Monster Minds

While I was still a graduate student at Princeton, I worked as a research assistant under John Wheeler. He gave me a problem to work on, and it got hard, and I wasn’t getting anywhere. So I went back to an idea that I had had earlier, at MIT. The idea was that electrons don’t act on themselves, they only act on other electrons.

There was this problem: When you shake an electron, it radiates energy and so there’s a loss. That means there must he a force on it. And there must he a different force when it’s charged than when it’s not charged. (If the force were exactly the same when it was charged and not charged, in one case it would lose energy, and in the other it wouldn’t. You can’t have two different answers to the same problem.)

The standard theory was that it was the electron acting on itself that made that force (called the force of radiation reaction), and I had only electrons acting on other electrons. So I was in some difficulty, I realized, by that time. (When I was at MIT, I got the idea without noticing the problem, but by the time I got to Princeton, I knew that problem.)

What I thought was: I’ll shake this electron. It will make some nearby electron shake, and the effect back from the nearby electron would be the origin of the force of radiation reaction. So I did some calculations and took them to Wheeler.

Wheeler, right away said, “Well, that isn’t right because it varies inversely as the square of the distance of the other electrons, whereas it should not depend on any of these variables at all. It’ll also depend inversely upon the mass of the other electron; it’ll be proportional to the charge on the other electron.”

What bothered me was, I thought he must have done the calculation. I only realized later that a man like Wheeler could immediately see all that stuff when you give him the problem. I had to calculate, but he could see.

Then he said, “And it’ll be delayed—the wave returns late—so all you’ve described is reflected light.”

“Oh! Of course,” I said.

“But wait,” he said. “Let’s suppose it returns by advanced waves—reactions backward in time—so it comes back at the right time. We saw the effect varied inversely as the square of the distance, but suppose there are a lot of electrons, all over space: the number is proportional to the square of the distance. So maybe we can make it all compensate.”

We found out we could do that. It came out very nicely, and fit very well. It was a classical theory that could be right, even though it differed from Maxwell’s standard, or Lorentz’s standard theory. It didn’t have any trouble with the infinity of self-action, and it was ingenious. It had actions and delays, forwards and backwards in time—we called it “half-advanced and half-retarded potentials.”

Wheeler and I thought the next problem was to turn to the quantum theory of electrodynamics, which had difficulties (I thought) with the self-action of the electron. We figured if we could get rid of the difficulty first in classical physics, and then make a quantum theory out of that, we could straighten out the quantum theory as well.

Now that we had got the classical theory right, Wheeler said, “Feynman, you’re a young fella—you should give a seminar on this. You need experience in giving talks. Meanwhile, I’ll work out the quantum theory part and give a seminar on that later.”

So it was to be my first technical talk, and Wheeler made arrangements with Eugene Wigner to put it on the regular seminar schedule.

A day or two before the talk I saw Wigner in the hail. “Feynman,” he said, “I think that work you’re doing with Wheeler is very interesting, so I’ve invited Russell to the seminar.” Henry Norris Russell, the famous, great astronomer of the day, was coming to the lecture!

Wigner went on. “I think Professor von Neumann would also he interested.” Johnny von Neumann was the greatest mathematician around. “And Professor Pauli is visiting from Switzerland, it so happens, so I’ve invited Professor Pauli to come”—Pauli was a very famous physicist—and by this time, I’m turning yellow. Finally, Wigner said, “Professor Einstein only rarely comes to our weekly seminars, but your work is so interesting that I’ve invited him specially, so he’s coming, too.”

By this time I must have turned green, because Wigner said, “No, no! Don’t worry! I’ll just warn you, though: If Professor Russell falls asleep—and he will undoubtedly fall asleep—it doesn’t mean that the seminar is bad; he falls asleep in all the seminars. On the other hand, if Professor Pauli is nodding all the time, and seems to be in agreement as the seminar goes along, pay no attention. Professor Pauli has palsy.”

I went back to Wheeler and named all the big, famous people who were coming to the talk he got me to give, and told him I was uneasy about it.

