The Diagram That Did The Sum
Feynman's diagrams are the one case in which a private picture became the calculus of a whole field. They are also the clearest warning about what a picture can make you believe, and — in the last forty years — the strangest evidence that the picture was hiding something simpler underneath.
At the end of March 1948, twenty-eight physicists met at the Pocono Manor Inn in Pennsylvania to sort out a crisis. Quantum electrodynamics, the theory of how light and electrons interact, gave the right answers at first approximation and infinite answers at the second; the infinities had been known since the 1930s and nobody had tamed them. Julian Schwinger spoke first, for most of a day, and showed a way through that was correct, complete and nearly impossible to follow. Richard Feynman spoke second, and it went badly. He drew pictures: lines for electrons, wavy lines for photons, points where they met. Each picture, he said, stood for a term in the calculation, and the rules for turning the picture into the term were simple enough to write on a card. Niels Bohr interrupted to explain that the uncertainty principle forbade talking about the paths of electrons. Dirac asked whether the scheme was unitary. Feynman did not know, and the meeting moved on.
Within three years the pictures were the standard method of the field, within ten they were the language in which particle physics thought, and they have not been displaced since. Schwinger, who never drew one in his life, put the outcome with characteristic acid: like the silicon chip of a later age, the Feynman diagram had brought computation to the masses.
A companion essay on this blog argues that mathematicians and physicists think in structures before they think in symbols, and cites the diagrams in passing. They deserve more than a passing mention, because they are three different things at once. They are the best case there is of a picture that became a formalism rather than being translated into one. They are the best cautionary tale about what a picture can smuggle in. And since 1986 they have become something nobody at Pocono could have anticipated: the notation that turned out to be hiding a simpler mathematical object, one that has no diagrams in it at all.
How a drawing is a sum
The idea is easier than its reputation. Quantum theory, in the form Feynman had rebuilt it, says that the probability of a process is found by adding up a contribution from every way the process could happen and squaring the total. For light and electrons the ways are indexed by how many times the particles interact — how many times an electron emits or absorbs a photon — and each interaction costs a factor of a small number, the fine-structure constant, about 1/137. So the answer is a series: a first term with no interactions, a second with one, a third with two, each smaller than the last by a factor of roughly a hundred.
A Feynman diagram is a picture of one way. Straight lines are electrons, wavy lines are photons, and a vertex where a straight line meets a wavy one is an interaction. Two electrons scattering off each other, at the simplest order, is a single diagram: two straight lines exchanging one wavy one. At the next order there are several: the photon can be exchanged twice, or one electron can emit and reabsorb a photon on its way, or the exchanged photon can briefly split into an electron–positron pair that recombines. The rules assign a mathematical factor to every line and every vertex, an integral to every closed loop, and say: draw every diagram with the right external legs and the given number of vertices, multiply out the factors, add.
This is not a picture of the calculation. It is the calculation, in a form that the hand and eye can check. The bookkeeping that had defeated the 1930s — which terms exist, which cancel, which are counted twice — is handled by the topology of the drawing. If you can enumerate the drawings you have enumerated the terms, and a drawing is much harder to get wrong than a page of integrals. Whole categories of error simply disappear.
The proof that it works is the most precisely confirmed prediction in science. Schwinger showed in 1948 that the electron's magnetic moment is not exactly the value Dirac's equation gives but larger by α/2π, about one part in a thousand — the one-loop diagram, a single photon emitted and reabsorbed. The two-loop correction requires seven diagrams; three loops, seventy-two; four loops, 891; five loops, 12,672, computed numerically by Aoyama, Kinoshita and Nio over more than a decade. The result agrees with the measured moment, as of the 2023 Harvard experiment, to about ten significant figures. Nobody did that with Schwinger's method. The diagrams are why it was possible.
A bus through Nebraska
What made the diagrams respectable was not Feynman. It was Freeman Dyson, a twenty-four-year-old Englishman who had spent the academic year at Cornell listening to Feynman and the summer at Michigan listening to Schwinger, and who understood both. In September 1948 he was on a Greyhound bus crossing Nebraska, and — by his own account, half asleep, with nothing to write on — saw how the two methods fitted: Feynman's pictures were a systematic way of generating exactly the terms that Schwinger's and Tomonaga's operator formalism produced, and the diagram rules could be derived from that formalism rather than guessed. He wrote it up in Princeton that autumn. The paper, The Radiation Theories of Tomonaga, Schwinger, and Feynman, appeared in February 1949 and made the diagrams a theorem instead of a trick. Feynman's own paper, setting out the rules as he thought of them, came out later that year.
