weyl, Space-Time and Conformal Geometry

Fig. 13. A set of three nonperiodic

33

tiles, by Robert Ammann

discovered by Robert Ammann not very long after I had found my pairs. In this

case, there is always eight-fold quasi-crystalline symmetry in the completed

assembly — which is also forbidden for an exact crystal (Fig. 14). There are also

three-dimensional analogues of the five-fold quasi-symmetric patterns, such as an arrangement that I had found quite early on, exhibiting regular icosahedral

quasi-symmetry (Fig. 15), in which interpenetrating dodecahedra meet each other along edges. The first finite set of solids which tile three-space only nonperiodically appears to be a set of four discovered by Robert Ammann in 1976.

Fig. 15. A configuration in three dimensions analogous to Fig. 5

A

Hermann

Weyl, Space-Time and Conformal Geometry

35

ae

Fig. 16. Nets for assembling Ammann’s four non-periodic solids; the spots must match

Nets for constructing these are shown in Fig. 16. The spots are to be matched

in the assembly.

Quasi-Crystals Suppose that the atoms of some quasi-crystalline substance were somehow arranged at the vertices of one of these patterns. How would we ascertain that

fact? One method might be to obtain a diffraction pattern from it by firing electrons at it. The diffraction pattern would be, in effect, the Fourier transform

of the original array of dots (vertices of the original pattern). Fourier transforms of these patterns were taken by MacKay in 1982, and later by Levine and

Steinhardt in 1984. They have the striking property that they consist of sharp

delta-function-like peaks (Bragg peaks) — the hallmark of an actual crystal —

except that the peaks for a quasi-crystal turn out to be dense, and they exhibit “impossible” non-crystallographic symmetries! Remarkably, actual electron diffraction patterns from actual substances (initially aluminium-manganese

alloy) were observed by Shechtman in 1983 (Shechtman et al. 1984) which have just such features. See Fig. 17 for a comparison between the experiment-

ally observed pattern and that calculated by Levine and Steinhardt. More

recently, electron micrograph pictures have been obtained of these substances

Roger Penrose

»

36

° 0

o jo

GC

Ol eo

Oo

6

O

°

to)

2

«

Ss

+

0

0

ve

©

=O

Oo

»

©

8

Ss

oe

b

So

O

&

4

)

6

6

Oo

GO

#6

oP

°) e@

&

°

°

oO ©

8s >

5)

© °

1

© o

Oo

0 5

*% -

(e)

fe)

5 0

@

as

2)

5

eG

s ©

cr) oF

°

3) Ss

=

“8

Ou

ey

a

ee °

°

Qe) a)

QO

oe Coe

q

©

ace

t

Oo

°o

° |

°

—

Fig. 17. An observed electron diffraction pattern (a) compared with a diffraction pattern calculated by Levine and Steinhard, 1984 (b). Both represent projections of the reciprocal

structure normal to the S-fold symmetry axis of icosahedral point symmetry. The electron diffraction pattern courtesy T. Ishimasa and H.-U. Nissen (compare Shechtman et al., 1984)

ae

ne

ae

a?

-—*

*

Fen e ©,

he

+ *

«Boma

*

a *s

*»*

*,

ia

’ Pr

o.

hae

“

a®

*

te ore

»

a ~

é

.

of

ee | ~~" e te g* "ne ’ ~*~

?

~~

*

a

*

a

% *¢

ets Os seeyeaehe yt

;

¢

as

*

Fig. 18. High resolution electron microscopic images of the structure of AI-Mn quasicrystal projected along the 5-fold axis of icosahedral point symmetry. The objective lens of the electron microscope has a defocus of — 50 +5 nm in (a)and —90

+5 nm in (b). Identical points

in both images are marked by arrows. The strong bright spots in (b) correspond to the vertices of the tiling with two rhombohedrons, while these points appear in (a) as dark centres of doughnut-shaped regions. In general, for the parameters used in Fig. (a), that micrograph should

give

optimum

correspondence

T. Ishimasa and H.-U. Nissen, ETH

with

Ziirich

the

atomic

structure.

Micrographs

courtesy

(Fig. 18) which appear to give some indication of the actual atomic locations,

although the exact interpretations remain somewhat obscure. Numerous pentagonal arrangements and alignments are evident, reminiscent of the patterns of Figs. 5, 10, 12. The balance of informed opinion appears to support the view that there is at least a very close relationship between such patterns and

the quasi-crystalline substances! found in nature.

' There appear to be at least three diflerent types of quasi-crystalline arrangements in nature the icosahedral, the five-fold (or ten-fold) with periodicity in one direction, and twelve-fold with periodicity in one direction. I learned about this last type on my visit to Zurich for the purpose of giving this lecture (Nissen et al. 1985). One can construct hierarchial twelve-fold tilings, but I had not known of such arrangements before seeing the experimental data! | do

not yet know how to force such arrangements with a finite set of tiles.

38

Roger Penrose

There is, as I see it, something extraordinary about this. For if one attempts to assemble such sets of tiles by hand, placing the pieces one by one where they will locally fit without much regard to the global hierarchical arrangement, then one is likely to go seriously wrong and come upon configurations which cannot be continued. That is to say, the correct assembly of these patterns depends, from time to time, upon a global examination of the assembled pattern at that stage. Whereas the rules for deciding whether or not a completed pattern has been correctly assembled are indeed local, the rules for correct piece-by-piece

assembly of such a pattern are non-local (so the mere fact that the pieces “fit” so far does

not ensure

that the

pattern

can

be continued

indefinitely).

So,

a

puzzling question is: How does Nature do it? This puzzle should be kept in

mind. I shall return to the matter at the end.

Weyl Space-Times Now I wish to turn to some completely different matters. What I have had to

say so far has not really had a very detailed connection with Hermann Weyl or his work, though there is much connection with its spirit. What I shall have to say next will have a great deal more to do with him directly. I want to say something about space-time structure according to Einstein’s general relativity. As far as I am aware, the very concept of a manifold — essential for any global

understanding — is due to Weyl. Riemann’s idea of a Riemann surface made a start towards this, but the general n-dimensional global concept appears to be due to Weyl. The idea of a manifold has very broad application within mathematics, and is certainly crucial to any global understanding of space-time structure (Weyl 1922).

I want also to refer to a specific solution (or class of solutions) of the Einstein equations referred to as the Weyl (or Weyl-Levi-Civita) solution (Weyl 1917,

1919b, Levi-Civita 1917). The space-times which come under this heading are the static axi-symmetric vacuums (i.e., with vanishing Ricci tensor) where some axi-symmetric source (singularity) can be present. To construct such a solution,

one has just to solve the axi-symmetric Laplace equation with appropriate

sources (and then apply quadratures). A particular example is obtained from

the solution of the Laplace equation describing the field of two mass points at a distance along the axis. The space-time that one obtains has an interpretation as a pair of masses propped apart by a rod. One can also re-interpret the very

same metric expression in a different way, namely as two masses held apart by two fishing lines extending to infinity (and also in various other ways). How can it be that the very same metric expression can have two so very different

interpretations? The answer lies in how the angular variable is identified about

the axis (Fig. 19). In the case of the rod, the identification is chosen so as to

maintain regularity on the axis in the region beyond the masses, and this leads

to a “conical” singularity along the rod; while, in the case of the fishing lines, a

Hermann

a

Weyl, Space-Time and Conformal

Geometry

1

oy

ees

\

39

5

b

\

fishing line

a non - singular

1

axis

etry

SHAT Ses

rod

ea

ee

' Ng—— i

‘:

‘

identify

€

rod

=

sections

across conical

>

singularities

identify

fishing line Fig. 19. The same Weyl! metric represents two masses (a) propped apart by a rod (b) held apart by two fishing lines

different angular identification is chosen, which makes

the axis between

the

masses regular but introduces a “conical” singularity along the portions of the

axis extending beyond the masses (cf. Zipoy 1966). There are also various other examples in general relativity theory of the same type of phenomenon (e.g., the “C-metric” of Levi-Civita, cf. Kinnersley and Walker 1970).

One reason for my mentioning this phenomenon, is that it recently occurred

to me that some structure of this kind may supply the solution to a problem that has been worrying me for a long time. The problem arose in connection

with twistor theory, but its exact nature or significance will not concern us here. What

the problem

required

was a way

of “exponentiating” a relative sheaf

cohomology group element to obtain some sort of deformation of a complex

manifold. It has been known for a long time (Kodaira and Spencer 1958) that

ordinary

deformations

of complex

manifolds

exponentiation of ordinary sheaf cohomology

can

be

obtained

from

the

group elements — in fact elements

of H'(X, @), where @ is the sheaf of holomorphic vector fields on some complex

manifold X. (Incidentally, the original idea of cohomology itself seems to have

been largely another of Weyl’s innovations!) What is needed here is some analogous procedure for exponentiating elements of a relative sheaf coho-

mology group — explicitly, elements of H*y(X, @), where Y is a closed subset

(normally a submanifold) of X. In the case of H'(X, @), we may think in terms of a Cech representative. This provides us with a holomorphic vector field on

40

Roger Penrose

Fig. 21. A Cech representative for H¥(X, @); vector fields defined on triple overlaps, but zero unless a set from the covering of Y

is included

each intersection of each pair of open sets of some covering of X. By displacing the sets of the covering relative to one another by finite amounts along the

vector fields, we obtain

the required

deformation

(Fig. 20). In the case of

H?,(X, @), a Cech representative provides vector fields on triple overlaps of

open sets of a covering of X, where at least one of the sets must in each case belong to a given covering of some neighbourhood of Y in X. A similar displacement to that given above, of the open sets along the vector fields, now

gives a deformation not of X but ofa neighbourhood of Y in X (Fig. 21). Thus,

we have a local deformation at Y which does not extend to the whole of X,

reminiscent of the local deformation of the strut in the above Weyl space-time which replaces the strut by a non-singular region, but which region does not extend to the whole of the space-time.

Hermann

Weyl, Space-Time and Conformal Geometry

41

Spinors My next topic is spinor theory. I gained my first real understanding of spinors from lectures given in Cambridge by Paul Dirac. One thing that particularly caught my attention was a very elegant demonstration that he gave

which

shows

that

a continuous

rotation

through

4x

can

be

deformed,

continuously, to no rotation at all. I learnt afterwards that this demonstration

was due originally to Hermann Weyl. It proceeds as follows: Consider two cones, of equal semi-angle «, one being fixed in space and the other constrained to roll on the first. When « is very small, the moving cone rolls once completely around the fixed one, returning to its starting position, and consequently executing a motion which is, in effect, a continuous rotation through 4x. We

now allow the angle « to increase continuously from 0 to 3, keeping the axis of the fixed cone fixed. This gives a continuous succession of closed motions. When « nears 3, the motion of the rolling cone is reduced to a mere wobble.

Thus

we

have

an

explicit

realization

of the

deformation

continuous 4z rotation to no motion at all. Weyl made fundamental contributions to the theory

which

takes

of spinors.

a

With

Richard Brauer (1935) he co-authored a seminal paper which based the full n-

dimensional discussion on a study of Clifford algebras. While the spinor concept had originally been introduced generally by Cartan, it is the Clifford

algebra approach

of Brauer and

Weyl

which is the one most

followed by

modern mathematicians and physicists. (There is, indeed, a considerable interest among physicists in higher-dimensional spinors.) In a good deal of what follows I shall need to adopt the notation of two-

component spinors (van der Waerden

1929, see Penrose and Rindler 1984).

