Physics of Scale Activities

Alexander Polyakov interview
 

Interview with Alexander Polyakov, 6 February 2003

Interview recorded in Princeton, New Jersey.
Interview conducted by PoS collaborators: Babak Ashrafi and Sam Schweber.
Edited by Alberto A. Martínez and Silvan S. Schweber.

PoS

    It would be very helpful to us if you would tell us a little about your background; going back to your youth, or if you want, about your family background. What got you interested in physics and mathematics?

AP

    My father was a professor of literature. My mother was a director of a big chemical lab (both are retired now). When I was about ten years old, I was making ham radios and had my own transmitter. And then, I realized that designing the radios was more interesting than building them. This way, I came to physics and then I started participating in some Olympiades and I was admitted into the Moscow Physical Technical Institute rather early. It was 1961.

PoS

    Is that when you entered your graduate studies?

AP

    It was when I entered this Moscow Physical Technical Institute as an undergraduate. And then it was lucky that I had Arkadii Migdal as a teacher --- a wonderful and a great man. At that time, a remarkable paper was published by [Yoishiro] Nambu and [Giovanni] Jona-Lasinio. There were actually two papers. One was by Nambu and Jona-Lasinio and another was by Anatoly Larkin and Valentin Vaks. These people suggested the analogy between superconductivity and elementary particle physics. They basically introduced the idea of spontaneous symmetry breaking into particle physics. And Migdal motivated me and my friend, his son -- it was Sasha Migdal, -- he made us become interested in these matters and I think that from this time on I always was thinking in terms of both quantum field theory and condensed matter.

PoS

    When you came to Moscow, what was your background? How much quantum mechanics did you know?

AP

    You mean, when I entered the university? Well, when I entered (I was 15 years old), Sasha Migdal and I passed the first of the [Lev] Landau exams. This was the only one occasion when I met Landau, just before his accident. I think I had some elements of quantum mechanics and I certainly had no problems with calculus at that time; I knew differential equations to some extent. We passed the mathematics exam actually. Then we were more or less ready to pass other exams like mechanics and quantum mechanics, through some background we had...

PoS

    Mostly from the textbook of Lev Landau and Evgeny Mikhailovich Lifshitz?

AP

    Yes. That was the basic textbook. That was the great event of my life before university. I tried to read many popular books. And I took some freshman courses in physics, and I never really understood them and they didn't engage me. At some point, I bought a second-hand copy of Landau and Lifshitz's Mechanics and that was just a revelation. I still think it's a great book.

PoS

    And the equivalent in mathematics would have been?

AP

    Actually there was no equivalent in mathematics, I guess. I mastered without much difficulty some standard textbooks. There was a good course in calculus, several volumes by [Vladimir] Smirnov, but it did not really matter that much. There was also a nice book by Zeldovich that was called Higher Mathematics for Beginners or something like that.

PoS

    And when you came to the university, did you take courses? Did you attend lectures?

AP

    I didn't attend lectures, as a matter of fact. I passed exams, but I realized very soon that I'm a free bird. And I never attended lectures. I still don't.

PoS

    Who were the people that came to university with you? Who were your cohorts, who were the other students?

AP

    I was friends with Sasha Migdal and Sergey Gourwits who were also my age. We were younger than the other students.

PoS

    Vladimir Naumovich Gribov is older than you?

AP

    Oh Gribov was much older. Gribov was considered a great man, he was compared to Landau, while we were just undergraduates. At that time, he was in his thirties, it was probably the highest point in his career. So he was a great man, who by the way, was extremely critical of our attempts to deal with spontaneous symmetry breaking. That was also the case with other experts.

PoS

    So you were studying for the exams which went as far as quantum mechanics? Did they include relativistic quantum mechanics?

AP

    Yes, we kept studying it and using your book, by the way. During the second undergraduate year at the university we mastered, to some extent, field theory. There was a book which was extremely useful, I remember, I still think it's an important book. It's the book by [Alexei] Abrikosov, [Lev P.] Gor'kov and [Igor E.] Dzyaloshinskii. It's called Methods of Quantum Field Theory in Statistical Physics.

PoS

    And this work of Vaks, Larkin, Nambu and Jona-Lasinio, was it the first research topic that you picked?

