Interview with Kenneth G. Wilson, part II
So at Caltech, you were working with Jon Mathews. I don't know
what goes into your work on computers. Are you learning straightforward
things about how to do serious calculations on the computer?
It's learning programming. At that point, it was in Assembly
And nothing happened at Harvard. You got hooked up with someone
from MIT for a machine?
They had one of the first time-sharing systems at MIT. I was
able to get some time on it, but it was a joke. I mean I got more
done in terms of computing at Caltech when I could sign up for
some time and really work things out, whereas now, I had to submit
a program and the next day, I would get the results from just
a single run.
Did you learn something new about working with computers?
All that happened at MIT was I got incredibly frustrated. Because
of the slowness, nothing ever came out of that project. That was
probably part of what helped me to move from actual computation
to a lot of this conceptual idea of supposing that I had a computer
that was big enough, which in the end, proved much more fruitful.
That's the important part of the work I did through the 1960s,
working with this conceptual idea of distinguishing between something
that was large with finite degrees of freedom and therefore is
computable and something that has infinite degrees of freedom
and therefore is not.
And so you're thinking in terms of your momentum slices.
I'm thinking about various things. I'm doing the operator product
expansion, which is separating momenta, but in diagrams. Always
making approximations: the diagram has a finite number of momena
and then I would work out the approximation with some momenta
large and some momenta small. And it gets very complicated of
course when I have three or more momenta. There are all kinds
of relationships possible between three and four momenta. And
that's why it becomes much simpler once I go to position space.
In terms of computable solutions, what were the criteria for
something being computable? Finite time or polynomial time or...
Computable meant something that I could do within a finite amount
of time -- a reasonable approximation within a finite amount of
time. I was not concerned about polynomial or exponential increases
versus the number of degrees of freedom. All I wanted was to reduce
from an infinite number of degrees of freedom to a sequence of
steps each with a finite number of degrees of freedom.
So when you're thinking about what the degrees of freedom might
be, when is the next time you can get your hands on a computer
that can do something that you find useful with respect to quantum
field theory? Or just find something useful.
There was a small machine in Newman Lab and I learned how to
program it and I tried various things on it. But they were very
simple-minded calculations. Nothing of the kind that was worth
publishing. It sort of kept my hand in, continuing to learn what
you can do with a computer in practice as opposed to just in theory.
They were simple-minded things that involved only a few degrees
of freedom and I have no recollection anymore of what those problems
were that I actually programmed. But it sort of meant that I kept
up with the programming capabilities and at some point I learned
Fortran. I don't think it was on the machine there, but I did
get connected to the computing center and I learned how to program
in Fortran and so forth. But it all comes together at the point
when I start calculating approximations that I had set up.
When you come to Cornell, you start teaching. What did you teach?
The thing that really mattered to me in the beginning was teaching
quantum mechanics to graduate students. I was very happy teaching
quantum mechanics, because that was the closest I could get to
field theory and have something that made sense. I worried from
time to time about interpretation, the collapse of the wave function
and that kind of thing, but that was never a top priority... The
important thing was you could approximate the solutions for given
systems and there were all kinds of ways you could do the approximations.
The approximations worked, you could explain them, and everything
Including putting things on a computer in the quantum mechanics
At some point, I started putting things on the computer, but
I think that was not until afterwards. In the 1970s.
And you still have your notes on teaching the quantum mechanics
No. I don't remember to what extent I actually had notes.
Does anybody come to mind as to who took the course? We're now
Well, I taught it a number of times. I don't know if you can
get it or if the records exist of the years I taught it or who
the students were.
Somebody like John Negele might have taken your course?
Could be, but I do not remember.
Michal Johnson or somebody like that? Does that ring a bell or
I certainly won't remember. The one case I remember was when
I had a student named Kenneth Wilson.
How did he do? <laughter> So when did you put a field theory
on a lattice in the computer?
That comes later. Around 1976 or 1977.
So in that era, after you left Caltech, until you do this, was
there anything interesting that happened with you and the computer
other than conceptualizing what the degrees of freedom for discrete
field theory might be?
I think the first interesting thing is, of course, the paper
which the Nobel committee cited, which was the one-dimensional
integral equation. That I was able to solve on a PDP10 which had
just been installed at the Newman Lab at the synchrotron facility.