According to this notation, each four-dimensional vector or tensor index (denoted by a lower-case Latin letter) is replaced by a pair of two-dimensional

spinor indices (denoted by unprimed and primed capital Latin letters). Ina local orthonormal frame, we have a standard translation scheme for the components

of the spinor translation V4“ of the vector V*, namely

aoe

es

oe

bal

1 (ee Ve

= V2

Viiv?

ie

ye_yp3)°

Tensor indices are raised and lowered by using the (symmetric) tensors g*”, g,, of signature (+ ——-—), and spinor indices by using the skew Levi-Civita

objects ¢-°,.c*

5 fans ene

This notation enables us to write down the Weyl equation for the neutrino

(Weyl 1929a):

Va =0,

where 44’ stands for covariant derivative (a contraction taking place over the index

A).

Mathematicians

sometimes

refer to this equation

as the Dirac

42

Roger Penrose

equation, although the usual Dirac equation for the electron couples two such fields together via a mass term. Weyl suggested that the above equation might

hold for a massless particle —as it turned out, in fact, the neutrino — but this got

him into trouble with Pauli, the person who first postulated the neutrino. Pauli had insisted that Weyl’s equation was contrary to all the evidence of nature, for it was asymmetric

under the operation of space-reflection

— whereas all the

evidence from atomic spectra overwhelmingly pointed to a universal symmetry

in natural laws with respect to this operation. Unfortunately, Weyl did not live

to witness the experimental demonstration that nature is actually not, after all,

symmetrical under space reflection (Lee and Yang 1956, Wu et al. 1957). The Weyl! equation is now an accepted equation for the neutrino!

Conformal Geometry and Conformal Tensors Another of Weyl’s contributions to physics — probably his most important

such contribution — was the paper (Weyl 19184; cf. also Weyl 19295) in which he introduced the idea of a gauge theory. The important development

of gauge

theories has been thoroughly treated by Yang in his lecture, so I shall not need to discuss it here. But Weyl’s work contained a host of other important ideas related to this. One of these was the idea of a connection in differential geometry, as something which can be defined independently of a choice of metric on the space. Christoffel and Levi-Civita had, in effect, shown how a unique connection — or, rather, a covariant derivative operator — arises from a concept of metric defined on a space; but in Weyl’s original gauge theory there was only a conformal structure defined, so he had to allow that a more general

concept of connection could arise which is not dependent upon a particular choice of metric having been made (although in his specific case he required

some weaker form of compatibility with the conformal structure). The idea of

a local projective structure and of a projective connection seem also to have

arisen from this line of Weyl’s work (see Weyl 1921). Of prime concern for us here will be the various invariant objects that arise in conformal geometry — the geometry that is defined by a metric (or pseudo-

metric) which is given only up to a local scale factor, so that angles are well-

defined but not lengths. In the Lorentz-signature case (i.e., a space-time) the information in the conformal geometry is precisely the same as that contained in the light-cone structure of the space-time. This light-cone structure is, indeed, the physically most important part of the metric information, for it defines the

causal relationships between events in the space-time. Weyl seems to have been the

first

to

make

a

systematic

study

of conformal

manifolds

of higher

dimension, the two-dimensional case having been treated earlier by Riemann (Riemann surfaces).

The

basic invariant

curvature

object, in dimension

n>3

is the

Weyl

conformal tensor C,,.4 (see Weyl 1918b), where (with square brackets denoting

Hermann

Weyl, Space-Time and Conformal

Geometry

43

anti-symmetrization) Capt

= Rag

4

— n=?

Reel gy

2.

+ (n—1)(n—2)

Roe

Gui" >

the Ricci tensor R,, and scalar curvature R being defined from the Riemann

curvature tensor R,,.g by

Ry=Ra

and

R=R,',

respectively. In four or more dimensions, the vanishing of the Wey] tensor is a

necessary and sufficient condition for the space to be (locally) conformal to flat

space. The symmetry and trace-free properties of the Weyl tensor look, at first, a little complicated: Cabea = Ciabitea} = Ceaab »

Crabeya =0,

Cy

.=0.

However, these conditions are really just a statement of irreducibility under

local rotations (Lorentz transformations) and describe an object which is, in the

case of four-dimensional (Lorentzian) space-time, in essence remarkably simple. This simplicity is particularly brought out in the spinor formalism, where the Weyl tensor has a description Caapercepy =

ancnéa'p een + EanecnY ancy’ >

the 4-valent spinor ¥ 4gcp (with complex conjugate V 4-c:p,) being equivalent in information to the Weyl tensor, and totally symmetric: Vy = Y apco= ¥ (ascn):

(Round brackets denote symmetrization.) This spinor always decomposes in a

unique

fashion

¥ 4gcp=%4hgYcOp),

scheme (known to Cartan

Which

leads

to a simple

classification

1922, cf. also Penrose and Rindler 1986) in terms of

the coincidences between the four null directions associated with «4, B4, 4, 04-

Recall that Einstein’s field equations tell us that the Ricci tensor of space-

time is directly determined by the distribution of matter fields. The full Riemann

tensor may be viewed as being a sum of two terms, one of which is the Weyl

tensor and the other, a tensor with equivalent information to that in the Ricci

tensor. Thus, the Weyl

tensor describes precisely that part of the curvature

which is not directly (i.e., locally) described by the matter density: the Weyl

curvature describes the purely gravitational degrees of freedom in space-time. This turns out to be a very fruitful point of view. Rather than thinking of the metric tensor as being the object which describes the gravitational field, we use,

instead, the Wey] tensor, this being an object which gives us much more specific purely gravitational information about the space-time. It is remarkable that

44

Roger Penrose

this information can be encoded in so mathematically elegant object as ¥ scp,

sometimes referred to as the Weyl spinor. In empty

become

space (vanishing

Ricci tensor), the

VOY

Bianchi

identity equations

sacp=0

which has a striking similarity with the Weyl neutrino equation above, and also with the spinor form of the free-space Maxwell equations:

V4" 0 4p=0 (Laporte and Uhlenbeck 1931). Here, the spinor object ~4, is symmetric PaB=P AB)

and is defined from the Maxwell field tensor F,, by

F 44:pp

=P aneae + €spP

ap

These differential equations are all particular examples of the source-free

massless field equations

VAG

ap, r=9,

bap.

=%

as...)

the spin being one-half the number of indices. These equations conformally invariant under local rescalings of the metric Sap

2

Q*Zay,

.

Exp

QEgp,

Eg

are

all

‘

QE gin,

where we take bap.1 72

“bap.1-

Note that in the case of gravity (spin 2) we already have conformal invariance for the Weyl spinor. But the relationship with the above invariance is a little bit subtle because the Weyl spinor conformal invariance occurs with a different factor: YP asco> P asco:

The Weyl spinor is a conformal object and gravity exhibits some conformally invariant properties, but gravitational theory as a whole is not conformally

invariant.

Hermann

Weyl, Space-Time and Conformal Geometry

45

There is also conformal invariance for the massless field equation for spin 0,

provided that we take the appropriate form of the wave equation, namely

with 722-76,

where I have now

reverted to a general n-dimensional description. It is my

understanding that the above modified wave (or Laplace) operator and its

conformal invariance are due to Weyl. Recently Eastwood and Singer (1985)

have found a modification to the square of the wave operator, in four dimensions, which is conformally invariant, but with a different scaling from the above, namely

WVIV"—2R® +3ReVP, acting on @ with Qo.

A new general theory of conformally invariant differential operators has been

put forward by Eastwood and Rice (1986) and developed further by Baston

(1986). I feel sure that this theory is something that would have been close to

Weyl’s heart since the methods used depend, to a considerable degree, on the representation theory of semi-simple Lie groups.

Conformal Conformal

geometry

Structure of Space-times

is important

for general

relativity also for other

reasons. I have mentioned that the causal relations between events (that is, between space-time points) are determined merely by the conformal metric of

the space-time and do not require the metric scaling. Moreover, the concept ofa

null geodesic (or light ray) is also conformally invariant. (I imagine that this fact

was known to Weyl, but I do not know who first obtained it.) These properties are important for the global study of space-times. The proofs of singularitiy

theorems depend, to a very large extent, on a detailed study of the causal structure of space-times and on the geometric properties of null geodesics (cf.

Penrose 1972, Hawking and Ellis 1973). For a good proportion of the discussion, space-times are viewed as conformal geometries, and only now and then does the actual scaling of the metric play a role. Thinking of a space-time as

46

Roger Penrose

infinity

Fig. 22. Conformal

representation of the open Friedmann model

a conformal geometry is also useful in other ways. In particular, it provides a method of studying the structure of infinity by, in effect, transforming infinity to a finite region and, as one can frequently do, treating “points at infinity” as

perfectly ordinary boundary points to a smooth conformal manifold. This is very useful for the study of gravitational radiation, for example. It is often much more convenient to treat such matters by examining a little local conformal geometry than to have to consider complicated asymptotic limits (cf. Penrose and Rindler 1986).

This same kind of idea can also be used in the reverse sense. Rather than

“squashing space-time transform physically both types

infinity inwards” to a finite hypersurface, one can sometimes take a singularity and “stretch it outwards” by a similar infinite factor and the singularity into a smooth hypersurface with a well-defined, determined conformal geometry. (See Fig. 22 for an example having of conformal boundary at once.) In fact, this procedure only works

well for certain very “regular” types of singularity. However, a singularity of this type indeed occurs at the “big bang”

of a standard

Friedmann-Robertson-

Walker cosmological model (that is, of a spatially homogeneous and isotropic

space-time). Moreover, it is a remarkable observational fact that: in our actual universe, there is a very close accord with such models, and this agreement gets

better and better the farther we explore back in time (i.e., the closer and closer

we get to the big bang). Thus, this procedure is very appropriate for studying the early structure of the actual universe. The procedure yields particular insights

into the nature of cosmological horizons.

The Origin of the Second Law of Thermodynamics and

In the reverse direction in time, we have singularities inside black holes —

also

singularity

possibly

at a “big

possessed

by

some

crunch”,

the

cosmological

final

all-embracing

models.

However,

space-time such

final

Hermann Weyl, Space-Time and Conformal Geometry

47

singularities would not be expected to have the regularity of conformal structure that the initial big bang apparently had. Ina final singularity the Weyl

curvature would be expected to diverge to infinite values; whereas in the initial singularity the Weyl curvature seems to have been zero (or, at least, very small).

This distinction is related to the entropy content of the singularity, and to the Second Law of Thermodynamics. The Second Law states that the entropy of the

universe increases with time. Thus, if the universe begins and ends with a singularity, the entropy in the initial singularity must be much lower than that

in the final singularity. The distinction lies in the type of geometry which can

occur near the singularity. As far as the matter content of the universe is concerned, one can assume effective maximization of the entropy both at the initial and final singularities. At least, that is the way that the calculations are performed at the big bang: thermal equilibrium for the matter is assumed (superimposed on an expanding background), and some excellent agreement with observational fact is obtained on the basis of that assumption. The lowness

of the initial entropy was not a feature of the matter content of the universe, but

was a feature of the very special geometry that occurred at the big bang. In effect, the gravitational degrees of freedom —i.e., the Weyl curvature components — were (for some unknown reason) set at zero (or small) in the initial state. The big bang seems to have been a remarkably “regular” type of singularity, of closely Friedmann-Robertson-Walker form, with zero (or extraordinarity small) Weyl

curvature.