Then the diagrams spread, and the manner of it is the subject of David Kaiser's 2005 book Drawing Theories Apart, which is one of the best studies there is of how a way of thinking moves. The diagrams did not travel through the journals. Feynman's 1949 papers were terse, the rules were incomplete on the page, and readers who had only the printed version routinely got them wrong. They travelled through people. Dyson taught them to the postdocs at the Institute for Advanced Study; the postdocs took jobs; each department that acquired one acquired the diagrams within a year or two, and departments without one did not. Kaiser traces the map of who could draw a Feynman diagram in 1953 and it is a map of who had shared an office with someone who could. Japan, cut off from the postdoc network, learned them from print and developed a slightly different dialect. A picture had to be learned as a skill, by watching, the way one learns to play an instrument.
And as it spread it changed. By the mid-1950s the diagrams had been carried into nuclear physics and into the theory of solids, where the lines stood for mesons or for excitations in a crystal, and where the series they organised was not small and could not be trusted term by term. The notation outran its original meaning and kept working anyway, which is either a sign that it was tracking something deeper than electrodynamics or a sign that a good notation is dangerous, and the next section is about the second possibility.
Virtual particles and the series that does not converge
A Feynman diagram looks like a picture of an event. Two electrons approach, one emits a photon, the other absorbs it, they recoil. Time runs up the page, or left to right, and the internal wavy line looks like a thing that travelled from one to the other. The diagrams are drawn that way because that is how Feynman thought — he was, by his own description, someone who saw spacetime pictures, and the diagrams were the pictures. The positron appears in them as an electron travelling backwards in time, which is not a metaphor in the formalism: it is exactly how the rules treat it, and it was one of the things Feynman was proudest of.
But the diagram is not a picture of an event. It is one term in a sum, and only the sum has physical meaning. The internal photon in the exchange diagram is off-shell: the rules integrate over every possible energy and momentum it could have, including combinations no real photon can possess, and the contributions from all of those are added. Nothing travelled. The word for such internal lines is virtual particle, and a great deal of physics popularisation has been built on taking the word literally — on the vacuum seething with particles that flicker in and out of existence, on forces carried by messengers. There is a defensible reading of some of that language. There is no defensible reading on which the picture in the diagram is a movie of what happened. A single diagram is no more an event than a single term in the expansion of sin x is an angle.
The second thing the picture hides is worse. In 1952 Dyson gave a short argument that the series the diagrams organise does not converge. If it converged for small positive α it would converge for small negative α; but a world with negative α, in which like charges attract, is unstable in a way that makes the theory meaningless; so the series cannot converge at all. It is an asymptotic series: the first terms get closer to the answer, then, far out, the terms start growing without bound. For electrodynamics the turn comes at about the 137th order, so no calculation ever done is anywhere near it. For the strong interaction the coupling is not small and the turn is close. The diagrams, in other words, are a tool for computing the early terms of an expansion that is not, strictly, the theory. Weierstrass's nowhere-differentiable function showed the nineteenth century that a picture in analysis can lie; the Feynman diagram is the twentieth century's version, and it lies more persuasively because it computes so well.
Two hundred and twenty diagrams and one line
The strangest chapter began in 1986, and it began with a calculation nobody wanted to do. The strong interaction is carried by gluons, which unlike photons interact with each other, so the diagrams for gluon scattering multiply much faster. For a process with four gluons there are four tree-level diagrams. For five, twenty-five. For six, two hundred and twenty. Each is a page of algebra, and the pages must be added. Stephen Parke and Tomasz Taylor, at Fermilab, needed the six-gluon result for collider predictions, ground through the two hundred and twenty, and found that the sum, for a particular arrangement of the gluons' spins, was this:
One line. The brackets are simple two-component objects built from the gluons' momenta; the formula fits on a business card, and Parke and Taylor conjectured, correctly, that it holds for any number of gluons — the eight-gluon case, which would require 34,300 diagrams, and the ten-gluon case, which would require over ten million, collapse to the same expression with a longer denominator. They ended their paper with a challenge to the string theorists to prove it, and Frits Berends and Walter Giele did prove it, in 1988, by a recursion that never draws a diagram.
