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Irving Ezra Segal (1918–1998)

John C. Baez, Edwin F. Beschler, Leonard Gross, , , Michèle Vergne, and Arthur S. Wightman

Irving Segal died suddenly on August 30, 1998, After the war while taking an evening walk. He was seventy-nine Segal spent two and was vigorously engaged in research. years at the Insti- Born on September 13, 1918, in the Bronx, he tute for Advanced grew up in Trenton and received his A.B. from Study, where he Princeton in 1937. What must it have been like to held the first of the be a member of the Jewish quota at Princeton in three Guggenheim the 1930s? He told me once that a fellow under- Fellowships that he graduate offered him money to take an exam in his was to win. Other stead and was surprised when Irving turned him honors included down. election to the Na- He received his Ph.D. from Yale in 1940. His the- tional Academy of sis was written under the nominal direction of Sciences in 1973 , who suggested that Segal continue his and the Humboldt Award in 1981. At and Tamarkin’s investigation of the ideal theory the University of of the algebra of Laplace-Stieltjes transforms ab- Chicago from 1948 solutely convergent in a fixed half-plane. But, Segal to 1960, he had fif- wrote, “For conceptual clarification and for other teen doctoral stu- reasons, an investigation of the group algebra of dents, and at MIT, a general [locally compact] abelian group was of where he was pro- interest.” And the thesis was not restricted to fessor from 1960 abelian groups. on, formally retir- Segal was an instructor at Harvard in 1941, and ing in 1989, he had twenty-five. Segal’s mathe- then war work—first at Princeton and later in the matical ancestry runs from Hille and Marcel Riesz army at the Aberdeen Proving Ground—prevented through Fejér and Schwarz to Weierstrass. a full publication of the thesis until 1947. I had the great fortune to be one of Irving’s stu- Looking edgewise at a bound journal volume, dents. After telling him what I intended to do in one perceives a band spectrum for the articles— my thesis, I was embarrassed to learn from a fel- the darker the band, the more intensely has the ar- low student that one is supposed to ask for a topic. ticle been studied. Segal’s thesis acquired a dark But Irving never demurred; he gave me free rein band indeed. Together with M. H. Stone and I. M. and helped launch me on a career. I shall repeat Gelfand, he was one of the principal architects of here something I wrote on the occasion of his six- the application of algebraic methods to analysis, tieth birthday, since it recounts an early experience vastly simplifying and extending classical results that helped shape my mathematical life. His of . encouragement was strong when I was writing a

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thesis, and equally important was his total lack of ter is itself a deformation of the of the encouragement when I found a result unrelated to conformal group, and now we have reached the end anything beyond itself. One of the chief charac- of the road: this Lie algebra is rigid. teristics of Segal’s work is that his theorems are Segal’s vision was that the universe is the uni- part of theories, and this sense of the global na- versal cover M of the conformal compactification ture of mathematical research was one of the most of Minkowski space—Einstein’s spherical uni- valuable things that he imparted to his students. verse—with the universal cover of the conformal Segal had an extraordinary intuition for the es- group as symmetry group. He pursued this vision sential. The work of N. Wiener and of R. H. Cameron with passion and immense industry. In and W. T. Martin on was tied to it yields an alternative explanation of the a particular representation; in Segal’s hands, it be- as due to the difference between chronometric came a general theory of Gaussian integration on time and the time measured in an observatory. In . There is no orthogonally invariant quantum theory the compactness of space in Gaussian measure on an infinite-dimensional real the Einstein universe (it is S3) and a natural time Hilbert space, but Segal constructed the corre- cyclicity mollify the divergence problems. Together sponding algebra of random variables. And he in- with Zhengfang Zhou, Segal constructed quantum variably produced new concrete results that fol- electrodynamics and a nontrivial φ4 quantum lowed from his abstract constructions. Similarly, field on M. Here is a summary he wrote [2] in 1992: quantum theory—especially of systems of infi- Universal space-time is a natural nitely many degrees of freedom—was tied to par- candidate for the “bare” arena of the ticular representations by operators on some fundamental forces, being the maximal Hilbert space. It was Segal who realized that the ∗ 4-dimensional manifold having physi- structure of physical relevance was the C -alge- cally indicated properties of causality bra generated by the observables, a discovery that and symmetry. It is locally conformal was largely ignored at first and then became taken to Minkowski space, and globally con- for granted. These two developments were unified formal to the Einstein universe in a theory of algebraic integration that applies to E ∼ R1 × S3. The Einstein energy ex- commutative and noncommutative systems alike, ceeds that in the canonically imbedded with applications to stochastic processes, a Minkowski space, and the difference Plancherel formula for unimodular Type I locally has been proposed by the chronomet- compact groups, and implementability of canoni- ric theory to represent the redshift. Al- cal transforms in quantum systems of infinitely though this eliminates adjustable cos- many degrees of freedom. mological parameters, the directly In all his work Segal was a pioneer. To mention observable implications of this pro- one example not discussed elsewhere in this arti- posal have been statistically quite con- cle, Sergiu Klainerman, in accepting the Bôcher sistent with direct observations in ob- Prize (Notices, April 1999), credits Segal with being jective samples of redshifted sources. the first to point out the role of space-time in- These developments represent a math- equalities for nonlinear hyperbolic equations. ematical specification of proposals by In the 1960s Segal organized two conferences