“It’s all right,” he said. “Don’t worry. I’ll answer all the questions.”

So I prepared the talk, and when the day came, I went in and did something that young men who have had no experience in giving talks often do—I put too many equations up on the blackboard. You see, a young fella doesn’t know how to say, “Of course, that varies inversely, and this goes this way … because everybody listening already knows; they can see it. But he doesn’t know. He can only make it come out by actually doing the algebra—and therefore the reams of equations.

As I was writing these equations all over the blackboard ahead of time, Einstein came in and said pleasantly, “Hello, I’m coming to your seminar. But first, where is the tea?”

I told him, and continued writing the equations.

Then the time came to give the talk, and here are these monster minds in front of me, waiting! My first technical talk—and I have this audience! I mean they would put me through the wringer! I remember very clearly seeing my hands shaking as they were pulling out my notes from a brown envelope.

But then a miracle occurred, as it has occurred again and again in my life, and it’s very lucky for me: the moment I start to think about the physics, and have to concentrate on what I’m explaining, nothing else occupies my mind—I’m completely immune to being nervous. So after I started to go, I just didn’t know who was in the room. I was only explaining this idea, that’s all.

But then the end of the seminar came, and it was time for questions. First off, Pauli, who was sitting next to Einstein, gets up and says, “I do not sink dis teory can be right, because of dis, and dis, and dis,” and he turns to Einstein and says, “Don’t you agree, Professor Einstein?”

Einstein says, “Nooooooooooooo,” a nice, German sounding “No, “—very polite. “I find only that it would be very difficult to make a corresponding theory for gravitational interaction.” He meant for the general theory of relativity, which was his baby. He continued: “Since we have at this time not a great deal of experimental evidence, I am not absolutely sure of the correct gravitational theory.” Einstein appreciated that things might he different from what his theory stated; he was very tolerant of other ideas.

I wish I had remembered what Pauli said, because I discovered years later that the theory was not satisfactory when it came to making the quantum theory. It’s possible that that great man noticed the difficulty immediately and explained it to me in the question, but I was so relieved at not having to answer the questions that I didn’t really listen to them carefully. I do remember walking up the steps of Palmer Library with Pauli, who said to me, “What is Wheeler going to say about the quantum theory when he gives his talk?”

I said, “I don’t know. He hasn’t told me. He’s working it out himself.”

“Oh?” he said. “The man works and doesn’t tell his assistant what he’s doing ‘on the quantum theory?” He came closer to me and said in a low, secretive voice, “Wheeler will never give that seminar.”

And it’s true. Wheeler didn’t give the seminar. He thought it would he easy to work out the quantum part; he thought he had it, almost. But he didn’t. And by the time the seminar came around, he realized he didn’t know how to do it, and therefore didn’t have anything to say.

I never solved it, either—a quantum theory of half-advanced, half-retarded potentials—and I worked on it for years.

Mixing Paints

The reason why I say I’m “uncultured” or “anti-intellectual” probably goes all the way back to the time when I was in high school. I was always worried about being a sissy; I didn’t want to be too delicate. To me, no real man ever paid any attention to poetry and such things. How poetry ever got written—that never struck me! So I developed a negative attitude toward the guy who studies French literature, or studies too much music or poetry—all those “fancy” things. I admired better the steel-worker, the welder, or the machine shop man. I always thought the guy who worked in the machine shop and could make things, now he was a real guy! That was my attitude. To be a practical man was, to me, always somehow a positive virtue, and to be “cultured” or “intellectual” was not. The first was right, of course, but the second was crazy.

I still had this feeling when I was doing my graduate study at Princeton, as you’ll see. I used to eat often in a nice little restaurant called Papa’s Place. One day while I was eating there, a painter in his painting clothes came down from an upstairs room he’d been painting, and sat near me. Somehow we struck up a conversation and he started talking about how you’ve got to learn a lot to be in the painting business. “For example,” he said, “in this restaurant, what colors would you use to paint the walls, if you had the job to do?”