AP

    No, it was the second. The first, which was suggested to Sasha, myself and our friend Sergey Gourwits (we were 18 years old) by Arkadii Migdal, was a very nice problem actually. He asked us to find a surface energy of nuclei considered as a Fermi gas. And that was the problem that required what in mathematics is now called the methods of the heat kernel. We developed these methods without knowing that mathematicians were doing the same at the same time. And it was an extremely important first problem for us. I'm still amazed that he found such a problem which was just at the limit of our abilities. A very difficult thing to do, as I know now as a professor. The second problem was indeed this work on spontaneous symmetry breaking when we independently discovered the Higgs mechanism, in our own way.

    PoS

      While you were a student in Moscow, was there a sense that there were some problems that were more important than other problems? How does one see oneself as a theoretical physicist in the Soviet Union at that stage?

    AP

      I don't think there was much difference from the West in this respect. There were some acknowledged people who had strong opinions, and there were seminars where these opinions were expressed and it certainly influenced young people.

    PoS

      I meant it more in the following sense: You read Abrikosov, Dzyaloshinskii and Gor'kov. Was it considered as good theoretical physics as say, the book by N. N. Bogoliubov and D. V. Shirkov that deals with elementary particles and high energy?

    AP

      Well, it depends on the place of course. Basically Arkadii Migdal was very friendly with the top people, Gribov, Okun, Pomeranchuk. And through him, we got acquainted with these people and they expressed their opinions.... Both fields were considered important so there was no problem.
      Actually, people in the particle community had absolutely no understanding or sympathy for condensed matter problems; though they respected condensed matter, but they didn't think it could be useful in any way for their business. I remember giving a seminar on this paper and Gribov said; -- Gribov was very emotional -- he said, 'This is complete fiction,' to which Berestetskii commented: 'You are unfair. It is science fiction.' But since we were very young, these criticisms -- it was not so bad. People were rather kind, it was not anything personal. It's just their opinion...

    PoS

      So you would identify Gribov as dealing with high energy rather than condensed matter or statistical physics.

    AP

      Definitely.

    PoS

      What I'm really asking is in terms of status. Whether as a young physicist you saw a hierarchy... I would say that in this country, certainly high energy was considered as having a higher status than condensed matter at that stage.

    AP

      I don't think so. No.


    PoS

      But it sounds like the appreciation of Larkin's work wasn't universal.

    AP

      Oh no, it was not universal. It was close to zero. Migdal and ourselves would make a fair estimate. I don't know how it was in the West. It was better probably. It was not the question of the "status," people simply thought that these two fields don't and shouldn't overlap.

    PoS

      And so you learned all the things about superconductivity, Green's function, from Abrikosov, Dzyaloshinskii and Gorkov?

    AP

      Yes. There was also a very nice book on elementary particle physics on particle phenomenology written by Okun. Weak Interactions of Elementary Particles. Great book, very concise.

    PoS

      And were you reading Bogoliubov and Shirkov?

    AP

      I read this textbook. I used many textbooks simultaneously. It looked to me too formal. I was not very impressed by it. But a few topics were useful.

    PoS

      And the few topics were?

    AP

      I think they have an explanation of the renormalization group which I don't think was very good. Although through them I learned that there is a paper by [Murray] Gell-Mann and [Francis] Low and I read the paper.

    PoS

      At that time?

    AP

      At that time.

    PoS

      Which would be 1961, 1962?

    AP

      It was 1963. This paper I liked very much.

    PoS

      What was the impact of Landau, and Landau ghosts at the time?

    AP

      Now I will move to 1966-1967 when Gribov and Migdal... Well, that's a complicated story.... To answer in one word, the fact was that field theory was considered completely pathological. Not by me, because I was impressed by condensed matter things. But the general opinion was that field theory was generally no good. Just as it was here. Things started to be more concrete for me in 1966, 1967, when I started being interested in critical phenomena. I have to go back in time a little bit. In 1963, there was a paper published by [Alexander Z.] Patashinski and [Valery L.] Pokrovsky. The paper was very difficult to read, but it contained an important clue for me: it was very clear that the phase transition problem is completely equivalent to relativistic quantum field theory, that when you go to the critical point we're dealing with a relativistic field theory.

    PoS

      When did you learn about Patashinski and Pokrovsky? Did you read about it in 1963?

    AP

      It was a little later, 1965 or something like that.

    PoS

      Do you remember what the general reaction to the paper was? Was it a well-known paper?