Mach, Einstein, Minkowski, and Hub- at MIT that were the occasion of an initial break- ble and Tolman. They suggest that the through in constructive . The fundamental forces of Nature are con- extraordinary subsequent development, primar- formally invariant, but that the state of ily by James Glimm and Arthur Jaffe, was not along the Universe breaks the symmetry down lines that Segal favored—a viewpoint that he made to the Einstein isometry group. This painfully clear. provides an alternative to the Higgs The last thirty years of his professional life mechanism, and otherwise has impli- were dominated by a discovery he published in cations for particle , including 1951. In the last section of a wide-ranging article the elimination of ultraviolet diver- [1], Segal initiated the theory of deformations of gences in representative nonlinear Lie algebras. (Deformations became “contractions” quantum fields, the formulation of a in the physics literature and were “limiting cases” unified invariant interaction Lagrangian, in the article.) Classical mechanics is a limiting assignments of observed elementary case of as ~ → 0; the corre- particles to irreducible unitary posi- sponding commutative Lie algebra is a deforma- tive-energy representations of the con- tion of the Heisenberg algebra. Nonrelativistic me- formal group, and the correlation of chanics is a limiting case of relativistic mechanics the S-matrix with the action in E of as c →∞; the Lie algebra of the Galilei group is a the generator of the infinite cyclic cen- deformation of the Lie algebra of the inhomoge- ter of the simply-connected form of the neous Lorentz group. But Segal showed that the lat- conformal group.

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Why has this work not received an adequate eval- One day I noticed I had a counterexample to one uation? Part of the reason lies in Segal’s style of of Segal’s lemmas. I had never had a personal con- scientific exchange—at times it resembles that of versation with him, and my wrong impression was Giordano Bruno (later burned at the stake), who that he would not welcome one. It was at the urg- very shortly after his arrival in Geneva issued a ing of friends of mine that I finally mustered the pamphlet on Twenty Errors Committed by Profes- courage to go to his office and show him my coun- sor De la Faye in a Single Lesson. But part of the terexample. He graciously agreed that I was cor- fault lies with cosmologists and particle physi- rect. However, it was only a small matter. He had cists intent on defending turf. just neglected to add some rather natural hy- The time for polemics is past. Segal’s work on pothesis. As I was walking out the door he suddenly the Einstein universe as the arena for cosmology stopped me and asked, “What do you know about and particle physics is a vast unfinished edifice, Lie groups?” I replied that I knew something about constructed with a handful of collaborators. It is that subject, since I was currently taking a course rare for a mathematician to produce a life work that with Ed Spanier on Lie groups. Without saying a at the time can be fully and confidently evaluated word he went to his desk and started writing. He by no one, but the full impact of the work of Irv- then got up, handed me a paper, and said, “Okay, ing Ezra Segal will become known only to future here’s your Ph.D. thesis problem.” I was totally generations. stunned. This was the beginning of a period in my —Edward Nelson life when I could not say “no” to Irving Segal. Here I had walked into his office just to discuss some small matter about his course, and I walked out not only having a thesis advisor but also having a par- Bertram Kostant ticular thesis problem. I was a graduate student at Chicago in the early After that my graduate career radically changed. 1950s, and I became Irving Segal’s Ph.D. student Segal was very good to his students. Also, I began in the 1951–52 academic year. I want to say a lit- to know him quite well on a personal basis. There tle bit more about how that came about. To do so, was an intensity about Irving that resonated with I should say something about what Chicago was me. He had an apartment off the Midway and on like in the early 1950s. It always seemed to me that Friday nights held open house. These were the the graduate school environment at that time and only occasions I can remember at Chicago where place was unlike anything I have subsequently one could socially meet faculty members, visitors, seen throughout my career. The place was teem- and members of other departments. Segal would ing with students, and the intellectual atmosphere walk around joining small groups of his guests, be- was such that one was made to feel that doing coming a catalyst for good conversations. was the most important thing one It is an understatement to say that he affected could do with one’s life. Perhaps the person who the course my life took after I became his student. most contributed to this particular feeling in me Here are some details supporting that statement. was Irving Segal. It was through Irving that I got a two-year ap- But back to the story about how I became his pointment, starting in 1953, to the Institute for Ad- student. Frankly, Segal did not have a great repu- vanced Study (even before I began writing my the- tation as a teacher. However, Chicago’s graduate sis). This was a rare opportunity, since among education system was such that there were certain other things I met such luminaries as Einstein, courses in geometry, algebra, and analysis that von Neumann, and Hermann Weyl not very long be- one had to take. To fill the latter requirement, I fore their deaths. After my stay at the IAS I am sure found myself in Irving Segal’s course in measure it was due to Irving’s influence with W. Feller that theory. To my surprise and delight it turned out I received a one-year offer as Higgins Lecturer at to be a marvelous course. Segal worked very hard Princeton. After that I went to Berkeley on my on it. Each lecture was slowly and carefully deliv- own. But it was after only a few years, while still ered. There were typed notes, there was no wav- an assistant professor, that I received an offer of ing of hands, and every epsilon and delta was a full professorship at the . there. In fact, the whole set of notes produced a I can only imagine that this was engineered by Irv- book, which in my opinion was superior to Hal- ing and perhaps Adrian Albert. mos’s newly published book on measure theory. But leaving a rising Berkeley (and the beauty of California) to go to what I sensed was a declining Edward Nelson is professor of mathematics at Princeton Chicago was not terribly appealing to me. This University. His e-mail address is nelson@math. was my first “no” to Irving. A few years later Irv- princeton.edu. ing moved from Chicago to MIT. Not long after that, Bertram Kostant is professor emeritus of mathematics at I received a full professorship offer from MIT. By the Institute of Technology. His e-mail ad- this time (1961–62) my interest in Berkeley had al- dress is [email protected]. ready peaked. It would be painful to break a strong