Owing to the smallness of the gravitational coupling constant, it takes a

long while before the gravitational degrees of freedom — as measured by the Weyl tensor — can themselves become “thermalized”. Only gradually, as the

initially uniform matter in the universe begins to clump together owing to its

mutual gravitational attraction, does Weyl curvature begin to appear, largely in the spaces between the clumped regions of matter. Galaxies are formed with

their stars — those condensed lumps of matter which become “hot spots” of thermal imbalance. One of these hot spots (our sun) provides the thermal imbalance (or entropy sink) that is taken advantage of by the plant life of this

earth. We, in turn, take advantage of the low entropy in the plants in order to live. It is important to realize that it is gravitational clumping that enables the sun

to shine. Nuclear reactions actually slow down the sun’s inward collapse and

maintain the sun’s temperature at a value conducive to life. Entropy increases

in our solar system, just as it does everywhere else. Life merely takes advantage

of the comparative lowness of the entropy of the sun — and the lowness of that

entropy is due to the fact that the sun has been able to call upon the low

gravitational entropy of the uniform matter distribution from which the sun condensed. In a gravitational context, the uniform distributions (small Weyl curvature) tend to have low entropy, and the more clumpy that the distribution becomes (i.e., with larger and larger Weyl curvature) the higher will be the

gravitational contribution to the entropy. (This is owing to the universally attractive

nature

of the gravitational

field.) In a collapse

to a black

hole,

48

Roger Penrose

absolutely enormous values of entropy can be achieved (according to the Bekenstein-Hawking formula, cf. Wald 1984). Even larger values can be

achieved in a big crunch — the “ultimate black hole”. The final singularity has a

complicated Weyl curvature diverging to infinity.

Since it is the high entropy states which are to be regarded as “probable” states, the mystery that the Second Law presents us with is: why did the

universe start off in such a “highly improbable” initial state? We should also

take note of the particular way in which it seems to have been “improbable” namely that the Weyl curvature started at zero (or, at least, very small). Setting

aside, for the moment, the question as to what the reason might be, one can at

least formulate a conjecture (which I have referred to as the Weyl curvature

hypothesis, cf. Penrose 1986) to the effect: in any initial space-time singularity

the Weyl curvature tends to zero in a parallelly propagated frame as the singularity is approached from a future direction. From this hypothesis one

can, in effect, derive both the Second Law and the striking uniformity of the universe in the large.

Objective State-Vector Reduction? I should like to end on a somewhat more speculative note. It is not unreasonable to assume that the physics which governs the structure of spacetime singularities is some hitherto undiscovered theory of quantum gravity.

Since the Weyl curvature hypothesis refers to initial singularities and not to

final ones, such a quantum gravity theory would have to be time-asymmetric.

Now this is not a conventional view amongst theorists. But in my opinion, it isa

necessary ingredient in any ultimately successful theory of quantum gravity. In my view, also, the successful theory would need to involve a change in the very

rules of quantum mechanics. There is, indeed, something very unsatisfactory about

the

mechanics

evolution

strange

must

of

a

hybrid

way

be applied,

physical

in

with

system:

which

two

the

standard

mutually

deterministic

rules

of quantum

(say

Schrédinger)

incompatible

unitary

modes

of

evolution on the one hand ~ so long as no-one is “looking” — and probabilistic

non-unitary state-vector reduction on the other, whenever an “observation” is being made. In my opinion, some more comprehensive and uniform evolution

procedure is required which would incorporate both of these different types of evolution as limiting cases. In the absence of a theory of such an evolution procedure, one requires, at

least, some objective criterion for determining those circumstances under which unitary evolution is going to be an excellent approximation to the truth, and ? There

is a currently popular viewpoint that this uniformity can be deduced as a consequence

of the “inflationary

model

of the universe”.

“inprobable” starting assumptions deduction can be made.

must

be

However,

this is a fallacy, and

postulated

before

such

some

equally

a time-asymmetric

Hermann

Weyl, Space-Time and Conformal Geometry

49

those circumstances under which state-vector reduction is closely approximated. In my view, the distinction between these two cases lies in whether gravity — in the sense of sufficient change in the Weyl curvature — is effectively absent or present in the process under consideration. One clue to this can be gleaned if we consider the implications of the Weyl curvature hypothesis. Suppose we imagine an isolated container with reflecting walls, which has in it a large amount of matter. At certain times in its history, there will be a black hole in the container, while at other times not. We consider

the phase space

of the contents, and divide it into two

regions

B and

A

according to whether there is, or is not, a black hole in the container. The Weyl curvature hypothesis is assumed, and this precludes “white holes” — the timereverses of black holes — because their singularity structures are unsuitable; and

it also precludes space-time singularities of all other types than those arising from black holes. Black holes can form by a process of c

collapse, but

they can only disappear via the quantum process of Hawking evaporation. This has the implication that there are fewer flow lines leaving the region B than there are entering it (Fig. 23). This entails a violation of Liouville’s theorem (or its quantum analogue) in region B since there must be destruction of phasespace volume (convergence of flow lines) in its interior. This is not really so surprising, because information is lost at the black hole’s singularity. It is a loss rather than a gain because the singularity is of “final” rather than “initial” type — the latter being generally excluded because of the Weyl curvature hypothesis. There is, however, a problem with regard to region A. Here we have creation

of phase-space volume (production of new flow lines) to compensate for the loss in B; but there are no black holes or singularities in the container when

phase-space point is in

the

A! What, then can be responsible for this increase in flow

Fig. 23. Phase space for isolated container; phase-space volume lost in the black-hole region B is regained in A — perhaps because of objective state-vector reduction

50

Roger Penrose

lines at places in region A, where there appears to be no exotic physics taking place? My suggestion is that this is the phenomenon of quantum state-vector

reduction.

For

state-vector

reduction

is probabilistic,

not

deterministic:

different outcomes can result from the same input. Accordingly, several different flow lines in phase-space can branch out from a single flow-line input.

If this contention is correct, we see that state-vector reduction is “the other side

of the coin” from the Weyl curvature hypothesis, and that, indeed, state-vector

reduction must be a gravitational phenomenon.

Let us try to see whether this is at all plausible from the point of view of orders of magnitude. The idea will be that the linear superposition of states will cease to be maintained by nature as soon as the states become significantly

differently coupled in to the gravitational field. Imagine two different distributions of mass-energy initially occurring as part of a quantum linear superposition. When the gravitational fields of the two distributions become significantly different, I would contend, then linear superposition fails, statevector reduction takes place, and one or the other of the two states is singled out. What do I mean by “significantly different”? I mean that the difference between the two Weyl tensors (which can be assumed, at this approximation, to

be linear fields ona

flat space-time background) is a spin 2 field whose graviton

number count is at least one graviton.

Let us consider a particular example. Imagine a symmetrical atom which

decays, emitting an electron. As the electron leaves the atom, its wave function is

a spherically

symmetric

wave — a linear superposition

of all the possible

straight-line paths outwards from the atom. Let us suppose that some way out from the atom, and surrounding it, there is a cloud-chamber particle detector. When

the detector is entered, each possible electron

path causes a streak of

ionization. These different streaks again occur all at once, in linear superposition. The various centres of ionization begin to collect droplets around them. But

when

each

superposed

track

of droplets

reaches

a certain

size,

its

gravitational field will become significant — and at that point, I am suggesting, state-vector reduction objectively takes place, and the reality of the world

becomes just one of the tracks, rather than the linear superposition that it had been just a moment earlier. (Somewhat related ideas had been put forward

many years ago by Karolyhazy 1966, 1974.) At what droplet size should this happen, according to the ideas that I am

proposing? I have not yet been able to make an estimate for a whole track of droplets, but let us simplify the situation and consider just a single droplet. As an idealization, I assume that a spherical region of radius R condenses down to

a symmetrical droplet of radius r leaving a vacuum between the spheres, the mass of the droplet being m. There is Weyl curvature only in the vacuum region,

and one can estimate the graviton number arising, the formula being

7682°Gh_ where G

‘c~*m? log(R/r),

is the gravitational constant, c the speed of light and h is Planck’s

constant divided

by 2z. (I am

indebted

to A. Ashtekar

for supplying

the

Hermann

Weyl, Space-Time and Conformal Geometry

St

numerical coefficient and confirming the logarithmic expression.) This is of order unity, for a log term of order unity, when m~10~7

grams,

which is not totally unreasonable.

Let me now return to the “puzzle” that I referred to earlier concerning the

growth of quasi-crystals. If, indeed, they are accurate assemblys similar to the pentagonally quasi-symmetric tilings described in Figs. 5, 10, and 12 (and this is still not totally clear), then there appears to be something essentially non-local

about their formation. I should like to suggest that this formation is something

essentially quantum-mechanical. Rather than the atoms being laid down one-

by-one,

we

might

envisage

a quantum

superposition

of vast

numbers

of

different arrangements. One of these arrangements — say a quasi-crystalline one —may, as a whole, find favour for energetic reasons. So, when the size and energy distribution becomes right for a graviton count of order unity, one of these energetically favoured arrangements would, according to the ideas suggested above, be reduced out. There is, of course, a good deal of speculativeness in these suggestions. I

hope, nevertheless, that they may

provide some possible useful hints as to

directions for further developments in physical theory. Perhaps, moreover, they

give some idea of how varied and significant are the implications of just one of Hermann Weyl’s concepts, namely the Weyl conformal tensor. It is merely one

concept among so many that Wey]! has introduced, but it is one that has been particularly valuable for my own

attempts at understanding.

References Baston, R.J. (1986) The algebraic construction Thesis, University of Oxford

of invariant differential operators,

Brauer, R., Weyl, H. (1935) Spinors in n dimensions, Am. J. Math.

D. Phil.

57, 425-449

Berger, R. (1966) The undecidability of the domino problem, Mem. Amer. Math. Soc. no. 66

Cartan, E. (1922) Sur les équations de la gravitation d‘Einstein, J. Math.

141-203 (cf. p. 194)

Pures et Appl.

1,

Dyson, F.J. cited by S. Chandrasekhar, Physics Today 32, (1979), p. 27 Eastwood, M.G., Rice, J. Conformally invariant differential operators (to appear) Eastwood, M.G., Singer, M.A. (1985) A conformally invariant Maxwell gauge, Phys. Lett. 107A, 73-74 Gardner, M. (1977) Extraordinary nonperiodic tiling that enriches the theory of tiles, Sci. Amer. 236, no. 1, 110-121 Hawking, S.W., Ellis, G.F.R. (1973) The Large-Scale Structure of Space-Time (Cambridge University Press, Cambridge) Karolyhazy, F. (1966) Nuovo Cim. A 42, 390

Karolyhazy, F. (1974) Gravitation and quantum mechanics of macroscopic bodies, Magyar

Fizikai Polyoirat 12, 24 Kinnersley, W., Walker, M. (1970) Uniformly accelerating charged mass in general relativity, Phys. Rev. D 2, 1359

32

Roger Penrose

Kodaira, K., Spencer, D.C. (1958) On deformations of complex analytic structures I, II, Ann.