And there was hardly anybody using it, so I had it all to myself.
The integral equation was beautifully matched to the capabilities
of the PDP10 at that time.
I had an empty, time-shared machine. I was getting results instantly
and it was just computing one-dimensional integrals. For the PDP10,
that was not a big deal. So I was getting fast turnaround, learning
very quickly what was working, what was not working.
Were you actively keeping an eye out for what machines were available?
I was always aware of what was available. In the early 1970s,
Jeffrey Chester had organized a link to the CDC6600 at Berkeley
and I was part of the group that worked on that. And I did some
computations. I think, in fact, I used that for the numerical
study of the two-dimensional Ising model with a decimation transformation
of [Leo] Kadanoff. I never published that, but that was one of
the things I worked on. Most importantly, I used the CDC machine
computations on the Kondo problem.
So, should we go back now to 1966 or so?
I go to Cornell in 1963.
That's after I spend a year at CERN. I don't think I did anything
on a computer at CERN.
I arrived at CERN in the January of 1962. And I left January
1963. I then spent eight or nine months just wandering around
Europe and then I came to Cornell. And if you're interested, I
got started on the field theory thrust, in terms of looking at
high energy behaviors of field theory when I was recovering from
stomach problems in a hospital in Dubrovnik.
Because you ate something?
Yes, I ate something I shouldn't have. Took a long time to get
that straightened out.
So in 1963, you're still looking at fixed source meson theory
when you go to Cornell?
One strand is that I continue to work on fixed source, and one
strand I was doing the high momentum analysis on diagrams.
And who were the people you talked to at Cornell? Tom Kinoshita?
Who were the people closest to you? Don Yennie is there....
Yennie is there. Of course, Yennie does QED stuff. [Peter] Caruthers
tried to take an interest in what I was doing and there was some
exchange there. Kinoshita's there and then [Kurt] Gottfried comes.
And [Hans] Bethe's there. I didn't talk that much to Bethe, but
I was always going out to lunch with the other high energy people...
And Kinoshita at that stage was not yet involved in computing?
He does infrareds and infrared divergences and things like that.
I don't remember at what point Kinoshita starts his heavy computation
Are you teaching field theory at this point in the early 1960s?
Certainly at some point, I teach the field theory course, but
what I remember more fondly is teaching quantum mechanics. Quantum
mechanics made sense.
Would you have taught renormalization group if you had taught
Probably not. I doubt that I ever gave a course that was really
on the work that I was doing. Certainly not until the big paper
on renormalization group was out. Of course, then I did some special
You mentioned that in 1966, you go through [Lars] Onsager's solution
as given by Lieb, Mattis, and Schultz.
Could we ask what brings you to listen to Widom and gets you
to know some of the people in chemistry?
What must have happened was that Widom must have given a seminar
which was somehow on the schedule for physicists. I have no memory
of what caused me to be present at that seminar. Clearly a very
And when you first meet Michael Fisher...?
I mean I probably met him from the time that he came to Cornell,
because he was as much a physicist as a chemist. But I do not
remember what the process was that caused me to attend Widom’s
seminar For some reason, I start getting interested . And of course
the key thing that's fixed in my mind is going to that seminar.
And do you hear of the 1965 conference at the National Bureau
of Standards on phase transitions which brings together lots of
I may well have heard about it, but it...
Didn't stand out. And just in that sense also, how much of the
work on liquid helium are you aware of?
I paid no attention at all to that.
So then around 1966, when you do the Onsager solution, it's an
isolated event. It's not because you're looking at critical phenomena,
or because you've been talking to Widom or someone.
I'm looking at it precisely because it's just becoming apparent
to me that I have to think about critical phenomena and that it
just doesn't make sense to think about critical phenomena and
not know Onsager's solution.
Why is it more apparent to you that you need to know about critical
Because I'm seeing there's the same kind of problems of multiple
At the beginnings of the 1960s, critical phenomena was coming
to be at the center of interest of a large part of the theoretical
physics community. You say you learn Onsager actually from a paper
by Schultz, Mattis, and Lieb. Is that in the Reviews of Modern
Physics? [Note KGW presumes that this is right but asks that we
I remember that what happens sometime around this time is that
I start translating my ideas just into the language of statistical
mechanics and somebody says Kadanoff has already done that. So
I suspect that I did not read Mattis, Lieb, and Schultz until
after I know about Kadanoff's paper.