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tie I had developed with , but theorem determining exactly when invariant cones Irving convinced me that mathematically Boston in semisimple Lie algebras exist. A closer study of was the place to be. “The winters are not that bad. such cones is today an active subject. This elliptic Sure, it snows a lot, but you can learn to cross- element is at the heart of Segal’s cosmological the- country ski.” I accepted the offer from MIT. But this ory. What he is saying is that it is the elliptic ele- began the period when I found it easier to say ment that should be used to determine the energy “no” to Irving. of an electromagnetic wave, and not the nilpotent One of the reasons our work went separate element. There was no and no expansion ways was that Irving really focused only on those of the universe. The redshift is not a Doppler ef- aspects of mathematics, and in particular only on fect. It is accounted for by the difference between those aspects of , that he the elliptic and nilpotent elements—negligible lo- felt dealt directly with physical theory. He ignored cally, but significant at great distances. Although the revolution brought about by Harish-Chandra his cosmological theory has thus far attracted very and I. M. Gelfand. Besides becoming interested in few supporters, there is clearly much that is un- that development, I also became interested in satisfactory in the widely accepted big bang the- geometry and other areas that it seemed Irving ory. I have it from a highly reliable but unnamed found easy to ignore. Irving was single-mindedly source that there is a growing group of cosmolo- driven to find the right mathematical models to de- gists who have come to believe that the correct un- scribe certain physical theories, such as cosmol- derstanding of the redshift is some sort of fusion ogy and quantum field theory. In his later life, I of the Doppler effect and Irving’s theory. So it is think, cosmology superseded quantum field the- not impossible that Irving could turn out to be cor- ory. At the heart of the cosmology theory that rect after all. Segal developed was the 15-dimensional Irving Segal was a unique individual who af- SU(2, 2), referred to by physicists as the “confor- fected the lives and thoughts of a large number of mal group”. He focused all of his attention on this people, certainly including me. With his passing I group. There are certain properties of this group think the world is a poorer place. that he felt were at the heart of understanding im- portant things. Irving often pointed to certain phenomena that turned out to be the tips of icebergs. For example, Edwin F. Beschler he was fascinated by the fact that the conformal I first met Irving when, as acquisitions editor for group stabilized the solutions of the wave equa- Academic Press, I was seeking someone to estab- tion even though the wave operator did not com- lish a journal in the field of functional analysis. This mute with the group. He asked me about this, and was in the early 1960s when the boom in special- it seemed indeed to be an interesting question. I ized journals was about to begin. Irving’s name, thought about it and wrote a paper called “Quasi- along with that of Ralph Phillips, with whom I had invariant differential operators” which made con- also spoken, was among the most often mentioned. nections between a number of things, including in- When I approached him, his response was incisive tertwining operators on Verma modules. The latter and immediate—almost as if he had anticipated the subject was carried to deeper levels by the Gelfand question. With a clear understanding of editorial school and eventually led to the Kazhdan-Lusztig autonomy and assurance of support from Ralph theory, an important development in modern Lie and at least one other colleague, he agreed to un- theory. dertake the task. In short order he had brought Paul Another aspect of the conformal group that Malliavin into the group, an agreement was reached fascinated him was that its Lie algebra has elements within a few months of the first discussions, and that, for many representations, have a nonnegative the first issue of the Journal of Functional Analy- spectrum. From his perspective this could make sis (JFA) appeared not more than a year later. Irv- them candidates to represent energy in physical ing was not one to procrastinate. applications. One particular nilpotent element, in The concomitance of the three editors’ views and the representation of SU(2, 2) associated with so- the firm leadership provided by Irving was re- lutions of Maxwell’s equations, defines the stan- markable. Through the next twenty years, editor- dard operator to determine the frequencies of ial board meetings consisted of a get-together of light waves. But Irving focused on another ele- the four of us for coffee or tea every four years at ment with a nonnegative spectrum, an element the International Congress of Mathematicians (I that was elliptic and not nilpotent, but closely re- missed one or two), with agreement that everything lated to the above-mentioned nilpotent element via a theorem of Morosov. This elliptic element has Edwin F. Beschler is retired and works part-time as an af- beautiful mathematical properties, like generat- filiate member of Moseley Associates, Inc., a firm that of- ing an invariant cone. This is the tip of another ice- fers management consulting to the publishing industry. berg. I became involved in this study, producing a His e-mail address is [email protected].

