An Encomium of Hermann Weyl∗

R Sridharan∗∗

∗The text of the Presidential Address (General) delivered at the 76th Annual Conference ofthe Indian Mathematical Society held at the Sardar Vallabhbhai National Institute of Technology(SVNIT), Surat-395 007, Gujarat, during December 27 - 30, 2010.

∗∗I dedicate this talk, which, though I believe, consists merely of “tanquam folium a ventorapitur et quasi scintilla in arundinete” (leaves caught by the wind, sparks in a brush wood)with great respect to the “Master Builder” Prof K Chandrasekharan. I also want to take thisopportunity to remember with gratitude my good old days at the Tata Institute with my learnedcolleagues M.S. Narasimhan and C.S. Seshadri and wish both of them well.

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“We are but dwarfs mounted on the shoulders of giants, so that we can see moreand further than they; yet not by virtue of the keenness of our eyesight, nor throughthe tallness of our stature, but because we are raised and borne aloft upon that giantmass.”

– Bernard of Chartres, 12th century (Translation by Dr.Poole)

“To read history is to be present as it were in every age, to extend and stretchlife back-ward from the womb, and thus exhort from unwilling fate a certain foregoneimmortality.”

– Milton

“History is a rebirth of life, that without its continued effort would vanish andlose its vital force. Without a historical hermeneutic, without the art of continuedinterpretation contained in history, human life would be a very poor thing. It wouldbe restricted to a single moment of time, it would have no past and therefore nofuture, for the thought of the future and the thought of the past depend on eachother.”

– Ernest Cassirer

I should perhaps begin my talk on Hermann Weyl with a brief mention of hisfamily and his life. He was born on the 9th of November 1885 at Elmshorn inGermany. His father Ludwig Weyl was a bank Director and his mother was AnnaWeyl-Dieck. Hermann studied in a Gymnasium at Altona from 1895 till 1904. Heentered the university of Gottingen in 1904 and finished his doctoral work in 1908under the guidance of David Hilbert. He became a Privat dozent in Gottingen andmarried Helene Joseph in 1913. He became a Professor of Mathematics at theETH, Zurich during the Winter semester of 1913. He went back to Gottingen as asuccessor of Hilbert in 1930 but left Gottingen in 1933 (with the advent of Nazism inGermany) to become a Professor at the Institute for Advanced Study at Princeton.His wife Helene died in 1948 and he married Mrs Ellen Bar, nee Lohnstein (wife ofProfessor Richard Bar who died in 1940). Hermann Weyl passed away on December8, 1955, in Zurich.

Hermann Weyl is the most distinguished mathematician of the 20th century, witha penetrating vision not only in mathematics and physics but was universal in hisoutlook, for whom, to use his own words “the problems of mathematics are notisolated problems in a vacuum; there pulses in them the life of ideas which realisethemselves in concreto through our human endeavours in our historical existence, but

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forming an indissoluble whole, transcend any particular science”. His all pervasivevision in Science was matched by his deep insight into philosophical problems. Hewas literate in the widest sense of the term, he had an immense love for literature(specially German literature), a distinctive connoisseur of art and poetry. It is hardto imagine that such a many sided personality existed in the not so distant a past,when one looks at how fragmented is the world of knowledge of the present day, theworld of specialists, the tower of Babel.

It is of some comfort to note that Weyl himself felt the threat of specialisationapproaching, when one reads his preface to the first edition (1939) of his book onThe Classical Groups. Here is what he says “The stringent precision attainable formathematical thought has led many authors to a mode of writing which must givethe reader the impression of being shut up in a brightly illuminated cell where everydetail sticks out with the same dazzling clarity, but without relief. I prefer theopen landscape under a clear sky with its depth of perspective, where the wealthof sharply defined nearby details gradually fades away towards the horizon”. C NYang, who quotes this in his article on the decisive contribution of Weyl to Physicsin the Centenary Volume (I am going to discuss this volume later in this talk) alsoadds, “Indeed, this perhaps states very clearly Weyl’s intellectual preference, whichhad a determining influence on the style of his work in mathematics and physics”.

