← quantum · contents

§ Phase xiii · Feynman Diagrams · Chapter 01

Vertices and propagators

In the spring of 1948, at a small motel in the Pocono Mountains, a thirty-year-old American physicist drew squiggles on the blackboard and lost half the room. Three years later those squiggles were the most powerful calculational engine the field had ever produced. Today every paper in particle physics, every prediction of the Standard Model, every collision plot from the LHC, starts as a sketch of lines and vertices. This chapter is about why the sketches work, and what each pencil stroke is worth in numbers.

In late March 1948, a few dozen of the best theoretical physicists in the world gathered at a quiet inn in the Pocono Mountains of Pennsylvania. Robert Oppenheimer had organized the meeting. The atomic bomb was three years behind them, and the bigger problem of quantum electrodynamics (QED) was now in front. QED was the theory that combined Maxwell’s equations with quantum mechanics, and it was, on paper, the most accurate theory of physics ever attempted. It was also broken. Every honest calculation past the leading order produced infinity. The integrals diverged. The answers were wrong. Something had to give.

Julian Schwinger spoke first. He was twenty-nine, a wunderkind from New York, already a professor at Harvard, and he was about to spend eight hours at the blackboard. His presentation was a tour de force of operator algebra: a Lorentz-invariant reformulation of QED, line by careful line, in which the troublesome infinities were isolated, identified, and absorbed into redefinitions of the electron’s mass and charge. The technique was called renormalization. By lunchtime, Schwinger’s audience was both impressed and exhausted. By the time he finished, in the late afternoon, half the room could no longer follow him.

Then it was Feynman’s turn. He was nervous, prone to clowning, and he chose to skip the formalism. He drew pictures. Squiggly lines for photons, straight lines for electrons, dots at junctions where the lines met. Each picture, he claimed, stood for a single number, a single contribution to the probability that some process would happen. To compute a real prediction you drew every picture you could think of, added their numbers up, and squared the result. That was it. That was the whole theory. Some of the audience thought he was joking. Niels Bohr stood up and lectured him on the Pauli principle. Paul Dirac asked a single icy question about unitarity. Edward Teller asked another. Feynman, in his own words, “got the willies” and sat down before he was finished. He would later say it was the worst talk he ever gave.

What Feynman could not communicate at Pocono, but had already understood, was a translation table. On one side of the table lived a brutal, divergent, operator-laden expression in quantum field theory: an integral over field configurations weighted by the action, expanded in powers of the coupling constant, term by ugly term. On the other side of the table lived a small zoo of pictures, plus a recipe for turning any picture into the number it represented. The two sides agreed. Anything Schwinger could compute by his operator algebra, Feynman could compute by drawing diagrams and copying integrals off them. The diagrams were not a metaphor. They were a stenography for the algebra, and the algebra was the physics. Where Schwinger needed thirty pages of equations, Feynman needed half a sheet of doodles.

To see what a diagram is for, think for a moment about the simplest thing two electrons can do. They approach each other, they repel, they fly apart. We learned in chapter ii why: like charges repel through the electromagnetic field. In quantum field theory the field is not a continuous push at a distance; it is the photon, and the photon is exchanged. One electron emits a photon. The other electron absorbs it. Both electrons recoil. The act of emission and the act of absorption each happen at a single spacetime point, what we will call a vertex. The photon, between emission and absorption, propagates from one vertex to the other, what we will call a propagator. Feynman’s first diagram for electron-electron scattering is literally two parallel electron lines connected by a single photon rung, like an H tipped on its side. That picture is the entire story at lowest order. Drawing it is the calculation.

Sit with that wording for a moment, because it tells you the entire game. A Feynman diagram is “a pictorial representation of the mathematical expressions” of QFT. The picture is not a physical movie of what happens. There is no little ball with a tail flying between the electrons; you cannot photograph a virtual photon, and asking what the photon “really” looks like during the exchange is a category mistake. The diagram is a tag, a way of organizing one term in a long perturbation series. Each line in the picture corresponds to a propagator, the mathematical object that tells you how a particle’s quantum amplitude evolves between two spacetime points. Each vertex corresponds to a coupling, the mathematical object that tells you how strongly the particles talk to each other. You read the diagram off, line by line, vertex by vertex, multiplying as you go. You integrate over whatever is not fixed. The result is a complex number, the amplitude. Square it (in the right sense) and you get a probability.

