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This article is the last in a four-part series on quantum electrodynamics. You can read the previous article here. You might also want to read our interview with Freeman Dyson.

How do you work out whether a beam of light will reflect off a mirror in exactly the right way to, for example, make a camera work? You might draw a picture to understand what is happening, write down some equations, do some calculations, and out pops the result. This is how physics is usually done, and has been since the time of Newton. Equations, calculation, result.

"Feynman skipped all that," says Freeman Dyson, a physicist at the Institute of Advanced Study in Princeton. "He just wrote down the pictures and then wrote down the answers. There were no equations."

Richard Feynman

Richard Feynman (1918-1988).

Dyson is talking about Richard Feynman, recalling his time at Cornell University in the late 1940s. At the time, quantum electrodynamics (QED), a newly developed theory to describe the interaction of light and matter, was in deep trouble. Calculating even the simplest interactions between electrons and photons was so complicated, it frightened even the most accomplished physicists. There were other problems too, throwing the whole edifice of QED into doubt (see the previous article). Feynman's stick-like figures, now known as Feynman diagrams, came to the rescue, and they turned out to become a ubiquitous tool in physics.

"[Feynman was] a great guy," recalls Dyson. "The joy of Feynman was that he was totally outspoken. He always said exactly what he thought about you or about anything else. If I wanted to go and talk with Feynman, I would walk into the room and he would say 'Get out, don't you see I'm busy.' Another time I'd come in and he would be very friendly, so I'd know that he really welcomed me. I enjoyed him very much because he was a real performer. He just loved to perform and he had to have an audience."

Many tales have been told of Feynman's famous irreverence and perhaps it was the same irreverence that enabled him to skip the formalities and think in pictures. As an example, think of two electrons scattering off each other. Naively you would think of them as tiny billiard balls: if you know the speed and direction of travel of both of them, you can work out if they will meet and where they will end up at any given time after the collision. But in quantum physics things are not that simple. Electrons behave both like particles and like waves: it is impossible to determine both their location and their momentum to the same degree of accuracy, they don't travel along straight lines and you can't even tell two electrons apart. All you can do is work out the probability that two electrons will scatter in a particular way (see our article on Schrödinger's equation for an introduction to quantum mechanics).

Electron-electron scattering

Figure 1: This image is a Feynman diagram of electron-electron scattering.

What is more, electrons scatter, not by colliding, but by exchanging virtual photons. For example, one electron can emit a virtual photon and the other one can absorb it. Absorption and emission alter each particle’s speed and direction: that’s the scattering. Calculating the probability that two electrons start out at two points, $x_1$ and $x_2$, in spacetime and after scattering end up at points $x_3$ and $x_4$ involves the probability that the first electron travels to the point $x_5$ emits the photon there, and then travels to $x_3$, that the other electron travels to the point $x_6$ where it absorbs it and then travels to $x_4$, and the probability that the virtual photon makes the journey from $x_5$ to $x_6$. What is more, the points $x_5$ and $x_6$ could be anywhere in spacetime, since we can't be certain of the electrons' trajectories.

(This example is borrowed from Feynman's book QED - the strange theory of light and matter.)

Taking all this into account gives an unwieldy expression for the probability of our scattering event, involving double integrals and many different terms to take account of the different probabilities. All to capture this supposedly simple scenario.

Electron-electron scattering

This image shows all the ways in which two electrons can scatter by exchanging two photons. Image adapted from a figure in Richard Feynman's article Space-time approach to quantum electrodynamics. Copyright (2013) by The American Physical Society.

And this isn't all. Two electrons can scatter by trading any number of virtual photons in complicated ways. For example, an emitted photon could turn into a pair made up of a virtual electron and a virtual anti-electron (usually called positron), which then annihilate each other to form a new photon, which gets absorbed by the second electron. All the possible interactions need to be taken into account and each comes with a long and complicated mathematical expression that even the most diligent accountant would find hard to keep track off. The scope for mistakes, omissions and double counting is huge.

