Enrico Fermi’s network
Enrico Fermi’s collaborators and students
This image shows Enrico Fermi’s network of collaborators. This network is called a graph and each node of the graph (a dot) represents a scientist. Two scientists are connected if they are co-authors of the same scientific work. The data concerning the collaborations were obtained from the Scopus database (where scientific articles are catalogued) in which the co-authors (first neighbours) and the co-authors of the co-authors (second neighbours) of Fermi are classified. This data has been integrated with the complete list of Fermi’s papers obtained by Francesco Guerra and Nadia Robotti.
The weight of the connections (represented by the thickness of the connecting lines) between two nodes is initially set equal to the number of works co-authored by the respective scientists. Subsequently, to highlight the nearest neighbours in an image, the weight of the connections of the second neighbours and of the connections between first neighbours is scaled by a factor proportional to Fermi’s collaborations with his first neighbours. The colour of the nodes represents the various communities in the network, identified using a modularity maximisation algorithm, which is a way of quantifying the division of the network into communities.
One can see, on the left in purple around Amaldi’s name, the group of boys from Via Panisperna (Segrè, Pontecorvo, Rasetti, D’Agostino, plus Enrico Persico) and then the Italian school that originated from Amaldi after the war (Touschek, Bernardini, Pizzella, Cabibbo). Also, in green at the top left, is the group of Italian scientists who worked in the United States and Italy (Conversi, Pancini, Piccioni, Rossi). For the rest, there are the collaborators from the United States: from Anderson, who worked on the Chicago Atomic Pile and then on the Manhattan Project, to Chandrasekhar, the great Indian astrophysicist, Bethe, Metropolis, Feynman, Dyson, Steinberger, Teller and Szilard …. A very long list of the greatest American and European physicists of the first half of the 20th century, many of them Nobel Prize winners. Five of Fermi’s direct students were awarded the Nobel Prize: Chamberlain, Friedman, Lee, Segrè and Steinberger. Two others, Cronin and Yang also won the Nobel Prize and, although they were not officially Fermi’s students, both credit him with inspiring and guiding them. Many other students and collaborators went on to important and influential careers in the same field.
The memory of Freeman Dyson
Freeman Dyson was a physicist and mathematician who made pioneering contributions in a variety of fields during his distinguished career spanning more than 70 years. Born on 15 December 1923 in England to a musician but science-loving father and a law-educated mother, he grew up immersed in the Dyson household’s extensive book collection. His older sister remembered him as a child surrounded by encyclopaedias and sheets of paper, which he used to do all sorts of calculations. He loved playing with numbers, for example, at the age of four he tried to calculate the number of atoms in the sun. Reconfirming his interest, he later said: “Science was exciting because it was full of numbers that I could calculate”.
So I had great enthusiasm and decided to go to Chicago to show these results to Fermi and tell him how well we were doing. We wanted to have Fermi’s blessing for our efforts, because he had really been the first to move in this area, and it would be a good opportunity for me to meet Fermi. Anyway, I arranged with Hans Bethe to go to Chicago and tell him what we were doing. So I arrived in Chicago and knocked on Fermi’s door, and he was very kind. I went in and he said, “Yes?” and I showed him the graphs on which the results of our experiment were plotted, our theoretical numbers and Fermi’s experimental numbers, and the agreement was on the whole quite good. And Fermi almost didn’t look at these graphs, he just put them on the desk, took a brief look at them and said, “I’m not very impressed with what you’ve done.” And he said, “When you do a theoretical calculation, you know, there are two ways of doing it. Either you have to have a clear physical model in mind, or you have to have a rigorous mathematical basis. You have neither.” And so it was, with a couple of sentences he dismissed the whole subject. So I asked him: “Well, what do you think about numerical agreement”, and he said: “How many parameters did you use for the fit? How many free parameters are there in your method?” So I counted down. It turned out to be four. And he said: “You know, Johnny von Neumann used to say: ‘With four parameters I can fit an elephant, and with five I can make it move its trunk. I don’t find numerical agreement very impressive either.”
So I said: “Thank you very much for your help” and said goodbye. There was nothing more to say. The whole discussion lasted maybe 10 or 15 minutes. And I went back to Cornell to tell the team the bad news. That was another watershed in my life, and I think it was deeply helpful what Fermi did. He had an incredible insight. He could immediately recognise what was good and what was bad.
I mean, we could have worked on these calculations for five years if Fermi hadn’t given us an alt, and as it was, Fermi was absolutely right because in the end, of course, it turned out that the theory on which we based the whole calculation was an illusion. In fact there is no such thing as a pseudo-scalar theory of pions. In fact 10 years or so later quarks were invented and the whole theory of strong interactions was totally transformed into a theory of quarks, and it is only when you represent the pion as a system composed of two quarks that you can start to have a real physical theory.