But was it Widom's talk that pointed you in that direction that
made you aware?
It starts with Widom's talk. I mean that's what sticks in my
mind as sort of the first catalytic event....
Can we ask what is it about Widom's talk that was so striking
What is striking about it is that he makes all these conclusions
but based on assumptions which he cannot justify.
And the conclusions being homogeneous equations of state?
So it's the homogeneity that strikes you.
Well, he makes a set of assumptions. He says, "Well, if
this is such, this is what we conclude." And I see no way
of justifying the assumptions. So I'm thinking about it, and I
don't know at what point it becomes apparent to me that what is
important is that you have this correlation length going to infinity.
That here you have this system with an essentially infinite set
of scales just as you have in the field theory. And of course,
it's at the critical point that the correlation length goes to
infinity, so that's what I should be interested in. And then there's
Kadanoff's paper, which is again based on assumptions where he
has no mathematics to fill in the details, but where he spells
out the process by which you would deal with the infinite scales.
And I'm pretty sure I don't read Mattis, Lieb, and Schultz until
I learned about Kadanoff's paper. But it's all in that same period.
So with Widom, you're introduced to equations of state which
have certain homogeneity properties -- it's certainly clear that
this should be derivable from some model of field theory. Kadanoff
introduces you to magnetic systems, and he can seemingly get you
an explanation for scaling. You don't know how to justify the
steps in Kadanoff.
So, just to get the collection of the inputs at this point, it's
Widom's talk, there's the Onsager solution, there's Kadanoff's
preprint... What else? Does that cover it? When you turn your
attention to critical phenomena, what things do you take in?
Those were the things I remember taking in. I mean there's the
concepts of scaling where you have an exponent and relations between
exponents, which is of course what they generalize beyond the
classical exponents. Oh, and there's Landau theory.
Since I don't remember exactly, can Johnson's talk about dilatation
invariance and high energy behavior play a role? Is that already
something that had come out of that stage?
As far I remember, the period of time where I'm working with
Johnson and Baker is the period before this.
Yes, but I'm asking for the specific connection between scale
invariance, and later dilatation invariance, and high energy behavior.
Is that paper out already? Is that something you knew about and
talked to Ken about?
Well, I surely knew about it from the time it came out, but we'd
have to look at the exact date. What I remember is the focus on
dilatation invariance comes a bit later.
But scale, I mean the Thirring model would...
I would have to sort of correlate in my mind the things I learned
from the Thirring model and things I learned from scaling in critical
phenomena. And then when I take that back to field theory I focus
more directly... I mean that's what gives me the confidence to
understand enough of what's going on here to start making hypotheses
so that I could redo the operator products at short distance,
and that leads to my 1969 paper.
Were you involved in an active discussion with Widom and Fisher
at this point or was it...
I certainly had an active discussion with Widom at the time of
the seminar. I kept probing him by saying, "Why do you make
this assumption?" But whether there's much conversation except
right after the seminar itself, I don't know.
Is it clear to you at that stage that since all the arguments
of Widom's are essentially thermodynamic they were not specific
to any type of model, is that something you're aware of and find
I'm very aware, and I don't know how this awareness builds, but
I certainly became very aware they're talking about the exponent
relations and expecting them to hold across a wide variety of
systems. But I think my initial interest was more on the simplest
model, especially the Ising model, and not on the issue of universality.
Because it's thermodynamics and not specific to any models? "Thermodynamic" being
statements about systems where you don't specify them and you're
not specifying anything....
Well, I'm pretty sure that I don't do any thinking except in
the framework of statistical mechanics, I mean exp(beta H) and
that kind of thing, and only the simplest cases such as the Ising
model. I wanted the simplicity most of all, to avoid the complexity
of the field theories I was studying.
And therefore, Kadanoff is a link to a specific model which allows
you to go back to field theory.
Right. Because that's the thing that connects to field theory,
it is that kind of model.
Is there interest in explaining scaling? Or is there interest
in what looks to you like the methodological similarities to what
you've already been doing?
I mean what I'm interested in is learning how to think about
the problem and always there's the question of how to calculate
And calculate implies a specific model.
