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ORAL HISTORIES

Interviewed by

Katherine Sopka

Location

Lyman Laboratory of Physics

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Interview of Arthur Jaffe by Katherine Sopka on 1977 February 15, Niels Bohr Library & Archives, American Institute of Physics, College Park, MD USA, www.aip.org/history-programs/niels-bohr-library/oral-histories/31284

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Professor Jaffe discusses his perspective on developments in physics at Harvard since he came there as an Assistant Professor in 1967. This interview was done in the interest of compiling a history of the Harvard University Physics Dept. in the mid-20th century.

Transcript

This is Katherine Sopka speaking. I am visiting today, the 15th of February, 1977 with Professor Arthur Jaffe in his office in the Lyman Laboratory of Physics. In the interest of compiling a history of the physics department in recent decades, Professor Jaffe has kindly consented to share with me today his perspective on developments in physics at Harvard since he came here as an Assistant Professor in 1967. Professor Jaffe, perhaps we can best begin by asking you to comment upon your pre-Harvard background and upon the circumstances leading up to your coming to Harvard.

Oh, this is the pause button.

Yeah, this is the pause button, and the two down record.

Right. Okay.

Alright, anytime.

Well, I guess it will take me a little while to get into this, because I'm a little nervous speaking with a tape recorder in front of me.

Just start and tell us a little about your background. I already know a few things about you simply from American Men of Science, where I found out that you were born in New York City in 1937. And did you grow up in New York City?

In the suburb of New York, Palm [?], New York. [phone rings...] I grew up in Palm, New York, which is a suburb of New York City, and my parents moved there when I was about one year old from New York. I went to Princeton as an undergraduate in 1955, and I guess I chose between going to Princeton and Caltech by some random process. I wasn't sure what I wanted to do at that time. In fact, physics was very far from my mind. I started as an undergraduate studying chemistry and in fact got my Bachelor's degree in Experimental Chemistry.

Oh. Had you gone to a public high school? You had gone to Pallum High School?

Pallum High School, yes, and then went from there to Princeton. And mainly because I wanted to go to Caltech and my uncle convinced me that it would be better to go to a liberal arts college and thought Princeton was a good idea. [???]. My father always was interested in my studying medicine because he's a very distinguished doctor, and in fact one time had been offered a chair at the Harvard Medical School. But in my senior year at Princeton I decided that I would drop all those ideas and decided to study mathematics. [???] at that time I got a scholarship to go to England for two years and went to Claire College Cambridge where I took a second Bachelor's degree taking the examinations my first year in mathematics.

Oh. Presumably you had taken a fair amount of mathematics at Princeton then as an undergraduate. Or not?

Not any more than ordinary, but I did have one professor, Don [?; John?] Spencer, who encouraged me a great deal at Princeton and helped me go in that direction, and also a person whom I got to know, an historian there, Charles Gillespie, who was responsible for my going to Claire College Cambridge because he was a friend of the new Master, Eric Ashby [?], who had just gone to Claire College, and thought since his friend was at Claire that it was probably a good place for me to go as well. And he made a perfect choice of college, because I found it a very friendly place and was extremely happy there compared to some of my compatriots who went to Cambridge [???] other places and didn't enjoy it as much.

I see. How long did you spend then there?

I spent two years at Cambridge. The first year [???], formally an undergraduate for both years. In fact it was a little funny, but I decided to inform the Marshall Commission in the summer of 1959 that I was no longer interested in studying chemistry, although my scholarship was to be a student in chemistry at Cambridge. And they told me if I wanted to do mathematics and if that was alright with Cambridge they advised that I perhaps should go ahead with it anyway. But they couldn't really give official permission at the time, and in fact it might take them over a year to give [???]. So —

You certainly can't sit around for a year waiting for permission.

Well, I just went ahead and did what I wanted. So at that time I got involved during my second year at Cambridge in studying Fermi [?] surfaces, because Ron Depart [?], who at that time had just become professor of physics at the university and was interested in the connection between the topology of Fermi surfaces and magneto resistance that were inspired by his own research and research in Russia, Lifshitz and Pechanski [?], suggested that I work on a problem which I found quite challenging and in fact wrote a paper which I never got around to publishing. The people Cambridge encouraged me to. And in fact I was going to stay as a graduate student at Cambridge. But at the same time I had a number of friends there who encouraged me to read some other papers in theoretical physics, and I found Arthur Whiteman's [?] work and work related to that by Ray Seose [?] and Zurich [?] to be quite interesting, and so in the end I decided to go back to Princeton and do my Ph.D. with Arthur Whiteman, even though as an undergraduate I had never heard his name.