Two things need explaining here, and the diagrams cannot explain either. The first is where the simplicity comes from. Two hundred and twenty diagrams' worth of terms cancel against each other almost completely, and the cancellation is invisible diagram by diagram; it happens only in the sum. The second is why the diagrams are so bad at showing it. The answer, worked out over the following twenty years, is that the Feynman rules are built to make two principles manifest at every step — locality, that interactions happen at points, and unitarity, that probabilities add to one — and that the price of keeping both visible in every term is an enormous redundancy. Each diagram carries unphysical pieces that are there only to cancel against unphysical pieces in others. The notation is honest about the principles and, in consequence, blind to the answer.
The methods that replaced it, in the hands of Zvi Bern, Lance Dixon, David Kosower and others through the 1990s and of Ruth Britto, Freddy Cachazo, Bo Feng and Edward Witten in 2005, work only with physical quantities — on-shell particles, momenta that a real particle could have — and build complicated amplitudes recursively out of simpler ones, without ever passing through the sum over diagrams. They are faster by orders of magnitude, they are now what the collider calculations actually use, and they have the peculiar feature that locality and unitarity are not put in. They come out.
The volume of a polytope
In 2013 Nima Arkani-Hamed and Jaroslav Trnka took the last step, for a simplified theory that physicists use as a laboratory — a supersymmetric cousin of the gluon theory, in the planar limit, which is not the real world but is close enough to the real world's structure to be worth studying. They showed that in that theory the scattering amplitude is the volume of a geometric object. Not a sum over diagrams; not a recursion; a single region in a high-dimensional space, defined by positivity conditions, whose volume is the answer. They called it the amplituhedron. Locality and unitarity, the two principles that every Feynman diagram was built to display, are theorems about the shape of the region. They are consequences of the geometry, not inputs to it.
Whether this survives the trip to real theories is open; the construction is known to be special, and the corresponding objects for the actual strong interaction, and for gravity, are partly or wholly unknown. But the direction of travel is clear enough to state. The diagrams were a notation that made a physical picture — particles interacting at points, in sequence — the organising principle of the calculation. The picture was extraordinarily productive and, taken literally, false. Underneath it, invisible to anyone drawing diagrams, was an object with no particles and no points and no sequence in it, whose shape encoded everything the diagrams computed and explained the cancellations the diagrams could not. Parke and Taylor found the object's shadow by brute force in 1986. It took twenty-seven years to see what was casting it.
Two kinds of picture
The series this essay accompanies argues that mathematics is a matter of made vocabulary and found structure, and that researchers grasp structure before they have words for it. The diagrams are that argument's best physical example and its most awkward one, and both halves matter.
They are the best example because they are a picture that did not need translating. Feynman thought in them; Dyson proved they were the theory; the field learned them as a skill; and for seventy-five years the drawing has been the computation. Nothing in Hadamard's survey or Thurston's list of derivatives shows the structural representation and the certified formalism coinciding so completely. The diagrams are the case where the picture in the head and the proof on the page were, for once, the same object.
They are the awkward example because the picture was the wrong one. The diagrams organise the theory around a story — things travelling, meeting, exchanging — and the story is what made them usable and what made them mislead. Parke and Taylor's one-line formula, and everything since, says that the found structure was not the story at all. It was a shape, and the story was the made part: a scaffolding that let people compute while hiding what they were computing. The cancellations that take two hundred and twenty pages to see are the shape asserting itself through a notation that was not built to show it.
So there are two kinds of picture, and the difference between them is the difference the whole series has been circling. A picture of a process is a narrative aid: it organises a calculation around a sequence of events, it is easy to think with, and it commits you to things — virtual particles, paths, an order of events — that the theory does not contain. A picture of a shape is a representation of the structure itself: harder to arrive at, indifferent to sequence, and true. The history of the amplitudes programme is the replacement of the first kind by the second, and it is, on the reading this blog has been defending, what discovery looks like from the inside: a made picture that worked, then the structure it was hiding, then a picture of that.