    AP

      It was a well-known paper. The general reaction was that the paper had some problems. Which was true. However, it was a big step forward because it was a first scaling solution, which by the way would be exactly correct in supersymmetric theories. But in normal field theory, without supersymmetry, it was not correct and that was noted.... The point was that they thought that the equations of field theory predicted the value of the anomalous dimensions. Later, in 1965, they dropped this approach and developed a phenomenological theory of scaling, as did [Leo P.] Kadanoff and [Benjamin] Widom in which the critical indices were some unknown parameters.
      And then, Sasha Migdal and myself explained what happens in field theory, which, when treated properly, supported the above phenomenological theories and gave some new results. Our papers were published in 1968. A little later, in 1969, there was also a very important paper by Larkin and Khmelnitski who looked at the four-dimensional problem of critical phenomenon. They used the Gell-Mann-Low renormalization group equations to solve this problem and they found, probably for the first time, a very concrete singularity for the specific heat and other quantities. So basically the picture of critical phenomena started to become very clear at that time.

    PoS

      Was it common to have one foot in field theory and one foot in statistical mechanics?

    AP

      No, it was not common at all. I remember telling someone that I wanted to learn about elementary particles by studying boiling water, and getting a strange look. It was considered a completely crazy remark. Completely pathological. So it was not common at all and I think very few people did this. I actually don't know who else.... Sasha was thinking like that. Larkin actually switched to condensed matter probably because of this very hostile reception to their attempt in particle physics. And a little later, when we met Ken Wilson, we were on the same wavelength.

    PoS

      In looking at your paper, 'Microscopic description of critical phenomena,' a question comes to mind: How widely known was it that for almost any kind of system you could actually get to an expression for the free energy or for the Gibbs free energy which looks like Landau-Ginzburg? I mean, what was in your appendix. How widely known was that? Or is that something which you could derive and that made it clearer why you...

    AP

      I think it was -- how shall I put it -- very off-mainstream, the whole thing.

    PoS

      How did you end up in this almost unique position?

    AP

      Well, I don't know. Probably because the first push was made by Arkadii Migdal who advised us to study this paper by Larkin and Vaks on spontaneous symmetry breaking and then this work on spontaneous symmetry breaking actually forced me to learn both subjects, because spontaneous symmetry breaking occurs in both cases. So that was "a frozen accident," and then it was more or less a natural development.

    PoS

      Was Migdal thinking in this way?

    AP

      No, he was not thinking this way, but he had some great intuition. He really had the feeling that this way of thinking -- this spontaneous symmetry breaking idea -- has a great future. Although he didn't work on it himself.

    PoS

      Who were you talking to? Were you talking a lot to Patashinski and Pokrovsky?

    AP

      Yes. That was after 1967... I think in 1965 or 1964, after this advice from Migdal, Sasha and I talked with Larkin a lot and we learned many things from Larkin.

    PoS

      Where was Larkin?

    AP

      He was at the Kurchatov Institute for Atomic Energy. And later he joined the Landau Institute, and later we joined the Landau Institute.

    AP

      He was in Moscow then?

    AP

      He was in Moscow. He was 35 years old. And then later I had many enlightening discussions with Pokrovsky and Patashinsky who, although we corrected their work, were very enthusiastic about it. And very supportive.

    PoS

      Did you hear of the conference held in Dubna in 1966, I believe, that [David] Pines and John Bardeen were at? Where they met Patashinski and Pokrovsky.

    AP

      I don't think I was at this conference... No, I wasn't there. I remember another conference in 1968, when I met Kadanoff.

    PoS

      And explicit notions of scaling become clear after you read Kadanoff, 1966 or earlier?

    AP

      Well, I think that actually Patashinski and Pokrovsky had a pretty clear idea about it. It was independent of Kadanoff, I think. The idea that you have correlation functions which are scale invariant is contained very explicitly in some papers of Patashinski and Pokrovsky.

    PoS

      And the distinction between magnetic systems, which is what Kadanoff looks at in terms of scaling, and what happens in something like a liquid gas transition where....

    AP

      Universality in critical phenomenon, that was actually quite clear to Larkin, and to Patashinski and Pokrovsky, and to Migdal and myself. I think the most important achievement of Patashinski and Pokrovsky in this old, formally incorrect paper was that they realized that this was an infrared problem for which all ultraviolet details are irrelevant.