Yes. I'm interested in taking a specific model, if it looks like
a field theory because I always want to have something that I
can take back to field theory.
If you were to characterize somebody like Widom – a physical
chemist -- coming and telling you these things and a physicist
coming and telling you these things, what would be the difference?
at least in terms of his way of thinking about the problem and
your way. Did he give some thought to go beyond the thermodynamic
description and try to see what does it say about the microscopic
models to try to explain his equation of state or the homogeneity
of his equation of state?
My impression of Widom's work is that he was less focused on
actually digging into the statistical mechanics than Michael Fisher
And how much field theory did Michael Fisher know during these
He knew the statistical mechanics cold, but I don't know that
he had much field theory background.
I think he says that he didn't know any field theory at the time.
So at Aspen, people thought that Kadanoff had done this? And you
looked at Kadanoff, and what did you see?
I saw the same problem as with Widom. If you could have some
way of connecting the parameters of the adjacent scales, then
everything could work out. But I wanted to know what that connection
was, so you could do calculations. Of course I don't figure out
anything practical that one could do with Kadanoff’s “decimation
procedure” until five or six years later.
What Kadanoff indicates to you is that if I take something like
an Ising model, and do this decimation of going to bigger and
bigger cells, therefore getting scaling.... that I could justify
going from one to the next and telling you that the interactions
are the same at the different levels, that that would give me
Widom back for that particular system. And that's clear to you
from reading Kadanoff.
He just postulates, of course, that decimation would come back
to the same Hamiltonian, and one of the things I do work through
at some point is doing decimation on two-dimensional Hamiltonian,
but which, of course, produces infinite sets of couplings, not
just two as Kadanoff assumed. But it turns out to be simple enough
that I could keep a finite set of those couplings and get some
quite good numbers for two-dimensional Ising model exponents.
And the justification for just keeping the finite number of new
terms that are generated?. Does it become clear that that's the
Well, you have to make sure that you can keep iterating and it
doesn't just become more and more of a mess. And I was able in
the end to set up a computer program in which it appeared to become
reasonably stable, get reasonably accurate exponents in the two-dimensional
case where, of course, I can only keep a finite number of couplings
and the ones that are more complicated have to be forgotten, but
there was no way I could do that in more than two dimensions because
of the computational demands of the problem (there were too many
interactions to be tracked in the three dimensional case to be
feasible on computers in the 1970s.
Was it clear already to you at that stage how to compute critical
exponents in a two-dimensional case?
No. I had to go through the experience of working and seeing
how that worked to really understand the way the exponents come
out when you have a more complicated fixed point than just Kadanoff's
ad hoc fixed point. I did not realize how to set up a renormalization
group combination for the 2-d Ising model until after 1972.
So if I understand you correctly, what Kadanoff did for you was
to present you with a question, that here's a method that works,
now how do you justify the assumptions. What would it have added
to your thinking at the time?
I think it sort of helped to gel the thinking, because I was
certainly thinking in that direction already, because that's why
people pointed me to his paper. But I think it was helpful to
have him have written down something even if he didn't have justification
for it. Because you have to keep thinking these things through
and getting a clearer and clearer picture, and I think it was
helpful in that sense. But it wasn't as if he had done something
that was such that I had never thought of thinking in that direction,
because I was already moving in that direction.
Is there any way you can describe for us or help us understand
or just list the steps to tell how your work changed between reading
Kadanoff [in 1966] and 1970 when Widom asked you to look at DiCastro
and Jona-Lasinio? What was going on in your work?
Clearly, I was just continuing to try to understand how to think
about the field theory. And I was understanding that the critical
phenomena problem was simpler in the sense that the individual
degrees of freedom you had to deal with were much simpler in the
case of the Ising model than in the field theory. And also, when
you're doing field theory, the parameters are all given to you
and you can't change them, whereas in statistical mechanics, you
get a chance of varying the parameters and seeing what happens.
It had sort of given me confidence and understanding that even
if I couldn't do the calculation, I was becoming more familiar
with the idea of what would happen when you have anomalous exponents,
because the critical exponents were clearly anomalous. I was drawing
on the scaling theory in critical phenomena and to give me the
confidence to say that the main thing I needed to change from
my perturbative work on operator products was simply to allow
the dimensions of the operators to vary. And moreover, the dimensional
analysis would work the way I had set it up before, as long as
I replaced canonical and known (free field) dimensions by anomalous
and unknown ones.
