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ORAL HISTORIES

Interviewed by

David Zierler

Interview date

Location

Video conference

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Interview of Arthur Jaffe by David Zierler on June 14, 2021,

Niels Bohr Library & Archives, American Institute of Physics,

College Park, MD USA,

www.aip.org/history-programs/niels-bohr-library/oral-histories/48170

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Interview with Arthur Jaffe, the Landon Clay Professor of Mathematics and Theoretical Science at Harvard University. Jaffe discusses his childhood in New York, where his father was a physician. He shares memories of life during World War II and his affinity for building model airplanes and radios. Jaffe recalls the factors that led him to pursue his undergraduate degree at Princeton, where he began as a chemistry major but switched to physics. He recounts how he learned about the work of Arthur Wightman, leading him to continue at Princeton for his graduate studies. Jaffe describes his work on bosonic field theories and his time at a summer program in Montenegro. He discusses his move to Stanford and his work in the theory group at SLAC under Sidney Drell. Jaffe recalls the beginnings of his collaboration with James Glimm, as well as his move to Harvard. He explains his role in forming the Clay Mathematics Institute at Harvard and discusses his involvement in the International Association of Mathematical Physics and the American Mathematical Society. Jaffe shares his take on topics such as superstring theory, supersymmetry, and the four-dimensional problem, and reflects more broadly on changes he has seen in the field of mathematics over the years.

Transcript

Okay. This is David Zierler, Oral Historian for the American Institute of Physics. It is June 14th, 2021. I am delighted to be here with Professor Arthur Jaffe. Arthur, it's great to see you. Thank you for joining me today.

Great to be here.

Arthur, to start, would you please tell me your title and institutional affiliation?

I am a professor at Harvard University. I have the chair of the Landon Clay Professor of Mathematics and Theoretical Science.

Alright, so let's unpack that a little bit. First, who is or was Landon T. Clay?

Landon T. Clay was an investor who donated a chair to the Harvard Mathematics Department, and I was the second occupant of that chair.

Now, the affiliation, "and Theoretical Science." Is your home department the department of math, or is this a dual appointment?

My home department was the department of physics. I came to Harvard in 1967, became full professor in 1970 as Professor of Physics. And in 1973, I joined the mathematics department as a courtesy, and eventually in 1987 I became chairman of the department. But two years before that, upon the retirement of George Mackey in 1985, I was given the Landon Clay chair.

So, overall, in terms of your teaching, in terms of your interaction with students, have you been more involved in the department of physics, the department of math, or has it been about evenly split over the course of your career?

Oh, it's gone back and forth, just as my career has touched many different bases and many different subjects. I wouldn't want to say that I'm more physics or mathematics. I certainly have training in both. I started as Professor of Physics. I've been President of the American Mathematical Society, maybe the only person in that position who started in physics. And I do what I do. I like to enjoy understanding the world. I think of mathematics and physics actually as the same thing.

A nomenclature question that might add a little more detail to this, what do you see as the difference between physical mathematics and mathematical physics?

Oh, I don't understand what those terms mean. For me, mathematics is a way of understanding things precisely through proving theorems. Physics actually grew out of experiment and verification in the laboratory, but eventually something called theoretical physics came along which had to do with understanding laws of physics connecting with mathematics. The way that traditionally people in physics and mathematics work is that it's very similar at the initial stage, but in mathematics, people try to get things completely pinned down and prove theorems. In physics, that's not always the case, and people tend to take ideas and work with them at a very early stage. But in mathematical physics, the subject that I've worked in, we like to understand mathematics through physics, and physics through mathematics. We see that the two subjects, which historically were the same, are really very closely aligned together. Probably more closely than any other two sciences to each other.

Arthur, just for a snapshot in time, circa June 2021, what are you working on these days, and more broadly, what's interesting in the field for you?

Right now, I'm interested in basically two big things. Of course, I've never lost my interest in quantum field theory, a subject that I worked on for a very long time, and also the relations between quantum field theory and statistical physics. But starting around 2015 I had a postdoctoral fellow, Zhengwei Liu, who had a background in mathematics of planar algebras, and was a student of Vaughan Jones. He started to work with me as a postdoctoral fellow. Zhengwei had this pictorial view of mathematics which came out of planar algebras, but also his whole way of thinking about things was in terms of pictures. He taught me planar algebras and the importance of pictures beyond Feynman diagrams, and I taught him something from physics. We put them together and started to work together on understanding some physical phenomena and mathematical open problems through pictures. That's been my love for the last five or six years. Oh, and I should mention that there's a second thing. I have another postdoctoral fellow who developed in his PhD thesis in Berlin, at the Free University, a new platform for drug discovery. I'm very interested in that and I'm pursuing that with some colleagues at the Harvard Medical School. That relates to the picture language because the pictures turned out to have gotten me into something related to quantum computing, and we're very interested to apply ideas from quantum computing to this drug discovery program.

I was going to ask exactly on that topic, the ways in which, both, your interest in how quantum computing can advance your research, and conversely, how you see your research as potentially advancing the reality of quantum computing?

Well, the closest that my research at the moment has come to that is that the big problem that's not understood in quantum computing is how to do error correction. We think that the research that I've been doing with Zhengwei, and other people now sheds some light on that, and hopefully we could try to implement those ideas with some experimental physicists at Harvard.

Arthur, a more contemporary question, how has your research been affected one way or the other over this past year and a half in the pandemic?

Well, I haven't been to my office for 15 months. This has been a big problem. I really enjoy teaching on Zoom, because for small classes you can have very good interaction with the students. I have a panel with everybody's face and name. That leads to a lot of potential back-and-forth. But for research, being isolated has been a big problem. All the random interactions that are unplanned have stopped occurring. So, it's been tremendously hard and time consuming to try to keep things going from the point of view of research. Also, there is the fact that my research group, which before the pandemic comprised of about 10-12 people. Some of them came from China, and went back to get renewal or new visas, and now travel is impossible. So, there are many problems that have been created by the pandemic, by politics, and by other forces which have made research a problem.

What are you most looking forward to getting back to as life returns, slowly but surely, to normal?

Well, since I was vaccinated several months ago, I've begun to see some friends. I hadn't done that for a long time. I really enjoy meeting other people in person. I'm looking forward to getting back to some random interaction with my scientific colleagues that isn't planned, because all these occasional meetings in the hallway where you say something and it leads to a very interesting outcome, or you can run down the hallway and ask a question from a colleague -- it's a very different situation and very different evolution of ideas from what has gone on over the past year and a half.

Well, Arthur, let's take it all the way back to the beginning. Let's go back to Pelham, and even before. Let's start first with your parents. Tell me about them.

My father was a physician, a bone pathologist. He was well-known in his field. In fact, he discovered a number of diseases. When Queen Elizabeth was made queen, he was the coronation lecturer of the Royal Society of Medicine. My mother was a housekeeper. She did many volunteer jobs, but she never worked after I was born. But that trip to England was very important for me, because it connected me with a number of people that my father knew and interacted with. My parents, of course, would have liked me to be a physician, and I thought as an undergraduate I would do that. That's why I started out by studying chemistry. In my junior or senior year at Princeton, I decided that wasn't what I was going to do, and even though I continued and applied to medical school -- I was accepted with a telegram at Harvard, then I had to really make a choice, and told my parents I've decided I wanted to study science. They were somewhat disappointed, but that's the way things were.

Arthur, how many generations back does your family go in the United States?

I don't really know. I know my parents were born in the United States, and I think their parents were, but I don't know much about this history.

What was your father's educational trajectory? What was his training in?

My father was trained as a doctor. I don't know if his separate training was as an undergraduate and then as a medical student. I'm not sure if they weren't combined, but I do know that when he graduated from medical school, he must have done fairly well because he was made, for his second job, directory of the laboratory of a hospital in New York, and he stayed in that job his entire life. It's very unusual that you have a first position as the director of a laboratory, and it's also now very usual that people stay in the same job. I know he was even later offered a job as, I think, chairman of the department of pathology at Harvard, and turned it down because he wanted to stay where he was.

Did your father involve you at all in your work? In other words, did you have an understanding, even as a small boy, what it meant to be a scientist?

He occasionally took me to his lab. I didn't really understand what was going on. He often showed me X-ray pictures and tried to explain the various things in the photographs. He actually liked medical photography and had a number of friends who were extremely good medical photographers. He had been a consultant to the Eastman Kodak Company, and would often go to Rochester and come back with stories about how good the people who worked there were. He thought they were the best photographers that he had come across. So, he involved me sometimes in the photographic work. We had a dark room at home, and I often developed and printed pictures for him of medical things. So, he was very interested in that, and got me interested in photography when I was a young kid.

Arthur, what memories as a young boy do you have of the United States being World War II? What stands out in your mind?