That's an interesting odyssey.

So that's more or less how I got into physics. So I had three different degrees, a B.A. in Chemistry, a B.A. in Mathematics, and a Ph.D. in Physics.

I see. In all likelihood it all hangs together now, in retrospect, as being probably a good way to gain the various competencies.

That's right. Right, in fact I still retain much of each of these experiences. On the other hand I find it very hard to remember a great deal of elementary chemistry. Okay. So I did my doctoral dissertation at Princeton with Arthur Whiteman, and I had the fortune at that time to be at Princeton as a graduate student in an era where there were many other extremely good young students who were contemporaries, and I think I learned more — [phone interruption...] I was in the middle of saying something, but I don't remember what.

You were talking about the other stimulating graduate students at the time at Princeton.

Oh, yes, I was about to say I think that I feel I learned as much if not more from my other graduate students as a student than I did from the professors at Princeton, although the faculty was full of distinguished people there. It was the way of life at Princeton the students spent a great deal of time in Palmer [?] Laboratory up on the top floor, talked to each other hours on end. Princeton, as you know, is a small town where there is very little else to do but work, and we spent a lot of time teaching each other physics, mathematics, and I look back with quite pleasant memories of that time.

What was the title of your thesis that you did write with Arthur Whiteman?

Oh, I don't remember the exact title.

What was the topic?

But the topic of my thesis is one which I came back to later on and have been more or less working on ever since, and I can — perhaps we could come back to that question and if you want to go over my research interests I could work that in at that point.

Sure. During your experience at Princeton, was there the close cooperation between the mathematics and physics departments that were sort of epitomized in the existence of Fine Hall and the Fine Hall Library, or was that—?

Oh yes, that was very much the cornerstone of my time there. Since I left Princeton of course Fine Hall has been deserted and there's the new Fine Hall and the new library which is much bigger and has many more books, but is in many ways much less convenient that they have all set up. The nice thing about Princeton was that the physics and mathematics goings were connected, and the Common Room in Fine Hall was a meeting place for both. The Common Room was just at the base of the stairs leading up to the library, and had mailboxes in the mathematics department, and the mathematics and physics library on the top floor at Fine Hall was with its wood paneling and leather easy chairs the place where you could find physics or mathematics students any hour of the day or night.

I remember being impressed the first time I went into the Fine Hall Library, was the fact that the stained glass windows had the integral of PDQ [?] worked in, like a theological symbol.

Right. Right. Yes, those windows were — I don't remember who designed those windows, but there's a long story around them based on the windows there. The cooperation between physics and mathematics was not only expressed by the students who talked to each other all the time, but also by the faculty members. There were several faculty members in both departments — Arthur Whiteman who I worked with, as well as Valentine Bargman [?] and I guess Eugene Wigner [?] was in both departments as well. And there was at the same time in the mathematics department Edward Nelson, who was quite interested in physics, did many things related to physics, and of course in physics John Wheeler, who had some interests in mathematics. So there was a great deal of common ground. There were common seminars, and also seminars involving both people at the university and the institute. So, in fact I didn't realize as a student that was so unique, communication between mathematics and physics. In fact it's hard to think of any other university where there is that close contact, which is made by a bond of common interest.

It looks as though you made an ideal choice for your own particular interests in Princeton.

Right. Well, it turned out that way. So what should we go over now?

Well, I realize that after you got your Doctor's degree you stayed on for an extra year.

That's right.

At Princeton. And then you went out to Stanford for a year, and then came back to the institute so that you —

Right. Well, for part of that same year, that's true. After finishing up my thesis, I was a little unsure where I wanted to go, and since I was working on problems that were closely related to things that other people around Princeton were interested and since I got a postdoctoral fellowship that allowed me to do whatever I wanted, I decided to spend that year in Princeton. And after that I received numerous jobs offers, I don't remember how many, from different places, and was offered an assistant professor's job at MIT at Stanford, at Berkeley, New York University, Stony Brook, Maryland. I don't know all of them. It seemed every place. It was back in the good days when you didn't apply for jobs but just offers of jobs came every day in the mail.

