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Wilson interview, part II

### PoS

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?

### KGW

It's learning programming. At that point, it was in Assembly language.

### PoS

And nothing happened at Harvard. You got hooked up with someone from MIT for a machine?

### KGW

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.

### PoS

Did you learn something new about working with computers?

### KGW

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.

### PoS

And so you're thinking in terms of your momentum slices.

### KGW

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.

### PoS

In terms of computable solutions, what were the criteria for something being computable? Finite time or polynomial time or...

### KGW

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.

### PoS

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.

### KGW

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.

### PoS

When you come to Cornell, you start teaching. What did you teach?

### KGW

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 made sense.

### PoS

Including putting things on a computer in the quantum mechanics
course?

### KGW

At some point, I started putting things on the computer, but I think that was not until afterwards. In the 1970s.

### PoS

And you still have your notes on teaching the quantum mechanics course?

### KGW

No. I don't remember to what extent I actually had notes.

### PoS

Does anybody come to mind as to who took the course? We're now talking 1966/1967.

### KGW

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.

### PoS

Somebody like John Negele might have taken your course?

### KGW

Could be, but I do not remember.

### PoS

Michal Johnson or somebody like that? Does that ring a bell or not?

### KGW

I certainly won't remember. The one case I remember was when I had a student named Kenneth Wilson.

### PoS

How did he do? <laughter> So when did you put a field theory on a lattice in the computer?

### KGW

That comes later. Around 1976 or 1977.

### PoS

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?

### KGW

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.

### PoS

Which means what?

### KGW

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.

### PoS

Were you actively keeping an eye out for what machines were available?

### KGW

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.

### PoS

So, should we go back now to 1966 or so?

### KGW

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.

### PoS

Because you ate something?

### KGW

Yes, I ate something I shouldn't have. Took a long time to get that straightened out.

### PoS

So in 1963, you're still looking at fixed source meson theory when you go to Cornell?

### KGW

One strand is that I continue to work on fixed source, and one strand I was doing the high momentum analysis on diagrams.

### PoS

And who were the people you talked to at Cornell? Tom Kinoshita? Who were the people closest to you? Don Yennie is there....

### KGW

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...

### PoS

And Kinoshita at that stage was not yet involved in computing? He does infrareds and infrared divergences and things like that.

### KGW

I don't remember at what point Kinoshita starts his heavy computation work.

### PoS

Are you teaching field theory at this point in the early 1960s?

### KGW

Certainly at some point, I teach the field theory course, but what I remember more fondly is teaching quantum mechanics. Quantum mechanics made sense.

### PoS

Would you have taught renormalization group if you had taught field theory?

### KGW

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 topics courses.

### PoS

You mentioned that in 1966, you go through [Lars] Onsager's solution as given by Lieb, Mattis, and Schultz.

### PoS

Could we ask what brings you to listen to Widom and gets you to know some of the people in chemistry?

### KGW

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 important event.

### PoS

And when you first meet Michael Fisher...?

### KGW

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.

### PoS

And do you hear of the 1965 conference at the National Bureau of Standards on phase transitions which brings together lots of people?

### KGW

I may well have heard about it, but it...

### PoS

Didn't stand out. And just in that sense also, how much of the work on liquid helium are you aware of?

### KGW

I paid no attention at all to that.

None. OK.

### PoS

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.

### KGW

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.

### PoS

Why is it more apparent to you that you need to know about critical phenomena?

### KGW

Because I'm seeing there's the same kind of problems of multiple length scales.

### PoS

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 verify it]

### KGW

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.

### PoS

But was it Widom's talk that pointed you in that direction that made you aware?

### KGW

It starts with Widom's talk. I mean that's what sticks in my mind as sort of the first catalytic event....

### PoS

Can we ask what is it about Widom's talk that was so striking to you?

### KGW

What is striking about it is that he makes all these conclusions but based on assumptions which he cannot justify.

### PoS

And the conclusions being homogeneous equations of state?

Yes.

### PoS

So it's the homogeneity that strikes you.

### KGW

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.

### PoS

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.

### PoS

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?

### KGW

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.

### PoS

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?

### KGW

As far I remember, the period of time where I'm working with Johnson and Baker is the period before this.

### PoS

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?

### KGW

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.