Well, I was pretty young. A number of things stand out. We used to go to Vermont where my parents rented various homes over the summer. I remember one house, where we went a number of times, had a telephone that you would turn the crank. It was a party line, and if you wanted to talk, you'd take it over. We ate canned food that sometimes had something in the bottom of the can that would heat the can if you opened part of the can. I remember the rationing. We had ration stamps for food and other things. But the thing probably that sticks most in my mind was the fact that we always worried whether the U.S. was going to be attacked. There were constant drills at school, and when also we were at home, there were sirens that would go off, and we were supposed to try to find cover. I don't link if the U.S. were attacked, if it would have helped very much. But those were things that stick in my memory. We didn't have a car until after the war, though I had some aunts and uncles who had cars. They often came, and on the weekends, we would do things together with them. I remember the announcement that the U.S. had dropped the atomic bomb was a really big shock. The other thing, before that, I remember I was at home and I turned on the radio and I heard that President Roosevelt had died, and then my mother came home, and I told her, and she was extremely upset. But these are just little bits of memories that stick in my mind.

Arthur, growing up, was your family more secular, or was religion part of your household?

It was pretty secular.

And did you go to public schools throughout?

Yes, I went to high school in the town where I grew up, Pelham, New York. My parents always suggested that I think of going to prep school, but I told them I had a lot of friends in my high school, and I'm not sure I wanted to go away. Actually, some of my best friends then left and went to places like Exeter, but I stayed in high school until I went to Princeton.

Growing up, what kinds of interests or activities did you involve yourself in that might have suggested an innate interest or talent in math and science?

Well, I'm not sure any of the things I did as a kid would lead to mathematics or science. You know, I had a very ordinary life as a kid. I built model airplanes, and often I would go to a club where we could fly them on the weekend in some big fields. These airplanes were tethered on wires to some controller. That was a lot of fun. I also built radios. So, that was another hobby, and once when I was careless, I got an electrical shock that sent me across the room, but it didn't have a long-term effect, I think. Although maybe it made me a little crazy. And then I also was involved with music as a kid. My parents had a grand piano, and they wanted me to play it, so they had various music teachers. For a time, I also played the clarinet. In high school, I briefly had a jazz band. But I didn't practice the piano enough. I really enjoyed the piano, but I didn't practice, and one of the teachers would come to our home, and my mother enjoyed talking with him. They'd always talk for a long time and have a coffee or tea together, but he would admonish me for not spending sufficient time practicing on the piano. Eventually, I gave it up, but I tried it again as a graduate student, but really then didn't have time either to make it work. I've been very lucky later in life to get to know some of the top musicians in the world. That's been a tremendous joy.

Would you say that your high school offered a strong curriculum in math and science?

Oh, I don't have any way to judge that. I don't think I learned that much. I certainly never learned anything about calculus in high school. For science, it got me interested and I really enjoyed it, but I'm not sure that the experience in high school was so strong. I was very lucky at Princeton to be put into the honors classes in physics and mathematics. There, I had tremendous teachers, and I really learned what the subjects were about.

Did you have a sense that you needed more education than your high school could offer, taking classes at community college, tutors, things like that?

No. I never did. In fact, I'm not sure at the time I really knew that there were those opportunities. I spent most of my spare time building things.

What kinds of undergraduate schools did you apply to?

I don't remember all the places that I applied, but I do remember that when I was going to go to college, my first choice was probably Caltech. I flew out to California to visit Caltech, which was interesting. It was the first time I'd flown alone and went across the country. The airplane I was on had an engine problem, so it was a little bit of an emergency in Chicago. I really enjoyed my trip, and was going to do that, but I had an uncle who was a history teacher, and his wife was my mother's sister and was a mathematics teacher. This uncle was well-known as a college advisor in his school. He actually suffered polio as a boy, and was still in a wheelchair, but he was a tremendously interesting person, and active, and even drove his car with some extra gadgets to control it by hand. He convinced me that it would be better to go to Princeton than Caltech because he thought that it left more options open if I decided I wanted to do something different.

Meaning, it would not be a strictly technical education.

That's right.

And was the plan chemistry from the beginning, no matter where you went to school?

I'm not sure. I think so, but I don't really remember.

What were your impressions of Princeton when you first arrived?

Well, I had gone there to visit when I was in high school, and I really enjoyed that, but when I got to Princeton -- I think the first two years at Princeton were not very happy times. I met some really brilliant people, and I must say that at Princeton I enjoyed learning from my friends. That's something I didn't have the opportunity to do so much at high school. I really enjoyed being around a lot of extremely talented and brilliant people. I, in some cases, thought I learned more from my friends than from the teachers. So, that was the part of Princeton I really enjoyed, and I got to know more and more in the last two years. So, by the time I graduated from Princeton, I thought that was the best place in the world.

When did you declare the major in chemistry? Was it right away, or did you take a more general approach, and then focused later on?

I don't remember. I think we had to declare a major in the second year, but I'm not sure.

How much physics education did you get at Princeton?

I didn't get that much. I, of course, took elementary physics courses and mathematics courses. Actually, my roommates were all in physics or mathematics. So, they had discussions, and we talked about that a lot, but my undergraduate education in physics and mathematics was pretty spare. So, as a senior -- actually, I must have done a good job as a junior, because my advisor, Dave Garwin, recommended me to the Harvard Society of Fellows. At that time, it was open to undergraduates or graduate students, but you couldn't work on a degree while you were a junior fellow. Now, it's just a postdoctoral fellowship. I got to the stage of an interview, and this was really exciting because I came to Cambridge, and my host was Dudley Herschbach, who was a junior fellow. Dudley had been a student of E. B. Wilson, father of Ken Wilson. I was at dinner between Dudley and Ed Purcell, who was famous for NMR. But we talked about other things, and he was a really interesting person.

So, I thought this society was great, but just about the same time, I was told that I'd had this Marshall Scholarship to go to England if I wanted, and because of my parents' trip, I had really become excited about studying abroad. So, I withdrew from the Society of Fellows, and I decided to go to Cambridge, where one of my teachers, Charles Gillispie, who was a historian of science at Princeton, he arranged that I'd be admitted to Clare College, because he was friends with the new master. That worked out wonderfully, and just that summer I decided to switch from chemistry and study mathematics.

Was there a particular professor, or interaction, or course that stands out that helped you make that switch?

There was one professor who encouraged me a lot, and that was Donald Spencer. He had been a student at Cambridge, and he did his PhD with Littlewood. He told me all about his experiences. He encouraged me a great deal both then and later when I came back to Princeton as a graduate student. He was actually involved with quite a few undergraduates and helped their careers. For me, Don Spencer was a really special person, and later, actually, was the reason I went to Stanford for a year. Don Spencer left Princeton. He was upset that Princeton hadn't appointed Kodaira as professor. He and Kodaira did really famous work in analysis and geometry, and Kodaira was always a research professor, but he was also by that time a national treasure in Japan. And yet, Princeton said that he couldn’t teach and therefore didn't appoint him to the regular faculty. So, Spencer, I guess, it was during the time I was finishing graduate school, quit Princeton and became chairman of the math department at Stanford, and took Kodaira there with him. When I finished my PhD, I expected I'd go to Berkeley. They had offered me a job, and also a job for my friend Oscar Lanford. The two of us wanted to be close to each other to talk to each other. So, we thought we'd both go to Berkeley. We were both offered jobs as assistant professor in mathematics. I went out to look for a place to live and I told Don Spencer, who was at Stanford, that I was going to look for a place in Berkeley. He asked when I was coming, and I said I'd rent a car and drive down and see him in Stanford. And he said, "No, let me pick you up." And he drove me down to Stanford, and I never got to Berkeley.

Arthur, what were your considerations for staying on at Princeton for graduate school?

Well, I didn't stay on. I was in Cambridge, England, for two years. It was only at Cambridge I learned about the work of Arthur Wightman, whom I hadn't even heard of as an undergraduate at Princeton. But friends of mine in Cambridge gave me some of his papers, and I read them, and I got really interested in what was going on in axiomatic quantum field theory. And I decided I really wanted to go back to Princeton, although generally Princeton didn't accept undergraduates as graduate students, but since I was switching departments, it seemed to work out.

What was Wightman writing on during those years? What was captivating to you specifically?

Well, his Les Houches lectures were about representations of the Lorentz group. So, Wigner had started this whole subject with a 1939 paper on the representations, and this had been continued by Wightman in context of field theory, and he also had lectures on PCT theorem, and spin and statics, which ultimately became the book, *PCT, Spin and Statistics, and All That*, that he wrote with Ray Streater. That book was actually being written my first year as a graduate student -- I guess, first and second year. The third year, Wightman took Oscar and me to Paris with him when he was on sabbatical. I spent a good deal of the fall that year proofreading the book with Oscar. We sort of had a competition who could find the most typos or mistakes. So, that was the era of axiomatic field theory being set down as text.

Now, did you join the department on the basis that you would be Wightman's student, or was that only cemented later on?