It was nice while that lasted.

Right. And I decided, since my old teacher, Don Spencer, had gone out to Stanford and was then chairman of the mathematics department there, and when I went out to visit at Stanford he spent so much time with me and expressed such interest in my coming to Stanford that I thought Stanford would be a nice place to go. In fact I often had thought about going to California. And my good friend Oscar Lanford [?] who finished up his thesis a year after I did was going to go to Berkeley, and so I thought that were would be an opportunity to have some contact with him. And we wouldn't be in the same department so we wouldn't be competing with each other in our closely related fields, and it seemed to be ideal. So I went out to Stanford and enjoyed that year extremely much, although I spent a couple of months out of the year back at the institute at Princeton. However Stanford did turn out, though I was both in the mathematics department at Stanford and at the Stanford Linear Accelerator Center, it did turn out for me to be quite scientifically isolated, because there were very few people I found on the west coast aside from my friend Oscar Lanford in Berkeley who had much actual interest — not interest, I guess there were many people who had interest in the type of thing I did, but who actually worked on problems and were current on things in a way that I could discuss physics and mathematics with profit with them. [interruption of someone knocking on door...]

Sure. [tape off, then back on...]

I found the distance from Stanford to Berkeley a little bigger than I had at first imagined, and Oscar Lanford and I did not see each other so much, so when out of the blue in the fall of my first year at Stanford, my year at Stanford I received a letter from the physics department at Harvard offering me a job in the east as Assistant Professor, I took it very seriously. And while I had liked Stanford very much and I thought it would be a very pleasant place to stay, I felt that scientific isolation would probably be detrimental to my long term work and for that reason I decided to come to Harvard, though I had never been in Cambridge before for any extended length of time. But I always had a great respect for physics and mathematics that was done here.

Did you come directly then from Stanford to Harvard?

Right. Well, I was at the Institute for Advanced Study and then I went back to Stanford for the summer, and then I came directly here.

Oh, I see, I understand. Yes.

So I was at Stanford for a year, and out of that year I spent a few months at Princeton at the institute.

I see.

And so, and I've been at Harvard ever since with the exception of leaves here and there. Half a year at the Currant [?] Institute, half a year at Princeton, half a year in Paris, and I guess that's it.

Well, I guess that you're not feeling any difficulties of isolation now that you are in this —

Well, another thing that happened this year I was at Stanford is that I started collaboration with Jim Glim [?] that summer just before coming to Harvard, which has lasted for ten years since then and has produced any number of papers that we have written jointly, and has been perhaps one of the most fruitful collaborations, scientific collaborations that has gone on. What happened was that we invited Jim, who was interested in work similar to what I was doing and was at that time at MIT to spend a month at Stanford, and during that month and with my plan to come to Cambridge we talked to each other a great deal and started some joint work which we continued when I came to Cambridge and while he was at MIT and I was at Harvard we actually lived more or less around the corner from each other in the Radcliffe area, so that was a very easy and fruitful start to our collaboration. After a year Jim went off to New York to the Currant Institute, and he's been in New York there at Rockefeller University ever since, but we maintain contact and have written a tremendous number of papers together.

And New York isn't so far away [???].

That's right. And so I don't know if you want to get into the scientific aspects of things at this point or if you want to —

Yes, I think that it would be appropriate now to ask you about the kinds of topics that you like to work on and the approach that you take which might be different from others; in other words, your own research interests and I use the word "style."

Right. Well, I guess most of the research I've done in physics has been related to two topics: Quantum Field Theory as a part of Quantum Mechanics, and Statistical Mechanics, and relations between these two things. Now these are two very broad areas, and the particular aspect of Quantum Field Theory in which I work is something that more or less started about the time that I was a graduate student. Maybe I should sketch a little bit of the history. In fact I was just explaining it to somebody here yesterday [???].

Well sure, that would be fun.

Notes on the [???], but we all know that quantum mechanics and relativity were two of the most successful theories of the early 20th century in physics, and the great success of quantum mechanics, when combined with special relativity, led Heisenberg [?] and Durac [?] to propose the Quantum Theory of Fields. On the other hand, the combination of these very two simple principles immediately led to difficulties. The infamous — [interruption of knock on door; tape off, than back on...]