### PoS

But scale, I mean the Thirring model would...

### KGW

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.

### PoS

Were you involved in an active discussion with Widom and Fisher at this point or was it...

### KGW

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.

### PoS

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 striking?

### KGW

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.

### PoS

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....

### KGW

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.

### PoS

And therefore, Kadanoff is a link to a specific model which allows you to go back to field theory.

### KGW

Right. Because that's the thing that connects to field theory, it is that kind of model.

### PoS

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?

### KGW

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 it.

### PoS

And calculate implies a specific model.

### KGW

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.

### PoS

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?

### KGW

My impression of Widom's work is that he was less focused on actually digging into the statistical mechanics than Michael Fisher was.

### PoS

And how much field theory did Michael Fisher know during these initial discussions?

### KGW

He knew the statistical mechanics cold, but I don't know that he had much field theory background.

### PoS

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?

### KGW

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.

### PoS

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.

### KGW

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.

### PoS

And the justification for just keeping the finite number of new terms that are generated?. Does it become clear that that's the problem to...

### KGW

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.

### PoS

Was it clear already to you at that stage how to compute critical exponents in a two-dimensional case?

### KGW

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.

### PoS

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?

### KGW

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.

### PoS

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?

### KGW

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.

### PoS

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?

### KGW

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.

### PoS

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?

### KGW

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.

### PoS

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?

### KGW

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 else matters.

### PoS

So you're not looking at experimental reviews or things like that on critical phenomena.

### KGW

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.

### PoS

At some point, Carlo DiCastro comes to give a talk.

### PoS

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?

### KGW

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.

### PoS

Did Michael understand it? I mean given that he lacked the background in field theory....

### PoS

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 of homogeneity?

### KGW

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.

### PoS

Except that they were trying to justify getting Widom, starting from a microscopic model. That was certainly their aim.

### KGW

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.

### PoS

And then you need dimensional regularization to give you the insight to set up the epsilon expansion, in your counterfactual history.

### KGW

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 expansion calculations.

### PoS

Do you remember at what point you read the Lev Landau and Vitaly Lazarevich Ginzburg paper that specified the criterion....

### KGW

I'm not sure I ever actually read that paper.

### PoS

Right. I sort of got the feeling that this was a cursory reference from your Nobel lecture that you had subsequently been made aware of.

### PoS

But you were aware of A. A. Migdal and Alexander Polyakov in 1969?

### KGW

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.

### PoS

And this was before 1970.

### KGW

It was right around 1970.

PoS
But by the time that you see the Varenna lectures, you were certainly aware that it was 1970.....

Right.

### PoS

Were there any events in this development before we get to the 1970 Widom seminar that we should talk about?

### KGW

That's a good question, but I don't remember the timing of these things.

### PoS

Dimensional regularization... when did you learn about that?

### KGW

I certainly learned about dimensional regularization essentially as soon as it happened, but I don't remember when it was.

### PoS

So, there's the talk by DiCastro and then in 1970, Widom and Fisher have this seminar and they invite you to talk about....

### KGW

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.

### PoS

So he specifically asked you to talk about DiCastro and Jona-Lasinio's paper?

### KGW

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.

### PoS

These lectures were in Rockefeller rather than at Newman Lab?

### KGW

I assume they were over in Baker actually. They were not in Newman.

### PoS

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.

Widom might.

### PoS

But you mentioned someone else on the phone when we were talking…

### KGW

Did I mention Michael Fisher or ....

### PoS

Is David Nelson already active as a graduate student at that stage?

### KGW

Oh, did I mention Joe Serene or someone like that?

Could be.

### KGW

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 know.
I might have suggested John Wilkins.

### PoS

Yes, that's right. John Wilkins.

### KGW

If Ben Widom himself doesn't know, then Wilkins is the person to ask.

### PoS

He's at Cornell?

### KGW

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.

### PoS

Do you have any notes besides the paper with Arthur Wightman's notes as referee of that period?

### KGW

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 Ising model.

### PoS

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.

### KGW

There would be very little in the lecture notes that didn't get into those two papers.

### PoS

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 get.

### KGW

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.

### PoS

Did people understand the lecture? How did Widom and the other folks react?

### KGW

It's not clear that they understand the lectures. Have you tried to interview Widom?

Not yet.

### KGW

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.