Feynman would probably not have minded. He was interested in what computed, and the amplituhedron computes. But it is worth noticing that the thing that finally explained his diagrams is exactly the kind of object the first essay in this series found underneath the complex numbers — something rigid, with no free parameters, whose properties are forced — and that physicists describe it, without embarrassment, as beautiful.
Where this could go
Kaiser's map, redrawn. Drawing Theories Apart traces the diagrams' spread through personal contact to about 1960. The same study for on-shell methods and the amplituhedron — which spread through a small number of institutions, summer schools and one very influential set of lectures — has not been done, and the pattern is likely the same: a way of thinking that travels through people, not papers.
Gravity as a square. Bern, Carrasco and Johansson found in 2008 that the amplitudes of gravity can be obtained from those of the gluon theory by, roughly, squaring — the “double copy” — a relation invisible in either theory's diagrams and unexplained at the level of the underlying equations. It is the Parke–Taylor story again, one theory up, and it is the most concrete evidence that the shape, whatever it is, is shared.
The divergent series, taken seriously. Dyson's 1952 argument is two paragraphs long and most physicists have never read it. What an asymptotic series with a finite radius of usefulness means for the claim that quantum electrodynamics is “the most accurate theory ever” — and the resurgence programme that tries to make sense of the terms beyond the turn — is a piece in its own right, and it connects to the third essay of the series on what counts as a proof.
The popular picture. A catalogue of what the public has been told virtual particles are, from Hawking radiation explanations to the Casimir effect, set against what the formalism supports, would be a useful and slightly unkind essay. The diagrams are the source of most of it.
Machines and the shape. The AI provers of the third essay work with symbols and have no pictures of either kind. The amplitudes programme is a natural test of whether that matters: an on-shell recursion is exactly the sort of structure a symbolic system should find, and the amplituhedron exactly the sort it should not. Nobody has tried.
- The Bus at Coutances — how mathematicians and physicists actually think; Part V of Found or Made.
- Invented, Then Unavoidable — imaginary numbers, rigidity, and why physicists trust beauty; Part I of the series.
- What a Proof Is Now — certificates, explanations, and proofs no one has read; Part III.
- The Interference Machine — what a quantum computer actually does.
- Schwinger, “On quantum-electrodynamics and the magnetic moment of the electron”, Physical Review, 1948.
- Dyson, “The radiation theories of Tomonaga, Schwinger, and Feynman”, Physical Review, 1949; “The S matrix in quantum electrodynamics”, 1949.
- Feynman, “Space-time approach to quantum electrodynamics”, Physical Review, 1949.
- Dyson, “Divergence of perturbation theory in quantum electrodynamics”, Physical Review, 1952.
- Schwinger, “Renormalization theory of quantum electrodynamics: an individual view”, in The Birth of Particle Physics, 1983.
- Parke & Taylor, “An amplitude for n gluon scattering”, Physical Review Letters, 1986; Berends & Giele, “Recursive calculations for processes with n gluons”, Nuclear Physics B, 1988.
- Schweber, QED and the Men Who Made It, 1994.
- Bern, Dixon, Dunbar & Kosower, “One-loop n-point gauge theory amplitudes, unitarity and collinear limits”, 1994.
- Witten, “Perturbative gauge theory as a string theory in twistor space”, 2004; Britto, Cachazo, Feng & Witten, “Direct proof of tree-level recursion relation in Yang–Mills theory”, Physical Review Letters, 2005.
- Kaiser, Drawing Theories Apart: The Dispersion of Feynman Diagrams in Postwar Physics, 2005.
- Bern, Carrasco & Johansson, “New relations for gauge-theory amplitudes”, Physical Review D, 2008.
- Aoyama, Kinoshita & Nio, “Revised and improved value of the QED tenth-order electron anomalous magnetic moment”, Physical Review D, 2018.
- Arkani-Hamed & Trnka, “The amplituhedron”, Journal of High Energy Physics, 2014.
- Fan, Myers, Sukra & Gabrielse, “Measurement of the electron magnetic moment”, Physical Review Letters, 2023.
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