You were beginning to discuss your own research and you were giving a little summary of how quantum field theory came into begin.

Oh, right. So when quantum mechanics and relativity were combined, the immediate thing that appeared in calculations were divergences; these famous, or infamous, infinities whereby some rules of calculation by the 1940s there were devised rules of calculation of dividing infinity by infinity and subtracting infinity from infinity and coming out with some numbering. And it was amazing that when Lamb and Redford [?] measured the shift in the energy levels of hydrogen that it was supposedly due to the effect of quantum field theory over the Durac equation that this agreed perfectly with the numbers arrived at by these cooked up rules. And therefore everybody in the world believed that quantum field theory had a great deal of truth in it, and yet was not clear whether there was a theory in the sense that we're used to in physics; that is, in the sense of some underlying logical or mathematical theory. And that problem remained more or less open from the invention of quantum field theory up until very recent times. So that was the area which Arthur Whiteman suggested that I direct my interest toward when I was a graduate student, and the way I like to think of it is that people who study quantum field theory have followed more or less three lines. And the first line I would call the heuristic quantum field theory, which goes back to this early work I have been describing up to now. And about 1955 people decided, a small group of people decided that they would ask what could be concluded just from the basic principles of quantum mechanics and relativity, causality and stability of a ground state. About the supposed solutions to quantum field theory — and this subject grew up called axiomatic [?] field theory. Axiomatic, which is really a bad name, but the one that has become attached to it because people wanted to assume axioms, namely the existence of a quantum mechanics and its special relativity, co-variants under special relativity. And people like Hogg [?], Whiteman, Layman [?], Sevancik [?], Zimmerman [?], later people like Ruell [?] and Hepp [?] and also earlier Yost [?] worked in this subject and found a number of very interesting things. For instance, the connection between spin [?] and statistics which for non-interacting particles had been established in 1940 by Pauling [?], was provable within the framework of the axiomatic field theory just as a consequence of relativity, quantum mechanics, and stability of the ground state. So this subject of axiomatic field theory included many other interesting consequences — the PCT Theorem and later Scattering Theory, to give a particle interpretation to quantum fields. But all in all, it was a very unsatisfactory subject from the point of view that there were no known examples of quantum fields because of these famous difficulties that I alluded to before, nickeristic [?] field theory. The only known example satisfying these simple requirements of relativity and quantum mechanics were not interacting particles; they were so-called free fields. And so about 1965 or in the 1960s the subject which now I like to call Constructive Field Theory, started up, in which people tried not to look at consequences of the basic principles but rather construct models or construct examples. And that was the step in this direction was something that I worked on in my Ph.D. thesis. It wasn't the first attempt to do such things. A number of people had looked at approximate quantum field theories before. And in my thesis what I did was to take the simplest type of quantum field theory that I could imagine, the quantum field theory coming from quantization of the nonlinear wave equation in some polynomial self-interaction — polynomial in the field giving the energy density of interaction — and asking whether in approximate form that quantum field theory could be built. And I found that by applying appropriate mathematical tools that that field theory could be more or less completely understood and reduced to the problem of in principle of diagonalizing certain simple operators on standard Hilbert [?] space. But this thesis was very incomplete, because the approximation which I took in studying these quantum fields was one in which the infinite numbers of degrees of freedom of quantum field theory, Schrodinger [?] equation with an infinite number of degrees of freedom is what you study in quantum field theory, was reduced in the approximation to a finite number of degrees of freedom. And that was clearly only the first step, because I had not dealt with the problem of regaining the infinite number of degrees of freedom. And this is a problem which some of the things I did in my thesis were prototypes