Well, I joined the department as a first-year graduate student. You weren't assigned to a particular person unless -- I don't remember how that official assignment was made. In my first year at Princeton as a graduate student, Arthur was on leave. So, I didn't meet him right away, but he was at the Institute for Advanced Study and occasionally came to the university. Don Spencer, I remember, had recommended me to Arthur Wightman. So, Wightman knew who I was, and once I told this story. I was sitting in the common room of the math department. The math department was connected to the physics department, and the joint library for the two departments was at the top of some stairs. At the bottom of the stairs was the common room, and it made the common room a very popular place where people would sit, talk, play games, and discuss mathematics or physics. But sitting in the common room, you could also look down the hallway of this Fine Hall, which was a wood-paneled hall. Once, I saw Spencer and Wightman walking together, and they were talking about me. They didn't see me there, but although I was very shy after that, I knew that I could go and talk to Arthur Wightman, so I did, and somehow, I became his student.

Arthur, who were some of the other professors who you would consider mentors, or who exerted a strong intellectual influence on you during graduate school?

Oh, there were so many good people at Princeton at the time. There was a tradition of everybody in theoretical physics going to tea on one afternoon a week. At that tea, generally most of the professors came, and the students came, and there was a lot of interaction. I certainly had a great deal of interaction with Valentine Bargmann, and Sam Treiman. Also, Murph Goldberger. I got to know Robert Naumann. He was peripheral in my interests, but he was somebody who was always around, and liked to talk. He knew me as an undergraduate and had tried to convince me that the future was in biophysics, or biomedicine, and that was a good thing to study. But he was very interesting. And of course, Don Spencer. Who else? So, the person who had the most influence on me as a graduate student aside from Arthur Wightman was Ed Nelson. I took every course that Nelson gave and got to know him. He was a wonderful person.

What was it like to be Wightman's graduate student? Did you work closely with him? Was he approachable? Was he hands on with your research?

I did not work closely with Wightman at all. Wightman had a number of students, and I always felt awkward taking his time. So, I didn't interact with him that much, but I appreciated any interaction we had. He gave me direction, but we never actually worked together on a problem. But he would point out what was important and what wasn't important. That was extremely helpful. He had two students who worked more or less simultaneously on the start of constructive field theory, Oscar Lanford and me. I worked on bosonic field theories, and Oscar worked on the Yukawa interaction, boson-fermion. We had desks in the same room, which I guess it was called 430 in the laboratory, and there were about 10 students scattered around at different desks. There was a lot of interaction with other students and postdocs in the room. This also made a lot of long-term friendships. I remember I got to know David Cassel, and he was working with Cronin, Fitch, and Turlay on this CP experiment. I took a course from Jim Cronin, and I remember in his course that the students all had to give an oral report at the end on some topic, and I was assigned the topic of explaining CP invariance. We didn't understand at the time that they were sitting on this famous experiment and trying to understand what it meant. They held off publishing for a year, it turned out later, because they wanted to understand the theory behind the experiment. That eventually ended up with their Nobel Prize.

Arthur, how parochial was your physics word as a graduate student? In other words, were you aware of people like Murray Gell-Mann, things that were going on at places like Harvard and Stanford, or was it entirely a Princeton reality for you?

Oh, no. We were aware of quite a lot of things going on in the world. Of course, we knew about Gell-Mann. I guess, I was lucky, because of my experience abroad, I was very prone to like to travel. During my first two years as a graduate student -- well, in 1961, before going to Princeton, I was still in Cambridge, and I went to a summer school in the beautiful Montenegro village, Herceg Novi. This was really important for my development, because I had a friend at Cambridge, Gavin Wraith, who introduced me to the work of Kurt Symanzik. Gavin had actually even done something that made Symanzik take notice. And he told me about this school; although I wasn't admitted as a student, I could go as an observer, and met Symanzik there, and many other people. In fact, going to Herceg Novi, I took the train from Cambridge to London, and then London to Paris. This was a long trip, but in Paris I went immediately on the Orient Express expecting to go to Trieste, from which I'd take a boat to Herceg Novi on the coast. When we arrived in the middle of the night at [inaudible], they announced there was an Italian strike, so the train wouldn't go any further for a while. Eventually, we got to Milan, and at that point, I decided maybe I'd stay in Milan for a day or so, for a stop, having not slept all night. When I had got on the train in Paris, I hadn't realized that you really have to have a reservation, or you might not have a seat. That meant I would have had to stay standing in the hallway for most of the day. So, I went up and down the hall of this train looking for an empty seat, and everything was occupied. Finally I saw one empty seat and sat down. It turned out that the person sitting next to me was going to the same meeting in Herceg Novi. That was Maurice Jacob. He was then at the École Normale. He was going as a lecturer. And we got to be friends for the rest of his life. He became a member of the theory group in CERN, and eventually became the diplomatic representative for many years. His son became a professor at Harvard.

What was the intellectual process of developing your thesis research?

Well, at the time, Arthur Wightman was really intent on knowing if there were some examples of his axioms. The axioms had led to the spin and statistics theorem, which was an abstract result generalizing what Pauli-Lüders-Zumino had discovered for spin and statistic and PCT as symmetries of free-field theories. But Wightman, Jost, and their collaborators showed the symmetries occurred in all theories satisfying the Wightman axioms. Those were the two most fundamental results of axiomatic field theory. However, there were no examples of field theories that satisfied these axioms except for free-fields and some two-dimensional models. But these models had a problem because their scattering theory was basically trivial. So, the big question at the time was, did nonlinear fields compatible with special relativity and quantum theory exist? Oscar and I were given that problem, which we didn't solve as students, but we worked toward that end. Both of us studied field theories with approximations, and showed that those field theories could be given a firm basis. But the singular limits that you would find for removing the cutoffs that we put in our examples only became possible later.

How mathematical would you say your thesis research was?

Oh, it was very mathematical. It was on the frontier of both the theoretical physics and mathematics. We had to develop, both Oscar and I, some new mathematical methods to deal with our problems.

Was there anything happening at the institute that was relevant for you during graduate school?

Oh, of course. I often went to the theoretical physics seminars. During my -- I think it was my first year – Res Jost was visiting at the Institute. This was also extremely important because he and Arthur Wightman were close. They had a seminar. Also, I went to Oppenheimer's seminar, which was extremely interesting. I remember Dirac coming and giving a talk about the history of the relativistic wave equation. This was a big event. Dirac never called his equation the Dirac equation. So, this story I tell a lot, but I'll tell it here too. I remember at the end of Dirac's talk, Oppenheimer asked, "Professor Dirac, at what point did you realize that the relativistic wave equation for the electron was correct?" Dirac said, "I was working at home. I had to go to the university to check the spectrum of hydrogen. But as I was getting my bicycle out of the garage, I realized this equation was so beautiful, it had to be true."

Arthur, I wonder if you could reflect a little more on that, about the connections you saw between beauty and truth in this discovery.

Well, that's been something that's haunted me over the years. I think there's a great deal to say about beauty and truth of ideas. It's, I think, somewhat different in physics and mathematics. In mathematics, beauty is often connected with simplicity of ideas and elegance of proofs. While, in physics, beauty is in the eyes of the experimental physicists, where it turns out to be right. But I'm intrigued by the fact that the most beautiful theories in mathematics seem to be connected to physics. I'm disappointed at the moment that we haven't found supersymmetry in nature, but I wouldn't be surprised if it comes back in some way in the future.

Arthur, more broadly, was there anything going on experimentally that was useful for you during graduate school?

Well, of course, as I said before, the discovery of CP violation occurred when I was a student, and I knew the people involved. Dave Cassel was a student; Cronin and Fitch, both of whom were my teachers. As a graduate student in physics, I had to do an experiment, and this was a major challenge. I think what it showed me was that it was really difficult to do experiments well. I knew that. That's one reason I quit chemistry, because I felt I wasn't good enough in the lab. I had an idea as a senior to trap free radicals in a crystal lattice, and they had not been isolated before. My advisor didn't think it would work, and it didn't work, so my thesis was a bust. I guess it was a junior thesis, and it impressed Garwin, because a year later, somebody made this work at the National Bureau of Standards. So, he thought that maybe I did have a good idea. But I couldn't implement it in the lab, and that's why lab work was always a challenge to me. So, Oscar Lanford and I were paired up to do an experiment on optical pumping, which had been discovered not so many years before. We were supposed to make this experiment work, and we worked on this for months, setting up the equipment. We needed a very steady electric current, and we somehow got, I don't know, 15 or 20 automobile batteries and put them together. We just could not see the effect. Dave Wilkinson was in charge of the course, so finally we went to Dave Wilkinson, and we said, "Could you please come and look at what we've done? We just don't know how to make this work, and we can't see the effect, but we think we've done a lot of work, and it's taking all our time." And so, we made an appointment. That day Dave Wilkinson came, and we showed him what we had done, and we saw the effect! Oscar and I were absolutely sure that Wilkinson had come into the lab the night before and made the equipment work so we could show it to him.

Besides Wightman, who else was on your thesis committee?

Ed Nelson and Bob Dicke.

Ah. Did you interact much with Dicke?

Well, I took his course. That was really beautiful. I didn't interact that much with him aside from that.

Any memorable questions from the oral defense?