Can I rephrase the question slightly differently? From Kadanoff,
what you get is that by virtue of the iteration of the decimation
and the assumption that the Hamiltonian remains the same (basically
only nearest neighbor interactions) you get a clear notion of
a fixed point and an understanding of that. When is there a transition
and how does operator product expansion come into play when you
start thinking of taking off degrees of freedom, and you start
thinking in terms of spaces of Hamiltonians rather than the parameters
which exists in the Hamiltonian that Kadanoff talks about?
Those don't really connect that much I think. I mean I have the
operator product expansion and I use the statistical mechanics
to build the confidence that I should simply put in the anomalous
dimensions and keep the rest of the analysis as I had it before.
But that doesn't help me to deal with the question of how to actually
generate a workable renormalization group transformation. So on
the question of generating a transformation, I think the other
input was the momentum slice models and the whole struggle to
get rid of the slicing. Because when I come to 1971, I was still
trying to do momentum slices, but do it in a way that allows me
to pretend that I don't really have gaps between the slices. By
1971 I was ready to say: I'm going to divide the whole momentum
continuum and to divide it up into slices so that their average
momenta are separated by a factor of 2 and I'm going to try to
make the simplest approximations I can, based on separating low
momentum from high momentum at least qualitatively. This is the
strategy in my 1971 paper. But this does not refer at all to the
operator product expansion.
You told us you had a conversation with Widom after this seminar,
but after you start looking at Kadanoff's paper it changed your
thinking between 1966 and 1970. Did you start new conversations
with Widom, Kadanoff, or people in that circle?
I'm sure that I had conversations with Widom from time to time.
(I do not remember whether I interacted with Kadanoff.) Certainly,
at some point I start learning about high temperature expansions
and about [Cyril] Domb and so forth, and at some point I learn
to program those expansions. I don't remember when I did that.
That was probably one of the things I did just to help me continue
to learn the statistical mechanics and I'm sure things like that
I talked about, especially with Michael. So, there is a period
of time when I learn the technology of doing high temperature
expansions and actually since it's computing, I set up programs
and I learned to calculate a few more terms than anybody had calculated
on some of the expansions.
You mentioned here in your Nobel lecture that you're not always
reading what everybody else is doing in physics, but are you at
this point reading the reviews of Kadanoff or of [Peter] Heller
or the lectures that Fisher is giving everywhere? Are you collecting
these materials on critical phenomena or are you focused specifically
on your problem?
I know I want to work on the Ising model and the simplest models
and that's all that I need. If I can do the Ising model, nothing
So you're not looking at experimental reviews or things like
that on critical phenomena.
I'm certainly not spending a lot of time on experiment. I'm sort
of getting to know more about it because I'm talking to people
and such issues come up. But all this material was organized for
me by Michael, I don't need to be concerned about that.
At some point, Carlo DiCastro comes to give a talk.
In the fall of 1969, DiCastro comes to Cornell. I think he's
invited by Widom and he meets you and he talks about his paper
with Giovanni Jona-Lasinio on renormalization group. …
Did you go there?
Yes, I certainly know that Michael and I had a discussion about
Jona-Lasinio and DiCastro and the question was, was there something
here that we needed to know about? They had the same problem as
others in statistical mechanics in that they were making the same
kinds of assumptions that everybody else was. They couldn't put
in something that justifies the assumptions. When Michael and
I talked about it, we couldn't figure out how this would help
us at all.
Did Michael understand it? I mean given that he lacked the background
in field theory....
Presumably, he did not know about Gell-Mann and Low and that
sort of thing. It's not until you have the epsilon expansion that
you see that using field theory renormalization group can actually
do anything beyond just getting you to the same scaling relationships
that Michael and Ben Widom had already gotten to. My comment now
is very different than it was at that time because there is a
question here about counterfactual history. It is a reasonable
question about counterfactual history to ask if I had disappeared
from the scene before 1971, what would have happened? I think
that in one way or another, the work on the renormalization group
of DiCastro and Jona-Lasinio, plus the work on dimensional regularization
in field theory, which certainly had nothing to do with me, would
have inspired to pull these two ideas together and realize that
you could do the epsilon expansion. There might be no formal renormalization
group of the kind I discussed, but you would still have Kadanoff’s
work. However, that work might have proved a dead end because
nobody would be able to figure out how to take it any further;
but you would have the field theory renormalization group method
of Gell-Mann and Low combined with the epsilon expansion. I think
that's a reasonable hypothesis, and then of course, in that case,
Jona-Lasinio and DiCastro would have had an absolutely crucial
role in that counterfactual history.