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was fine. In between, the JFA worked Ph.D. Students of Irving Segal smoothly and efficiently, and it was Isadore M. Singer, Chicago (1950) John Chadam, MIT (1965) always a pleasure to deal with Irv- Henry A. Dye Jr., Chicago (1950) Jan M. Chaiken, MIT (1966) ing and the board. In my tenure I do Joseph M. Cook, Chicago (1951) Robert R. Kallman, MIT (1968) not recall a single problem that was Ernest A. Michael, Chicago (1951) Michael Weinless, MIT (1968) not handled fairly and expeditiously. If there were editorial problems of Ernest L. Griffin Jr., Chicago (1952) Michael J. J. Lennon, MIT (1969) which we at the publishers were un- Jacob Feldman, Chicago (1954) Niels Skovhus Poulsen, MIT (1970) aware (and one suspects they arose, Bertram Kostant, Chicago (1954) Tomas P. Schonbek, MIT (1970) as they surely do in even the best- Lester E. Dubins, Chicago (1955) Arthur Lieberman, MIT (1971) ordered groups of researchers), it Edward Nelson, Chicago (1955) Abel Klein, MIT (1971) was another mark of Irving’s style that he settled them with the least Brian Abrahamson, Chicago (1957) Stephen Berman, MIT (1972) amount of fuss possible. He man- Ray A. Kunze, Chicago (1957) Steven Robbins, MIT (1973) aged to create, from my point of W. Forrest Stinespring, Chicago Edmund G. Lee, MIT (1975) view, a model of that peculiar mix (1957) Hans Plesner Jakobsen, MIT (1976) of autocracy and democracy re- Robert J. Blattner, Chicago (1957) Bent Ørsted, MIT (1976) quired to make a journal work. The Leonard Gross, Chicago (1958) Thomas P. Branson, MIT (1979) model served me well in the fol- lowing years, though I cannot say I David Shale, Chicago (1960) Mark A. Kon, MIT (1979) was often able to replicate it. Walter A. Strauss, MIT (1962) Stephen M. Paneitz, MIT (1980) On a more personal note, I re- Roe W. Goodman, MIT (1963) Derrick C. Niederman, MIT (1981) member with nostalgic amusement Matthew Hackman, MIT (1963) John C. Baez, MIT (1986) my arrival, along with Irving and at A. Robert Brodsky, MIT (1965) Jan Pedersen, MIT (1991) least another one hundred or so members of the AMS, at Shmeretvyo Richard B. Lavine, MIT (1965) Airport in Moscow in 1966. In those days one did not learn the name of ferred to be barefoot whenever possible, I was one’s hotel assignment until arrival at the airport. somewhat intimidated by his appearance. He was Our group found itself lined up in front of a small always impeccably dressed in a suit, he wore a table, staffed by two Intourist employees with a goatee shaved short in a no-nonsense sort of way, smattering of English and armed with a ledger and he made up for his lack of height by an erect book in which was inscribed each of our groups’ posture and commanding manner. But I decided names, in Cyrillic—and I suspect not even in that to work with him because of all the pure mathe- alphabet’s order. The procedure was that the first matics faculty, he seemed the most passionate person in line pronounced a name and then a about physics, not just as a source of mathemat- search through the list was conducted, attempting ics problems, but as an end in itself. to find a reasonable match. It was obvious after the I wanted to work on quantum gravity, but at MIT first two or three such searches that the process would take all night. Rising above the growing din everyone interested in this subject was working on of complaints was Irving’s voice, coming from far superstrings, for which I had little taste. Segal him- back in the queue as he approached the table, self found Einstein’s equations too ill-behaved to protesting something like “NO, NO, NO!! This will bother trying to quantize them. The lack of a con- never do!” Irving firmly commandeered the book, served energy, the tendency for solutions to de- began at the top of the list, and called out the velop singularities—these qualities convinced him name of the first person on the list, then the sec- that general relativity was fatally flawed. My ar- ond, and so on. The Intourist employees were star- guments in favor of general relativity failed to con- tled and, I think, uncertain as to whether to be vince him, so I wound up working on one of his angry or simply amazed. They apparently had specialties, the mathematical foundations of quan- never seen such a performance nor imagined such tum field theory. a procedure. Irving was in charge, and the sense I learned a lot and successfully completed a of gratitude among the group was palpable. He was thesis, but I did not have much success proving re- not able to save us from a six-hour wait in our ally interesting theorems. Later, as a postdoc, I hotel’s lobby for room assignments, but I know he decided that quantum field theory was too hard saved us an equal amount of time at the airport. for me, so I worked with Segal and Zhengfang Zhou on classical field theory, i.e., nonlinear wave John C. Baez equations, another of Segal’s specialties. The three

I met Irving Segal in 1982 shortly after I came John C. Baez is professor of mathematics at the Univer- to MIT in order to get my Ph.D. in mathematics. As sity of California, Riverside. His e-mail address is a slouching, scruffy graduate student who pre- [email protected].