for the approach to the problem which I later carried through to the end, much of it in collaboration with Jim Glim, and some with other people, to give the first example of a quantum field theory with interaction. The first example actually came out in 1968, two-dimensional quantum field theory, and the fact that it satisfied all these axioms of Whiteman was only established bit by bit, and finally the last bit put in place in 1972. And since that time the subject has more or less grown from only a handful of workers who were interested in these questions in the 1960s to a considerable number of young people who have come into the subject and contributed to it, so now it's a very big and active field. On the other hand, the questions that people are studying now have changed over these ten years and there are many different sorts of questions which I could come back to perhaps afterwards, but let's stop for a minute. I really have to organize my — [tape off; then back on...] Okay. Before continuing further with these ideas, I would like to mention that during a couple of years, including the year I was at Stanford, I devoted much of my time to trying to understand in quantum field theory a completely different question, namely how to extend the axioms of Whiteman to take into account fields that were sufficiently singular to possibly be adequate to describe non-renormalizable quantum field theory. And this was a line of research which I wrote a few papers in, but I seem to understand the basic problem and which other people have carried forward in many directions since. And I don't want to say anything else about it. Coming back to the construction of examples, in the summer of 1968 I spent a 3-month period in Zurich and during that time was a visiting professor at the UTH. I gave lectures there on the first model of quantum field theory in the course, and there was a very insidious student named Conrad Osterwalder [?] who wrote up the notes for my lectures. Somewhat later he came to the United States and was actually the second of my postdoctoral fellows here at Harvard, the first Robert Shrodder [?], whom I also met during this same visit in Zurich and who had just finished his Ph.D. and came to this country a year later to work with me. These two people overlapped for about a year and a half at Harvard — and during that time did some extremely important work on quantum field theory. In fact, the most important work in the axiomatic direction of quantum field theory since the original papers of Whiteman, Land [?], Sebacic [?], Zimmerman and Hogg. And what Osterwalder and Shrodder discovered here working with me was that there was a very beautiful connection which goes back to Feynman [?] in his Ph.D. thesis in Princeton between statistical mechanics and quantum field theory. And Feynman proposed that the classical statistical mechanics of a lagrogen [?] could be interpreted as the quantum mechanics of that same lagrogen, the so-called Feynman history of Feynman Path Integrals. And these made no mathematical sense in the way that Feynman proposed them in his thesis in 1942, but in 1952 Mark Katz, who was the mathematician at that time at Cornell, realized and presented at the 2nd Berkeley Symposium on Probability Theory the formula representing solutions not to the Schrodinger equation but to the heat diffusion equation where you placed [?] time by [???] times at the time, representing these solutions as an integration over Weiner path; that is, using the standard mathematical notion of Weiner Integration or integration over continuous functions. He gave an explicit integral representation for the solution to the diffusion equation and he showed that in that case the ideas of Feynman were exactly right. Now, in the early 1960s a German physicist, Kurt Semancic [?], had proposed some similar representations for quantum field theory, and the work of Osterwalder and Shrodder on axioms for Euclidian Greens Functions [?] puts in beautiful perspective the connection for quantum field theory which could be understood in the case of ordinary quantum mechanics as the analytic continuation from the Feynman-Katz integral representation for solution to the heat equation to the corresponding solution for the Schrodinger equation. This construction of Osterwalder and Shrodder has become a standard tool now in quantum field theory and is the basis for the functional integral interpretation of quantum field which in the last few years has become very popular. But physicists even though they may not completely understand the connection between the functional integrals and the Hamiltonian quantum mechanics. But it's there, and there is a very simple and precise mathematical connection that can be made between the two which is an abstraction of the older ideas.

