
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. MartÃnez 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]
JonaLasinio. There were actually two papers. One was by Nambu and JonaLasinio
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 secondhand 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 JonaLasinio, 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
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
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
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] GellMann and
[Francis] Low and I read the paper.
PoS
AP
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
19661967 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 wellknown paper?
AP
It was a wellknown 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 fourdimensional problem of critical
phenomenon. They used the GellMannLow 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 LandauGinzburg? 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 offmainstream, 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
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
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 Smatrix we show that all the poles
corresponding to the Goldstone particles disappear from the Smatrix 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
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 npoint 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
JonaLasinio 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
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 f^{4}
(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 Wilsontype 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
PoS
In 1968/1969 after you
finish this work, you said before that you saw immediately that GellMann 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 nonrenormalizable
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
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
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
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
AP
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
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 nontrivial. 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. ZinnJustin 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 4epsilon
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 threepoint function explicitly. I
discovered it in 1969. You have scale invariance that determines the twopoint
function explicitly, it's just a power. Conformal symmetry determines the
threepoint function explicitly and then you can plug it into Dysonlike
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 WilsonFisher 4epsilon 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 humanmade, they are invented by
people. The other type reflects some "preestablished 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
semisimple 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 socalled 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 JonaLasinio,
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 JonaLasinio 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 selfpresentation 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
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
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
[end of the interview]