It is perhaps appropriate here to mention a book titled Paul Dirac, The Manand His Work published by the Cambridge University Press, which is based onfour lectures given by some eminent physicists in 1995, celebrating Dirac’s memory.Michael Atiyah in his article titled The Dirac Equation and Geometry in this volumediscusses the geometry underlying the spin representation and shows how the Diracoperator plays a very important role in the work of Seiberg-Witten and others.At the end of his article, while underscoring the fact that the spin representationand the Dirac equation still play an important role in geometry leading to a betterunderstanding of the mysterious role of the spinors, gives the following quotationfrom Hermann Weyl’s Classical Groups which, as usual, is intriguing and highlysuggestive, by which Atiyah says, he was“bemused”. Here goes the quotation: “Onlywith spinors we strike that level in the theory of representations (of the orthogonalgroup) on which Euclid himself, flourishing ruler and compass, so deftly moves inthe realm of geometric figures. In some way, Euclidean geometry must be deeplyconnected with the existence of the spin representation.”

The mention of the Centenary Volume of Hermann Weyl above leads me to makea few remarks as to how the idea of writing this article came to me. I generally donot like making autobiographical remarks, and I shall be very brief. When I was agraduate student at Columbia University in New york with Professor Eilenberg as

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my thesis adviser, I also got interested in Quantum mechanics and began buyingbooks on this subject, published by Dover, from the Columbia University bookstoreand among the books I bought was Theory of Groups and Quantum Mechanics andI came under its spell immediately. It is not that I understood all the physics andmathematics it contained (it is quite a formidable book to read!), but began browsingthrough it from time to time. Just to indicate one instance of its recondite but verysuggestive and attractive language, I would mention for example how Weyl beginswith the metaphysical principle that in whatever way one counts a finite aggregate,the number of its elements is an invariant, before he gives a beautiful proof of theJordan-Holder theorem as an illustration! I carried this book with me to Chicagoduring the summer of ’59, to attend, at the suggestion of Eilenberg, a summer schoolon Homological Algebra and ended up feeling depressed mathematically, while atChicago. To feel better, I was leafing through the book of Weyl and was struckby a remark of his in the book that the Heisenberg commutation relation was themathematical principle that the differentiation operator and the multiplication bythe variable have as their commutator the identity operator. Indeed this one remarkled me to the idea which, in turn led me to my doctoral thesis which I finished withina few weeks after my return to Columbia University (of course with the greatestsupport which Prof Eilenberg was ever willing to give me).

I received as gifts in 1968 from Professor Eckmann the Selecta of Weyl’s pa-pers edited by him as well as the four volume set of the complete works of Weylpainstakingly edited (with a lovely introduction) by Professor Chandrasekharan. Ishould recall here Chandrasekharan’s great tribute to Weyl in his Preface to thesevolumes that “Whatever he touched, he adorned”. (One perhaps may recall thatthese very words were used by Fenelon about Cicero and by Samuel Johnson aboutOliver Goldsmith!). In 1986, Professor Chandrasekharan sent me a copy of a slenderand beautiful volume edited by him, titled Hermann Weyl (1885-1985), containingthe three scientific lectures, one by Yang, one by Penrose and the third by ArmandBorel, touching on some aspects of Weyl’s work and their tremendous influence forfurther work in physics and mathematics, tributes paid to the memory of Weyl bythe Dr Ursprung, President of the E.T.H., and others from Zurich and lovely remi-niscences of Michael Weyl about his father Hermann Weyl. The volume was editedby Chandrasekharan and as is to be expected, contains a beautiful preface by Chan-drasekharan himself where he rightly points out, talking about the scientific lectures:“ The themes chosen represent only a fraction of Weyl’s mathematical interests. Butthey give us more than a glimpse of the mighty effulgence of Weyl’s mind... No oneelse bounds among the peaks of mathematics with quite such dazzling aplomb”.Chandrasekharan emphasizes that for Weyl, “The world of ideas and concepts wasas real as the world of human beings. The force, steadiness, the comprehensiveness

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and versatility of his intellect, were matched by his generosity, sympathy and sup-port for striving young researchers spread around the globe.” Chandrasekharan endshis short preface by saying that Weyl’s memory will remain a source of inspirationfor succeeding generations.

During the rest of my talk, I would like to give, not in detail the contents of thevolume, but only pick a few snippets which bring out the pre-eminence of Weyl as aphysicist, mathematician and philosopher with a penchant for literature and poetry.The opening address of Dr Ursprung, president of the ETH, Zurich, begins with aquotation from Weyl (from his book Gruppentheorie und Quanten Mechanik), inwhich Weyl says that he acted merely as a messenger between physics and mathe-matics. He stresses Weyl’s vision of unity of thinking and outlook; he also quotesfrom Weyl’s lecture at Columbia University’s bicentennial celebration in 1954. Weylsays in his lecture “Doubts about the methodical unity of natural sciences havebeen raised. This seems unjustified to me. Following Galileo, one may describe themethod of science in general terms as a combination of passive observation refinedby active experiment with that symbolic construction to which theories ultimatelyreduce. Physics is the paragon... I do not suggest that we are safe against surprisesin the future development of science. Not long ago we had a pretty startling one inthe transition from classical to quantum physics. Similar future breaks may greatlyaffect the epistemological interpretation as this one did with the notion of causality;but there are no signs that the basic method itself, symbolic construction combinedwith experience will change”.