In Feynman's mind, however, the double integral above turned into a simple diagram, shown in figure 1. At first sight this looks like a picture of a real physical process, but it is not. The horizontal axis represents space (we're assuming the particles move in one dimension), the vertical one represents time. Thus, a particle standing still for a few seconds (which in actual fact it never does, but let's assume so for a moment), would be represented by a vertical line which represents the passage of time. What is more, the straight and wriggly lines don't represent real trajectories of particles, but only probabilities that particles are first at one point in space and time and then another. (If you would like to find out more, Quantum diaries has a great introduction to Feynman diagrams. For a fascinating account of their dispersion and use in physics see David Kaiser's book Drawing theories apart)

Yet, from such a picture, physicists could easily read off the maths. "Every time you see a straight line in a Feynman diagram it has exactly and uniquely this expression in your corresponding equation," says David Kaiser, a historian of physics at MIT. "When you see a wiggly line, it has exactly this expression. It becomes shockingly simple once you learn a few quick rules." All the other possible interactions, involving more than one virtual particle, come with similar diagrams. The number of them grows large very rapidly as you consider more and more virtual particles, but it's a lot easier to keep track of than when you're only looking at the maths.

Feynman's brilliant diagrams have since become an indispensable part of physicists' toolkits, so much so that people sometimes mistake them for literal depictions of reality, rather than just drawings on paper that serve as shorthands for equations. "I love this mnemonic of, 'Let me take a magnifying glass and look at nature and what I will see is Feynman diagrams,'"says David Kaiser. "That's a wonderful slippage. We are drawing pictures as if it's the same as a baseball tossed through space and time. Feynman was remarkably untroubled by that confusion. He would say, in effect, that this is how it makes sense to him to think about it. And he would often speak in very anthropomorphic terms, 'I'm this electron. I am sitting here and I'm getting bashed by this photon and I'm being knocked off course.'"

Feynman's intuitive approach didn't obscure his physical insight, but it did hamper communication. When Feynman first presented his diagrams at a conference, the audience wasn't impressed. He wasn't able to explain exactly how the straight and wriggly lines should be translated into equations and show that his pictures didn't obscure more complicated situations. "Feynman in some sense had the right rules, but it wasn't clear how general they were," says Kaiser. "Were they really coming from the heart of quantum field theory or were they just representing special cases?"

Freeman Dyson

Freeman Dyson in 2005. Image: Jacob Appelbaum.

It was Freeman Dyson who eventually tied up the rules. "Dyson showed that if we take quantum field theory as the first effort to unite quantum mechanics and special relativity, then that is uniquely fixing what those translation rules should be," says Kaiser. "So it was Dyson who spelled them out systematically and then showed that they would indeed hold. Then applying the rules became remarkably straightforward."

Feynman's diagrams helped with one problem that plagued QED: keeping track of all the complicated ways that any number of these virtual particles could be involved, and taking account of these in the calculations. But there was also another problem. Even a single virtual particle could come with any amount of energy. Taking this unlimited energy into account meant that calculations involving just one virtual particle returned infinity as the answer. Schwinger and, independently, Tomonaga had provided a solution to this puzzle: rather than trying to get at the bare particle, the particle on its own, one should only ever consider it together with its virtual entourage and use effective quantities in calculations (see the previous article). Feynman had himself developed a similar approach.

The problem was, however, that all three had only been able to tame the infinities in calculations that involved one or two virtual particles. It wasn't clear the methods would work for greater numbers. Armed with a mathematicians' understanding of the techniques of quantum physics, Dyson came to the rescue, showing that the approach of Feynman, Schwinger and Tomonaga would work for any number of virtual particles. "I had only the tools of quantum field theory, which [others] didn't have," says Dyson. "I was able to put them all together and demonstrate it was all quite simple. It was really an amazing piece of luck. From being a humble student, I suddenly became a big shot."

Feynman's diagrams and Dyson's policing of the theory spurred a mood of optimism: calculations became easier, infinities could be tamed, and the puzzle of QED seemed cracked. The next challenge was to apply the same ideas to the other forces of nature. But this, it turned out, was a whole different story. We will explore it in another series of articles coming up on Plus soon.

Further reading

David Kaiser's book Drawing theories apart explores the development of Feynman diagrams and contains an excellent introduction to the topics discussed in this article.

You can buy the book and help Plus at the same time by clicking on the link on the left to purchase from, and the link to the right to purchase from Plus will earn a small commission from your purchase.

About this article

Marianne Freiberger and Rachel Thomas are Editors of Plus. They interviewed Freeman Dyson and David Kaiser in February 2013. They are hugely grateful to Jeremy Butterfield, a philosopher of physics at the University of Cambridge, and Nazim Bouatta, a Postdoctoral Fellow in Foundations of Physics at the University of Cambridge, for their many patient explanations and help in writing this article.

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