I said I didn’t know, and he said, “You have a dark band up to such-and-such a height, because, you see, people who sit at the tables rub their elbows against the walls, so you don’t want a nice, white wall there. It gets dirty too easily. But above that, you do want it white to give a feeling of cleanliness to the restaurant.”

The guy seemed to know what he was doing, and I was sitting there, hanging on his words, when he said, “And you also have to know about colors—how to get different colors when you mix the paint. For example, what colors would you mix to get yellow?”

I didn’t know how to get yellow by mixing paints. If it’s light, you mix green and red, but I knew he was talking paints. So I said, “I don’t know how you get yellow without using yellow.”

“Well,” he said, “if you mix red and white, you’ll get yellow.”

“Are you sure you don’t mean pink?”

“No,” he said, “you’ll get yellow”—and I believed that he got yellow, because he was a professional painter, and I always admired guys like that. But I still wondered how he did it.

I got an idea. “It must be some kind of chemical change. Were you using some special kind of pigments that make a chemical change?”

“No,” he said, “any old pigments will work. You go down to the five-and-ten and get some paint—just a regular can of red paint and a regular can of white paint—and I’ll mix ‘em, and I’ll show how you get yellow.”

At this juncture I was thinking, “Something is crazy. I know enough about paints to know you won’t get yellow, but he must know that you do get yellow, and therefore something interesting happens. I’ve got to see what it is!”

So I said, “OK, I’ll get the paints.”

The painter went back upstairs to finish his painting job, and the restaurant owner came over and said to me, “What’s the idea of arguing with that man? The man is a painter; he’s been a painter all his life, and he says he gets yellow. So why argue with him?”

I felt embarrassed. I didn’t know what to say. Finally I said, “All my life, I’ve been studying light. And I think that with red and white you can’t get yellow—you can only get pink.”

So I went to the five-and-ten and got the paint, and brought it back to the restaurant. The painter came down from upstairs, and the restaurant owner was there too. I put the cans of paint on an old chair, and the painter began to mix the paint. He put a little more red, he put a little more white—it still looked pink to me—and he mixed some more. Then he mumbled something like, “I used to have a little tube of yellow here to sharpen it up a bit—then this’ll be yellow.”

“Oh!” I said. “Of course! You add yellow, and you can get yellow, but you couldn’t do it without the yellow.”

The painter went back upstairs to paint.

The restaurant owner said, “That guy has his nerve, arguing with a guy who’s studied light all his life!”

But that shows you how much I trusted these “real guys.” The painter had told me so much stuff that was reasonable that I was ready to give a certain chance that there was an odd phenomenon I didn’t know. I was expecting pink, but my set of thoughts were, “The only way to get yellow will be something new and interesting, and I’ve got to see this.”

I’ve very often made mistakes in my physics by thinking the theory isn’t as good as it really is, thinking that there are lots of complications that are going to spoil it—an attitude that anything can happen, in spite of what you’re pretty sure should happen.

A Different Box of Tools

At the Princeton graduate school, the physics department and the math department shared a common lounge, and every day at four o’clock we would have tea. It was a way of relaxing in the afternoon, in addition to imitating an English college. People would sit around playing Go, or discussing theorems. In those days topology was the big thing.

I still remember a guy sitting on the couch, thinking very hard, and another guy standing in front of him, saying, “And therefore such-and-such is true.”

“Why is that?” the guy on the couch asks.

“It’s trivial! It’s trivial!” the standing guy says, and he rapidly reels off a series of logical steps: “First you assume thus-and-so, then we have Kerchoff’s this-and-that; then there’s Waffenstoffer’s Theorem, and we substitute this and construct that. Now you put the vector which goes around here and then thus-and-so …” The guy on the couch is struggling to understand all this stuff, which goes on at high speed for about fifteen minutes!

Finally the standing guy comes out the other end, and the guy on the couch says, “Yeah, yeah. It’s trivial.”