In the way that things happened when DiCastro and Jona-Lasinio's
paper came out, and when DiCastro gave his talk at Cornell, it
didn't have an obvious impact, that it was an arguable formulation
but one that offered a new insight for you or for Fisher.
Were you convinced that whatever they had done, and this is now
for the magnetic system, had in fact given a convincing proof
It was not a proof. It was just like Kadanoff’s work. It
offered conclusions based on assumptions that had no justification.
It was not a proof. Moreover, the actual history involves my using
the momentum slice concept to arrive at the approximate recursion
formula published in 1971, which is not a Gell-Mann Low type formulation
because there is a function to compute at each step, not just
a single coupling constant. The function can be expanded
in a Taylor series, which defines an infinite set of couplings,
all of which change at each step of the recursion formula. Then,
at Michael's urging, I work out what happens near four dimensions
for the approximate recursion formula, and find that d-4 acts
as a small parameter. Knowing this it is then trivial, given my
field theoretic training, to construct the beginning of the epsilon
expansion for critical exponents.
In contrast, DiCastro and Jona-Lasinio were not able to compute
anything such as critical exponents, as well as not able to prove
anything either. But in the counterfactual history, which assumes
that I do not contribute, my belief is that someone would be inspired
by the DiCastro and Jona-Lasinio work and the dimensional regularization
work to explore dimensions above three, and thereby discover that
perturbation theory is legitimate near four dimensions. In this
counterfactual, the epsilon expansion emerges, with a major credit
to DiCastro and Jona-Lasinio, but even more credit to whomever
is first to explore near four dimensions. But the renormalization
group formalism with infinite numbers of couplings either does
not emerge at all, or else emerges at some later date, perhaps
by someone inspired by Kadanoff's ideas to try to construct a
calculable version of his transformation.
I have discussed a counterfactual here because I think such counterfactuals
are needed to help identify work such as that of DiCastro and
Jona-Lasinio that merits respect precisely because it might have
played a key role in subsequent developments. I do not think the
present practice of honoring only the developments that played
a key role in actual practice is really fair to all parties, nor
does it provide adequate incentives to scientists to explore a
variety of directions that could be important.
Except that they were trying to justify getting Widom, starting
from a microscopic model. That was certainly their aim.
That was their aim, but all they did not accomplish it. They
had to assume it. .... You see, the trouble was they were working
with the Gell-Mann/Low version with just one coupling constant.
And there was just no way they could justify that. I mean I knew
that the problem was “how do you get from an infinite set
of couplings, which you could justify, but you couldn't do anything
with, to just one” (I didn't even know yet how to really
set them up with an infinite set of couplings). But in going from
an infinite set of couplings to just one, they were assuming their
answer, namely, a one-parameter Gell-Mann/Low renormalization
group. I mean, you could justify it when you had the field theory
perturbation expansion, but they didn't have any field theory
perturbation expansion. So until there was the epsilon expansion,
there was no basis for DiCastro and Jona-Lasinio’s assumptions.
And then you need dimensional regularization to give you the
insight to set up the epsilon expansion, in your counterfactual
Well, to give you the insight that you should have fractional
dimensions. But I'm just saying, when you look at the way the
whole science process works, you find out that you have the analogue
of an army of ants trying things with varying degrees of justification,
but they're just trying things. It seems to me a reasonable prediction
that somebody, once there was dimensional regularization that
was out there and published, and Jona-Lasinio and DiCastro was
out there and published, somebody would have figured out that
they needed to apply this to the fractional dimension and somebody
would have tried it and whoever did would have discovered it works.
I mean that it was not rocket science to do the lowest order epsilon
Do you remember at what point you read the Lev Landau and Vitaly
Lazarevich Ginzburg paper that specified the criterion....