    PoS

      Can you describe what you did with this suggestion of Arkadii Migdal when you looked at the spontaneous symmetry breaking, what did you do with that? How did it come into your research?

    AP

      Well, what we did we published the paper with Sasha Migdal in 1965... (I don't remember the precise date when it was published), but we had problems with the referees. It took more than a year to get it published. The paper contained the following statement, the following result. If we have a gauge theory with a spontaneously broken symmetry, then the gauge bosons become massive and there are no mass zero particles. That was a very explicit statement. And we proved it I think in a nice way. If we actually look at the S-matrix we show that all the poles corresponding to the Goldstone particles disappear from the S-matrix and there are only massive particles. That was another reason why this was very useful for our development. That was a first acquaintance with gauge fields.

    PoS

      At what point did you begin to change your mind? You said that it was pretty universally agreed upon that field theory was a dead end, and yet you were pursuing field theory.

    AP

      I think it was basically from the very beginning. I think that probably my opinions became very strong that field theory was the right way to do things; they became strong in 1967 when my paper on critical phenomenon was written. And then there was the next paper in which I showed that relativistic field theories with anomalous dimensions were consistent. I introduced a method which was the equivalent to operator product expansions. That was also done by Leo Kadanoff and Ken Wilson, probably a little earlier. There was no doubt in my mind that field theory was consistent. So I can say with certainty that in 1967, that I was completely certain that field theory was right.

    PoS

      Do you mean field theory in condensed matter systems...

    AP

      No, field theory in general. And therefore, I started at the same time the project describing elementary particles in terms of field theory. And my thinking at that time was that field theory should have a scale invariant fixed point and that the Landau problem is avoided by looking at field theory, which is conformally invariant in the ultraviolet. And well, I developed a rather nice practical correct picture of e+ -e- annihilation, and I showed that in scale invariant theories, it persists by forming jets. And there's a cascade process of the type of Kolmogorov turbulence. I was also at the time very interested in turbulence as I am now. This Kolmogorov cascade in elementary particle physics with jets which produce small jets etc, that was actually in my paper that was published I guess in 1970. But long before that, I was more or less convinced that particles should be described by field theory.

    PoS

      So you learned field theory from say, S. S. Schweber's book, Bogoliubov and Shirkov, and other particle physics approaches. Your understanding of field theory and your confidence in it grew in critical phenomena and condensed matter physics and you brought the idea of a fixed point back to particle physics......

    AP

      Yes, exactly.

    PoS

      Tell us more about the period of following 1965 and 1966.

    AP

      Well, in 1965/1966 there was this work on spontaneous symmetry breaking. Then in 1967, there was the paper that corrected Patashinski and Pokrovsky, which showed that at the fixed point you expect some dynamic anomalous dimension, not simple numbers. But then there was a next work in which I developed this subject further and introduced the operator product expansion without knowing that Wilson and Kadanoff did the same. And then I applied the whole thing to particle physics. I had a series of papers which analyzed e+ -e- annihilation and deep inelastic scattering. I can provide you with the references to the papers if you need them. And one of the predictions was that even in a scale invariant theory, the Bjorken scaling does not exist. That is because you have many anomalous dimensions of various operators, and that changes the exact Bjorken scaling in a definite way. I actually made this announcement at the Kiev conference in 1970.

    PoS

      At what point did you become aware of the work of 1965 of Ben Widom, Leo P. Kadanoff, and Michael Fisher?

    AP

      Oh yes, I was certainly aware of it. Before I published the work in 1967 on Patashinski and Pokrovsky, there were these papers which you mention which I read and also the papers by Patashinski and Pokrovsky. The latter were phenomenological papers with scaling, which I found actually easier to read because they were very concrete which I could check directly using field theory. They simply stated things very clearly, I think the clearest paper was by Patashinski and Pokrovsky. They stated that the n-point correlation functions must be scale invariant. Just practically in these words, so it was a very concrete statement.
        Actually I should have mentioned that the technique I used at that time heavily relied on the previous work by Gribov and Migdal on Reggeons. The history is very entangled. Gribov in 1966 developed a wonderful Reggeon calculus and then there was a problem. Those Reggeons were strongly interacting in the infrared. And Gribov and Migdal started analyzing this infrared interaction and found some nice methods which I used later for critical phenomena.

      PoS

        Did you talk at all to the people in the West?