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of us wrote some papers together and also coau- of others I recall his writing concerned von Neu- thored a book [3] summarizing Segal’s work on mann and Wiener. quantum fields. Thus I spent about six years in In his later years Segal spent most of his time close contact with him and came to know him on an alternative to the big bang cosmology in rather well. which were to be explained, not by the We would typically discuss mathematics in his expansion of the universe, but by an effect of con- office, taking turns scribbling equations on the formal geometry. According to him, his theory blackboard. He had a devastating way of express- predicted a quadratic redshift-distance relation ing doubt when my reasoning failed to convince instead of the usual linear one. He spent a lot of time statistically analyzing redshift-brightness him. Without saying a word, he would gradually data for quasars and galaxies and wrote papers raise his eyebrows higher and higher as I spoke. claiming they supported his theory. Most as- As they slowly climbed up his forehead, it became tronomers disagreed. ever more difficult to keep up the momentum of I thought long and hard about his derivation of my reasoning. When I finally lost the thread of what the quadratic redshift-distance law from his the- I was saying, he would interrupt and point out my ory, and it never seemed right to me. At first I error as he saw it. Being stubborn, I would not al- hoped I was making a mistake, so I tried to get him ways accept these criticisms. As he was even more to explain this derivation. His explanation did not stubborn, our discussions sometimes became quite convince me. Later I tried to explain what I thought heated. Zhengfang Zhou served as a calming in- was his error. He became quite angry. When I re- fluence when he was around. alized we would never see eye-to-eye on this sub- Segal’s office was a cozy, lived-in place, cluttered ject, I tried to avoid it. But this was very difficult, with decades of accumulated papers. He had a and our relationship became strained. I am sad to couch where sometimes he would take short naps. say that I eventually wound up avoiding him. He also made coffee in his office, refusing to touch Despite this, I remain very fond of Segal, because the stuff served in the mathematics department he had a real passion for understanding the uni- lounge. He took coffee very seriously, grinding the verse. He did not believe in God and was suspicious beans in his office, using only distilled water, and of all forms of organized religion. The quest for heating it to a precisely optimized temperature. (He perfection that some express through religion he claimed to have done a study to determine this op- expressed through . He could timal temperature.) He often let me work on his never take it lightly! computer while he worked at his desk or type- writer. Sometimes when he wanted to prove a the- orem, he made a great show of setting a kitchen timer, allowing himself no more than thirty min- Arthur S. Wightman utes to get the job done. This was but one of many I first encountered Irving Segal in the winter of ways he emphasized the importance of a busi- 1946–47 when he spoke in a seminar in (old) Fine nesslike attitude. When I passed my thesis de- Hall at Princeton on the results of his forthcom- fense, the first thing he said was “Good, now we ing paper on postulates for general quantum me- can get back to work.” He never slacked off; he chanics. I was only partly prepared for the grand often came to the office on weekends, and his re- sweep of the talk and the paper that followed. I had tirement seemed not to slow him down in the least. studied ’s book Mathematical People who failed to understand the essentially Foundations of Quantum Mechanics in the Dover prickly nature of Segal’s relationship to the world reprint of the German edition of 1932, available would sometimes misinterpret his actions. For ex- during the war, and had heard of the work of ample, he recently wrote a review of ’s Gelfand and Naimark on C∗-algebras discussed in Noncommutative Geometry for the Bulletin of the a mathematics seminar in New Haven, but I did not AMS. While largely positive, the review contained know of the existence of von Neumann’s paper on a number of serious criticisms. For example, he ex- an algebraic generalization of the quantum me- pressed disappointment that Connes, with all his chanical formalism, which Segal mentioned as mastery of analysis, still treated quantum field being most closely connected with his work. Both theory the way most particle physicists do, using papers can be regarded in retrospect as part of a perturbative Lagrangian methods rather than the mathematical reaction to the physical discoveries more rigorous framework of algebraic quantum of the quantum mechanical revolution of field theory pioneered by Segal and others. Some 1925–1927. They had a twofold motivation: on the mathematicians were greatly upset by these criti- one hand, to distill the essence of the mathemat- cisms. What they perhaps failed to understand was that merely by writing the review, Segal was Arthur S. Wightman is professor emeritus of mathemat- saying that Connes’s work was of the highest cal- ics and physics at . His e-mail ad- iber! Indeed, the only other articles about the work dress is [email protected].