I want to add on my own here that Weyl had an immense faith in the symboliccontent of science, he had in fact a real admiration for Ernest Cassirer (who in hisphilosophy emphasised the capability of humans for symbolic constructions). IndeedWeyl quotes in the beginning of his talk, Cassirer’s attractive characterisation of manas an animal symbolicum, but Weyl finds that Cassirer himself is finally unable togive very convincing arguments.

Dr Ursprung also quotes the famous statement of Weyl to the students atGottingen, telling them why he chose to come back to Germany to teach the youngergeneration, quitting his position in Zurich (in 1930). ’Wer erkennt, den, verlangtnach Rede’ (He who gains knowledge desires to communicate). To sum up, Dr Ur-sprung’s address is indeed a beautiful sketch of Herman Weyl as an eminent scientistwith a profound perception of the structures of theoretical physics.

The three scientific talks by Yang, Penrose and A Borel bring out in their owncharacteristic and individual manner some of the detailed facets of science whichWeyl had dealt with and the profound influence that they have had for futureresearch. It would of course be impossible for me even to attempt a summary of the

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contents of these lectures. I shall however pick a few leads, which I choose more orless at random.

The first of the three centenary lectures by Yang deals with insights into HermanWeyl’s contribution to physics. The lecture begins with the interesting piece of workthat Weyl did as a young man, the impetus coming through his attending a lecture ofLorentz, where Lorentz raised a problem originating in the radiation theory of Jeans.Weyl solved this problem by using the method of integral equations, a techniquedeveloped by his teacher, Hilbert. This work had several offshoots in the future (seefor instance the paper of Mark Kac, “Can One Hear the Shape of a Drum?” in theAmerican Mathematical Monthly Vol. 73, no.4, Part 2).

Yang quotes a passage from Hermann Weyl in his 1930 edition of Gruppen The-orie und Quanten Mechanik, which in retrospect was prophetic: “The fundamentalproblem of the proton and the electron has been discussed in its relation to the sym-metry properties of the quantum laws with respect to the interchange of right andleft, past and future and positive and negative electricity. At present no solution ofthe problem seems in sight: I fear the clouds hanging over this part of the subjectwill roll together to form a new crisis in quantum mechanics”.

This quotation deals with questions of symmetry which Yang describes in detailin his talk, but I shall restrict myself only to one aspect: the break in symmetry ofleft and right which was proved later by Yang (jointly with Lee). This question wasalready raised in Hermann Weyl’s Philosophy of Mathematics and Natural Sciences,first published in 1926 in German with an English translation in 1949, where Weyltalks about left-right symmetry (parity) “Left and right. Were I to name the mostfundamental mathematical facts, I should probably begin with a fact (F1) that thecounting of a set of elements leads to the same number in whatever order one picksup these elements and mention as a second fact (F2) that among the permutationsof n ≥ 2 things, one can distinguish between the even and odd ones. The evenpermutations form a subgroup of index two within the subgroup of all permutations.The first fact lies at the bottom of the fundamental notion of dimensionality, andthe second of that of sense (orientation)”.

Indeed, as Yang remarks, left-right symmetry or parity conservation was such anatural and useful concept for physicists that it had always been taken for granted asa sacred law of nature (Remarkably, in 1957, less than two years after Weyl’s death,Yang and Lee disproved the parity conservation). Yang remarks that Wu, Ambler,Hayward, Hoppes and Hudson found by experiments that the left-right symmetrywas after all not exactly observed by the laws of physics. The violation was slightbut observable if one knows where to look for.

It’s interesting that, in his very beautiful book of Weyl, Symmetry, one reads the

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following passage: “For, in contrast to the orient, occidental art, like life itself, isinclined to mitigate, to loosen, to modify, even to break strict symmetry. But seldomis asymmetry merely the absence of symmetry. Even in asymmetric designs, one feelssymmetry as a norm from which one deviates under the influence of forces of non-formal character. I think the riders from the famous Etruscan tomb of Tricliniumat Corneto provide a good example.”

Riders of Triclinium

Yang ends his lecture by saying that “if Weyl were to come back today, he wouldfind that amidst the very exciting, complicated and detailed developments in bothphysics and mathematics, there are fundamental things that he would feel very muchat home with. He had helped to create them”.