I see.

So that's one line of work that went on here. And another line of work was this construction of models. I was very lucky to continue to have extremely good postdoctoral people here, and also some extremely good graduate students. And among them were Yurt Verlik [?], who is now an assistant professor at Princeton; Conrad Osterwalder, whom I mentioned, who has just been awarded the chair for mathematics in Zurich; —

Oh, how interesting.

— and Robert Shrodder, who has gone back and now a professor at the University of Berlin; and Erhardt Siler [?], who is now an assistant professor at Princeton; Joel Feldman [?], who is at this time at MIT as a Moore [?] instructor in the department of mathematics and will next year be at the University of British Columbia in Vancouver; Tom Spencer, who spent a year here in 1974 or 5 and is now at Rockefeller University; John — Now, in the scientific direction, these first models that were constructed that I described before actually were quite simplified in a way that's extremely important, and I should have mentioned earlier, and that most important simplification is that the three-dimensional space in which we live is replaced by one-dimensional space. And this in statistical mechanics is something that is quite familiar — the Eising [?] Model, which was solved in closed form by Lenz [?] and Eising in one dimension in the 1920s or before that. What was it?

It was early in the century, as I recall.

1911, I guess it was. Yeah. But the two-dimensionalizing model was only solved in closed form by Ansheiger [?] in 1944; that is, with the free energy.

That's quite a deltoidei [?].

Right. Although it was known before that, based on work of Piles [?], that low temperatures, the two and higher dimensionalizing models, had a phase transition. But the three-dimensionalizing model or the two-dimensionalizing model in non-zero external magnetic field has never been solved in closed form, although there are many calculations on it. The abstract proof of Piles of existence of phase transition applies. Actually this proof was incorrect, and it was only in 1964 that Robert Griffiths [?] independently Divershin [?] in Moscow showed that the argument could be made complete. And there's an interesting story about that which connects up with quantum field theory and the work that I have been doing. But one thing that was of interest to people who studied quantum field theory was this question of whether, since statistical mechanics and quantum mechanics were connected, whether a phenomena like phase transitions that we know occur in statistical mechanics could occur in quantum field theory. Now there was a heuristic physical picture of this and the physicists believed it and associated certain phase transitions with what's known as symmetry right now [?]. And I set out with Jim Glim and Tom Spencer to establish this actually early in 1975. Well, the story is a little more complicated. We actually began to study it about two years earlier, but there was an announcement from Moscow that Divershin and his co-worker Menlos [?] had solved the problem. And in fact they published a paper in which they announced the solution to the problem in 1973. And so everybody else stopped thinking about it, because they were two very respected scientists, and when two of the best Russian mathematicians say they have solved something you don't generally, when you don't know exactly how to do it, keep on working on it — especially on a difficult problem. But after a year and a half when there was still no detailed version of their paper appearing, and when no information could be got from them or anybody else going to Moscow about what they had actually done —

Oh, they announced that they had solved it; they didn't publish the solution!

Right. People began to think that perhaps they hadn't really done it.

I see.

And so we decided to go back to work on that problem, and in fact it turned out not to be so terribly hard just to prove that there were phase transitions and the simplistic sample of quantum field theory and the space transition was associated with the symmetry breaking. But it was a little ironic. We got the proof and presented it at a conference in June of 1975 in Marseilles. In fact I gave the talk there on our paper. And at that conference somebody reported they had just come from a meeting with Poland [?] a little earlier in the year where there were some lectures given by Divershin on the Russian work, and they gave me a copy of the lecture notes, and I looked through the 60 pages of typed material and could only find that they felt the problem was so difficult that it was far too difficult to put it into their 60 pages of lectures and so would present the proof somewhere else. But at that time we had our 30-page proof.

Oh. It must have made you feel even better.

[???] in Marseilles and sent a copy off to Moscow. And the next year we spent taking that further to develop an expansion in the 2-phase region about mean field theory. And it was nice that this expansion then gave conversion, gave a way to convert — gave a way to approximate the two pure phases by their mean field values and give a systematic converging expansion in the coupling [???] and away from the mean Hill [?] values, the first end terms in the expansion just being the usual proturbation expansion, [???] all the errors in the expansion with deviate and proturbation period [?].

Excuse me a moment. This is —

Yeah. [tape off, then back on...]

Okay.

So, we developed this expansion about the end field theory, and one nice story associated with that and the whole question of base transitions was associated with an incident in the fall of 1975 when Jeffrey Goldstone [?] was visiting at MIT. Now the proposal that there were phase transitions of this type in field theory was first made by Nambu [?] and Goldstone and Yono Lasinio [?] and other people. Goldstone in the early 1960s, and Goldstone is one of these people I thought it would be nice to tell that we had actually proved that that was what happened, and he then related a story that when he wrote his paper proposing this in the early 1960s the famous Rudolph Piles invited him to give a seminar. But when he drew the picture of what was happening on the blackboard, Piles immediately stopped him and said that the ground state which Goldstone said would be degenerate could not be degenerate because there must be quantum mechanical tunneling from one ground state to the other. And therefore the ground state would be unique. And Goldstone related that he was not allowed to complete his seminar because Piles didn't believe at that point that the argument could be correct. And so I think it's an ironic twist of history that one very important ingredient to show that the ground state in this quantum field theory actually is degenerate goes back to the 1936 argument of Piles himself to show the existence of phase transitions in the Eising model.

Oh.