      AP

        Well, as I said, the first time I met people from the West was in 1968 when I met Leo Kadanoff in Moscow. I think he was more or less interested in the work that I was doing. Then in 1970 Ken Wilson came to Moscow, and that was very nice because we spent a lot of time talking together.

      PoS

        How about Giovanni Jona-Lasinio and Carlo DiCastro?

      AP

        I don't think I met them then. Or do you mean their work? Oh, let me try to remember, their renormalization group paper on critical phenomena. I certainly knew this work, but I didn't feel it was practical, or somehow my thinking was in a different way. It didn't influence me very much. I can't remember if I knew it before or after I published this paper, but in any case, it didn't really influence me.

      PoS

        I want to clarify what you get from Patashinski and Pokrovsky. The notion of scaling at least the way it was initially introduced by Kadanoff is primarily for spin systems. Does it bother you when you read the paper that somehow Kadanoff assigns to the spin block again essentially two values when there's a whole range of possibilities? How does one justify this strong correlation which would say they all pointed the same either up or down? That's certainly not obvious from Kadanoff, right? He asserts it.

      AP

        Right.

      PoS

        And to say that all the spins within that block would be correlated so that you can only have....

      AP

        Plus or minus... Well, you know, I didn't really care very much at that time about this introduction of block spins, rescaling and so on. I thought it was all very qualitative and I thought it might be right, but I didn't think it was the way to go in concrete terms. I was wrong about this. At the same time it didn't bother me at all. I knew that I could replace those discrete spins by some f4 (phi to the fourth) field theory. In field theory the field actually changes from minus infinity to plus infinity, and then you can make these changes introducing the corrective variable block spins, or something, you will get just another field. My thinking was very different at that time. It was not renormalization group in the Wilson-type thinking. Actually, I still think that renormalization group methods are useful sometimes but it's not the only way to look at the system. An example: conformal field theory.

      PoS

        What about the reaction that you mentioned? Gribov, for example was very skeptical... Do you remember as you started applying these methods to e+ -e- annihilation, what was the general reaction?

      AP

        Even more skeptical. Critical phenomenon, he was saying, it was OK. It was not our field, but e+ -e- annihilation... I think I had a very hard time. No one took it seriously.

      PoS

        Were you talking to experimentalists, such as Alexander Voronel or people in the West?

      AP

        In the West? No. I became friendly with some of the western theorists, though. At the Kiev conference, I was discussing these matters with David Gross and then I discussed it with Ken Wilson who came to Moscow for about a month in the 1970s. David, as I said, was skeptical but very interested and he was very intense as usual. When he was young, he was even more intense than now, and more aggressive. Which was just what I wanted, because the worst thing is when people don't react at all. And Ken, I think, was also interested. He independently suggested that some elementary particles interactions are described by a scale invariant theory. But he did not consider it e+ -e- annihilation.
          I remember that it was quite a shock to me when I saw his paper in the Physical Review because when I was doing this I thought that no one was thinking in this direction. And it was quite a shock to see that he was proposing the same theory, but he did not consider this e plus and e minus annihilation and did not consider deep inelastic scattering.

          PoS

            What about experimentalists?

          AP

            Experimentalists in Moscow didn't do e+ -e- annihilation.

          PoS

            But in critical phenomena in general?

          AP

            Oh, in critical phenomena, I had nice contacts. There were wonderful experimentalists in critical phenomena in Russia. Voronel, who became a Refusnik. He later left Russia and went to Israel. He was a Jewish activist and a very colorful person. We discussed with him the physics of critical phenomena and he was very positive about all these developments. I think it was his and Pokrovsky's suggestion that the next thing for me to study was the dynamics of critical phenomena. That was because of the direct influence of experimentalists, because they were interested in dynamics and not only in thermodynamics. That was a very hard subject and it was not a very successful paper, because I really didn't get the key point. But there were a few nice formulae in this paper.

          PoS

            Was that the most important outcome of your conversation with the experimentalists? That one?

          AP

            Yes.

          PoS

            In 1968/1969 after you finish this work, you said before that you saw immediately that Gell-Mann and Low noted the possibility of a fixed point. The classification of the various terms in the Hamiltonian as relevant, irrelevant, and marginal, are these things clear to you at that stage?

