Lunch with George Advisor, Marshall Stone
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occasionally one-on-one. I have always been im- them on: my latest student, Hadi Salmasian, has pressed by the independence of his views—he used these same ideas to take the line of work came to his own conclusions and advanced them further and show that what had seemed perhaps with conviction born of long thought—and by his ad hoc constructions for classical groups could scholarship—he carefully studied relevant papers be seen as a natural part of the representation in mathematics or physics and took them in- theory of any semisimple or reductive group. to account, sometimes accepting, sometimes not, George’s body may have given up the ghost, but according to what seemed right. I particularly his spirit and his mathematics will be with us for remember a short but highly referenced oral dis- a long time to come. sertation on the Higgs Boson, delivered for my sole benefit. I forget when George took to referring to me as Arthur Jaffe his grandstudent, but a particularly memorable oc- casion when he did so was in introducing me to his Lunch with George advisor, Marshall Stone. The Stone-von Neumann Theorem, which originated as a mathematical Background characterization of the Heisenberg canonical com- mutation relations, was reinterpreted by Mackey I was delighted to see that the program of the as a classification theorem for the unitary repre- 2007 New Orleans AMS meeting listed me cor- sentations of certain nilpotent groups. Both Cal rectly as a student; in fact I have been a student Moore and I have found new interpretations and of George Mackey practically all my mathemati- applications for it, and now my students use it cal life. George loved interesting and provoking in their work. In fall 2004 I attended a program mathematical conversations, and we had many at the Newton Institute on quantum informa- over lunch, explaining my congenial title. tion theory (QIT). There I learned that QIT had Most of our spurred new interest in Hilbert space geometry. individual meet- One topic that had attracted substantial atten- ings began at tion was mutually unbiased bases. Two bases {uj } the Harvard Fac- and {vj }, 1 ≤ j ≤ dim H, of a finite-dimensional ulty Club. George Hilbert space H, are called mutually unbiased if walked there from the inner product of uj with vk has absolute val- working at home 1 1 2 to meet for our ue dim H , independent of j and k. A number of luncheon, and I of- constructions of such bases had been given, and ten watched him some relations to group theory had been found. pass the reading The topic attracted me, and in thinking about it, I was amazed and delighted to see that George’s room windows. Photo courtesy of Ann Mackey. Generally our con- work on induced representations, systems of im- Mackey in Harvard office. primitivity, and the Heisenberg group combined versations engaged to give a natural and highly effective theory and us so we continued construction of large families of mutually unbi- afterward in one of our offices, which for years ased bases. It seemed quite wonderful that ideas adjoined each other in the mathematics library. that George had introduced to clarify the founda- Some other occasions also provided opportunity tions of quantum mechanics would have such a for conversation: thirty years ago the department satisfying application to this very different aspect met over lunch at the Faculty Club. Frequent- of the subject. I presented my preprint on the sub- ly we also exchanged invitations for dinner at ject to George, but at that time his health was in each other’s home. Both customs had declined decline, and I am afraid he was not able to share significantly in recent years. Another central fix- my pleasure at this unexpected application. ture revolved about the mathematics colloquium, I hadn’t expected the strange-seeming ideas which for years George organized at Harvard. in George’s notes for that reading course to im- George and Lars Alfhors invariably attended the pinge on my research. I had quite different, more dinner, and for many years a party followed in algebraic and geometric, ideas about how to ap- someone’s home. George also made sure that proach representation theory. But impinge they each participant paid their exact share of the bill, did. When I was struggling to understand some a role that could not mask the generous side of qualitative properties of unitary representations his character. of classical Lie groups, I found that the ideas from that course were exactly what I needed. And I am Arthur Jaffe is the Landon T. Clay Professor of Mathe- extremely happy not only to have used them (and matics and Theoretical Science at Harvard University. His to have had them to use!) but also to have passed email address is [email protected]. August 2007 Notices of the AMS 833 George also enjoyed lunch