And by combining this old picture from 1936 with many of the techniques we developed to study quantum field theory mathematically, we could show that that's actually what is going on. Well I met Piles this past January when I was out in California. He was visiting at UCLA, and I asked him if this story was true, and he completely denies it. He says of course [???] would be degenerate [???] in that system, and he couldn't remember the seminar with Goldstone. [laughing] So I don't know if this story is true or not, but in any case it's pretty amusing.

Well, did Goldstone ever get around to publishing these results that apparently he was presenting in a seminar that—?

Oh yes. He had already published them I think at the time, in 1961.

Oh, I see.

It's a well-known paper.

So it won't turn out to be one of those sad cases in history by somebody is turned aside and never been [???] [laughter] [???], "But I was saying that, and somebody stopped me."

No.

Because we do have those.

Right.

From various times.

But in any case, the fact that the physics of these quantum fields might [???] the questions of base transitions and the spectrum of particles, Eigen values of the Hamiltonian, are becoming more and more the question of whether or not they are bound states and resonances and what this scattering looks like, and becoming more and more the questions that we're looking at now seems to show that after ten years of constructive field theory we're really — and with this connection between quantum field theory and statistical mechanics, we're now dissolving the barriers between mathematical physics as it existed in the past little while studying these questions, and more heuristic theoretical physics. And it turns out that what looked like the most difficult problems on quantum field theory for the future are the very same or closely related problems to the ones that the more heuristic theoretical physicists are now worried about. And so I think the boundaries have really dissolved, and it's more or less one subject looked at from many different points of views. Complementary points of view were often very useful in solving things.

That's an encouraging note. In general then, you feel that the communications among the theoretical physicists in general and say the local group of theoretical physicists are opening up rather than —

Oh, absolutely, yes.

That's good.

And I'm especially interested in discussing things with people like Sydney Coleman [?] and Shelley Boshaw [?] and the questions also Steve Wanberg [?] to some lesser extent, and also the many body [?] theorists like Hal Brinjunger [?], people like Nelson Martin, because actually all these subjects seem to be one subject, and I'm interested in the questions of tracking of quarks, as many people around here are at the present time. But I'm not sure exactly what I'll do, because I'm also trying to write a book and perhaps that's a good way to leave this subject where there are many other bright young people and look for something else to do. I'll be a pioneer in some different area.

Well, it looks as though there's no dearth of interesting topics to turn your attention to. In general, do you find yourself in what kinds of approaches you are happy with, do they tend to be more characterized by applying mathematical techniques, or by using mathematics and physics?

Well, I would say really neither one exactly. It turns out that these problems of the existence of quantum field theory have actually led to a great deal of interesting new mathematics. And perhaps one reason they hadn't been worked on so much earlier was that the mathematical development wasn't at the stage where it was right for this type of work. Well, I think the real reason was probably that people never tried very hard — tried very hard. Many mathematicians are interested in physics, but very few are willing to learn physics, and I think that's the mistake that most mathematicians who work in physics make, that you can do good work in mathematical physics without learning physics. But what happens of course is like in any of the historical cases of work on mathematical physics, the mathematics that you need really isn't just there and available, but it's necessary to develop some of it as you go along. In fact we had to build up some logical structure in which to understand these problems and to develop the theory more or less of that structure more or less from the beginning. And so the problem was made somewhat more difficult by not having the complete framework at our fingertips which we could apply, but more interesting as well, and I think that actually physics has in the past always been very important for the development of new directions in mathematics, and in this case has been up to this point in the study of quantum field theory for interesting connections between physics and algebras of operators with probability theory with equations in an infinite number of variables have already been made. And now even topology seems to interact with the so-called gauge fields that are of interest at the present time to describe the unified theory of elementary particles proposed around here by people like Weinberg [?] and Boshops [?].

I see.

So I would not say that mathematical physics as I view it as applying mathematics, but rather understanding physics and developing mathematics at the same time.

I see. Well, very well. It certainly seems as though many of the labels that are put on things are inhibiting and barrier-forming, whereas really it's a total subject with no sharp barriers, that different perspectives are useful and helpful if they're in context.

