Physics and Feynman’s Diagrams In the hands of a postwar generation, a tool intended to lead quantum electrodynamics out of a decades-long morass helped transform physics David Kaiser eorge Gamow, the wisecracking theoretical tools, known as the Feynman diagram. Since Gphysicist who helped invent the Big Bang the middle of the 20th century, theoretical model of the universe, was fond of explaining physicists have increasingly turned to this tool what he liked best about his line of work: He to help them undertake critical calculations. could lie down on a couch and close his eyes, Feynman diagrams have revolutionized nearly and no one would be able to tell whether he was every aspect of theoretical physics. Of course, working or not. A fine gag, but a bad model for no tool ever applies itself, much less interprets thinking about the day-to-day work that theo- the results of its usage and draws scientific retical physicists do. For too long, physicists, conclusions. Once the Feynman diagram ap- historians and philosophers took Gamow’s joke peared in the physics toolkit, physicists had to quite seriously. Research in theory, we were learn how to use it to accomplish and inform told, concerns abstract thought wholly sepa- their calculations. My research has therefore rated from anything like labor, activity or skill. focused on the work that was required to make David Kaiser is the Leo Marx Theories, worldviews or paradigms seemed the Feynman diagrams the tool of choice. Associate Professor of the His- appropriate units of analysis, and the challenge The American theoretical physicist Rich- tory of Science in the Program lay in charting the birth and conceptual devel- ard Feynman first introduced his diagrams in Science, Technology, and opment of particular ideas. in the late 1940s as a bookkeeping device for Society at the Massachusetts In the accounts that resulted from such stud- simplifying lengthy calculations in one area of Institute of Technology and ies, the skilled manipulation of tools played physics—quantum electrodynamics, or QED, a lecturer in MIT’s Depart- little role. Ideas, embodied in texts, traveled the quantum-mechanical description of elec- ment of Physics. His physics easily from theorist to theorist, shorn of the ma- tromagnetic forces. Soon the diagrams gained research focuses on early-uni- verse cosmology, working at terial constraints that encumbered experimental adherents throughout the fields of nuclear and the interface of particle physics physicists (tied as they were to their electron particle physics. Not long thereafter, other the- and gravitation. His historical microscopes, accelerators or bubble chambers). orists adopted—and subtly adapted—Feyn- interests center on changes in The age-old trope of minds versus hands has man diagrams for solving many-body prob- American physics after World been at play in our account of progress in phys- lems in solid-state theory. By the end of the War II, looking especially at ics, which pictures a purely cognitive realm of 1960s, some physicists even used versions how the postwar generation of ideas separated from a manual realm of action. of Feynman’s line drawings for calculations graduate students was trained. This depiction of what theorists do, I am in gravitational physics. With the diagrams’ This article is based on his convinced, obscures a great deal more than it aid, entire new calculational vistas opened for forthcoming book, Drawing clarifies. Since at least the middle of the 20th physicists. Theorists learned to calculate things Theories Apart: The Disper- sion of Feynman Diagrams century, most theorists have not spent their that many had barely dreamed possible before in Postwar Physics (Uni- days (nor, indeed, their nights) in some philos- World War II. It might be said that physics can versity of Chicago Press). He opher’s dreamworld of disembodied concepts; progress no faster than physicists’ ability to has also edited Pedagogy rather, their main task has been to calculate. calculate. Thus, in the same way that comput- and the Practice of Science: Theorists tinker with models and estimate er-enabled computation might today be said Historical and Contem- effects, always trying to reduce the inchoate to be enabling a genomic revolution, Feynman porary Perspectives (MIT confusion of experimental and observational diagrams helped to transform the way physi- Press, 2005). Honors include evidence and mathematical possibility into cists saw the world, and their place in it. the Leroy Apker Award from tractable representations. Calculational tools the American Physical Society mediate between various kinds of representa- Stuck in the Mud and the Levitan Prize in the Humanities from MIT. Ad- tions of the natural world and provide the cur- Feynman introduced his novel diagrams in dress: Building E51-185, MIT, rency of everyday work. a private, invitation-only meeting at the Po- 77 Massachusetts Avenue, In my research I have adopted a tool’s-eye cono Manor Inn in rural