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ical structure of quantum mechanics and, on the was not able to absorb much of this. His knowl- other, to state its principles in a form that might edge base was much more sophisticated than mine. make it possible to go beyond quantum mechan- His ideas came forth quickly. Even after I returned ics. The latter was surely a prime impulse of P. Jor- to Yale I received letters from him raising inter- dan in the paper that led, via the joint investiga- esting questions close to my area of expertise. In tion of Jordan, von Neumann, and Wigner, to retrospect I realize that he was driven not only by von Neumann’s paper. his sense of duty to provide his intellectual prog- As Segal pointed out, the most conspicuous ex- eny with plenty of food for the mind but also by ample of a system of observables satisfying his pos- his single-minded determination to solve one of the tulates is the set of all self-adjoint elements of a big problems of mathematical physics: the exis- C∗-algebra. He left as an open problem to prove tence of interacting quantum fields. Although whether up to isomorphism these were the only much of his work may seem to many mathemati- such. In the decade after the appearance of the pos- cians to be motivated simply by the usual aes- tulates this problem was studied by a number of thetic considerations—and is certainly justified authors, of whom I will mention only two: David by the intrinsic beauty of his ideas—Irving told me Lowdenslager and Seymour Sherman. These au- a few years ago that all of his work was aimed in thors constructed a rich class of examples of Segal one way or another at understanding quantum systems of observables that are not isomorphic to physics. the self-adjoint elements of a C∗-algebra. Fur- Among the many papers of Irving’s that formed thermore, they gave necessary and sufficient con- the core of my mathematical education, one group ditions for Segal systems that such an isomor- of his papers influenced my own work and the work phism hold, thereby completing Segal’s postulate of my students in two distinct directions. Irving’s system. Segal’s insight that the C∗-algebra is the papers [4, 5, 6] were aimed at understanding object with physical meaning and not any partic- the mathematical structure of the Hilbert spaces ular representation of it on a Hilbert space is now associated to a variable number of identical quan- a commonplace of mathematical physics. tum mechanical particles. Although a significant Meanwhile, in physics Rudolf Haag was strug- part of the problems he addressed pertained to gling to understand how the fact that physical integration over an infinite-dimensional Hilbert measurements take place in bounded regions of space, some of the ideas of these papers are most space-time should affect the structure of algebras easily understood in finite dimensions. Let pt and n n generated by observables. He worked originally µt, for t>0, be the heat kernels on R and C re- with algebras of unbounded operators, but over the spectively. One need only write out the convolu- 2 n course of a decade, in part in joint work with Huz- tion pt ∗ f to see that if f is in L (R ,pt (x) dx), ihiro Araki and Daniel Kastler, he came to the con- then pt ∗ f has an analytic continuation, h, to the n clusion that algebras of bounded operators, and entire complex space C . The map St : f 7→ h, the in particular C∗-algebras, provided a language Segal-Bargmann transform, is a unitary operator 2 Rn H 2 best suited to the expression of the ideas of quan- from L ( ,pt (x) dx) onto the space t tum field theory, which is what Segal maintained consisting of holomorphic functions in 2 n in the first place. In the hands of Haag, Doplicher, L (C ,µt (z) dxdy). Furthermore, the Taylor coef- and Roberts localization in space-time and the ficients at 0 of the holomorphic function h may Gelfand-Naimark-Segal construction led to a sys- be assembled so as to define an element α of the tematic theory of superselection rules. This was the space of all symmetric tensors over the dual space ∗ beginning of a profound theory created over three (Cn) . The Taylor map T : h 7→ α is also unitary, H 2 decades and summarized in Haag’s 1992 book. this time with domain t and range equal to the “Fock space” Ft consisting of those symmetric tensors with a finite (t dependent) norm. Now the 2 n overall unitary map TSt : L (R ,pt (x) dx) →Ft can Leonard Gross be described in many other ways: there are Her- mite polynomials lurking in these maps. But the Irving Segal always had lots and lots of ideas. I description of these maps given above provides a remember when, in 1958, I returned for a few days stepping stone to some recent extensions. Just a to the mother institution, the University of Chicago, few years ago Brian Hall generalized the Segal- for my Ph.D. exam after being away for almost a Bargmann transform, replacing Rn by a compact, year. At the end of the visit, as Irving drove me back connected, simply connected Lie group and Cn by to the bus station, he used every minute to pro- the complexification of the group. Soon afterward, vide me with a goodly supply of ideas to keep me Bruce Driver proved that the Taylor map in that busy after I went back out into the wilderness. I context is also a unitary map in a natural way. A Leonard Gross is professor of mathematics at Cornell survey of these theorems and their link to the University. His e-mail address is gross@math. work of Segal, Bargmann, Cameron and Martin, cornell.edu. and P. Krée is given in [2].