Math. 67, 328-401, 403-466

G.E. (1931) Application of Spinor Analysis to the Maxwell and Dirac Laporte,O., Uhlenbeck,

Equations, Phys. Rev. 37, 1380-1552

Lee, T.D., Yang, C.N. (1956) Question of parity conservation in weak interactions, Phys. Rev.

104, 254-258

Levi-Civita, T. (1917) ds? Einsteiniani in campi

Rend. Acc. Lincei 26, 307

Newtoniani,

Levine, D., Steinhardt, P. (1984) Quasicrystals: a new class of ordered structures, Phys. Rev.

Lett. 53, 2477

MacKay, A.L. (1982) Crystallography and the Penrose pattern, Physica, 114A, 609 Nissen, H.-U. (1985) Material + Struktur-Analyse, Dec. 85, p. 21; T. Ishimasa, H.-U. Nissen,

Fukano, Phys. Rev. Lett. 55 (July 1985), 511-513 (1972) Techniques of Differential Topology in Relativity, CBMS Regional Conf. Appl. Math., No. 7 (S.LA.M., Philadelphia) (1974) The rdle of aesthetics in pure and applied mathematical research, Bull. Inst.

and Y. Penrose, R. Ser. in Penrose,R.

Math. Applications 10, no. 7/8, 266-271

Penrose, R, (1979) Pentaplexity, Mathematical Intelligencer 2, 32-37 Penrose, R. (1986) Gravity and State- Vector Reduction, in Quantum Concepts in Space and Time, eds. R. Penrose and C.J. Isham (Oxford University Press, Oxford) 126-146 Penrose, R., Rindler, W. (1984) Spinors and Space-Time, Vol. 1: Two-Spinor Calculus and Relativistic Fields (Cambridge University Press, Cambridge)

Penrose, R., Rindler, W. (1986) Spinors and Space-Time, Vol. 2: Spinor and Twistor Methods in Space-Time Geometry (Cambridge University Press, Cambridge)

for tilings of the plane, Invent.

Robinson, R.M. (1971) Undecidability and nonperiodicity

Math. 12, 177-209 Shechtman, D., Blech, I., Gratias, D., Cahn, J.W. (1984) Metallic phase with long-range orientational order and no translational symmetry, Phys.

van der Waerden, B.L. (1929) Spinoranalyse Nachr. Akad.

100-109

Rev. Lett.

53, 1951

Wiss. Gétting. Math.-Physik K1.,

Wald, R.M. (1984) General Relativity (Chicago University Press, Chicago)

Wang, H. (1961) Proving theorems by pattern recognition — II, Bell System Tech. J. 40, 141

Weyl, H. (1917) Zur Gravitationstheorie, Annalen der Phys. 54, 117; (1919) 59, 185 Weyl, H. (1918a) Gravitation und Electrizitat, Sitz. Ber. Preuss. Ak. Wiss. 465-480 Weyl, H. (1918b) Reine Infinitesimalgeometrie, Math. Zeit. 2, 384-411 Weyl, H. (1919a) Eine neue Erweiterung der Relativitatstheorie, Annalen

101-133

Weyl,

H. (1919b)

Bemerkung

iiber die axialsymmetrischen

Gravitationsgleichungen, Annalen der Phys. 59, 185-188

Lésungen

der

der

Phys.

59,

Einsteinschen

Weyl, H. (1921) Zur Infinitesimalgeometrie: Einordnung der projektiven und konformen Auffassung, Nachrichten der Kéniglichen Gesellschaft der Wissenschaften zu Gottingen Math.

Phys. Klasse, 99-112

Weyl, H. (1922) Space-Time-Matter (Methuen, London)

Weyl, H. (1929a) Elektron und Gravitation I, Z. Phys. 56, 330-352 Weyl, H. (1929b) Gravitation and the electron, Proc. Nat. Acad. Sci. 15, 323-334 Weyl, H. (1952) Symmetry

(Princeton

University Press, Princeton)

Wu, CS., Ambler, E., Hayward, R., Hoppes, D., Hudson, R. (1957) Experimental test of parity conservation in beta decay, Phys. Rev. 105, 1413-1415; Further experiments on beta decay

of polarized nuclei, Phys. Rev. 106, 1361-1363

Zipoy, D.M. (1966) Topology of some spheroidal metrics, J. Math. Phys. 7, 1137

Mathematical Institute, Oxford, U-K. and

Rice University, Houston, Texas, U.S.A.

(Received 3 March 1986)

Hermann Weyl and Lie Groups Armand Borel During the first thirteen years or so of his scientific career, H. Weyl was concerned mostly with analysis, function theory, differential geometry and

relativity theory. His interest in representations of Lie groups or Lie algebras! and Invariant Theory grew out of problems raised by the mathematical underpinning of relativity theory:

“...but for myself I can say that the wish to understand what really is the

mathematical substance behind the formal apparatus of relativity theory led me to the study of representations and invariants of groups;” [W 147: p. 400]. This involvement, at first somewhat incidental, turned

rapidly into a major one. Weyl soon mastered, furthered and combined existing

techniques, and

contributions.

within two

years announced

in 1924 a number

of basic

The first serious encounter with Lie group theory arose out of a problem considered in the 4th edition of ,,Raum Zeit, Materie“ [W I] on the nature of the

metric in space-time, an analogue in this context of the Helmholtz-Lie space problem, which aims at characterizing the orthogonal group by means of some

general mobility axioms. As you know, the mathematical framework of general relativity is a 4-manifold,

say M,

endowed

with

a Riemannian

metric

(of

Lorentz type). The latter, in particular, assigns to x € M a quadratic form on the tangent space T,(M) at M. Riemann himself had already alluded to the possibility of considering metrics associated to biquadratic forms or more general functions. H. Wey] investigated, for manifolds of arbitrary dimension n, whether it was possible to start from a general notion of congruence, defined at each point by a closed subgroup G of automorphisms of the tangent space at x, belonging to a given conjugacy class of closed subgroups of GL,(R),? and deduce from some general axioms that it would be conjugate to the orthogonal

group ofa (possibly indefinite) metric. This would then prove that the notion of congruence was associated to a Riemannian metric of some index. The problem was reduced to showing that the complexification of the Lie algebra of G was * A Hermann Weyl Centenary Lecture, ETH Ziirich, November 7, 1985.

54

Armand

Borel

conjugate in gl,(C) to that of the complex orthogonal group. [Originally, it dealt with real Lie algebras. However, since only the quadratic character of the metric, and not the index, was at issue, it was sufficient to consider their

n=2, 3 and,

complexifications.]* In [WI], H. Weyl stated he could prove it for

shortly afterwards, published a general proofin [ W 49]. It was a rather delicate

and long case-by-case argument, which H. Wey] himself likened to mathematical tightrope dancing (,,mathematische Seiltanzerei*). E. Cartan read about the problem in the French

published

in 1922, and

lost no time

in providing

translation

a general

of [WI],

proof in the

framework of his determination of the irreducible representations of simple Lie algebras [C 65]. It was more general and more natural than Weyl’s, even than Weyl’s somewhat streamlined argument in [W II]. The comparison between

them appears to have been a strong incentive for Weyl to delve into E. Cartan’s

work. Whether he had other reasons I do not know; at any rate he did so not

long after, with considerable enthusiasm, as he was moved to write later on to E. Cartan (3.22.25): »seit der Bekanntschaft mit der allgemeinen Relativitatstheorie hat mich

nichts so ergriffen und mit Begeisterung erfiillt wie das Studium Ihrer Arbeiten tiber die kontinuierlichen Gruppen.* [W I] makes ample use of tensor calculus. A student of Weyl considered the

problem of showing that the usual tensors form a family characterised by some

natural conditions. This eventually amounted to proving that all the Lie group homomorphisms

elements

of GL,(R)*

in GL,(R)

with

into

GL,(R),

where

positive determinant,

GL,(R)*

is the group

are compositions

of

of inner

automorphisms, passage to the contragredient, and sums of maps M+>|det M|* (a€R)[Wn].* About thirty years later, that former student told me that, at the

time, Wey] was clearly broadening his interest in representations of semisimple

Lie groups, and had suggested that he work further along those lines. He even felt he might have shared some of the excitement to come, had he done so, but he had preferred to go back to his main interest, analysis,” and became indeed a well-known analyst: I was talking about Alexander Weinstein.

Another push into Lie groups came, in a way, again from the tensor calculus

in [WI], but for a completely different reason. In 1923 E. Study, a wellknown expert in Invariant theory for over thirty years, published a book on Invariant Theory [St2]. In a long foreword, he complained that Invariant Theory, in particular the so-called symbolic method to generate invariants, had been all but forgotten and that several mathematicians did less by other methods than would be possible using it. Among those was H. Weyl, identified by a quote,

criticized for his treatment of tensors. Apparently H. Weyl was stung by this.

This can already be seen by the rather sharply worded footnote in his answer to Study, which makes up the first part of [W 60], but it was also well remembered

about 25 years later by one of his colleagues here at the time, M. Plancherel, who mentioned it to me then as an example of the extraordinary ability H. Weyl had, shared only by J. von Neumann among the mathematicians he had known,

to get into a new subject and bring an important contribution to it within a few

Hermann

Weyl and Lie Groups

55

months. In fact H. Weyl published two papers on Invariant Theory in 1924,

[W 60: 1], [W 63], which

brings me to the achievements already alluded to

before. For the sake of the discussion I shall divide them and more generally Weyl’s

output in this area into two parts, one concerned with linear representations of semisimple

Lie groups, complex

or compact,

and

semisimple

Lie algebras,

which operates with Lie algebra techniques and transcendental means, and is tied up to the real or complex

numbers,

and

one concerned

with

Invariant

Theory and representations of classical groups, initiated by [W 60, W 63], in which Weyl wears an algebraist’s hat. This will be convenient to me as an organisational principle, but is, of course, to some extent artificial and should

not be construed as a sharp division. In the following years, H. Weyl wrote a number of papers on both, but remained actively interested for a longer time in the second one, and I shall discuss it later.

At that time, the outstanding contribution to the former one was the work of

E. Cartan which H. Weyl was discovering [C 5, C 37]. A second stimulus was

provided by two papers of I. Schur [Sc 2] about representations of the special

orthogonal group SO,, (as well as of the full orthogonal group O,, but I shall limit myself to the former) and invariants for SO,(C), in which he, in particular, extended the theory of characters and orthogonality relations known for finite groups by the work of Frobenius and Schur done at the turn of the century. For later reference also, let me recall briefly some features of the latter.