Pennsylvania during Cambridge, MA 02139. Inter- view of theoretical physics, focusing in par- the spring of 1948. Twenty-eight theorists had net: [email protected] ticular on one of theorists’ most important gathered at the inn for several days of intense 156 American Scientist, Volume 93 © 2005 Sigma Xi, The Scientific Research Society. Reproduction with permission only. Contact [email protected]. Santa Barbara News-Press Figure 1. Feynman diagrams were invented in 1948 to help physicists find their way out of a morass of calculations troubling a field of theory called QED, or quantum electrodynamics. Since then, they have filled blackboards around the world as essential bookkeeping devices in the calculation-rich realm of theoretical physics. Here David Gross (center), in a newspaper photograph taken shortly after he was awarded the 2004 Nobel Prize in Physics with H. David Politzer and Frank Wilczek, uses a diagram to discuss recent results in perturbative QCD (quantum chromodynamics) motivated by string theory with 1999 Nobelist Gerardus ’t Hooft (facing Gross at far right) and postdoctoral researchers Mi- chael Haack and Marcus Berg at the University of California, Santa Barbara. Gross, Politzer and Wilczek’s 1973 discovery paved the way for physicists to use the diagrams successfully in QCD. discussions. Most of the young theorists were scatter?—theorists could scrape together rea- preoccupied with the problems of QED. And sonable answers with rough-and-ready ap- those problems were, in the understated lan- proximations. But as soon as they tried to push guage of physics, nontrivial. their calculations further, to refine their starting QED explains the force of electromagne- approximations, the equations broke down. tism—the physical force that causes like charg- The problem was that the force-carrying virtual es to repel each other and opposite charges to photons could borrow any amount of energy attract—at the quantum-mechanical level. In whatsoever, even infinite energy, as long as they QED, electrons and other fundamental par- paid it back quickly enough. Infinities began ticles exchange virtual photons—ghostlike cropping up throughout the theorists’ equa- particles of light—which serve as carriers of tions, and their calculations kept returning in- this force. A virtual particle is one that has bor- finity as an answer, rather than the finite quan- rowed energy from the vacuum, briefly shim- tity needed to answer the question at hand. mering into existence literally from nothing. A second problem lurked within theorists’ Virtual particles must pay back the borrowed attempts to calculate with QED: The formal- energy quickly, popping out of existence again, ism was notoriously cumbersome, an algebraic on a time scale set by Werner Heisenberg’s un- nightmare of distinct terms to track and evalu- certainty principle. ate. In principle, electrons could interact with Two terrific problems marred physicists’ ef- each other by shooting any number of virtual forts to make QED calculations. First, as they photons back and forth. The more photons in had known since the early 1930s, QED pro- the fray, the more complicated the correspond- duced unphysical infinities, rather than finite ing equations, and yet the quantum-mechani- answers, when pushed beyond its simplest cal calculation depended on tracking each sce- approximations. When posing what seemed nario and adding up all the contributions. like straightforward questions—for instance, All hope was not lost, at least at first. Heisen- what is the probability that two electrons will berg, Wolfgang Pauli, Paul Dirac and the other © 2005 Sigma Xi, The Scientific Research Society. Reproduction www.americanscientist.org 2005 March–April 157 with permission only. Contact [email protected]. interwar architects of QED knew that they could approximate this infinitely complicated calcula- tion because the charge of the electron (e) is so small: e2~1/137, in appropriate units. The charge of the electrons governed how strong their in- teractions would be with the force-carrying photons: Every time a pair of electrons traded another photon back and forth, the equations describing the exchange picked up another fac- tor of this small number, e2 (see facing page). So a scenario in which the electrons traded only one photon would “weigh in” with the factor e2, whereas electrons trading two photons would carry the much smaller factor e4. This event, that is, would make a contribution to the full calcula- tion that was less than one one-hundredth the contribution of the single-photon exchange. The term corresponding to an exchange of three pho- tons (with a factor of e6) would be ten thousand times smaller than the one-photon-exchange term, and so on. Although the full calculations extended in principle to include an infinite num- ber of separate contributions, in practice any given calculation could be truncated after only Figure 2. Richard Feynman and other physicists gathered in June 1947 at Shelter a few terms.