at the “long table” ago, the theme “The Mathematical Theory of Ele- in the Faculty Club, where a group of regulars mentary Particles” represented more dream than gathered weekly. Occasionally I joined him there reality. or more recently at the American Academy of I knew George’s excellent book on the math- Arts and Sciences, near the Harvard campus. I ematical foundations of quantum theory, so I could count on meeting George at those places looked forward to meeting him and to discussing without planning in advance. Through these in- the laws of particle physics and quantum field teractions my informal teacher became one of my theory. George was forty-nine, and I was still a best Harvard friends. So it was natural that our student at Princeton. Perhaps the youngest per- conversations ultimately led to pleasant evenings son at the meeting, I arrived in awe among many at 25 Coolidge Hill Road, where Alice and George experts whose work I had come to admire. George were gracious and generous hosts, and on other and I enjoyed a number of interesting interac- occasions to 27 Lancaster Street. tions on that occasion, including our first lunch While the main topic of our luncheons focused together. on mathematics, it was usual that the topic of Our paths crossed again two years later, on- conversation veered to a variety of other subjects, ly weeks before my moving from Stanford to including social questions of the time and even Harvard. That summer we both attended the to novels by David Lodge or Anthony Trollope. “Rochester Conference”, which brought together George seemed to come up with a viewpoint on particle physicists every couple of years. Return- any topic somewhat orthogonal to mine or to ing in 1967 to the University of Rochester where other companions, but one that he defended both the series began, the organizers made an attempt with glee as well as success. to involve some mathematicians as well. George began as a student of physics and The Rochester hosts prepared the proceedings found ideas in physics central to his mathemat- in style. Not only do they include the lectures, but ics. Yet George could be called a “quantum field they also include transcripts of the extemporane- theory skeptic”. He never worked directly on this ous discussions afterward. Today those informal subject, and he remained unsure whether quan- interchanges remain of interest, providing far tum mechanics could be shown to be compatible better insights into the thinking of the time than with special relativity in the framework of the the prepared lectures that precede them. The Wightman (or any other) axioms for quantum discussion following the lecture by Arthur Wight- fields. man includes comments by George Mackey, Irving When we began to interact, the possibility to Segal, Klaus Hepp, Rudolf Haag, Stanley Mandel- give a mathematical foundation to any complete stam, Eugene Wigner, C.-N. Yang, and Richard example of a relativistic, nonlinear quantum field Feynman. It is hard to imagine that diverse a appeared far beyond reach. Yet during the first ten years of our acquaintance these mathemat- spectrum of scientists, from mathematicians to ical questions underwent a dramatic transition, physicists, sitting in the same lecture hall—much and the first examples fell into place. George less discussing a lecture among themselves! and I discussed this work many times, reviewing Reading the text with hindsight, I am struck by how models of quantum field theory in two- and how the remarks of Mackey and of Feynman hit three-dimensional Minkowski space-time could be the bull’s-eye. George’s comments from the point achieved. While this problem still remains open in of view of ergodic theory apply to the physical four dimensions, our understanding and intuition picture of the vacuum. Feynman’s attitude about have advanced to the point that suggests one may mathematics has been characterized by “It is a find a positive answer for Yang-Mills theory. Yet theorem that a mathematician cannot prove a George remained unsure about whether this cul- nontrivial theorem, as every proved theorem is mination of the program is possible, rightfully trivial,” in Surely You’re Joking, Mr. Feynman. questioning whether a more sophisticated con- Yet in Rochester, Feynman was intent to know cept of space-time would revolutionize our view whether quantum electrodynamics could be (or of physics. had been) put on a solid mathematical footing. Despite this skepticism, George’s deep in- Today we think it unlikely, unlike the situation sights, especially those in ergodic theory, con- for Yang-Mills theory. nected in uncanny ways to the ongoing progress in quantum field theory throughout his lifetime. Harvard George chaired the mathematics department Early Encounters when I arrived at Harvard in 1967, and from I first met George face-to-face during a conference that time we saw each other frequently.