Let G be a finite group, G the set of equivalence classes of irreducible complex representations of G, and F(G) the space of complex valued functions on G. On F(G) there is a natural finite dimensional Hilbert space structure, with

scalar product given by

(1)

(LI=IG'

Y fOdge)

geG

(LgeF@)

where |G| is the order ofG. Given an irreducible representation 1: G>GL,(C) by matrices (aj(g)), let V, be the vector subspace of F(G) generated by the

coefficients (a'). It depends only on the equivalence class [7] € G of z(and can be defined more intrinsically) and will also be labelled by [7]. Moreover, we have the orthogonal decomposition

(2)

FG)= neG @Y,,

and the aj form a basis of V, (orthogonal if x is unitary). The space F(G) is a Gx Gmodule via left and right translations, and the V, are the irreducible G x G submodules. If E, is a representation space for x, then V,~EndE,, hence is

isomorphic to E,.®E,

as

a Gx G module, where n* is the contragredient

representation to z. As a G-module under right (or left) translations F(G) is the

56

Armand

Borel

regular representation of G, and V, is isomorphic to the direct sum of d, copies of x, (or x*), where d, is the degree of z. Let y, be the character of z.° It belongs to the space

(3)

CG)={feFGlfixyx

=f)

(&, yeG)}

of class functions on G; we have the orthogonality relations

(4)

IGG

and the x, (7G)

te )= Sun

(tn

EG)

form an orthogonal basis of C(G).

A very simple consequence of the orthogonality relations is that the average over G of the character of a given representation x gives the dimension of the

space of fixed vectors.

The key for the extension of these results to orthogonal groups was provided by a paper of Hurwitz [Hu]. The main concern there was the

invariant problem for SL,(C) and SO,(C): given a finite dimensional

holo-

morphic representation of one of these groups on a space E, show that the invariant polynomials on E form a finitely generated algebra. The known averaging procedure for finite groups to prove this could not be directly applied since SO,(C) or SL,(C) were not “bounded”, but it could be to their “bounded” subgroups SO, and SU,, and that was sufficient, since a holomorphic function on SO,(C) [resp. SL,(C)] is completely determined by its restriction to SO,,

(resp. SU,), in the same way as a holomorphic function on an connected open set ofC is determined by its restriction to a line. This was the first instance of what H. Weyl

first called the “unitary restriction” (,,unitare Beschrankung*),

and later [W VI] “unitarian trick”.

I. Schur, whose initial motivation was also Invariant Theory, extended the

orthogonality relations to SO,, drew the consequence about the average of a character to compute the dimension of the spaces of invariants in a representation of SO,(C), using Hurwitz’s integration device, and then went over the

determination of all the continuous irreducible representations of SO,, their

characters and dimensions. He also checked that these representations were in fact analytic and, even more, rational in the sense that their coefficients could be expressed as polynomials in the entries of the elements of SO,. Also full reducibility of finite dimensional representations could be established as in the

finite group case, by the construction of an invariant positive non-degenerate hermitian form, integration.

H. Weyl

where

was now

the averaging

over

the

group

ready to strike. He first extended

was

replaced

by

an

Schur’s method

to

SL,(C) and the symplectic group Sp,,(C) [W 62] and then almost immediately

afterwards combined the Hurwitz-Schur and the Cartan approaches in an extraordinary synthesis, announced first in the form ofa letter to Schur [W 61].

Until Weyl came on the scene, neither did Cartan know about the work of

Hermann

Weyl and Lie Groups

a7

Hurwitz and Schur nor did Schur about E. Cartan’s, as can be seen from the

introduction to [Sc 2] in the latter case, from a letter of E. Cartan to H. Weyl (3.1.25) in the former case. I. Schur expressed his admiration for the results of H. Weyl and shortly afterwards suggested that Weyl write them up and publish

them in Math. Zeitschrift, which was soon done [W 68]. In these papers

H. Weyl

first discusses separately the series of classical

groups: SL,(C), SO,(C), and Sp,,(C), and then sets up the general theory. A first

main goal is to prove the full reducibility of the finite dimensional represen-

tations of a complex semisimple Lie algebra, a problem which E. Cartan had

hardly alluded to in print before 1925. To be more precise, E. Cartan had in

[C 37] given in principle a construction of all irreducible representations of a given simple algebra’ and had just not considered more general ones. However,

H. Wey! pointed out that, as far as he could see, an argument of E. Cartan at one important point could be justified only if full reducibility were available.® The key point was to show that the “unitary restriction” could be applied in

the general situation. This was done in two steps: first H. Weyl showed that a

given complex semi-simple Lie algebra g has a “compact real” form q,, i-e., a

real Lie subalgebra such that g = 9, ®@gC, on which the restriction of the Killing

form is negative nondegenerate.° For instance, if g=sl,(C) one can take for q, the Lie algebra of SU,,. To this effect, H. Weyl had first to outline the general

theory of semisimple Lie algebras, for which the only sources until then were the

papers of W. Killing and Cartan’s Thesis, all extremely hard to read, and then

prove the existence of g, by a subtle argument using the constants of structure. Already this exposition, which among other things, stressed the importance ofa

finite reflection group (S), later called the Weyl group, was a landmark and for

many years the standard reference. But there it was really only preliminary material. Identify g to a subalgebra of gl(g) by the adjoint representation, which

is possible since g, being semisimple, is in particular centerless, and let G° be the

complex subgroup of GL(qg) with Lie algebra g. It leaves invariant the Killing

form K, defined by .#4(x, y) =tr(adx Oady), which is nondegenerate by a result

of E. Cartan. Let then G? be the real Lie subgroup of G° (viewed now as a real Lie group) with Lie algebra g,. Since the restriction K,, of K, to g, is negative

nondegenerate, it can be viewed as a subgroup of the orthogonal group of K,,,

hence is compact.!° This is a situation to which the Schur-Hurwitz device can be applied,

therefore any finite dimensional representation z of g which integrates to one of

G° is fully reducible. However, in general, a representation z of g will integrate

to one not of G°, but of some covering group G, of G°. In the latter, there is a

closed real analytic subgroup G,,,, with Lie algebra g,, and the Schur-Hurwitz

method will be applicable only if G,,_, is itself compact, i.e., is a finite covering of

G°, which is equivalent to G,,,, having a finite center. This H. Weyl shows at one

stroke for all possible G,,,, by proving more

generally that G? has a

finite

fundamental group, i.e. that its universal covering has finite center.'! This first

of all yields the full reducibility, but also sets the stage to extend the character

theory of Schur for SO,, to all compact semi-simple groups. All this is by now so

58

Armand

Borel

standard that a detailed summary is surely superfluous. I shall content myself with some remarks mainly to comment on some of the work which arose from

it. Let then K be a compact connected semisimple Lie group and T a maximal torus.

It is unique

up to conjugacy,

and

Weyl

shows

that

it meets

every

conjugacy class. Its Lie algebra t is a Cartan subalgebra of the Lie algebra f of K. Its character group X(T) is a free abelian group of rank equal to /=dimt. A root

is defined globally as a non-trivial character of T in fe =f @) C with respect to

the

adjoint

representation

of K.

The

Weyl

group

W

R

is the

group

of

automorphisms of X(T) induced by inner automorphisms of K. It is generated by the reflections s,, where ~ is a root and s, the unique involution of X(T) having a fixed point set of corank one, mapping « to — and leaving stable the

set of roots. Since T meets every conjugacy class in K, it suffices to describe the

restrictions to T of the irreducible characters of K. At first these are finite sums of characters of T, i.e. trigonometric polynomials, invariant under W. To describe them, we consider the trigonometric sums A(A)=

X

wewW

(detw)w-4,

(AE X(T)).

The sum A(,) is skew invariant with respect to W, equal to zero if / is fixed under

a reflection s, in W and depends only on the orbit W - / of2. Fix a closed convex cone C in X(T)gp=X(T) @®R which is a fundamental domain for W, call 2 Zz

dominant if it belongs to C and say that a root « is positive if it lies in the half-

space bounded by the fixed point set of s, and containing C. Let 20 be the sum of the positive roots. We have Weyl’s denominator formula (5)

A(g)=

T]

a>0

(@'/?-«7 '?).

Weyl shows that for every continuous class function f on K we have

(6)

J f(kbdk= J f()u(ddt, a

£

where dk (resp. dt) is the invariant measure on K (resp. T) with mass 1 and (7)

L=|W|'|A(@)|?,

(|W of order of W),

a point which Schur had singled out in his praise of Weyl’s results. From this and the orthogonality relations for characters of K or of T Weyl deduces that every irreducible character is of the form

(8)

%n=AU, +e) A(o)*

where 1, is dominant.'*

He does not show

however,

but derives from

E. Cartan’s work, that every dominant / occurs in this way.'? I shall soon come

Hermann

Weyl and Lie Groups

)

back to this problem. He also deduces from (8) a formula for the degree of z. Both (8) and this formula were not at all to be seen from Cartan’s construction

of irreducible representations.

Among many things, these papers mark the birthdate of the systematic global theory of Lie groups. The original Lie theory, created in 1873, was in

principle local, but during these first fifty years, global considerations were not ruled out, although the main theorems

were local in character. However,

striking feature here was that algebraic statements

were proved

a

by global

arguments which, moreover, seemed unavoidable at the time.'* H. Weyl had

not bothered to define the concepts of Lie group or of universal covering (the latter being already familiar to him in the context of Riemann surfaces).'> He had just taken them for granted, but could of course lean on the examples of the

classical groups, which had been known global objects already in the early

stages of the Lie theory.

These papers had a profound impact on E. Cartan. He had first known the

results of H. Weyl through the announcements

[W 61, W 62], in which the

general case was only cursorily discussed. His first reaction [C 81] was to show that, given the Hurwitz device, the use of “analysis situs” could be avoided by

means of older results of his."® However, once the full papers were published,

his outlook changed and from then on, the global point of view and analysis situs were foremost in his mind. He began to supplement his earlier work on Lie

algebras with a systematic study of global properties of Lie groups. In [C 103, C113] he developed the geometry of singular elements in a compact semisimple

Lie group, used the Weyl group systematically, to the extent even of deriving some basic properties of compact Lie groups or Lie algebras from results on Euclidean

reflection groups.

The

scope

of these

investigations

was

further

increased when he began to look from this point of view at a problem originating in differential geometry he was involved with at the time: the study of a class of Riemannian manifolds he later (from [C117] onwards) called symmetric spaces. These spaces were originally, by definition, those in which the Riemannian

curvature

tensor

is invariant

under

parallelism.

They

are

locally Riemannian products of irreducible ones. Cartan had classified them

[C 93, C94] and seen, with considerable astonishment, that this classification

was essentially the same as that of the real forms of complex simple Lie algebras he had

carried

out

ten

years

before

[C38].

Up

to that

point,

Cartan’s

investigations had been really local, although this was a tacit rather than an

explicit assumption. In present day parlance we would say he had classified isomorphy classes of local irreducible symmetric spaces. However, he had soon

recognized that these spaces were also characterized by the condition that the local symmetry at a point x, i.e. the local homeomorphism which flips the geodesics through x, is isometric [C 93: Nr. 14], which of course led to his choice of terminology. There also Cartan adopted systematically a global point of view, put in the foreground a global version of this second condition (each

60

Armand

Borel

point is isolated fixed point of a global involutive isometry), and developed a theory of semisimple groups and symmetric spaces in which Lie group theory and differential geometry were beautifully combined (see, e.g., [(C 107, 116]). A

particularly striking example is his proof of the existence and conjugacy of maximal compact subgroups by means of a fixed-point theorem in Riemannian geometry [C116]. Nowadays, all this has been streamlined, in the sense that

group theoretical (resp. differential geometric) results have been given group

theoretical (resp. differential geometric) proofs. Such an evolution is unavoidable and has many advantages, but sometimes loses some of the freshness and suggestive power of the original approach. It seems to me still fascinating to

watch

[C105]:

Cartan

explore

this new

territory

and

display,

as he

once

put

it

“toute la variété des problémes que la Théorie des Groupes et la Geomeétrie,

en s’appuyant mutuellement l'une sur l'autre, permettent d’aborder et de resoudre.” In [C111] he also points out that, since the set of singular elements in a

compact semisimple group has codimension 3, H. Weyl’s homotopy argument for the finiteness of the fundamental group (cf. ') can also be pushed to show

that the second Betti number is always zero (in fact, what he sketches leads to a

proof that the second homotopy group is zero). He then sets up a program to

compute the Betti numbers of compact Lie groups or their homogeneous spaces by means of closed differential forms, conjecturing on that occasion the theorems which de Rham was soon going to prove. Using invariant differential forms, he reduced these computations to purely algebraic problems [C 118]. This led later to the cohomology of Lie algebras.