I'm not sure I ever actually read that paper.
Right. I sort of got the feeling that this was a cursory reference
from your Nobel lecture that you had subsequently been made aware
But you were aware of A. A. Migdal and Alexander Polyakov in
When I went on a couple of trips to Russia I got to meet Migdal
and Polyakov and then learned about what they were doing.
And this was before 1970.
It was right around 1970.
But by the time that you see the Varenna lectures, you were certainly
aware that it was 1970.....
Were there any events in this development before we get to the
1970 Widom seminar that we should talk about?
That's a good question, but I don't remember the timing of these
Dimensional regularization... when did you learn about that?
I certainly learned about dimensional regularization essentially
as soon as it happened, but I don't remember when it was.
So, there's the talk by DiCastro and then in 1970, Widom and
Fisher have this seminar and they invite you to talk about....
I was invited to talk at the seminar while Fisher was somewhere
else, so it was just Ben Widom and he asked me to talk. The peculiar
thing is that I managed to get things figured out enough between
the time he asked me to talk and the time I actually give the
talk, to be able to talk about what's coming out of the calculations
and giving some kind of basis for it. At the time I gave the talks,
I was actually doing the calculations.
So he specifically asked you to talk about DiCastro and Jona-Lasinio's
I don't remember what I was expected to do. All I remember was
I would talk about something... it may have been that paper, I
don't remember what the original plan was. The plan got much more
focused as soon as the calculations started working.
These lectures were in Rockefeller rather than at Newman Lab?
I assume they were over in Baker actually. They were not in Newman.
In Baker. I'm sorry.... in the Chemistry Building.
You mentioned the name that I wrote down at my desk of someone
who might have been there and had notes of one of these.
But you mentioned someone else on the phone when we were talking…
Did I mention Michael Fisher or ....
It was someone's name I didn't recognize immediately. I was going
to ask you about this person, but...
Is David Nelson already active as a graduate student at that
Oh, did I mention Joe Serene or someone like that?
I gave some lectures later, and Joe wrote up some lecture notes
for that, but whether he did for the earlier talk I don’t
I might have suggested John Wilkins.
Yes, that's right. John Wilkins.
If Ben Widom himself doesn't know, then Wilkins is the person
He's at Ohio State now. But he was at Cornell at the time and
he and I shortly afterwards had a joint student working on the
extending my work on the Kondo problem. And of course, that involved
the renormalization group ideas.
Do you have any notes besides the paper with Arthur Wightman's
notes as referee of that period?
I have some stuff, I mean not really notes, but just some of
the records of the calculations that I did. I don't know if I
still have the two-dimensional Ising calculations. I mean I certainly
have some records of the high temperature expansion I did on the
So there's a one paragraph description here of your lecture about
that seminar and how you applied the phase-space analysis and
Landau/Ginzburg model. And here are your two papers.
There would be very little in the lecture notes that didn't get
into those two papers.
I don't know if I can formulate the question well, but I was
wondering if you could tell us more about the process that went
on. There's the final results of the two papers and a short paragraph
here. I wonder if there is more to say. If you can recollect the
process that you went through in getting the results that you
All it required was getting this idea that there was something
I could do... that I could simplify things enough that I could
separate the scales and reduce it to that one-dimensional integral
equation with the concept of phase-space cells and what later
became the wavelet analysis. But I don't think the wavelets existed
at that time. But that you could take a field and reduce it to
these discrete variables that you could assign to phase space
cells and then once you had the phase space cell concept, that
I could have one big cell connected to smaller cells. And once
I had that concept, then it was quite straight-forward. I knew
how to set up the stuff on the computer, that was not a problem.
I don't think there was much more to it than was in those things.
Did people understand the lecture? How did Widom and the other
It's not clear that they understand the lectures. Have you tried
to interview Widom?
Widom, I would expect him to be reasonably straight-forward and
he's the person to ask. What I know is that Franz Wegner came
and visited us once he learned about those papers and for him,
it was no problem. But not very many other people were ready to
pick it up that quickly, because my equations required numerical
integration on a computer, and in 1971 not many researchers were
comfortable with numerical integration. And of course, the big
breakthrough in terms of the impact on the field was the epsilon
expansion, which could be computed analytically.
Continue reading part III of the interview.