Roger Penrose, in his article pays tribute to the scientific contributions of Weyl,by indulging, (to quote his own words), “in some flights of fancy of (his) own”,related to the work of Weyl. He first discusses some problems of Tiling the planewith not necessarily symmetric shapes, which is related to the work of Weyl onsymmetry and moves on to to the study of quasi-crystals. He notes that the ideaof the study of Riemannian manifolds is a direct offshoot of the earlier pioneeringwork of Weyl himself on problems of Space-Time-Matter and then discusses the socalled “Weyl-Levi-Civita connection” of the Einstein equations and problems thatarise in connection with Twistor theory; He refers to the beautiful work of Weyl (incollaboration with Richard Brauer) on spinors through Clifford algebras, which wasmotivated, in turn by the Dirac equation for the electron. He also refers to Weyl’sequation for the neutrino. Other areas like conformal geometry and the theory ofentropy which he discusses are also related to the work of Weyl.

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Borel’s article on the contributions of Hermann Weyl to Lie theory can be thoughtof as an exquisite mathematical tapestry full of intricate individual designs and yetpresenting a grand whole. Beginning with how Weyl’s interest in the study ofLie theory was an outcome of his preoccupation with the nature of Space-Time,Borel traces steadily and step-by-step as to how Weyl eventually ended up, with hisprofound understanding, with decisive results on the structure and representations ofLie groups and Lie algebras and how he grappled with problems on invariant theory,with a remarkably adept use of the work of I.Schur, E.Cartan and A.Hurwitz.

An interesting aspect of the complete reducibility of representations of semisim-ple algebras is that though Weyl’s proof uses global methods of integration, it wasleft to the physicist H.B.G. Casimir to provide an algebraic proof and this makesan interesting story. Indeed, in the representations of g = sl2(C) or equivalentlyso3(C), an important role is played by a polynomial of second degree which rep-resents the “square of the magnitude of the moment of momentum”, (given in thebook “Gruppen Theorie und Quanten mechanik”), which is the sum of the squaresof the infinitesimal rotations around the co-ordinate axes. It commutes with all theelements of g and is hence given by a scalar in any irreducible representation: thisyields an important quantum number j(j +1), in the representation of degree 2j +1(2j a natural number). Casimir, motivated by this, defined an analogous operatorfor an arbitrary semisimple Lie algebra and this has later come to be called theCasimir operator. I would like to include an interesting remark of Casimir himselfwhich he made in an interview of his with some American Physicists in 1963 inwhich he says that when he went to work with Pauli in Zurich during 1931-32, Paulisaid “Now there is one unsolved problem in group theory, which is a problem ofthe complete reducibility of representations which is not satisfactory, because it isproven by Weyl, but only with a kind of integration over group space and not bypurely algebraic methods”. ‘So, he put me to work on that and I was able to do iteasily for the rotation group and for a number of other cases for other groups andthen I wrote to van der Warden who completed the proof’, mentions Casimir.

Soon there was another algebraic proof, followed by a cohomological proof byJ.H.C Whitehead and at the suggestion of Weyl, Jacobson began his study of Liealgebras over arbitrary fields of characteristic 0.

To sum up, in the article of Borel, his aim, in his own words is “to give an ideaof Weyl’s work on Lie groups and of its repercussions (that were) felt in a broadrange of topics in analytical, differential geometric, topological or algebraic contextsand took many forms: general theorems or specific results in special cases, clear-cutstatements as well as sharply delineated suggestions or guiding principles mirroringthe many-sidedness of Weyl’s outlook and output”.

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“Not only much more than chance was needed to produce such a synthesis, butWeyl 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 oneelse was conversant with all of these; in fact, except for Schur, with hardly more thanone. Although I limited myself to a rather sharply circumscribed and quantitativelyminor part of Weyl’s work, this already provides a demonstration of, a practicallesson in, the unity of mathematics, given to us by a man whose mind was indeed ameeting ground for most mathematics and mathematical physics.”