We now come back to the problem of showing that every dominant character 4 occurs in (8) as the highest weight of an irreducible representation. Already in a footnote to [W 61], H. Weyl had stated that this “completeness”

can be proved by transcendental means. In [ W 68: III, §4] he is more precise but also more circumspect: the problem is to decompose the regular

representation of a compact group, but there are serious technical difficulties, and this method is maybe not really worth pursuing until simplifications are available since the result is known anyway from Cartan.'’ However, whatever

difficulties there were, they were soon surmounted in the paper written jointly with his student F. Peter [W73], completeness, but is very broadly

which not only proves the sought for conceived and is to be viewed as the

foundational paper for harmonic analysis on compact topological groups. In the simplest case, that of the circle group, this boiled down to the study of trigonometric series but even there, the group theoretical point of view was

new.'® We now refer for comparison to the earlier discussion of finite groups. Since G is now a compact Lie group, the authors replace F(G) by the space L(G) of square integrable functions with respect to a fixed invariant measure. It defines, via right translations, a unitary representation of G. The completeness problem is to show that the algebraic direct sum F of the spaces V,, defined as

Hermann Wey! and Lie Groups

61

above, with x running through the equivalence classes of irreducible finite dimensional continuous representations, is dense in L7(G), (Peter-Weyl theorem). Indeed, if that is the case, then the characters will form an orthonormal basis of the space of measurable class functions, and this will

imply easily that any expression A(A+@)A(g) ' (A20) does occur as a character in the right-hand side of (8). To prove this density, the authors introduce the convolution algebra C*(G) of the continuous functions on G. They show that any finite dimensional representation x yields also one of C*(G) by the formula

ma) = J oe(x)n(x)dx

G

and view x(a) as a Fourier coefficient of«. The convolution ~» by « is an integral operator of Hilbert-Schmidt type, with kernel k(x, y)=o(x-y '), which is self adjoint if «=% where &x)=&x '). This operator commutes with right translations hence its eigenspaces are invariant under G operating on the right.

The

authors

extension

consider

in particular

of E. Schmidt’s

theory

the operator

of eigenvalues

associated

and

to «*d.

eigenspaces

An

for such

integral operators on an interval shows that (« * @) * has a non-zero eigenvalue if a+0 and that the corresponding eigenspace is necessarily finite dimensional. All such eigenspaces are then contained in F. Using the operators so associated to an “approximate identity”, (a sequence of positive functions, whose supports tend to {1} and whose integral is one), they show that every finite dimensional irreducible representation of G occurs in this way (up to equivalence) and that

every continuous function is a uniform limit of elements of F. This implies in particular that F is dense. As a further consequence,

the finite dimensional

representations separate the elements of G and the continuous class functions separate the conjugacy classes in G. Apart from some technical simplifications,

such as the use of the spectral theorem for completely continuous operators (as done first in [Wi 1: § 21]), this is pretty much the way it is presented today and it

made a deep impression at the time it was published. Wey] himself viewed it as

one of the most interesting and surprising applications of integral equations [W 80: p. 196]. It was immediately extended to homogeneous spaces of

compact Lie groups by E. Cartan, with emphasis on symmetric spaces [C 117], thus supplying in particular a group theoretical framework

certain special functions,

H. Weyl [W 98]. H. Weyl

had

seen

such

as spherical harmonics,

that the same

approach

would

and

to the theory of

then

also yield

again

by

the main

approximation theorem in H. Bohr’s theory of almost periodic functions on the line [W 71, 72]. After Haar showed the existence of an invariant measure on any

locally compact group, and noted that this allowed one to generalize [ W 73] to

compact groups without further ado, J. von Neumann extended to groups

S. Bochner’s definition of Bohr’s almost periodic functions (cf. [Wi 1: §§ 33, 41]) and this led to what H. Weyl called the “culminating point of this trend of ideas” [W VI: p. 193], providing the natural domain of validity for the arguments of

62

Armand

Borel

the Peter-Weyl theory, but he pointed out immediately its limitations by quoting a result of Freudenthal, to the effect that a group whose points are separated by almost periodic functions is the product of a compact group by the additive group of a vector space, i.e. one does not get more than the two cases initially considered. Therefore, an extension of this theory to other noncompact groups, in particular non-compact semisimple groups, would have to

be based on quite different ideas and Weyl never tried his hand at it. Still his work has exerted a significant influence on its development. First the obvious one: the character

formula,

the

Peter-Weyl

theorem,

the use of a suitable

convolution algebra of functions have been for all a pattern, a model. The

results of Harish-Chandra on the discrete series for instance, albeit much harder to prove, bear a considerable formal analogy with them. But, less obviously maybe, Weyl was also of help via his work on differential equations [W 8], which gave Harish-Chandra a crucial hint in his quest for an explicit form of the Plancherel measure. In the simplest case, that of spherical functions for SL,(R), (or real rank-one groups) the problem reduces essentially to the spectral theory

of an ordinary differential equation on the line, with eigenfunctions depending

on a real parameter 4. It is the reading of [W 8] which suggested to HarishChandra that the measure should be the inverse of the square modulus of a function in J describing the asymptotic behaviour of the eigenfunctions [HC: IL, p. 212], and I remember well from seminar lectures and conversations that he never lost sight of that principle, which is confirmed by his results in the

general case as well.!?

Around 1927, H. Weyl got involved with the applications of group representations to quantum mechanics. His first paper [W 75] contains notably

some suggestions or heuristic arguments which also led to new developments in

unitary representations of non-compact Lie groups, but of a quite different nature from those mentioned above. Weyl proposes that the spectral theorem

should allow one to associate to an unbounded hermitian operator A on a Hilbert space H a one-parameter group

{expitA} (teR) of unitary transfor-

mations of H having iA as an infinitesimal generator, as is well known to be the

case

in

the

finite dimensional

case.

Heisenberg commutation relations

(9)

[P;,Qi]=5j.

Then,

given

[Pj PiJ=[0;,0)=0,

operators

satisfying

the

(1Sj,kSn),

Weyl views these relations as defining a Lie algebra and considers the associated group N of unitary transformations generated by the expitU, where

U runs through the real linear combination of the P; and Q,. To (9) correspond

commutator relations in N, which have since been known as the “Weyl form” or “integrated form” of (9), (and also prove that N is a “Heisenberg group”). He

then gives some heuristic arguments to prove the uniqueness (up to equiva-

lence) of an irreducible unitary representation of N with a given central character, out of which follows the uniqueness of the Schrddinger model for the

Hermann

Weyl and Lie Groups

63

canonical variables. Rigorous treatments of these suggestions led to the Stone theorem on one-parameter groups of unitary representations [So], which soon

became a foundational result in unitary representations of non-compact Lie groups, and to the Stone-von Neumann uniqueness theorem [Ne]

[So], itself a

fundamental result and the source of many further developments (for all this see

[Ho 4] and [M)).

Weyl soon provided a systematic and impressive exposition in his book

“Gruppentheorie und Quantenmechanik” [W V]. I shall not attempt to discuss its importance in physics and shall go on confining myself mostly to Lie groups.

As far as Weyl was concerned, the main mathematical contribution stemming

from it is the paper on spinors [W105], written jointly with R. Brauer. Infinitesimally, the spinor representations had already been described by

Cartan in 1913 [C37], by their weights. But [W 105] gave a global definition, based on the use of the Clifford algebra, itself suggested by Dirac’s formulation of the equations for the electron. However, the most unexpected fall-out

originated with a physicist, H. Casimir, and led to the first algebraic proof of the

complete reducibility theorem. In the representations of g=sl,(C), or equivalently so,(C), an important role in the quantum theoretic applications is played by a polynomial of second degree in the elements of g, which represents the “square of the magnitude of the moment of momentum” [W V], p. 156 (or p.179 in the English version), the sum of the squares of the infinitesimal rotations around the coordinate axes. It commutes with all of g, hence is given

by a scalar in any irreducible representation: this yields an important quantum number j(j+1), in the representation of degree 2)+1 (2)¢N). Casimir was struck

by

this commutation

property

and

defined

in

1931

an

analogous

operator for a general semisimple Lie algebra, called later on, and maybe in [WII] for the first time, the Casimir operator, and indicated how it would allow one to derive the Peter-Weyl theorem from results about self-adjoint elliptic operators [Cs]. A year later, he noticed that in the case of sl,, it could be

used to give a purely algebraic proof of full reducibility, which was later extended to the general case by B. L. van der Waerden, using the general

Casimir operator [CW]. As we shall soon see, H. Weyl was quite concerned at

the time with finding algebraic proofs of results obtained first in a transcendental way, but in a different context, and it seems that this problem was not anymore

of

much

interest

to him,*°

although

he had

concluded

his first

announcement [W 62] by saying that an algebraic proof would be desirable and had suggested, at the time of the lectures [ W 3], that it would be worthwhile to develop a purely algebraic theory of Lie algebras, valid at least over arbitrary fields of characteristic zero, a suggestion which was picked up by N. Jacobson (see [J 1]) and had a considerable impact on his research interests.

Although I have spoken at some length of the Math. Zeitschrift papers, I

have not yet exhausted their content and my survey has been incomplete on at

least two counts. Making up for it will provide a bridge towards the more algebraic aspects of Weyl’s work.

64

Armand

In Chapter

I of [W 68], devoted

Borel

to the representations

of SL,(C) and

GL,(C), Weyl not only combines Hurwitz-Schur and Cartan, but also relates the results to older ones going back to Schur’s Thesis [Sc 1]: After having determined all the holomorphic irreducible representations of SL,(C), he points

out that the matrix coefficients are in fact polynomials in the matrix entries, and

that these representations are the irreducible constituents of the representations of SL,(C) in the tensor algebra over C". They are therefore just the tensor spaces, described by means of symmetry conditions on the coefficients. In this he sees the “group theoretical foundation of tensor calculus,” a point important enough for him to make it the title of this Chapter and of the announcement [W 62]. Moreover, Schur had given a direct algebraic construction of those, setting up the well-known correspondence with representations of the symmetric groups, stated in terms of Young diagrams, which also yielded an

algebraic proof of full reducibility. This example of an algebraic treatment is one to which he will come back repeatedly and which he will other classical groups. It later became of even greater interest its applications to quantum theory [W V: Ch. V]. The second point is “Invariant Theory”. In broad terms its is, given a group Gand a finite dimensional representation of G V, to study the polynomials on V which are invariant under

try to extend to to him in view of general problem in a vector space G (or sometimes

only semi-invariant, i.e. multiplied by a constant under the action of a group

element). The questions which are usually asked are whether the ring of invariants is finitely generated (first main theorem) and if so, whether the ideal

of relations between elements in a generating set is finitely generated (second

main theorem). In concrete situations, one will want of course an explicit presentation of the ring of invariants in terms of generators and relations. One may also look for the dimension of the space of homogeneous invariants of a given dimension (the “counting of the number of invariants”). Such a formulation, however, where G and V are free, emerged at a later stage of the theory, as an abstraction from the classical invariant theory, to which I shall come in a moment, which focuses on very specific instances of these

questions.?!