I do remember that Dicke asked me a very nice question, but I don't remember what it was.

Arthur, what did you want to do after Princeton? What was available to you?

Well, I spent a year as a postdoc at Princeton. I got a postdoctoral fellowship given by the Air Force and the NAS. So the whole year I was at Princeton, and the next year, I went half a year to the Institute. I got several offers as assistant professor. In those days, you would get assistant professor offers right away. I think I could've gone to Berkeley, MIT, maybe Stanford. Those were, I think, the three offers I had. I was inclined to go to MIT, because being in Cambridge intrigued me, and I had turned down this opportunity to be at the Society of Fellows, but I like the people I met there. However, when I went to MIT, I was under the umbrella of Irving Segal, and he and Arthur Wightman were big competitors, and I felt a little uncomfortable. So, eventually, I ended up at Stanford, thinking I'd go to Berkeley. I think at one point I'd even accepted the job at MIT, and then I wrote them a little while later and said I changed my mind. That didn't put me in good stead there.

What was your appointment at Stanford? What department was this in?

I was jointly in the math department -- they called it Visiting Assistant Professor -- and I was a postdoctoral fellow at SLAC.

I was wondering about that, if you did have an affiliation at SLAC. Were you part of the theory group at SLAC?

I was part of the theory group under Sidney Drell. I shared an office -- I don't remember -- I think with Bob Rosner.

What was happening at SLAC during those years? What were some of the big projects that were exciting?

Well, that was the first year that SLAC opened. So, just getting the machine up and running was the initial project. In fact, the summer before that fall, there was a meeting at Stanford. I don't remember what meeting it was, but Arthur Wightman, Harry Lehmann, and Jost form Hamburg were there. I have a bunch of photographs of a tour that I was invited to by Arthur Wightman. I guess, Yang wanted to have a tour of SLAC, and the director was Panofsky. He gave a tour to Yang, and invited along with about six other people. I was really lucky to be one of them. This was before the machine was turned on -- we went down into the tunnel.

Wow. Did you interact with Bjorken at all?

Oh, of course, yes, but we never worked on anything together. But we talked a lot.

Arthur, was your time at Stanford initially an opportunity more to refine and expand your thesis research, or to take on new projects?

I started a new project, which was strictly localizable field theory. That was a way to extend Wightman field theory to the non-renormalizable fields, which had local singularities worse than polynomial of the type that was required in a Wightman field theory. So, I started that at Stanford. I also was just looking around at other problems. Also, at Stanford, Spencer organized a seminar in the math department, because he was very interested in learning about quantum field theory. He had very interesting people come to this seminar. It included Ralph Phillips, who was a professor of mathematics, well-known for his work on semigroups. Paul Cohen was interested in physics, but worked on Gödel's theorem, and was quite famous for his work in logic and mathematics. So, I gave all the seminars, and they were a really interested audience. We would always go to lunch afterwards at the Stanford faculty club. It was a tremendous environment, and very supportive, but it took a lot of time and effort to focus on those things. At the end of the year, I had gotten, out of the blue, an offer to go to Harvard as an assistant professor. And also, Don Spencer had decided to leave Stanford because Kodaira had at that point -- I think Spencer told me that if you're a national hero in Japan and you pass a certain age, you lose the title unless you live in Japan. So, he decided with his family to go back to Japan. Spencer spent a year at MIT and then went back to Princeton. He came to Cambridge, Mass., the same year I did.

Arthur, when did you first meet James Glimm, and subsequently, when did you realize that this would be such a fundamental collaboration?

Well, we first met sometime when I was a student, because Jim was interested in field theory, and he was doing really fantastic things. I know he came to Princeton to give a talk, and we also met at some conferences when I was a postdoctoral fellow. But we really didn't have much interaction until the summer at Stanford, before I came to Harvard. Don Spencer liked putting together big meetings, and at Stanford, he was organizing a tremendous summer meeting, I think for the whole summer. He always reached out to his younger colleagues and compatriots, and he asked me, was there somebody I would like to invite to the meeting? And I said, "Well, why don't we invite Jim Glimm?" So, Jim spent a month at Stanford, and that's when we started to collaborate and talk to each other really seriously. It turned out, when I went to Cambridge that fall, he was living more or less around the corner from me. So, we would start to talk to each other at each other's homes, and that's when we began to collaborate seriously.

What was the intellectual division of labor between you and Jim? What did you each bring to the table?

Well, he had a very solid mathematic background. He had done really beautiful work in C*-algebra, so he was good at analysis. And I had more training in physics, so we complimented each other, although I had also studied mathematics. I got my BA in mathematics from Cambridge University.

How long did you stay in Stanford at the end?

I was only in Stanford for part of one year.

Was the plan to stay longer, and then opportunity in Cambridge presented itself, or did you think that it was going to be for that short from the beginning?

I had no plan, but I was offered -- they said they would be very happy if I stayed at Stanford, and I could be on the faculty. But I did feel isolated because there were no experts at Stanford in what I was doing. Everybody was interested to learn about it, which was wonderful, but I really didn't have any people I could talk to that I felt were knowledgeable about the frontiers of the questions I was working on. Oscar was at Berkeley, and we thought that Stanford and Berkeley weren't so far apart, but we learned that it's a little further to drive and deal with that distance than we thought, and we didn't see each other that much. So, it was very appealing for me to go back to the East Coast, and that's why I decided to accept the job at Harvard.

And was that initial position a tenure-track assistant faculty line?

Tenure-track is a new concept at Harvard. We didn't have it at the time. So, no, there was no promise of promotion or anything. But Harvard was a small place. I knew, in experimental physics, that assistant professors were never promoted, but they just had a handful of theoretical people there. Sidney Coleman was in the department, Shelley Glashow had just come, and I thought there was a lot of opportunity. When I arrived, there were very interesting junior fellows -- David Gross, Roman Jackiw -- and they were in the offices not far from me. So, we spent a lot of time talking to each other, and I really found that a really interesting situation. Also, there was another assistant professor who came the same year I did, and we had been graduate students together at Princeton. In fact, the first year we had rented places in the same house, and that was Curt Callan. In fact, both of us ended up with offers to stay permanently at Harvard, as did another person who had visited Princeton has a graduate student, but had been a Berkeley student, Bert Halperin. He ended up also as professor at Harvard.

What year did you arrive at Harvard?

1967.

On the social side of things, when did campus really start to erupt with antiwar protests, and civil rights protests? When did that start?

About a year later.

Were you political at all? Were you involved in any of those movements?

No. I was totally apolitical. Actually, the faculty became very polarized in 1968. There was a conservative caucus and a liberal caucus, and I was sympathetic with both. I had friends in both, and I just tried to keep out of politics.

Was your plan to be -- even though it was in the department of mathematics -- to be really involved in what was happening in the world of physics as well?

I'm sorry. I didn't understand the question. You mean, when I went to Harvard?

Right, when you got to Harvard.

At Harvard, I was Assistant Professor of Physics.

Oh, it was in physics. Okay. So, let me reverse the question then. Did you want to be involved with the department of math from the beginning? Did you always want to have that crossover in your research?

Well, I knew that I was involved in both, and at Princeton, there were many people involved in both. So, I didn't realize how distinct the line was at Harvard, but I had friends that I knew in the math department. Especially, George Mackey, who was quite interested in mathematical physics. He was a good friend of Wightman, too. He might have been chairman of the department when I first when to Harvard. So, we talked a lot. We started to have lunch together on a regular basis. I never thought about what my ultimate -- I mean, at Harvard, in those days, things were very small and informal. It didn't really matter.

More broadly, what were some of the advances in quantum field theory in the late 1960s and early 1970s, and where did you see your contributions to those advances?

There was a lot of advance, of course, in quantum field theory in the early 1970s. That was the time when strong interactions were being understood. That was the time when unified field theory, Glashow-Weinberg-Salam, came into being. And renormalization, that was the time when 't Hooft and Veltman understood that field theory and perturbation theory could be renormalized. And I was sort of complementing that. I talked a lot -- I mean, when I first went to Harvard, it was very interesting that we used to, every Sunday, get together for lunch. A lot of the professors, including me, were not married, and even the ones that were, came with their family. This was organized often by Coleman. They liked a particular Chinese restaurant where we could all sit together and had a big table. It was out in Medford, so it was about a 20-minute drive from Cambridge. But we organized that practically every Sunday. Schwinger often came, Paul Martin, Glashow, Weinberg, sometimes Roy Glauber, and the postdocs came, the assistant professors came. Everybody came, and we were really one family. We weren't divided into different subjects. My complement to what was going on in high energy physics was that I was interested in whether relativity and quantum theory were mathematically compatible. In fact, this is still an open question. So, I worked on, and with my collaborators, solved part of that problem, but the fundamental problem is still open. Even special relativity we don't know is compatible with quantum theory in the four-dimensional world that we live in.

In the early 1970s, what was so exciting about renormalization? What was revolutionary in its approach?

What timeframe are you talking about?

The early 1970s. I know you wrote about renormalization in '71-'72.