          AP

            They were clear to Sasha Migdal and me, but in a clumsy way. We wrote down some scale invariant equations for the vertex functions, propagators, etc. And in all these equations, it was clear that in the scaling regime, you can drop the bare terms. So we showed that there is independence of the bare parameters. In this part of the subject, Ken had a more advanced but basically equivalent understanding. But certainly, there was no doubt about the universality, that ultraviolet details are irrelevant in the infrared domain. Although I don't think that many people understood that.

          PoS

            When you go back to quantum field theory, is there a new conception of what renormalizability means as a result of all of your work or you don't worry about non-renormalizable theories?

          AP

            I didn't worry at all. I thought that they were ok. That actually had some negative impact to some extent... When [Steven] Weinberg's theory of electroweak interactions appeared my reaction was the following. I noticed this paper in 1967. I had absolutely no doubts that this was a renormalizable theory from the very beginning for the following reason. It was very well known to me from critical phenomena that at small distances, there is no difference between broken phase and unbroken phases. So there was no doubt in my mind that it was a renormalizable theory (while even Weinberg himself wasn't sure), and so far this was the positive impact of critical phenomena. But then there was the negative impact. I thought- renormalizable, so what? I thought the theory may be interesting, but nothing special. So that was a mistake and it was induced by knowledge of critical phenomena.

          PoS

            There is a view that was made famous by Joseph Polchinski and others, that QCD, electroweak theory, the standard model, are low energy approximations, that it doesn't matter what the fundamental theory would be... that eventually at "low" energies you will always come out by virtue of symmetry and whatever else, to QCD, electroweak, quantum electrodynamics... Was something like that clear to you and if so when?

          AP

            I don't know. Well, it was in my 1970 paper, in a sense. In it I was writing explicitly that I imagined that there was some fundamental theory, with a lattice or something like that which in the lower energy with the long range approximation gives you the effective theory.

          PoS

            So you would say that this insight of talking about effective field theory is really something that condensed matter physicists knew.

          AP

            Yes, I would say so.

          PoS

            Did you use the work of Cyril Domb's group, or that of Fisher very much?

          AP

            I read the book by Fischer. He published some short book on critical phenomena in 1965 or something.

          PoS

            The Boulder Lectures?

          AP

            Could be. It must have been some lecture course. I used it to learn the subject. I still have this book. It was translated into Russian. I used it, and I certainly knew of these papers, but the most useful thing for me were the works by Patashinski and Pokrovsky. I knew they were not alone, but their presentation looked to me to be more suited for my field theoretic purposes.

          PoS

            Were you teaching this material in the 1960s or early 1970s?

          AP

            No, you see, good or bad thing, in Russia, I had a position with no duties at all. I didn't have any teaching, no salary also.

          PoS

            So how did you survive?

          AP

            I'm kidding. Of course I had some small salary. But that's a standard thing in Russia.

          PoS

            How about seminars? Summer school? Were you teaching what you were learning?

          AP

            Yes, I gave a lot of seminars actually. Also some summer schools, which actually were winter schools. That was extremely useful for me.

          PoS

            Would you happen to have saved any of these lectures that you gave?

          AP

            Actually, yes, it was published as a preprint and I think I have this preprint at home. It was 1971 when I lectured, the whole idea of deep inelastic scattering and of scale invariance in relativistic quantum field theories.

          PoS

            This is in Russian?

          AP

            No, it's in English.

          PoS

            This was where? At which summer school?

          AP

            It was in Yerevan. This particular lecture was in Yerevan, in Armenia in 1971. And I still have it. I can give you a copy.

          PoS

            One of the things that would be very useful would be your complete CV, which we don't have. We have a list of your publications, but not your CV, when you were where.

          AP

            The list of publications is quite incomplete, I just took it from the Internet. It's incomplete, but it's more or less accurate as far as late works are concerned, but there is practically nothing there of my early works.

          PoS

            Since we're concentrating on the early works it would be helpful.....

          AP

            Right, I should do this.

          PoS

            Do you remember your reaction to the Wilson papers? What did it add to your understanding?

          AP

            You see, there were two steps. First, he invented what he called a recursion formula and it was just some uncontrolled approximation for the renormalization group. This didn't impress me very much, though I already had met him at that time and knew that his works are always non-trivial. I studied them carefully and even found an interpretation of the recursion formula in terms of Feynman diagrams.
            But then there was a paper on the epsilon expansion and this time I was ready to kick myself, because that was an expansion that I could have discovered using my own conformal methods and had missed the opportunity. It was a great revelation. I had a lot of mixed feelings when I saw that work done so beautifully in 4 minus epsilon dimensions.