As remarked earlier, such problems were at the origin of the papers of Hurwitz [Hu] and Schur [Sc 2], and it was to be expected they would also be

very much in Weyl’s mind, even independently of the Study incident. Indeed, he points out at the end of [W 68] that the unitary restriction method now allows one to prove the first main theorem for all semisimple groups, which provides, for the first time, a natural group theoretical domain of validity for it. In addition, following Schur, he notes that the dimension of the space of fixed vectors in a given representation z is given by integrating over G the character 7, of the representation. He pursues this in [W 69, 70] where he also states that,

more generally, the multiplicity of an irreducible representation o in z is given by the integral | y,(g)z,(g)dg, where dg is the invariant measure with total mass

1, and applies this to a number of classical cases. Although he also stresses in [W 68] the superiority of the integration method over the traditional proce-

Hermann Weyl and Lie Groups

65

dures, based on differentiation operators of the kind of Cayley’s Q-process, he

soon became preoccupied with finding an algebraic framework for the main results of [W 68] pertaining to classical groups, which would encompass the classical invariant theory. We have now come close to the two main themes of this second, more

algebraic, part of Weyl’s work, which culminates in his book on Classical Groups [W VI]: “,..The task may be characterized precisely as follows: with respect to the

assigned group of linear transformations in the underlying vector space, to

decompose the space of tensors of given rank into its irreducible invariant

subspaces. ...Such is the problem which forms one of the mainstays of this book, and in accordance with the algebraic approach its solution is sought for not only in the field of real numbers on which analysis and physics fight their battles, but in an arbitrary field of characteristic zero. However, I have

made no attempt to include fields of prime characteristic.”

After having briefly explained that the determination of representations logically precedes the search for algebraic invariants, he goes on to say: “...My second aim, then, is to give a modern introduction to the theory of

invariants. It is high time for a rejuvenation of the classic invariant theory, which has fallen into an almost petrified state.” All this seems rather clear, but I dare say I am not the only one to have found

the book of rather difficult access. According to the introduction, the program is

first to decompose tensor representations and then to derive Invariant Theory. But Weyl apparently could not do this for all classical groups, algebraically and

in that order, so that the itinerary between the two is more sinuous, starting in

fact with invariant theory. It would seem also that, in spite of some occasional,

rather pungent, comments on the symbolic method, Weyl had found Invariant

Theory and some of its specific techniques of independent interest, since he

gives them prominent billing in [W VI] and had also devoted to them a course

here (of which a 13 page outline can be found in the Weyl Archives) and at the

Institute for Advanced Study [WIV]. In addition, Weyl could not realize his program fully algebraically and did not refrain from introducing and using the

integration method, whether to realize his immediate goals or in its own right.

This rather subtle interplay between various points of view does, of course, broaden the horizon of “the humble who want to learn” for whom the book is

“primarily meant” [W VI: viii], but does not make it easier for them to get a

clear picture of its organization. This somewhat tentative character may also

have been felt by Weyl when he wrote [W “At present I have come

117]:

to a certain end, or at least to a certain halting

point, from which it seems profitable to look back upon the track so far

pursued, and this is what I have tried to do in my recent book, The Classical Groups, their Invariants and Representations.”

Reflecting upon the greater finality of the transcendental results as compared to the algebraic ones, with the added wisdom of fifty or so more years, one may observe that in the former case Weyl already had the natural

66

Armand

Borel

framework and all the necessary tools at his disposal, but not so in the latter; in particular,

the point

of linear algebraic

of view

groups,

or the

use of the

universal enveloping algebras of Lie algebras, which became essential, were not

part of his vision of future developments, as described at the end of [W117]

where he forecast

“a similar book dealing comprehensively with the representations and invariants of all semi-simple Lie algebras in an arbitrary characteristic.” Ishall try shortly to give some idea of the content of [W VI] and some of the developments to which it has led. Before that, however, I should backtrack and say something about the already often mentioned Classical Invariant Theory and the earlier work of Weyl pertaining to it.

A typical example is the search of invariants of p vectors for G=SL,(C), SO,(C), i.e. of polynomials in the coordinates (x;;) (i=1,....nij=1,...,p) of p vectors x; (j=1,...,p) homogeneous in the coordinates of each x;, which are invariant under G. In other words, one is looking for the fixed vectors in the

tensor product of p copies of the polynomial algebra over C”. In the case of

SL,(C) or GL,(C) one wants more generally the invariant of p vectors and q covectors (y,) (k=1, ...,q). The first main theorem in this last case says that the ring of invariants is generated by the products ¢x,, y,>, to which one should add the determinants in n of the p vectors or g covectors in the case of SL,. For O,, there is no need to add covectors, the invariants of p vectors are generated by the scalar products (x;,x,). Also a generating set of relations between these

elements was given. Let me now limit myselfto the case of p vectors. To prove

such theorems one uses differentiation operators which commute with G, hence transform invariants to invariants, and which allow one to decrease the degree

in certain variables and to carry out induction proofs. The first ones are the polarisation operators D;, = ;x;,0/0x,. Another fundamental one is Cayley’s

operator

=x an lax (1), 15 -++s a OXe(n),n> Q=Z(sgno)6"/OX where o runs through the symmetric group S,, in n letters. This operator commutes with the D;, for i+j, but not with the D;;. Capelli [Ca 1] showed that H=det(x;;) det(0/0x;;) (for p=n) does commute with all D;; and gave an

expression for H as a determinant in those, the so-called Capelli identity, which is the main formal tool in much of the theory.** There are of course many

variants of this problem. One may, e.g., replace the identity representation of G

by a symmetric power, hence consider the invariants of p homogeneous forms of

a given degree. The “symbolic method” reduces in principle the form problem

to the search of multilinear invariants, i.e. to the fixed points in tensor powers of the identity

representation.

In the nineteenth

century,

such

procedures

to

generate invariants out of a given one were often used to try to check the validity of the first and second

main

theorems.

In that function, they were

discredited by Hilbert’s work on Invariant Theory, which “almost

kills the

Hermann Weyl and Lie Groups

67

whole subject. But its life lingers on, however flickering, during the next decades” [W VI: p. 27, 28]. In his answer to Study, in [W 60], Weyl goes back to Capelli, provides a new

proof of Capelli’s identity and then establishes the first main theorem for SL,, SO,, which were known, as well as for Sp,,, which had not been considered

before. He also discusses the determination of invariants for some non-simple

groups such as the group of euclidean motions or the subgroups of SL,, or GL,,

leaving invariant a strictly increasing sequence of proper subspaces (now called parabolic subgroups, see **). [W 63] centers on the use of the symbolic method.

After some contributions to the program outlined above, Weyl publishes his

book [W VI]. It starts with an exposition of the first and second main theorems for invariants of vectors and covectors for the classical groups. A central point of the book, or at any rate of its algebraic part, is the double commutant

theorem, proved first in [W 107], which is the main principle on which Weyl

organizes the discussion of the decomposition of tensor representations. Let A

be a subalgebra of the algebra End(£) of endomorphisms of a finite dimensional vector space E over a field K of characteristic zero and let A’ be its commuting algebra in EndV. Assume V is a fully reducible A-module. Then V is also fully

reducible under A’, and A is the commuting algebra of A’. In particular, V decomposes into a direct sum of A’ x A irreducible submodules. Any such is the

tensor product

U,@U4,

of an irreducible

A-module

by an irreducible

module, whence a correspondence between some irreducible representations

A

of A and o’ of A’, which is particularly nice of no o or o’ occurs twice in these

pairs. The assumption of full reducibility under A is in particular fulfilled if A is the enveloping algebra of a finite group (even in positive characteristic prime to

the order of that group).

The origin of this theorem, and its most perfect illustration, is the reciprocity

between

irreducible

representations

of the symmetric

groups

©, and the

irreducible subspaces of the tensor representations of GL,, discussed first in Schur’s Thesis [Sc 1] (for

It is operated

upon

by

K =C). Let V be the tensor product of p copies of K".

GL,,

via

the p-th

tensor

power

of the

identity

deduces

from

this full

representation and by S,, via the permutations of the factors and these operations commute. Let 4’ and A be the corresponding enveloping algebras.

Schur

proves

reducibility.

He

that

A’ is the centralizer

also shows

that

of A and

the correspondence

(¢,a’) is bijective and

determines the characters of the o’. After having discussed this case, Weyl goes

over to the orthogonal and symplectic groups. However, this has to be more

roundabout since there is no finite group on the other side. Weyl had made the

first steps in that direction in [W 96]; here he avails himself also of some results

of R. Brauer [Br 2]. He considers simultaneously the enveloping algebra of the

group

algebra

under

consideration

A’. None

of them

in a tensor

representation

is a priori known

and

its commuting

to be fully reducible, and the

interplay between information gained successively on each of them is rather subtle. An important fact is that the description of A’ is equivalent to the first

68

Armand

Borel

main theorem. However, in the case of gl,, this relationship conversely to prove the first main theorem [Br 2].

can

be used

Weyl’s next goal is the determination of the characters of the irreducible

constituents of the tensor representations. Lacking an algebraic method valid for all classical groups, he turns to the transcendental one of [ W 68], based on integration. For GL,, however, algebraic treatments were available. I already mentioned one by Schur [Sc 1]. Here Weyl follows a slightly earlier one due to

Frobenius. He then goes over to more general aspects of Invariant Theory, which he discusses both from the algebraic and the transcendental points of view.

A supplement

complements

to the main

published

text

in the second

first in [W122].

edition

includes

some

In particular, it points out that a

subalgebra of a matrix algebra over a subfield of R which is stable under transposition is fully reducible, which gives an algebraic proof of full reducibility in such cases.

Obviously not an easy book.” Its results were not as spectacular and clear cut as those of the Math. Zeitschr. and M. Annalen papers and, not surprisingly,

it did not have such an immediate impact, but this was really only a question of time, and its influence has been felt more and more during these last fifteen years

or so.

First of all, its treatment of Classical Invariant Theory became the standard

reference and made it available to potential users, whether specialists or not. This led to further applications but also to improvements of the theory, already

over C. Asa first example, [ABP] provides a new proof of the determination of the tensor invariants for O, directly from those for GL, bypassing the use of the Capelli identity. The application the authors had in mind was to a new proof

of the Atiyah-Singer

theorem

by the heat equation

method.