Right. Well, renormalization had predicted experiments which ultimately became the most accurate experiments ever done. So, everybody would say, this must be correct theory of nature. But it wasn't a theory, because there wasn't a completely logical theory. It was a set of rules to predict numbers, and these rules worked to phenomenal accuracy. At present, they work to something like 13 decimal places. So, there are no other measurements that have ever been done in the laboratory that agrees so precisely with theoretical predictions. Is it possible to have a mathematical theory of renormalization? People in the 1970s only studied it perturbatively. What I was interested in was to understand, even in the simplest examples, could you understand renormalization non-perturbatively, as an exact theory? This is what eventually Jim Glimm and I, and also Tom Spencer and our collaborators and students, showed could work. What we did was we showed it works in two-dimensional spacetime, and three-dimensional spacetime, and it's still an open problem in four-dimensional spacetime.

Arthur, I'm not sure where to root this question in the chronology, but do you have a specific memory of when computers became powerful enough to be relevant to the most important questions you were asking?

Well, I mean, this came at very different stages. Even when I was at Stanford, computers were being used to understand perturbation calculations in high energy physics. There was a postdoc there, Tony Hearn, who developed a computer program to do the algebra connected with Feynman diagrams, and this turned into symbolic manipulation, which eventually led to all the programs, including Mathematica. So, this was quite early that there were things that, even in perturbation theory, had to be done by computer. Whether computers could do much more complicated calculations, like lattice gauge theory, calculations of masses, that developed over a much longer period of time. I wouldn't say I could tell you exactly when that became important, but I know that Ken Wilson was a tremendous advocate of using computers to understand physics. He had a sort of religious fervor about spreading that idea, also using computers to communicate. I guess, this was a theme that entered all the time with my friends, but to use the computers took a tremendous number of manpower hours to develop the software. That became a whole specialty of its own with people who just were devoted to that. I didn't follow it very closely.

How did you get involved in research on quark confinement?

Oh, the work I did there was very theoretical. I was interested in an aspect of it that -- I guess it didn't fully pan out, but the confinement problem is really connected theoretically with the mass gap problem in gauge theory. Of course, I was fascinated by early discussions with Ken Wilson, because we had this -- in 1974, I had a friend, Pronob Mitter, whom I met in Paris. I was on sabbatical in Paris, and he was at the University of Paris. We decided that we'd try to organize a mathematical physics summer school in Cargèse. He was the main person. He knew Maurice Levy, who had started the Cargèse summer schools, and I had attended one which was very important for me at the tail end of the year I spent in Paris in 1984. So, I knew Cargèse, and I knew what a wonderful place it was. We put together the first school in 1976. Ken Wilson liked coming there, so we had many discussions about computers. I think the summer school met six times, and had many fantastic lecturers and students, and many students were inspired to go into their careers after attending it.

Arthur, tell me what it was like when you won the Dannie Heineman Prize in 1980.

Oh, well, that was tremendously gratifying, and I was very grateful to receive the prize. But still, the biggest questions had yet to be answered.

Like what? What comes to mind? What are those big questions?

Well, the biggest question is whether field theory works in four-dimensional spacetime. We know there are problems with classical electrodynamics in four-dimensional spacetime. So, is quantum electrodynamics really not a theory the way we except for asymptotic freedom not being satisfied? And is QCD a theory? That's a major question for physics. And that aside, putting in gravitation, I think string theory has a long way to meet the same level of understanding as quantum field theory.

That gets me to my next question. I'm curious in what ways the so-called Superstring Revolution in 1984 resonated with you.

Well, string theory has brought so many new connections between mathematics and physics that it's been extremely important and will remain important in the history of mathematics. So, how it is in describing physics is still open. As I said before, I'm really disappointed we haven't seen supersymmetry in nature, and I hope that will be the case. Maybe not in the way that supersymmetry is currently formulated. Maybe it will connect some way with noncommutative geometry. I don't know.

Where do you see this lack of seeing supersymmetry on the spectrum from, we're simply not operating at high enough energies, to there might be a problem theoretically with the way we're going about looking for it? Was string theory exciting for you, and did you want to see string theory blossom and become a bigger part of physics and math departments?

String theory is exciting for me. Unfortunately, I think there's been a bit of a chasm, which -- the axiomatic field theories, Arthur Wightman, Res Jost, and the school that they began, tried to do physics as if it were in a mathematics department, as if it traditional mathematics. And a lot of string theory is not done that way, although it has resulted in big revolutions in geometry, in pure mathematics, and also representation theory. Tremendous amounts have come out from symmetries that were suggested by string theory. There were these big oscillations in trends in mathematics. In the 1950s, the pendulum swung far too far, in my opinion, toward the Bourbaki school, and having a unified picture of all of mathematics that you could write down in a series of textbooks, and to try to take applications away from the main line of mathematics. I think integrating the applications in physics with pure mathematics -- and in string theory, maybe the pendulum swung maybe too far in the other direction, where people lost sight, sometimes, of the necessity to have mathematical proofs in the old-fashioned way. I would like to see the pendulum swing back more toward the middle.

Arthur, in the 1980s and 1990s, you spent a lot of time writing and presenting on the connections between math and physics, sometimes even to a broad audience. What were some of the key things that you wanted to convey in these discussions about the duality, or the value of having this approach from one to the other?

Well, I think the key thing on my mind was that it's not good to be isolated in your own world of thought, and that big advances in mathematics came from bringing these subjects which traditionally had been together, back together. I think, in terms of science, mathematics, and physics, probably the two over history, going back to the Greeks and Chinese, have been the closest together. I think that's a very vibrant history, and it's not only something from the past, but I would like to see it continue in the future.

By the mid 1990s, given all that you have done with quantum field theory, how had the field changed, and in what ways was it responsive to some of the really big questions, existential questions, even, in physics?

Well, I think that one aspect of the change in science has been an aspect of a change in life. In the generation where I grew up, people worked with very long-term goals. And then, people focused, as the competition increased, more on immediate gratification. So, it's very hard to find people today, in my mind, who develop the technical expertise that was necessary some years ago to solve the very fundamental problems, because they don't want to work on them for that period of time that it takes to get up to the level of depth that you need. So, I think, everything that's complicated isn't good, and everything that's good isn't complicated. But I feel that people do like to -- well, I was lucky. I came along in quantum field theory at a time where you could solve a lot of problems, and you could be very naive and have some new tools and go into the field and do something. Now it's much more difficult. The problems are more difficult.

Are they more difficult on the basis of, the more you understand about something, the more you understand you don't know about it?

Exactly.

I wonder what really hits home, in terms of conveying that point.

Well, just last week I gave a talk at a virtual conference in Istanbul. It was supposed to be an anniversary of a meeting they had 30 years ago. So, in preparing that talk, I actually went back further a meeting, the one I mentioned in Herceg Novi, but also one at Robert College in 1962. Robert College now is part of the University of Istanbul. And all these meetings had different themes, which today, if you look back, are very closely related. But people didn't understand that at all at the time. So, you put in quantum field theory, which Kurt Symanzik was just being to understand, constructive quantum field theory, and then in the 1960s, Tomita-Takasaki theory, operator algebra theory, the theory of the modular operator and the whole development of mathematics that led to Connes' classification of operator factors. And quantum field theory reflection positivity, which was developed at Harvard with my postdocs and Robert Schrader and Konrad Osterwalder. And these things are, it turns out, all related. They're related to other things going on at the time that people didn't realize were connected. So, for me, the unification of all these different strands of mathematics and physics knowledge are fascinating, and I think we have much, much more to learn because each of these subjects has gone off into very technical directions. Often, the people who understand one part of the subject don't realize the connection to another.

Arthur, how did you come up with the idea for the Clay Mathematics Institute, and what were some of your key motivations?

Well, I got to know Mr. Clay because I had met him once in 1970 at a dinner, and then afterwards I was appointed to the chair that he donated to the department. Over the course of time, because I had his chair, he wanted to get together for lunch. We had lunch every couple months or so. He told me a little about what was going on with him, and I told him what was going on with mathematics in the department, and so on. He was very interested in the department, but he had been in English major in college, so he really didn't know much about mathematics or science. But he felt that the department was at a very high level. When he explained at one of these lunches that he was being kicked out of his company, Eaton Vance, where he was CEO, and then he was planning to sell his investments in the company and start a foundation, I didn't give him any advice. I knew that if he didn't ask for the advice, he maybe wouldn't pay attention. But this went on over the course of much of a year. Finally, he did ask my advice, and I told him that what he was planning to do, he'd get his tax deduction, but maybe not the best use of his money, and if he wanted to do something for mathematics, I'd help him. And that was the end of that. But several months later, he called me on the telephone, and said that he decided to do something for mathematics. Could I write something down and meet him the next morning for breakfast? So, I put some ideas together and met him for breakfast, and eventually, about half a year later, the plans came into being. At that time he was on a war with Harvard. He was on the Board of Overseers, and he was trying to remove the president of Harvard. So, at the time, he did not want to have the Clay Institute be part of Harvard; but he would have been very happy to have it located at Harvard. I thought, well, if that would be possible, then eventually it would become part of Harvard. But the department was first reluctant, so the plans to it eventually became an independent organization.