Right. For instance historically the current status of group theory has been tremendously influenced by physics and the study of continuous groups really grew out of the work of Wigner in Princeton who [?] defined the theory of groups to first non-relativistic quantum mechanics and later relativistic, special relativity in quantum mechanics. And that led to tremendous work. On one hand the abstract group representations, the type of thing that George Mackey does here at Harvard, induced representations, and on the other hand the specific examples such as were done by the Indian mathematician Harris Chander [?] at Princeton at the institute and the Russian school by Gelfond [?] and his co-workers in Moscow.

Certainly the picture that you present of activities in your particular area of theoretical physics and here at Harvard contrasts greatly with what it was 50 years ago, and this was principally a laboratory. I always feel it rather amusing when I talk to a theoretical physicist and say I am sitting in his office in the Lyman Laboratory. [laughter] Because certainly the buildup here was laboratory activity, and the theory has, was slow getting started, but then has burgeoned tremendously in recent years. I wonder if you might want to make some comment on your perspective of the activity at Harvard both in the total picture of physics and in particular in theoretical physics, that when you came here —

Well, the past ten years, that is the time I've been at Harvard, have been a very exciting time in many different areas of physics, and I've been glad to be in a department where there are so many active people in different fields and so can keep abreast of things going on in many different directions. In fact it's quite a privilege to be here, and also to have such interesting students. I find the Harvard undergraduates tremendously interesting to teach. In fact I often enjoy teaching undergraduate courses more than graduate courses because the students are so enthusiastic. So I would not want to predict the future. I think that physics changes so rapidly that any prediction you make about the future is bound to be wrong. But I would just hope that the department at Harvard can preserve its excellence that we now have. It's going to be very difficult in the face of the economic situation in the world as we see it, and also when we remember that the department during the last ten years has expanded a great deal, especially with the increased number of junior appointments, and for that reason without the same increased number of senior staff positions available, a larger percentage of the junior people are going to be leaving. And we have been extremely lucky here at Harvard, both during the time I've been here to have some of the very best young people in the world, especially in theoretical physics where I am most familiar in our department. Lucky to get them at an early stage before they could be grabbed up by other universities, and I just hope that that situation continues in the future. It's hard when these people leave to take senior positions at other prestigious places to replace them, because each of them came with some special story and some degree of light [?].

Well, I thank you very much for what you've told me today, and is there anything else that you think we might touch upon? I understand from talking with other people in the department that everybody is very happy with the flourishing of theoretical physics, both the internal activity and I guess in a sense the image that Harvard is the place where people will come from all over the world really, even though just visiting, even if they can only come for a year, that it's the most desirable place to come.

Right. You can see all the visitors we have this year. In fact I have this year, although I have money from the National Science Foundation to pay one postdoctoral fellow, I also have one person from Belgium who has money from Belgium, one from Germany who has money from Germany and is bringing a German student with him here, two from Italy who have NATO fellowships who are coming here as postdoctoral fellows, and this is not unique in the department that in every theoretical field there seem to be a number of people coming with their own money, which is very unusual just to spend a year here. I think that that's something that we must not get complacent about, and I think it's very dangerous to say that Harvard is the place to go, because just by saying that you have the danger of destroying it and I think that we'll be very lucky if that is still true in five years. I certainly hope it is, and I'll try my best to keep it that way.

Well, are there other centers in the United States that you see drawing a comparable stream?

Not at this time, no, but certainly there are traditional centers for physics that are extremely good in the United States. Princeton is a traditional center that is extremely good; Stanford with the linear accelerator is a center for theoretical physics associated with high-energy experimental physics; Berkeley is certainly a center, but there are no large numbers of visitors at Berkeley the way there are here.

The other thing that seems to concern the department is the experimental side and the fact that it's difficult to maintain the tradition of the experimental excellence that has been here at Harvard. There seems to be more difficulty with that than the theoretical. It's doing fine, but the concern as to whether well let's see we become top heavy, but we not have a balance seems to be a concern. Is your own feeling that it's highly desirable to have a department where both experimentalists and theoreticians are comparably active?

Oh, of course. You want the best department that you can have.

Sure.

But that's much easier said than done.

Than done.

And I think that experimental physics has a real problem at Harvard. But it's a difficult problem, and there is no easy solution, and just saying that you are going to appoint some people in experimental physics does not solve the problem. Even making precipitous appointments can make the problem worse.

Well, I'm sure that the next decade and the next half century will contain lots of interesting surprises as we go through.

Right.

Well thank you again. It's been fine.

Well, it was a pleasure.