Feynman Diagrams: The science of doodling

By David Lindley

penguin diagram

All grown up. Today Feynman diagrams are also used to calculate properties of complicated interactions involving forces other than the electromagnetic. This "penguin" diagram shows quarks exchanging a virtual gluon (strong force carrier) and a virtual W particle (weak force carrier).

[photo credit:

Every popular explanation of particle physics is liberally illustrated with cartoon-like pictures of straight and wiggly lines representing electrons, photons, and quarks, interacting with one another. These so-called Feynman diagrams were introduced by Richard Feynman in the journal Physical Review in 1949, and they quickly became an essential tool for particle physicists. Early on, Feynman struggled to explain the meaning of the diagrams to his fellow physicists. But using them, he came up with easy answers to difficult problems in quantum mechanics and ultimately won a share of the Nobel Prize.

Through the 1940s, theoretical physicists were using new ideas from quantum mechanics to understand the electromagnetic properties of electrons, photons, and other elementary particles. But they ran into many difficulties, particularly in the form of calculations that produced infinite answers for such basic quantities as the mass of an electron. The charge on an electron generates an electromagnetic field, and forces between the charge and the field produce a kind of electromagnetic tension around the particle. That tension – somewhat like the tension in a squeezed or stretched rubber ball – corresponds to stored energy, and any form of energy, as Einstein showed with his formula E=mc2, has an equivalent mass. In this way, the "self-interaction" of the electron's field with its own charge adds to the particle's mass. All attempts to calculate this self-energy, however, yielded infinite results, and the appearance of such infinities stymied theorists' attempts to evaluate such basic processes as the scattering of an electron by a photon.

The answer to this difficulty was a trick known as "renormalization". Crudely put, the idea was to give the electron an infinite "bare" mass that would cancel the infinity arising from self-energy. The measured mass--the difference between these two infinities--could then be made finite. Performing this mathematical balancing act was challenging, but in 1948, Julian Schwinger of Harvard University calculated the self-energy as the sum of an infinite series of progressively smaller terms and showed that renormalization could tame the infinities arising from each term in the series. A Japanese theorist, Sin-Itiro Tomonaga, independently came up with a similar demonstration.

time space napkin self energy napkin

Simple Sketches. Left: A simple Feynman diagram representing an electron moving to the right, emitting a photon and then recoiling to the left. Right: A series of Feynman diagrams becoming more complex illustrate the possibilities of emitting and absorbing photons. The totality of these possibilities contributes to the self energy of the electron. The emitted photon can even briefly create an electron positron pair.

photo credit: Ken Cole

Richard Feynman, then a young professor at Cornell University, also solved the problem, but in an idiosyncratic way: he drew pictures, with smooth lines representing electrons and wavy lines representing photons. The vertical axis was time, with space on the horizontal. One contribution to the electron's self-energy, for example, was a photon emitted by the electron and then quickly reabsorbed by it--a vertical line with a wavy line that starts and ends on the electron line. This "virtual" photon can appear out of nowhere because its energy and lifetime are below the limit set by the uncertainty principle--meaning that the photon is strictly unobservable. Feynman drew additional diagrams representing more complicated self-interactions. For example, a virtual photon that springs temporarily from an electron can split into virtual electron-positron pair, which briefly appear before recombining back into a photon. And either the electron or the positron can sprout another virtual photon. And so on, and so on. For each diagram Feynman wrote down mathematical formulas giving its contribution to the self-energy, and he showed how to calculate the self-energy to any desired accuracy by adding in as many diagrams as were needed -- with the more complex diagrams yielding smaller contributions to the final answer.

At a conference in the Poconos Mountains, in Pennsylvania, in March 1948, Schwinger gave a marathon presentation that exhausted but impressed his audience. A frustrated Feynman, however, failed during his talk to convince the attending physicists of the soundness of his methods. Elder statesmen such as Paul Dirac and Niels Bohr concluded that the young American simply did not understand quantum mechanics.

But Feynman was able to solve complex problems far more quickly than those using the Schwinger formulation. Clarification came from Freeman Dyson, of the Institute for Advanced Study in Princeton, New Jersey, who showed in a classic paper that the Schwinger, Tomonaga, and Feynman methods were equivalent.

The first published Feynman diagram

Baby picture. The first published Feynman diagram shows the classic case of two electrons exchanging a virtual photon. Such diagrams are essential in particle physics, as they represent terms in the equations, as well as illustrating particle interactions schematically.

The first published example of what is now called a Feynman diagram appeared in Feynman's 1949 Physical Review article [see image labeled fig 1]. It depicted the simplest contribution to an electron-electron interaction, with a single virtual photon (wavy line) emitted by one electron and then absorbed by the other. In Feynman's imagination--and in the equations--this diagram also represented interactions in which the photon is emitted by one electron and travels back in time to be absorbed by the other, which is allowed within the Heisenberg time uncertainty. As before, more complex diagrams are easy to draw. The photon passing between the two particles can give rise to a virtual electron-positron pair, which can in turn produce more virtual photons. It turns out that each kind of diagram for a given process corresponds to some set of mathematical terms in the Schwinger-Tomonaga kind of calculation, and that adding in more diagrams is equivalent to calculating more of those terms.

Feynman's method was far more useful, however. Adding in more complex terms to the Schwinger-Tomonaga calculations became so mathematically difficult as to be practically impossible. But it was easy to draw more complex diagrams, then to use Feynman's methods to work out their contribution to the desired answer. So it isn't surprising that Feynman's diagrams were quickly adopted by the physics community, says Sylvan Schweber of Brandeis University in Waltham, Massachusetts. Feynman, Schwinger and Tomonaga shared the 1965 Nobel Prize in physics.


Twisting Feynman diagrams

A detailed history of Feynman diagrams