Arthur, a broad question, not exactly rooted in the chronology either, but I wonder if you reflect on the value, both from a service perspective and for your own research in serving in leadership roles both for the International Association of Mathematical Physics, and then later on for the American Mathematical Society. How would you compare and contrast those positions, those responsibilities?

Well, the International Association of Mathematical Physics is a professional association I've belonged to since it was officially founded in 1979. It was a pleasure to be asked to be president of the organization for my field. That is a small organization, and it was not a very time-consuming or big job. The American Mathematical Society is a completely different story. For the International Association of Mathematical Physics, at the time, the main thing they did was the have a meeting every three years. On the other hand, the American Mathematical Society is a huge organization. When I became president, it had 30,000 members. It had a staff of 245 FTEs. So, it was a monolith, and the executive director at the time I became president was John Ewing, who it turned out had come there just a couple of years before. I was his second president. He came during the term of Cathleen Morowitz, my predecessor. I had, during my year as president-elect, become involved in this problem that the University of Rochester wanted to downsize the mathematics department and make it a service department, and yet remain a major research university in science. The mathematics community was very upset by this, and I got involved because the president at that time was ill, and she was under treatment and could not deal with the situation, and asked if I would, as president-elect, take that over. Eventually, I learned a lot from this Rochester task force which I put together with Salah Baovendi, who was at the time very active in the Committee, the Profession of the Society. I think he was chair of that committee. He and I, over the course of the next four or five months, solved that problem.

And I learned so much from that that when I became president of the Society, I decided to maybe put that knowledge to use. Because I felt that government funding for mathematics was really at a low ebb. With quite a few other, especially those in natural science -- we decided it would be good to work together instead of competing with each other. Physics, mathematics, chemistry, and astronomy started talking, the presidents of the societies, talking to each other. We were lucky that Allan Bromley, who was President of the American Physical Society at the time, had been advisor to the President of the United States. So, he had good connections in Washington. Eventually, we put together a group of 106 scientific society presidents to attempt to have some effect. We did have an effect for a couple of years. But what I learned was that effects in Washington are generally very short term, and it is extremely difficult to have some long-term effect. So, we certainly brought some light the problem of the physical sciences, and in fact, we were joined eventually by the biologists and biochemists, who at first thought our ideas were far too modest, but then they became part of our coalition.

Arthur, I'm struck by your 2006 publication, the Millennium Grand Challenge in Mathematics. What was the grand challenge, and 15 years later, how much of the challenge remains?

Well, in my mind, the millennium problems were just a gambit to focus people on mathematics, and to get a lot of kids interested in becoming mathematicians. The grand challenge wasn't to solve the most important problems under the sky, but to talk about some old problems that were considered by mathematicians as really important. People had worked on them for some time, and they were still open. We eventually launched the millennium problems on the 100th anniversary of Hilbert's talk at the International Congress of Mathematicians in Paris. At that time, Hilbert talked about ten of them; over the next century the Hilbert problems became blueprints for major themes in mathematics. However, we weren't trying to do that 100 years later. We were just trying to refocus attention of people on mathematics. And I think for a while, we were very successful. We were so successful on the day after things were announced, we were in over 1,200 newspapers around the world, on the front page of Le Monde with a photograph. Many young people later told me that they decided to become mathematicians because they were fascinated by one of these problems. The fact that there were seven problems was an accident. When we got to seven problems, it was very difficult to either add a new one or replace one that was on the list. And we had a time constraint because at that point it was November 1999, and we decided we wanted to have a meeting in 2000. There were many time constraints because of the summer, and we had to prepare for it. So, I just said, well, it looks like we've decided what the problems are. That's it. A reporter once asked me, when they were announced, which problem would be solved first. I had no idea which one would be solved first, but I had a pretty good idea which one would be solved last.

That's great.

That was the one I knew most about, which was the Yang-Mills theory. I thought it was even more difficult than even the Riemann hypothesis.

Tell me about your work with Barry Simon and others to celebrate the life of Wightman.

Well, Arthur Wightman was a very special person in my life. He taught me so much. We had one article that he edited, and I contributed to, and he was one of Arthur's best-known students. What I learned most about Arthur which was so special -- I wrote in an essay which came out of the talk I gave at the Princeton memorial called *Nine Lessons from my Teacher, Arthur Wightman*. One of the most important lessons was to be modest but to do things well. Also one should pay attention for ideas that didn't at first seem to make so much sense, but come from very insightful people. So, the connections between physics and mathematics really are based on some ideas which seem pretty far out of the ordinary when they're first posed. You talked about renormalization, and renormalization, to many people, is crazy because you're redefining the parameters in an equation to be unobservable, infinite ones. But science evolves, and people have insights in different ways. So, I'm amazed by my friend, Vaughan Jones, who died last year. He was able to see knots in a very particular way in terms of certain algebras. At first, people didn't believe anything about it. It just sounded like a bunch of nonsense, but it turned out to be a fundamental revolution in mathematics. So, I think that goes on in physics and mathematics, too.

Arthur, when did you start to really think about some of the philosophical underpinnings in mathematical proofs? Was that gradual, or was there a particular period in your career when this was sort of front and center in your agenda?

This was very gradual. In fact, I'm not an expert on that, and I got involved in it in the early 1990s. I had been brought up, as I described, in a school of mathematical physics where you really try to do mathematics as mathematicians. I had my seminar at Harvard with a very wide spectrum of speakers, and now I have another seminar with a wide spectrum of speakers from a different point of view. But after one talk, I was a little upset because the speaker had these beautiful insights but was announcing theorems that didn't seem to be based on any notion of proof. I went home and satisfied myself by jotting down some thoughts on a page of text, and went off the next day to Virginia Tech, where I was giving a colloquium. At the party after the colloquium, I met Frank Quinn. While we were talking, I mentioned to him this experience. I said I had written this page down and I happened to have it with me, and he said, "Let me see it." So, I showed him what I had written down, and he said, "Oh, my goodness, we should collaborate on this. We should write an article." So we started to focus on these questions, and that resulted in our joint article on theoretical mathematics.

In all of your work in the political realm to increase support for mathematics in education, what has been most satisfying in seeing the fruits of these efforts, in the ways students have been involved, in the way they've been supported?

Well, I don't know if I'm really aware of all the ways this has happened, or how much it could be connected to these political activities. So, I don't want to answer that question, but I'm very happy that some people thought this could be helpful. How I got into these things, as I told you before, I'm not sure. This article with Frank Quinn that raised my level of awareness in people outside of my field. It generated many email liaisons with students. That's probably why I became AMS president, though the reason I agreed to run was I was sure I would lose the election. In fact, the person who lost the election -- it was a very sad thing. Fred Gehring had been very active in the society, and I thought there was no question that he was going to become president. The person who nominated Fred was a good friend of mine at MIT and was treasurer at the society. So, it seemed to be a done deal. Afterwards, Fred came to give a lecture at Harvard, so I invited him and my friend Frank Peterson for dinner. I said to Fred I hoped that he would be chair of this big committee in the AMS, and he said he was terribly sorry, but after this election he didn't want to really be very active in the AMS. I felt extremely bad about this, and decided that I would recommend after that a candidate to be president should be appointed to be chair of a major committee before the election. I didn't want the people to have a feeling, if they lost the election, that they weren't really welcome in the society. In fact, there were a couple people who lost the election who then were candidates again and were elected president.

Another brand question, Arthur. Have you seen undergraduates' motivations for pursuing math change over the years, the kinds of things they're interested in, the kinds of career ambitions they might have? Has that been more or less stable, or have things changed over the course of your teaching career?

Oh, things change all the time, and subjects become popular and less popular. I think subjects become popular when there's a lot of progress. And then people are attracted to work in that field, so I'm not surprised that people's interests change.

As you mentioned earlier, when Harvard was a much smaller place, when you first joined the faculty -- now, of course, that it's gotten bigger, you emphasize the smallness in a way to convey that the distinctions between math and physics and your research agenda were not necessarily hard to navigate. How has that changed over the years?

Well, from my point of view, the biggest change at Harvard was the primary focus of the university. We can apply it to mathematics and physics in a way that I'll talk about later, but I think at a higher level, if you're talking about the university as a whole, when I first joined Harvard, everything seemed -- number one priority, although there were many priorities, and of course a university does many things and has many different points of view, but the number one priority was always the intellect. That's all that counted most in any discussion. I think nowadays, the thing that counts most in any discussion is Harvard has become a business.

Not for the better, I assume.