Apart from the articles that I have briefly talked about in this lecture, theCentenary Volume also contains an Appendix which lists other activities that tookplace during the period, which were mainly three social gatherings, where therewere personal reminiscences of some people who had known and interacted withHermann Weyl at various levels, which bring out the great personal charm thatWeyl exerted on those who came to know him well. I shall particularly choose onesuch reminiscence, namely that of Michael Weyl, Hermann Weyl’s son, who bringsout very forcefully the qualities of Hermann Weyl as a person, who apart from beinga great mathematician and physicist was much more! Michael Weyl gives variousinstances to show the most passionate and wide literary tastes of Hermann Weyl:he was a real connoisseur of literature, in the art of linguistic expression, whichone clearly witnesses in all his writings. He had a distinct talent for quoting fromliterature - be it prose, poetry or philosophy. His language was limpid, a habit whichhe practised as a creed in all his speech and writing. Weyl had a deep relationshipto poetry. His poignant use of quotes culled out from poetry (and at times fromprose too) made cold mathematical reasonings look like lovable little essays. Heonce wrote ”Mathematics is not a rigid and rigidity producing schema as whichthe layman views it; rather, we find ourselves in it exactly that crossing point ofconstraint and freedom which is the very essence of man’s nature.” Michael Weylgives an example to show how spontaneous Hermann Weyl could be in finding aliterary quote even from obscure poets to make a mathematical point. In his verybeautiful book Symmetry , he quotes a poignant plea for symmetry made by a notwell known American poetess Anna Wickham, which runs as follows:

God, Thou great symmetry,Who put a biting lust in me

From whence my sorrows spring,For all the frittered days

That I have spent in shapeless waysGive me one perfect thing.

I should add that in the obituary of Emmy Noether that Hermann Weyl wrote,

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he very aptly quotes the American poetess Edna Vincent Millay, from her Dirgewithout music, the following verse:

Down, down, down into the darkness of the graveGently they go, the beautiful, the tender, the kind;Quietly they go, the intelligent, the witty, the brave

I know. But I do not approve. And I am not resigned.

Hermann Weyl was very much at home with the work of all the poets of Germanyand as Michael says, he loved to quote poetry to his family. But he also had aspecial kind corner for Swiss poets like C F Meyer and Gottfried Keller. One, forinstance would remember what Hermann Weyl wrote in the Preface to his bookon Classical Groups, the first book he wrote in English, which was not his mothertongue. He says and I quote ‘The Gods have imposed upon my writing the yokeof a foreign tongue that was not sung at my cradle - “Was das heissen will, weissjeder, Der im Traum pferdlos geritten” - I am tempted to say with Gottfried Keller.Nobody is more aware than myself of the attendant loss in vigour, ease and lucidityof expression’. For someone, who wrote in English for the first time, his very firststatement itself shows the inherent poet in him. Certainly Weyl rode a new horse,the English language; he rode almost as beautifully as the horse he grew up on, his“beloved German”, as those who had the pleasure of reading his books would testify.

I believe that the most touching remarks about Hermann Weyl are in MichaelWeyl’s reply to the various glorious tributes paid to the greatness of HermannWeyl. He mentions there, essentially the last thoughts of Hermann Weyl, which arefound in a formal lecture at Lausanne in 1954 and reprinted in an article entitledBesinnung und Erkentniss, which appeared in the 1955 volume of SchweizerischeHochschulzeitung in Zurich. Towards the end of Michael Weyl’s reply he mentionsthe poem Evening Song by Gottfried Keller and Hermann Weyl’s favourite verse,which begins with

‘Liebliches Jahr, wie Harfen und Floten,Mit wehenden Luften und Abendroten,

Endest du deine Bahn’.

and what Hermann Weyl says about his lectures on Symmetry (his “Swan Song”)at Princeton University: “I had the same feeling as that of a person after a long dayof hard work, now with the sun sinking and the twilight setting in, when he pipes afew notes from his flute”, goes a rough English translation.

To end then my talk, I emphasise that my eulogising Hermann Weyl as a greatscientist and a great human being endowed with extraordinary sensitivity, wisdom

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and intellect is to bring his following conviction to the notice of the scientists andteachers of this audience, which is succinctly and powerfully present in what he wrotewhen he was preparing for his Eranos lectures on “Science as a Symbolic Construc-tion of Man”. He writes “The effort to isolate facts in the midst of the muddyturbulent stream of our onflowing lives and the effort to find the adequate languagefor communicating facts to each other are creative human acts which must go handin hand”. Hermann Weyl was the generic model who found the “adequate language”and in so doing he turned dry scientific communication into a truly creative act. Hewas indeed a mathematician with the soul of a poet.

Acknowledgement: I am deeply indebted to Vinay for generously helping meto bring this article to the present form and Vijayalakshmi and Padma for theirvaluable help in correcting my typographical errors.

Prof. R. SridharanChennai Mathematical InstituteSIPCOT IT ParkPadur Post, Siruseri-603103E-mail: rsridhar@cmi.ac.in

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