          PoS

            How much did you become involved, or did you suggest to people to become involved, in different methods of computing critical exponents once you have the insight of 4 - epsilon? Do you follow what E. Brezin, J. C. Le Guillou and J. Zinn-Justin and others are doing in applying Feynman methods to calculate critical exponents...?

          AP

            Actually, at that point, I started to lose interest in the subject. At the point when the 4-epsilon expansion was developed. I realized that I had to move on to somewhere else. The thing that I considered important at this stage was conformal symmetry in critical phenomena, which determines the three-point function explicitly. I discovered it in 1969. You have scale invariance that determines the two-point function explicitly, it's just a power. Conformal symmetry determines the three-point function explicitly and then you can plug it into Dyson-like equations and get the equation for the critical exponents. I thought of using it for calculating different critical exponents, but never went very far before the Wilson-Fisher 4-epsilon expansion. It was possible to use this expansion with the conformal bootstrap but no new results followed.

          PoS

            I didn't get why your interest receded.

          AP

            Because it was clear that the main thing is done, the subject is more or less finished. Only the details remained to be worked out.

          PoS

            Do you still feel that way?

          AP

            Well, actually, not quite. Let me tell you what I think of the renormalization group. I think there are two types of useful equations. One type is human-made, they are invented by people. The other type reflects some "pre-established harmony." They can be discovered (uncovered) and not invented. Renormalization group is clearly a human made thing. It's clearly a smart way of calculating things but it doesn't have a breathtaking quality of, say, the Dirac equation.
            The example of the second kind is operator product expansions. They form some beautiful mathematical relations and I was dreaming in the 1970s to have some classification of fixed points based on the possible operator product expansions.
            The program was a little like classifying Lie algebras. In that case you start with the commutator relations which define the Lie algebra and then you classify all possible semi-simple algebras. You arrive at a stunningly beautiful theory (which was clearly discovered and not invented). I was working on that project in the 1970s and I still think it might have a chance. It was successful in two dimensions. We can classify possible fixed points in two dimensions using operator product expansions. That's what conformal field theories are about. And I think it's not excluded, that in 3 dimensions something like that is still possible. I was working for a while on this without much success in the 1970s and then I switched to other things.
            I think the epsilon expansion ended the subject in the practical sense. You can calculate more or less what you want with good accuracy but aesthetically the subject is not closed yet. It's possible that there will be classification of fixed points in three dimensions, based on string theory, similar to what we have in two dimensions. But that's just dreams.

          PoS

            Looking back on your own career after you left the Soviet Union as it developed in the 1970s, how would you look back upon the influence of having worked on this particular field in the 1960s and early 1970s? How did it shape you or what role did it play in the subsequent way that you do physics?

          AP

            Oh, it had a lot of influence. For instance, I started doing conformal field theory in 1969, and it was published in 1970. And then with my colleagues, in 1983, in connection with string theory, we developed conformal field theory in two dimensions. It was a direct continuation of this old work and of the thinking I was doing. In the 1970s, I was trying to solve this conformal bootstrap in three dimensions directly and that didn't work. But in 1983 we solved the 2d problem and it was useful in several areas.
            So I think that all these early works on spontaneous symmetry breaking and on critical phenomena formed my thinking. I can give you another example. When I was working on spontaneous symmetry breaking and learning superconductivity, I learned about vortices and I discussed with Larkin whether vortices are the poles of a Green's function. So I spent some time thinking about the status of these classical objects which we have in superconductors and in superfluids. And I was lucky to have great experts around at the Landau Institute. It played a crucial role when I worked on magnetic monopole solutions, on instantons and things like that.
            Another example: the instantons and quark confinement. There was a very remarkable physicist at our institute, Vadim Berezinsky. He developed a quantitative theory of condensation of vortices in two dimensions. Something similar was later done in the West by [David J.] Thouless and [J. M.] Kosterlitz. I studied his work very carefully. My work on quark confinement and lattice gauge theories is a direct generalization of Berezinsky's work to the gauge case. I'm still trying to follow condensed matter people but with less success.

          PoS

            Earlier you had mentioned that work on turbulence likewise had given you some insight. Can you amplify just a little on that?

