Irving Ezra Segal (1918–1998)

mem-segal.qxp 5/12/99 12:57 PM Page 659 Irving Ezra Segal (1918–1998) John C. Baez, Edwin F. Beschler, Leonard Gross, Bertram Kostant, Edward Nelson, 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 Einar Hille, 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- Irving Segal 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 harmonic analysis. encouragement was strong when I was writing a JUNE/JULY 1999 NOTICES OF THE AMS 659 mem-segal.qxp 5/12/99 12:57 PM Page 660 thesis, and equally important was his total lack of ter is itself a deformation of the Lie algebra 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 cosmology and W. T. Martin on Brownian motion was tied to it yields an alternative explanation of the redshift 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 Hilbert space. There is no orthogonally invariant quantum field 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 quantum field theory. 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 physics, 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 quantum mechanics 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. 660 NOTICES OF THE AMS VOLUME 46, NUMBER 6 mem-segal.qxp 5/12/99 12:57 PM Page 661 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.

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