So our whole physical basis was wrong, and so it was perfectly true that any experimental agreement we found was illusory, but it took Fermi to understand that, and he could see it without knowing about quarks – of course nobody had dreamt about quarks at that time – but he felt deep down that this theory was not good. And he was right. So he saved us perhaps five years of blind work and for that I am extremely grateful to him. But it was a difficult situation for us, especially because we had some graduate students involved in this project.
They were depending on this for their doctoral thesis, so it was difficult. I mean, I just had to say to the team, “Look, I’m sorry, but this work isn’t going anywhere, so all we can do is write up what we’ve done and publish it, but it won’t go any further, and you’d better find some other research.” So it was not a very pleasant experience, neither for the graduate students nor for me. But in the end, of course, it was for the good of all of us. That’s just the kind of genius that Fermi had, and I think that showed me very clearly that I was not a particle physicist, that I didn’t have that kind of instinct. I mean, that my talents are in mathematics and not fundamentally in physics.
So, when there is a theory that is well based in physics, as in the case of quantum electrodynamics, then I can succeed wonderfully well in using it, but I am not able to invent a new theory, and what was needed for the strong interactions was an invention, and clearly that is not my forte. And so, from that point on, I did not seriously try to solve the problem of strong interactions. From that point on I worked in the theoretical field of mathematical sciences and. I became interested in mathematical theories and was involved in Whietman’s project of trying to establish quantum theory axiomatically by deducing physical consequences from the axioms. This was something that I loved to do and which, in Fermi’s terminology, is based on solid mathematical foundations, whether or not it is correct from a physical point of view. That was my choice, and so I went with the mathematicians and not with the physicists.
The memory of Murray Gell-Mann
Murray Gell-Mann was a theoretical physicist who made immense contributions to elementary particle physics and is particularly known for his work on quarks, particles that are now considered the ultimate constituents of matter. The choice of the term quark is somewhat bizarre. Gell-Man had a similar-sounding word in mind that could be written as ‘KWORK’. He later noticed that James Joyce’s novel Finnegans Wake (The Wake for Finnegan) contained the phrase “Three quarks for Muster Mark!
He kept a small notebook with useful formulas, which were all exact, perfect. Every factor two correct, every sign correct, every dimension correct, everything was right. He never put anything in his little notebook, which he carried with him, without making sure that it was perfectly correct. And that allowed him to tackle a lot of practical problems on demand because he had worked them out beforehand and some of the critical formulae were in his little book. Most of the things he had thought of before in one way or another, most of the questions in physics were questions… they were… they were new forms of questions that he had already been answered. And so, if he was asked something, he would start at the top left of the blackboard and write line after line and then later, maybe at the bottom right or before that, he would get the answer and look it up. And the answer was always correct; he never made a mistake. One could not imagine a greater contrast with Viki Weisskopf. All Viki wrote was modulo a power of two, a pi power, an i power, the numerator and denominator could well be interchanged. I remember when in a nuclear physics lecture he tried to derive the resonance formula…the Breit–Wigner formula, which in atomic theory was called the Weisskopf–Wigner formula. Breit and Wigner generalised it to nuclear theory, but it was exactly the same formula as the Weisskopf–Wigner formula in atomic theory. Well, he got it all wrong, completely wrong. The… the numerator was in the denominator, the denominator was in the numerator, the sign was wrong, the values, there were factor twos and pi factors fluctuating. So he said ‘Well’, he said,’ this time I didn’t prepare, I have to admit I didn’t prepare this lesson, so… but next time I’ll come prepared and deduce it correctly”. Well, he came the next time and tried to derive it and failed again. Now, of course, every student in that class has learned how to derive the formula. So that was much more effective than Julian Schwinger’s linear presentation, which really didn’t leave you learning anything because he glossed over all the difficulties and just presented a very fluid picture of what was going on. He didn’t leave you with much idea of how to do it yourself, whereas Viki’s mistakes were very educational. Now Fermi derived everything exactly but he was not a formalist at all. He simply calculated things, usually using arithmetic, and got the answer by some trick that was very, very simple. However, it was a trick, and it was based on the fact that he had already solved the problem several times in different ways and had put the answers in his little notebook. So that it was a bit difficult to learn from him also because you would have to invent that trick, which was not necessarily so easy, if you had to do the problem yourself. He didn’t set up a problem in a general way so that you would immediately realise that you had solved it in the same way. He had his own little trick, which you would have to invent yourself to solve the problem yourself. Anyway, everything was fine. But if you asked him a question for which he did not know the answer, then things became much more difficult. He was not very happy about it and the discussion became difficult, and very interesting, but difficult. And of course I asked him quite often things that he didn’t know, questions that he didn’t really know the answer to, and as a result we had some very interesting discussions.