Well, it's different. So, I think that everything is looked at through a different lens, through a financial lens or a legal lens. It's not the intellect that is the number one -- of course, the intellect is present everywhere. Harvard is a great university, but it doesn't seem to me the intellect is the first priority for each administrator. Now, how about mathematics and physics? Well, as Harvard's grown, my research sphere has too. I used to talk to a few people. Now every professor is surrounded by a cloud of people, and it's difficult to penetrate that cloud. So, it does put different people in different places, and it's much harder to communicate outside your cloud, just because there are so many people. It takes so much time. There are so many more things to learn, so many more things to do, so much more time needed to apply for grants, and do all the paperwork that's necessary today. So, there's a different flavor as far as I'm concerned about being a professor at Harvard than there was when I first came.

When did you first meet Zhengwei Liu?

I met Zhengwei in 2015. I had the great fortune to have a research grant from the Templeton Religion Trust, and it allowed me to hire a postdoctoral fellow. So, I advertised and interviewed four people. I had decided to hire one of those, but I thought that I wanted to be sure that I had the very best person. So I knew that my friend Tom Spencer at the Institute for Advanced Study followed very closely all the young people, and in fact, read all the applications at the Institute. So, I called him up on the phone, and I told Tom what I was planning to do. I asked him if he thought that was a good choice. He said, he thought that was good. But maybe I'd want to take a look at this guy, Zhengwei Liu. He had not applied to Harvard. I don't know why.

In what ways was Zhengwei making waves already?

At that point, I didn't know anything, but I sent an email to his teacher, Vaughan Jones, and at that time, he was traveling in New Zealand. He sent back a short email, but it was so intriguing that I thought I have to find out about Zhengwei. In fact, Zhengwei had actually solved a major problem in subfactor theory. He had shown that a conjecture of Bisch and Haagerup that Jones was very inserted in too, was false. He had worked it out completely, and this was something that had been up in the air for quite a few years. And this was even not his thesis, but just something else he did as a graduate student. So, I called up Zhengwei, and I told him there was a position, and if he was interested in applying for it, he should come to Cambridge, and he could give a talk, and we could meet each other. And he did that, and I was very impressed by him, so I decided, in the end, to offer the job to him.

What did you recognize immediately about what you could teach each other?

Oh, I didn't, but I recognized that Zhengwei was quite brilliant. He seemed different from other people in that he was very independent, had his own way of thinking, was interested in a wide variety of problems, worked very hard, and he wanted to do something important. He had already done something important, but he wanted to do more.

What was that? What was obvious to do that was important?

Well, we didn't know, but he just wanted to make a name for himself. And that was very clear. So, Zhengwei came to Harvard, it was in August, and there were very few people around. So, we decided that I wanted to learn something about his subject, planar algebras, and I was interested at the time in parafermions, and had worked on that. So, I explained a little about parafermions to him, and he was also interested to know about quantum field theory. Actually, for about three weeks, we got together every day, and we spent most of the day just giving lectures to each other, off the cuff. We didn't prepare.

Arthur, in all of your work, why the blind spot in planar algebra? What does that suggest about the intellectual tradition you're coming from, your research focus? What might explain that gap?

It doesn't need any explanation. The world of mathematics is huge, and there are many parts of mathematics I don't know. So, the fact that I had maybe the world's expert, aside from Vaughan, on planar algebras sitting next to me, I could learn something. And I thought when he explained things to me, it was a fascinating subject. Then, I explained some things about what I was doing to him, and over the course of these three weeks, we realized we could put these two things together and come up with some new type of algebra we called planar para-algebra, which was an extension of planar algebra. Then, a student who was around, asked if he could sit in on some of our discussion, and he said, "Well, what you're doing on the blackboard reminds me of things I've learned in quantum information. Is there a connection?" We knew nothing about quantum information, and we found out it could be very useful in quantum information. That's how we started.

So, what were some of those bigger questions that were answered as a result of receiving this world class explanation in planar algebra?

Well, we eventually, through these planar para-algebras, developed a couple of what we thought were interesting new languages for quantum information. We found that we could have very intuitive ways to design quantum information protocols. We began my understanding the teleportation protocol, Bennett and all, but we found that it led to interesting new things. But also, the languages could be useful in pure mathematics. So, we just found that we opened a Pandora's box of interesting questions, and it led to something that we're very interested in now called quantum free analysis, which is a generalization of free analysis of these pictures, and others, and leads to certain analytic inequalities, which like in ordinary free analysis, lead to uncertainty principles that have been connected with physics, lead to new types of uncertainty principles which we thing are potentially useful in quantum information and other fields. So, we're excited about trying to develop this field of mathematics. Zhengwei has gone back to China. He went from his first postdoctoral fellowship to being full professor at Tsinghua University. It's a little difficult to collaborate now because of the distance and the lack of travel.

Conversely, after learning from him, what did you teach him?

I taught him about physics, and this led to the planar para-algebras, which is a way of looking at parafermions as a generalization of planar algebras. Parafermions is something that originated in mathematics a long time ago, perhaps in the work of Sylvester in the 1800s, but in physics, it became fashionable to talk about because of the commutation relations that come out of the parafermion algebra. So, I taught Zhengwei a lot about physics.

And as a result of this two-way-street tutoring, what were you able to accomplish as a result?

Well, we now have what we like to call the Quon Language, which is a mathematic picture language, which originally, we found for quantum information, but fits very beautifully into this subject of quantum Fourier analysis, which we're working on now.

What are some of the big goals with this analysis?

Well, Fourier analysis was developed over a couple of hundred years, and has applications in every field of mathematics, engineering, finance… you name it, you can find Fourier analysis. We think that the quantum aspect of free analysis will also be important.

And what are the big hopes, if it does turn out to be important?

Well, we hope to understand some limits on quantum computing. We hope to -- well, it's already surprisingly answered some frontier open quests in algebra, in pure mathematics. So, it's had an application there. That's work of Zhengwei and some collaborators at Tsinghua University.

You mentioned before that you remain excited about string theory. Is that to suggest that you're patient, that string theory will yield some explanations about how the real world works?

Oh, I don't know. For me, the ideas of string theory have been so important for mathematics that maybe some of the new mathematics coming out of that will come back in a different way in physics. I don't even know what string theory is because it's evolved so much over time. But ideas associated with geometry and algebra, representation theory, that are so important for string theory are certainly going to be in the future of physics. Exactly how that comes about, I'm not sure. Nobody knows. When we find it, it'll be a big discovery and revolution.

Tell me about your affiliation with the Chinese Academy of Sciences. How did that get started?

That was something that came through another friend of mine, Liming Ge. He's part time at the Chinese Academy of Science, and part time at the University of New Hampshire. For a number of years, we've had a seminar together. He was very interested in the type of work I did, although he started out as a student in operator algebras, related to subfactor theory, as a student of Dick Kadison at the University of Pennsylvania. But afterwards, changed to work on problems in number theory using his background in operator algebras, which is really exciting. That's another connection between different fields. In fact, he was the person who brought Zhengwei originally to the United States. That's another story, because Liming had been a professor at Peking University, and had a seminar for talented undergraduates, and took the best ones to be his graduate students at the University of New Hampshire. Zhengwei was one of those, so he went to New Hampshire, but there he met Emily Peters, who was a postdoctoral fellow. She had worked as a student with Vaughan Jones, and taught Zhengwei planar algebras. So, he immediately solved some problems of interest to Emily. So, she recommended Zhengwei to Vaughan when he decided to move from Berkeley to Vanderbilt, and Zhengwei became Vaughan's first student at Vanderbilt University.

Well, Arthur, now that we've worked right up to the present in our conversation, for the last part of our talk, I'd like to ask a few broadly retrospective questions about your work, and then we'll end looking to the future. So, I know, as you've emphasized, that you don't see the key distinctions in your research agenda between mathematics and physics. So, perhaps you'll reject the premise of the question, but I wonder if you've ever reflected on some of the overall value in having that dual approach, the ways in which your mathematical sensibilities advance the physics, and the ways that your physics sensibilities advance the math.

You mean, do we have different perspectives?

Most definitely, because just to state the obvious, many physicists don't have a strong background in math, and many mathematicians don't have a strong background in physics.

Right, and that is the consequence of the subjects moving apart. I think that really began at the advent of quantum theory in the 1920s, when people found it much more difficult to understand physics completely. Even though non-relativistic quantum mechanics was understood pretty much, when relativistic quantum mechanics and quantum field theory came into the focus, then people threw up their hands about mathematics and gave up. So, physicists were often discouraged from studying too much mathematics. I think the fact that the ideas of the subjects were different branches of the same thing, but developed independently, this was very healthy and good. And the fact that you can take ideas that are developed in one subject, and you see parallels in the other subject, they either may not be immediately applicable but could inspire things, that's tremendous. Where you get ideas is a big mystery, I'm sure, for psychologists, and to have a source coming from a different subject is wonderful, and that's happened so many times between physics and mathematics in both directions. So, I see the tension, and I'd like to bring the subjects back together, but I understand that you want to have free thinking in both worlds. The idea is not to constrain things to always be within the realm of proof, but I like to see in physics, people go a little bit further than what has become the practice in recent years.

Of course, it's a much bigger question that spans earlier from your career, and it's hard to know the road not traveled, but what do you think has been lost? Or another way of putting it, what might we understand today that we don't currently, had there not been this divergence in the fields earlier in the 20th century?