As a second

example, M. Artin [A] was led to consider invariants of r matrices and conjectured they would be generated by traces. This was soon deduced from the classical theory by Procesi [P 1], who more generally described the invariants

of p vectors, q covectors, and r matrices, results which were then used to determine the rational Waldhausen K-theory of simply connected spaces

[DHS]. As

a second

type of developments,

let me

discuss

some

in which

the

restriction on the characteristic of the groundfield was lifted, even beyond what

Weyl

could

envisage.

directions. On

Progress

was

made

the one hand, the first main

almost

simultaneously

theorem

was extended

in

two

to all

semisimple (even reductive) linear algebraic groups over an algebraically closed groundfield K of arbitrary characteristic.?+ In characteristic zero, this was

essentially Weyl’s theorem mentioned earlier (modulo a harmless reduction to C). But this proof was based on the full reducibility, which is false in positive characteristic, so it could not be extended directly. Motivated by his Geometric

Invariant Theory, D. Mumford proposed a weaker notion, later “geometric reductivity”: Given a rational representation G>GL(V)

called and a

Hermann Weyl and Lie Groups

pointwise fixed line DV,

69

there should exist a homogeneous hypersurface W

meeting D only at the origin and invariant under G. (In case of full reducibility,

W could be a hyperplane.) and conjectured that every semisimple algebraic

group would satisfy this condition. Soon after, M. Nagata showed that it would imply the first main theorem [Na]. The validity of the latter was then assured

when W. Haboush proved geometric reductivity in general [Ha], a few years

after Seshadri had established it for GL.

On the other hand, by introducing new methods in combinatorics, [DRS] gave a characteristic free proof of the first main theorem for GL,. This was quickly seized upon by de Concini and Procesi to extend the first and second main theorems to classical groups over

a commutative ring A, subject only to

the condition that a polynomial in n indeterminates with coefficients in A which is zero

on

A" is identically

zero

[CP].

algebraically closed groundfield or to Z.

In particular,

this applies

to any

At this point, although it is one more generation removed from Weyl, I feel it

is natural to mention a further and very extensive characteristic-free theory, in

which [CP] is an essential tool, the “standard monomial theory” of Seshadri et al. In its geometric form, it aims at giving canonical bases for spaces of sections

of homogeneous line bundles on flag varieties. As far as I know, it does not yet work in full generality but in a vast class of cases. In characteristic zero, when it does, it allows one to give a canonical basis of an irreducible representation;

more explicitly, it yields a combinatorial procedure to single out an irreducible representation with a given highest weight in the tensor product of fundamental representations in which Cartan looked for it (see ’), assuming this is done for

the fundamental ones which occur. This generalizes work by W. V. D. Hodge for SL,, who used it to give explicit equations for Schubert varieties and applied

it to enumerative geometry, and it has similar applications and goals in the general case. See the survey [Se] by Seshadri.

As my final item, I now turn to a development of which [W VI], or more

specifically the bicommutant theorem, has been namely R. Howe’s theory of reductive pairs.

In the early

seventies, some

remarkable

one

of the foster parents,

correspondences

were

set up

between certain families of irreducible unitary representations of some pairs of

real reductive groups. Knowing [ W VI], Howe was led to think that the proper

framework for such correspondence was a generalization of the situation of the

bicommutant theorem: start from a real semisimple group G and a so-called “reductive pair” (H, H’) in G, i.e. two reductive subgroups H, H’ each one of which is the centralizer of the other. Let now z be an irreducible representation of G, say first finite dimensional, to avoid analytical difficulties. Then it

decomposes into irreducible H x H’ submodules, each of which is a tensor product of an irreducible representation o of H by one o’ of H’, whence again

some relationship between the irreducible representations of H and H’ which occur. If now z

is unitary, infinite dimensional, then the decomposition into

H x H’ modules presents all kinds of difficulties. They become more manageable if one of the groups is compact. Also one may anyhow restrict one’s

70

Armand

Borel

attention to the closed irreducible subspaces. It turned out that the correspondences mentioned earlier could be given a natural explanation by viewing the two groups in question as a reductive pair in a symplectic group Sp,,(R) or

rather by going over to a two-fold covering of Sp,,(R) and taking for z the socalled oscillator or metaplectic representation. The choice of that particular representation had been suggested by Weil’s

use of it, over local and global

fields, in his group-theoretical treatment of 0-functions [Wi2]. Accordingly,

this principle has had many applications and variants to groups over local or

global fields. In the latter case it leads to correspondences built from 6-series

between spaces of automorphic forms. This is one of the most fruitful principles in representation theory of, and automorphic forms for, classical groups, which has suggested many problems and has been confirmed by many special results. See [Ho 3] for a survey of theorems and conjectures and [Ho 5] for further

results. At this point, we seem again to be far removed from Wey] and obviously

such developments are not direct offsprings of his work. But the importance of [W VI] as one of the main influences on the genesis of this general principle has

been stressed by Howe himself, who has also applied it to groups over finite

fields

[Ho1]

and

then

to

Invariant

Theory

itself

[Ho2].

There

Howe

generalizes the classical theory to the determination of “superinvariants,” i.e. of invariants in the tensor products of symmetric and exterior powers, and proposes a general recasting of the whole theory from that point of view.>

With this I conclude my attempt to give an idea of Weyl’s work on Lie

groups and of its repercussions. As you can see, those were felt ina broad range of topics in analytical, differential geometric, topological or algebraic contexts and took many forms: general theorems or specific results on special cases, clear

cut

statements

as well

as less sharply

delineated

suggestions

or guiding

principles, mirroring the many-sidedness of Weyl’s output and outlook. Early in this lecture, I quoted from a letter to Cartan in which Weyl expresses his admiration for Cartan’s work on continuous groups. He goes on to say, commenting on the results announced in [W 61]:

»Meinen

gegenwartigen

Anteil an dieser Theorie schatze ich gar nicht

besonders hoch; ich komme mir Treffpunkt von Ihnen und Schur.”

eigentlich

nur

vor

wie

der

zufallige

This is of course putting it very mildly. Not only much more than chance

was needed to produce such a synthesis, but Weyl had to be a meeting ground

for, and to combine, not only Schur and Cartan, but invariant theory, topology

and functional analysis as well. At that time, no one else was conversant with all

of these; in fact, except for Schur, with hardly more than one. Although I limited myself to a rather sharply circumscribed and quantitatively minor part of

Weyl’s work, this already provides a demonstration of, a practical lesson in, the unity of mathematics, given to us by a man whose mind was indeed a meeting

ground for most of mathematics and mathematical physics.

Hermann

Weyl and

Lie Groups

71

Notes ' (53) As a rule, I shall use current terminology rather than the one of the original papers and content myself with some occasional historical remarks

about the latter.

The term “Lie algebra” appears first in [WIII], and was suggested by N. Jacobson, but the concept was present very early in the theory. Until then, the usual terminology was “infinitesimal group,” or sometimes just “group,” (or

“abstract

group”

in

[WII]).

Obviously,

H. Weyl

felt

at

the

time

more

comfortable with the latter, since he reverts almost exclusively to it, after having

introduced “Lie algebra” as an alternate to infinitesimal group in a formal

definition, and later uses only “subgroup” and “invariant subgroup” for the

present-day “subalgebra” and “ideal.”

“Lie group” was introduced by E. Cartan around 1930 (see in particular [C 128]). For Lie (disregarding the distinction between local and global), and

until then, they were the “finite and continuous groups.” However, E. Cartan

uses there the latter expression for what we would call locally euclidean groups,

and requires twice differentiability for Lie groups.

2 (53) Some notation:

If K is a field, M,(K) is the algebra of n x n matrices with coefficients in K,

and GL,(K)

[resp. SL,(K)]

the group of invertible (resp. determinant one)

elements of M,(K). The transpose of X €M,,(K) is denoted ‘X. Sp,,,(K) is the symplectic group, i.e. the subgroup of elements of GL,,(K) leaving invariant the standard antisymmetric bilinear form 21 x;y, 4;—Xn+iThe terminology “symplectic group” is introduced in [W VI]. Earlier, for K=C, Weyl had used “complex group,” as a shorthand for group leaving a complex of lines invariant, which goes back to S. Lie (except for the fact that

Lie’s classical groups were groups of projective transformations). U,={AeGL,©)|'A-A=I}

is

0, ={AeGL,(R)|'A:

is the orthogonal

U,ASL,(C). O,,0SL,(R)

the

special

A=}

the

orthogonal

unitary

group

on

group

group,

O,,, +4 the

or

Lie

on

C”

and

SU,=

R" and

SO,=

subgroup

leaving 22x? — 2X", ,x7 invariant (n=p+4q). 0,(C)={AeGL,(C) |' A- A=J} and SO,(C) =SL,(C)n0,(C). The

Lie

algebra

of a complex

real

group

corresponding German letters: gl,; gl,(R), gl,(C), so, ...-

of GL,(R)

is denoted

by

the

%) (54) E. Cartan [C65] translated the problem in the formalism of affine connections he was developing at the time, which made Weyl’s transition from

half-philosophical considerations to the actual mathematical problem easier to

grasp

for

some.

In

present

day

terminology,

the

problem

can

be

stated

geometrically as follows: Let G be a closed subgroup of SL,(R). Let M be a smooth manifold of dimension n. Choose in a neighborhood U of a point x a

trivialisation

of the bundle

P of frames, i.e.

n smooth

everywhere

linearly

independent vector fields (e,) (or, equivalently, as Weyl and Cartan express it, a

set of n everywhere linearly independent one-forms «,). Let Q be the fibre

a

Armand Borel

bundle over U whose fiber at y is the set of G-transforms of {e,(y)}, where {e,(y)}

is used to identify the tangent space 7,(M) with R”. It is assumed that for every

choice of the e;’s, the principal G-bundle Q has one and only one torsion-free affine connection (the existence is Axiom I and the uniqueness Axiom II). Then the complexification of the Lie algebra g of G is conjugate in gl,(C) to so,(C).

Algebraically, this amounts to the following problem. before and (Ci,,)

Let G and g be as

(Si jsn;1Ssgl(V) if

V,={veV|x(hv=A(h) (heb)} £0. The space V is always the direct sum of the V,. The weights of all finite dimensional representations generate a lattice P in the smallest Q-subspace hg

of h* spanned by them, which is a Q-form of h*. A nonzero weight of the adjoint representation is a root. The roots generate a sublattice Q of P. Let bg be the real span of P. For each root « there is a unique automorphism s, of order 2 of bg leaving the set of roots stable, transforming x to — « and having a fixed point set of codimension one. Fix a set A of “simple roots”, ie. /=dimb linearly independent roots such that any other root is a integral combination of the ae€A with coefficients of the same sign, and call positive those with positive

coefficients. Introduce a partial ordering among the weights by saying that A= if A—is a positive linear combination of simple roots. Then /€ P is said to be dominant if A=s,/ for all simple «’s. Let P* be the set of dominant weights. [The group W of automorphisms of bg generated by the s, is one realization of the group (S) introduced by Weyl in [W 68] and called later the Weyl group. A

weight can also be defined as dominant if it belongs to a suitable fundamental

domain C of W, namely the intersection of the half spaces E, («€ A), where E, is the half-space bounded by the fixed point set of s, and containing «.]

Let x be irreducible. Cartan shows first it has a unique highest weight /,, i.e.

a weight which is > than any other weight. This weight has multiplicity one. The main result of [C 37] is that any J € P* is the highest weight of one and only

one (up to equivalence) irreducible representation. The dominant weights are

linear combinations with positive integral coefficients of the so-called fundamental ones «; (1