          AP

            Yes, you see there was a book published in 1968 by [Andrei Sergeevich] Monin and [Akiva Moiseevich] Yaglom on turbulence. I learned from this book that essentially the problem can be formulated in field theoretic terms. Again it's the problem of infrared behavior and there is a qualitative picture by Kolmogorov and people tried a lot to derive it from field theory. There was this so-called direct interaction approximation by [Richard H.] Kraichnan. He used the Wyld diagrams and so on. I learned all of that when I read this book in 1968/1969. And then in the 1970s, a couple of times, I tried to do something in turbulence, but didn't get very far. Never published. My picture of the deep inelastic scattering is inspired by turbulence and actually gives a first example of multifractality, later conjectured by other people to describe turbulence. And more recently, I published two papers on turbulence. I still keep thinking on this. If string theory comes to an impasse, I will switch to turbulence again. Escape route.

          PoS

            One of the things that our project aims to do is to get people like you involved in commenting on developments and the history of the field. If we send you things, would you be willing to comment on the materials. What is it that would interests you? What would be your interest in terms of helping to clarify the history...

          AP

            Yes. I'm certainly interested in history very much, although more ancient history. What kinds of... examples?

          PoS

            Well, we put up timelines and an analysis of, for example, what happened in Rome, with Jona-Lasinio, DiCastro, their coming to renormalization group methods to try to explain their formulation. We do the same for the Soviet/Russian developments. It would be of interest to have your comments...as you are a participant in the Soviet Union and you see what DiCastro and Jona-Lasinio doing.

          AP

            So you mean commenting on their work. But you know that the problem is that I certainly knew of their work at that time, but somehow, it did not excite me... So I can comment on things that really interested me like Patashinski and Pokrovsky.

          PoS

            Another thing that would be interesting would be to take your list of publications and make them complete in the early period and just ask you to tell a little about the ones that you think led....

          AP

            Actually, I have been asked to prepare my collected works, and I keep procrastinating now, for a couple of years already, but I plan to write down some short commentaries on the works so that...

          PoS

            Are you connected to the Internet? Because we put up the list of your publications that we had.

          AP

            It's very incomplete. That I can certainly send you.

          PoS

            We're doing the same for Ken Wilson; we've done it for Kadanoff to get reactions to what was happening as described by other people. And we're also talking to people like Voronel, to talk about the conferences he has attended. So these kinds of things, to get your comments on, things we don't know to ask, it would be very interesting. Another thing that would be of interest, which is more of a sociological character, is your impression of the different communities you have been a member of: the ones that you grew up with in Moscow and the way physics is practiced here where there is a much sharper division between condensed matter and high energy. Your thoughts on the conception of the self-presentation of physicists here and there and what it means to be a theorist, the contrast between Soviet/Russian as compared to...

          AP

            You know, in Russia, there was also a very sharp distinction: people who were doing particle physics didn't know much of what was going on in condensed matter.

          PoS

            But the status of people like Gor'kov, Abrikosov...

          AP

            Oh, it was very high.

          PoS

            It was very high-- of equal standing?

          AP

            Oh, yes. It was equal. I think it was the Landau tradition. Physics is a whole, so he was equally interested... You know, Abrikosov and [I. M.] Khalatnikov did this work on quantum electrodynamics with Landau. And then Khalatnikov went to superfluidity, and Abrikosov also did some work on quantum electrodynamics in the 1950s, so they were following their teacher. And to some extent, it was universal. Interestingly, the people who are most close to the particle experiment, people in ITEP [Institute for Theoretical and Experimental Physics] like Okun and Gribov they were focused on particles...
            But I confess I never had a right sociological perspective, neither in Russia nor in the US. A kind of blindness.

          PoS

            Before we go, one thing that's important. You mentioned this 1971 preprint of the lectures that you gave in Armenia. Are there other materials that you might have that would be interesting to show what different people were doing at different times? If you send us such materials, we would scan it and return it to you.

          AP

            I will make a copy of the Yerevan lectures and send it to you. Let's see.... published materials. Of course, I have tons of my notes, from these times, but I will take a look. If some of my scientific diaries are appropriate for this...

          PoS

            Are they in Russian?

          AP

            Actually, they're mixed. I was training myself in English so I was writing a lot in English. I think it was mostly in English. I don't remember.

          PoS

            Thank you.

          [end of the interview]