We physicists were laughing, trying to figure them out. We decided that “trivial” means “proved.” So we joked with the mathematicians: “We have a new theorem—that mathematicians can prove only trivial theorems, because every theorem that’s proved is trivial.”

The mathematicians didn’t like that theorem, and I teased them about it. I said there are never any surprises—that the mathematicians only prove things that are obvious.

Topology was not at all obvious to the mathematicians. There were all kinds of weird possibilities that were “counterintuitive.” Then I got an idea. I challenged them: “I bet there isn’t a single theorem that you can tell me—what the assumptions are and what the theorem is in terms I can understand—where I can’t tell you right away whether it’s true or false.”

It often went like this: They would explain to me, “You’ve got an orange, OK? Now you cut the orange into a finite number of pieces, put it back together, and it’s as big as the sun. True or false?”

“No holes?”

“No holes.”

“Impossible! There ain’t no such a thing.”

“Ha! We got him! Everybody gather around! It’s So-and-so’s theorem of immeasurable measure!”

Just when they think they’ve got me, I remind them, “But you said an orange! You can’t cut the orange peel any thinner than the atoms.”

“But we have the condition of continuity: We can keep on cutting!”

“No, you said an orange, so I assumed that you meant a real orange.”

So I always won. If I guessed it right, great. If I guessed it wrong, there was always something I could find in their simplification that they left out.

Actually, there was a certain amount of genuine quality to my guesses. I had a scheme, which I still use today when somebody is explaining something that I’m trying to understand: I keep making up examples. For instance, the mathematicians would come in with a terrific theorem, and they’re all excited. As they’re telling me the conditions of the theorem, I construct something which fits all the conditions. You know, you have a set (one ball)—disjoint (two halls). Then the balls turn colors, grow hairs, or whatever, in my head as they put more conditions on. Finally they state the theorem, which is some dumb thing about the ball which isn’t true for my hairy green ball thing, so I say, “False!”

If it’s true, they get all excited, and I let them go on for a while. Then I point out my counterexample.

“Oh. We forgot to tell you that it’s Class 2 Hausdorff homomorphic.”

“Well, then,” I say, “It’s trivial! It’s trivial!” By that time I know which way it goes, even though I don’t know what Hausdorff homomorphic means.

I guessed right most of the time because although the mathematicians thought their topology theorems were counterintuitive, they weren’t really as difficult as they looked. You can get used to the funny properties of this ultra-fine cutting business and do a pretty good job of guessing how it will come out.

Although I gave the mathematicians a lot of trouble, they were always very kind to me. They were a happy hunch of boys who were developing things, and they were terrifically excited about it. They would discuss their “trivial” theorems, and always try to explain something to you if you asked a simple question.

Paul Olum and I shared a bathroom. We got to be good friends, and he tried to teach me mathematics. He got me up to homotopy groups, and at that point I gave up. But the things below that I understood fairly well.

One thing I never did learn was contour integration. I had learned to do integrals by various methods shown in a book that my high school physics teacher Mr. Bader had given me.

One day he told me to stay after class. “Feynman,” he said, “you talk too much and you make too much noise. I know why. You’re bored. So I’m going to give you a book. You go up there in the back, in the corner, and study this book, and when you know everything that’s in this book, you can talk again.”

So every physics class, I paid no attention to what was going on with Pascal’s Law, or whatever they were doing. I was up in the back with this book: Advanced Calculus, by Woods. Bader knew I had studied Calculus for the Practical Man a little bit, so he gave me the real works—it was for a junior or senior course in college. It had Fourier series, Bessel functions, determinants, elliptic functions—all kinds of wonderful stuff that I didn’t know anything about.

That book also showed how to differentiate parameters under the integral sign—it’s a certain operation. It turns out that’s not taught very much in the universities; they don’t emphasize it. But I caught on how to use that method, and I used that one damn tool again and again. So because I was self-taught using that book, I had peculiar methods of doing integrals.

The result was, when guys at MIT or Princeton had trouble doing a certain integral, it was because they couldn’t do it with the standard methods they had learned in school.