That type of question is impossible to answer. I think it's a miracle that we are where we are today. It's like what Richard Feynman said, "A mathematician cannot prove a nontrivial theorem. Because once you understand it, it's trivial”. It often takes years to get something to the point where you see through it minutes, and you can write down in one page what took hundreds of pages, or even thousands, to get to. So, I think that different points of view, of course, are valuable. Mutual understanding between the different subjects is also valuable, and sometimes that goes away, like in current politics. This politicization of science, I find, is very discouraging, and to be discouraged. I'm not talking about politicization of not teaching evolution, but politicization in the fact that some people in one field say the other field is worthless. That's not a good idea.

As you said before, there's a certain fluidity in the field, when I was asking about the motivations of students in terms of what they study. When have you seen, over the course of your careers, the most optimism that mathematics and physics were talking to each other, and when have you been least optimistic about that?

I think there's so many times -- I can't assign a maximum to that function. Let me look at it historically. I looked back at a conference I attended in 1967 in Rochester, and there was an attempt to bring together some mathematicians and physicists, and you had -- in those days, you not only had the talks given at conferences and the conference proceedings, but you had the discussions. I found the discussion after the talks more interesting than the talks. I looked at a talk by Arthur Wightman, and I saw there was a discussion afterwards by George Mackey, Richard Feynman, Eugene Wigner. They were all trying to understand the same thing, and they were all interested in what the other person was saying about their point of view.

Feynman had this public persona where he was very anti mathematical, but privately, it was just the opposite. He was one of the few people, actually, who'd been four-time winner of the Putnam Competition as a mathematician, and himself, he was a fantastic person in that subject, but he had this persona that he went along with this popular way people looked at him. So, he pursued that whenever he could. I was very optimistic looking at these conference proceedings seeing that that long time ago, there was a lot of interaction between specialists in the two subjects. I'm optimistic today when I got to a colloquium with some mathematical colleague to a physics lecture, and I find interesting ideas coming from both sides. I'm optimistic when I see some major problem like the Poincaré conjectures solved with ideas coming from physics. Entropy was the missing piece where Richard Hamilton, for years, had studied Ricci flows on manifolds, but could not prove the Poincaré conjecture because he didn't have the entropy ideas that Perlman introduced. So, you asked me when I'm most optimistic. Those were most optimistic times. Did you ask me least optimistic, too?

Yes, because there is that fluidity. It ebbs and flows over the years.

I'm least optimistic when I hear my dean talk about having a search committee for a new appointment in a particular field which is so narrow that you can't think of it in very broad terms. That's when I'm least optimistic. I'm least optimistic when people talk about -- I don't know -- compartmentalization of science. And I'm most optimistic when somebody like my current postdoc, Christoph Gorgulla, comes across with some ideas that are in one field, and maybe could be quite important in another, one that people think is very far away.

Arthur, what have been some of the most satisfying intellectual moments in your career, either a eureka moment, or when you realize that you had solved a really difficult equation, or just the enjoyment of a really productive collaboration? What sticks out in your memory?

I think my long-time collaboration with Jim Glimm was really something that was very fulfilling for me. We haven't worked together since 1987, when we published the second edition of our book. But I don't know many people who have worked together with some other scientists for such a long time. We were singled out by some Boston psychiatrist who was doing a study of scientific collaborations. I think it’s interesting to be paid to see a psychiatrist, so that was fulfilling. It was very fulfilling to understand that you could make relativity and quantum theory compatible, if only in the first two-dimensional models, because that was a problem that was open for so many years. People had been announcing they would work on it and solve it, and it just never got solved. Finally, it was possible to do it. That was so fulfilling.

Why, specifically?

Because it was a problem that was obviously central to science; it could be answered “yes” or “no” and the answer was “yes!” It was an obstacle for so many years. It had been talked about by my student friends ever since I started to study mathematical physics. And finally, it was possible to get an answer. So, we were very happy.

Arthur, conversely, what sticks out in your memory on problems that have gnawed away at you, that no matter what you do, you always hit a theoretical or conceptual wall?

I think the four-dimensional gauge theory is really a problem I'd love to solve, but I now know that I can't devote the concentration of thought that I used to. The solution of dimensional quantum field theory went on over a period of four years; to develop the many ideas took total concentration. Now I have so many administrative and other duties to distract me. So, I think I have to leave that problem for somebody younger. In the west, young people have a big advantage, because young people are often shielded from administration. Now, at my age, I get involved in many administrative problems, if only the administration of grants, which is a tremendously time-consuming thing. Young people don't realize that not being head of a department or head of a society, you have a real opportunity to focus completely on your teaching and your research, which the research side of it is so important. So, I see this with my friend Zhengwei. He's now in China where the system is different, and young stars are given a lot of administrative responsibilities.

Obviously, if you knew the answer to the four-dimensional problem, you would have solved it, but if you just had to take a hunch, what's it going to take to solve this, and when might it happen? What's the breakthrough that the field is waiting for?

Well, my hunch is that at a compactified spacetime, it's actually possible to solve the problem. It just would take a tremendous amount of focus and work. I think for compact gauge group, SU(2), SU(3), using the lattice gauge theory approach, it probably can be done. The difficulty with getting a problem in infinite volume is that it involves not only some physics effects that aren't understood, but probably needs new mathematics as well. So, that's a much more difficult question. But in my mind, if you could solve the problem in finite volume, where the mass gap would be assured by the compactness of space, it would be a major breakthrough. That deserves, in itself, recognition of the millennium problem, but it's not the way the problem was stated.

To broaden that out, once the four-dimensional problem is solved, and we'll be optimistic and say "once" and not "if," what larger questions will that yield? What will be understood, or what will the field be able to explore as a result?

Well, as a result, you'll be able to throw out the concept that you need an effective theory in order to have compatibility between quantum mechanics and relativity. The fact that the simplest assumptions are sufficient to make a mathematical theory work. So, this is the beauty and elegance of simplicity. There, you would have established that. Maybe you wouldn't have a full theory of physics, but I'm not sure we'll ever have a full theory of physics. There are many things in nature that we can only hope to understand, but some extremely fundamental question which has been open since the advent of relativity and quantum theory, so back to when quantum theory -- probably you'd want to go to the 1926 version of quantum theory rather than the 1905 version of relativity -- but these are really longstanding questions. I think it's one of the most fundamental and interesting unresolved questions in theoretical science.

As you plan the four-dimensional problem squarely at the big questions about merging general relativity and quantum mechanics, I'm not sure if you're a fan of such melodramatic phrases, but do you see this getting us closer to the so-called theory of everything?

I think the theory of everything is a little overblown. We don't understand the discreteness of spacetime. We don't understand how to build quantum theory into spacetime. And that's why noncommutative geometry fascinates me so much, because it's a step in that direction. I think it goes beyond string theory, it goes beyond bringing together relativity and quantum theory. Maybe we'll eventually understand how wormholes fit into the picture, how black holes are not just something for Star Trek. I don't know. I think there are so many big, interesting questions in physics, but they need to be understood at a very fundamental level, and I'm hoping that advocating for this fundamental understanding of relativity and quantum theory, even special relativity, not general relativity, is such a -- up to now, we've only understood parts of physics. Understanding a theory of everything, I don't know. It's a nice phrase, but I'm not sure we're really capable of that. We don't understand a theory of all logic. We compartmentalize logic in hierarchies. I think physics may fall in the same category.

Arthur, where do you see advances in computation, experimentation, observation, as getting us closer to these big-scale goals?

Well, the old physics looks to experiments in the lab; new physics looks to computer experiments. You could think of mathematic experiments, the analog of experiments are mathematical proofs. So, proofs are the laboratory for mathematics. I'm not sure we'll ever have a perfect way to do it, but it's -- I don't know. That's a good problem for the future.

On that note, Arthur, looking to the future. For my last question, what do you want to accomplish, for however long you want to remain active in the field? There are such big questions out there, but with time being a valuable resource, and the way that you're pulled into administrative responsibilities, best case scenario, when you can only focus on the science, what are you most optimistic about accomplishing?

Well, first I would like to inspire somebody to solve the relativity and quantum mechanics problem. I would certainly like to find a young person who would devote a number of years and solve it. Second ,I would also like to do something that might help the world in terms of this drug discovery program. If I can complete that project in any way, that would be a very interesting thing for me. Third, I'm at the age where most people have retired, and yet, I have a very active research group in quantum physics and quantum information, and I'd like to able to show that the quantum error correction methods that we've understood through Quon could be helpful for building a quantum computer. There are all these projects that are unfinished that I hope to see them finished.

That's to say that in inspiring the next generation, you're optimistic that they can tackle these problems.

Well, I always meet exciting people, and they have to really concentrate on solving serious problems. History shows that if you work hard, and you persevere, then you can do something good.

Arthur it's been a great pleasure spending this time with you. I'm so glad we were able to engage on this, and I'd like to thank you so much.

Well, thank you.