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Actually, Irving focused primarily on the infinite- this dumped me into devastating thoughts. I dimensional versions of these isomorphisms over stopped working. A friend in the , linear spaces: one must replace Rn by an infinite- Graciela Chichilnisky, persuaded me, rightly, to dimensional real Hilbert space, H. There are sub- move from France to the U.S. Segal was the at- stantial problems in giving orthogonally invariant tractive presence at MIT. Of course, attractive also 2 meaning to L (H,pt (x) dx) when H is infinite di- were the other giant figures in the field of group mensional. For example, if one chooses an ortho- representations—Sigurdur Helgason and Bert normal basis of H, and thereby identifies H with Kostant. But Segal had a special talent for making 2 l , then the measure pt (x) dx, when n = ∞, has a one feel wanted. In front of him I felt unimportant ∞ clear interpretation as a product measure on R . and little, but I felt that my work was needed. He But the subset l2 has measure zero. There is no soon became essential to me. I remember some- way to interpret the expression for pt (x) dx as a thing he said to me: “You do not need to have countably additive measure on H. One always many friends. One is enough.” So I had two needs to choose some kind of enlargement of H friends—my friend Graciela and him—and this on which the measure will sit. was more than enough to make life worth living Typically, all really useful enlargements have again. He was fascinating for me, an immense spir- some orthogonal noninvariance built in. The clas- itual power in a tiny body. He was passionately in- sical example is that of Wiener space C, consist- terested in his ideas; intellectual work was ranked ing of all continuous real-valued functions on [0, 1] by him above all other activities. Maybe in a more vanishing at 0. C carries a natural probability radical manner, all other work, especially all tra- measure, namely Wiener measure. The subspace ditional feminine duties, were considered as no 0 C consisting of absolutely continuous functions work at all, just pleasurable distractions. To his with square integrable derivative is a Hilbert space. credit, unlike most of us, he would also take care 0 When H = C , the proper interpretation of the in- of all material issues. In fact, nothing seemed to formal expression pt (x) dx (for t =1) is precisely be difficult for him. He cared for others, especially Wiener measure on C. While C is recoverable from children, with great pleasure. My daughter, when 0 the Hilbert space C as the completion with respect she was a little girl, loved to come into his office, 0 to the supremum norm on C , this norm is not or- get an orange, and start a spirited conversation with thogonally invariant. him. My father came to the U.S. to visit me. Segal In order to emphasize the central role of the kindly invited us to his house. What a shock for Hilbert space H, as opposed to the accidental form my father to see a man of his age, and a profes- of some convenient ambient measure space, Irv- sor, serving him. But he was not doing so as an ing gave a definition of integration over H with the obligation due to liberal beliefs. He was just happy help of an equivalence class of measure spaces. Al- taking care of others. Indeed, there was something though the theorems and technology in these pa- highly charming in him that nobody could resist. pers have influenced much mathematical activity, With Hugo Rossi I had done some work before the slightly complicated, though orthogonally in- 1974 on certain special unitary representations of variant, meaning that he gave to the expression semisimple Lie groups. This work was of high in- pt (x) dx has not been as widely adopted. This terest for Segal’s cosmological theory. My past writer, strongly influenced by Irving’s view of the work gained immediate significance. Segal had primacy of H in infinite-dimensional Gaussian in- projects where my contribution was impatiently ex- tegration theory but forced by my foray into infi- pected. Due to his influence and strong will, I was nite-dimensional potential theory to have the mea- able to work again. In fact, I remember those first sure pt (x) dx live in some Banach space, abstracted years at MIT as one of the most happy periods in the ordinary Wiener space: if one completes the my life. I would again work and work and work and Hilbert space H with respect to a second, extremely report on my work almost every day to Segal. Hans weak norm, then the completion will support a Jakobsen and Bent Ørsted were Danish students measure that can, in a precise sense, be inter- of Irving. They were bright, friendly, and amusing. preted as the measure pt (x) dx. The influence of Birgit Speh, a student of Bert Kostant, was also very one part of mathematics on another is quite visi- often with us. Later on, there was also Steve Paneitz, ble here: few of the probabilists who are the cur- who died tragically when swimming with Segal in rent users of these abstract Wiener spaces have an a lake. We would all meet regularly for an infor- interest in or knowledge of the quantum field the- mal seminar in his office. We could also go for long ory problems that led Irving to study these struc- walks along the Charles River or have dinner at his tures. home. As if prepared by a genie of fairy tales, sud- Michèle Vergne denly in his house there was a dinner ready for

I met Irving Segal at the meeting of the Ameri- Michèle Vergne is director of research at the CNRS, Unité can Mathematical Society in Williamstown in 1973. de Recherche de École Polytechnique in Paris. Her e-mail One year later my mother committed suicide, and address is [email protected].

