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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
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as
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Ou
ey
a
ee °
°
Qe) a)
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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
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*,
ia
’ Pr
o.
hae
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a ~
é
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of
ee | ~~" e te g* "ne ’ ~*~
?
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*
a
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a
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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