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everybody, prepared by him, while we discussed Vogan were consulted as experts on all these is- cosmology passionately. sues. The model that Segal proposed for space-time There was a friendly competition among us. is a model where space is finite but the time infi- Work was the most important part of my life. But, nite. This space-time, the 3-dimensional sphere for following Segal’s example, I found nothing more space variables and the real line for the time vari- pleasurable in life than thinking and working. In able, can be equipped at each point with the cone fact, between 1975 and 1981 all my work on rep- of possible directions of the future. The group of resentations was inspired by Segal’s demands. I was causal transformations of this manifold is the uni- not, properly speaking, working for him; I was versal cover of the identity component of the in- pursuing my own research, but I was cheered up definite orthogonal group O(2, 4), a 15-dimen- by the pleasant idea that it was useful to him. Now sional symmetry group. This group, the conformal many mathematicians continue to work following group, became my favorite group. It contains the paths Segal opened on positive energy represen- Poincaré group of symmetries of the usual tations, semigroups of causal transformations, Minkowski space. Segal’s cosmological space is and decomposition of representations related to deduced from the Minkowski space by compacti- the metaplectic representation. fication of the space variables. Let us call the in- Spoken at the memorial service: Once again I am finitesimal generator of time translation in Segal’s here at MIT. But this time, Irving, I will not be able space the Segal energy. A representation of the con- to knock at your door; enter your office; see you, formal group (more precisely of its universal cover) a very small man welcoming me warmly behind has positive energy if Segal’s energy has positive huge piles of papers. You would start an impish discrete eigenvalues. Many questions were raised conversation about mathematicians, colleagues, by Segal for describing all positive energy repre- astronomers, life. It would be highly amusing. You sentations, their tensor products, the description had posted proudly in your office a drawing done of their K-types, etc. Segal’s work is sometimes by your daughter, Karen, representing you as a lit- highly conceptual, as are his fundamental discov- tle devil. It was quite true to life; indeed, you loved eries of the metaplectic representation or of the to be provocative. To be sad or depressed was a abstract Plancherel theorem, and sometimes very form of weakness, to be sick was not allowed, to applied and concrete. In particular, Segal’s work led be unsure of myself was to draw your fire on me. to a detailed study of representations of the con- In front of me, you considered women with open formal group. contempt, maybe just to know how I would react. It was challenging for me to apply my knowl- But you certainly were influential in attracting me edge of small representations to this special group. to MIT and took me onto the board of editors of It was not easy to obtain concrete results as needed your journal. Today I am very sad. I also feel that and to recognize well-known physical equations in I did not always behave right towards you. I loved my purely mathematical world. In these projects you, but it was not easy to be oneself and stand in everybody around Segal was adding his or her own front of you. contribution to Segal’s work. We were all working Dear Irving, you had the strong power to influ- incredibly hard. Many results were obtained in a ence the directions of people’s lives. You have small amount of time. Results obtained in the par- given to many people the love of research. You em- ticular example of the conformal group had impact ployed your charm to develop the creativity of for the general theory of representations. The in- those around you. I am very grateful to you. You variance of the wave equation under the confor- gave me essential help when my life was darkened mal group had a fundamental significance for Irv- by tragic sorrows. You made me a better mathe- ing Segal. It was proved by Bert Kostant. Masaki matician. You taught me that friendship was sa- Kashiwara and I showed that invariance of the cred. If I needed you, you would always be there. wave equation implied invariance of the Maxwell And today, you are here. equation. Hans Jakobsen proved the unitarity of the representation of the conformal group in the References space of solutions of the Maxwell equation. Birgit [1] JOHN C. BAEZ, IRVING E. SEGAL, and ZHENGFANG ZHOU, In- Speh described the list of K-types of some of the troduction to algebraic and constructive quantum positive energy representations. This list of field theory, Princeton Series in Physics, Princeton asymptotic directions in K-types led to the general Univ. Press, Princeton, NJ, 1992. concept of singular support of a representation. [2] L. GROSS and P. MALLIAVIN, Hall’s transform and the Segal-Bargmann map, Ito’s Stochastic Calculus and Bent Ørsted studied relations of these represen- (Ikeda, Watanabe, Fukushima, tations to nilpotent orbits contained in an invari- Kunita, eds.), Springer-Verlag, Tokyo, 1996, pp. ant convex cone. Steve Paneitz studied the image 73–116. of some of the nilpotent orbits of the conformal [3] I. E. SEGAL, A class of operator algebras which are de- group under the moment map. He also classified termined by groups, Duke Math. J. 18 (1951), all possible invariant cones. Bert Kostant and David 221–265.

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[4] ——— , Tensor algebras over Hilbert spaces, Trans. Amer. Math. Soc. 81 (1956), 106–134. [5] ——— , Mathematical characterization of the physi- cal vacuum for a linear Bose–Einstein field, Illinois J. Math. 6 (1962), 500–523. [6] ——— , The complex wave representation of the free boson field, Topics in Functional Analysis: Essays Dedicated to M. G. Krein on the Occasion of His 70th Birthday, Adv. in Math. Suppl. Stud. 3 (I. Gohberg and M. Kac, eds.), Academic Press, , 1978, pp. 321–344. [7] ——— , Fundamental physics in universal space-time, Physics on Manifolds (Paris, 1992), Kluwer, Dor- drecht, 1994, pp. 253–264.

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