Here is the simplest vertex in nature: the place where a single electron, a single positron, and a single photon meet. In QED there is only one vertex. Everything you can ever compute in QED is built from this one piece, repeated and connected.

Every line that enters or leaves a vertex carries a momentum, an energy, a spin. The vertex enforces that everything coming in equals everything going out: total energy is conserved, total momentum is conserved, total electric charge is conserved. Those conservation laws are not extra rules tacked on; they fall out of the mathematical structure of the vertex itself, which inherits them from translation symmetry and gauge symmetry of the underlying field equations. The vertex is the local seam where the conservation laws are stitched.

A vertex on its own is not a physical process; you need lines coming out of the vertex and going into another vertex, or off to detectors. Those lines, the legs of the diagram, are the propagators. A propagator is a piece of mathematics that answers a sharp question: if I put a particle here with momentum p, what is the quantum amplitude that it shows up over there with the same momentum p? For a free particle, the answer is a famous expression that we will write out in a moment. For a particle that briefly carries a momentum it would not be allowed to have in free flight (an off-shell or “virtual” particle, the kind that lives only inside a diagram, between two vertices), the propagator is the same expression evaluated at that off-shell momentum. There is no contradiction. The intermediate particle never has to be measured; it only contributes to a sum.

So now we have all the pieces. A diagram is a graph with two kinds of edges, straight (matter) and wavy (force), meeting at vertices. Every edge carries a propagator factor. Every vertex carries a coupling factor. Some edges run from a vertex to a detector or a source, the external legs. Some edges run between two vertices, the internal legs. The external legs are the physical particles you can prepare and measure. The internal legs are the virtual particles, the off-shell intermediates whose only existence is as terms in your sum. You write down the diagram, you copy off the factors, you multiply, you integrate over any unfixed internal momenta. You get a complex number. That number is the amplitude for that one diagram.

ℒ = −(1/4) Fᵤᵥ Fᵘᵛ ,   Fᵤᵥ = ∂ᵤ Aᵥ − ∂ᵥ Aᵤ

Dᵤᵥ(k) = −gᵤᵥ ÷ (k² + iε)

There is one more rule, the rule that makes the whole apparatus into a theory rather than a doodle: every legal diagram for a given process must be drawn, the amplitude for each must be computed, and the amplitudes must be added together. Then, and only then, do you square. This is the Born rule from chapter v applied to the quantum field theory: amplitudes add, probabilities are the squared moduli of the sum. In QED two electrons can scatter via the one-photon exchange we already drew; they can also scatter by exchanging two photons (a more complicated diagram with extra vertices and an internal loop). The two-photon exchange is smaller by a factor of α ≈ 1/137, but it is not zero, and ignoring it costs you precision. You sum the two amplitudes, then square. You do not sum two probabilities.

The Feynman rules (write down every diagram at a given order in α, copy off propagators and vertices, integrate over loop momenta, sum, square) reduced QED to a finite, mechanical procedure. With Dyson’s proof of renormalizability, every divergent integral could be absorbed into a redefinition of the electron’s mass and charge, and the remaining numbers were finite and predictive. By the early 1950s a graduate student could compute the leading correction to the magnetic moment of the electron in an afternoon, where six years earlier the same calculation had defeated Heisenberg and Pauli for a decade. By 1965, when the Nobel committee finally rang Stockholm, the calculation had been pushed to many higher orders, and the prediction agreed with experiment to one part in a billion. Today it has been pushed to roughly twelve significant figures, the most precise agreement between theory and measurement in the history of science.

That precision is what makes the diagrams worth learning. They are not pretty pictures used to motivate hand-waving stories about virtual particles. They are an exact, calculational engine: the recipe by which a quantum field theory produces a number you can carry to a particle detector and compare against. Every prediction of the Standard Model, every cross-section at the LHC, every constraint on physics beyond the Standard Model, has a Feynman diagram on the back of an envelope at the start of it. The diagrams outlived their inventors. They will outlive us. They are the bookkeeping of the universe, in Feynman’s own line: “the rules of calculation, as I conceive them, are part of the description of nature.”

In the next two chapters we will use this machinery. First we will draw the diagrams for two electrons bouncing off each other (Møller scattering), copy off the numbers, and watch a real cross-section fall out of pure pencil work. Then we will do the same for beta decay: a neutron emits a W boson, the W decays into an electron and an antineutrino, and a free particle of one kind becomes three particles of three other kinds. The vertex changes from QED’s −ieγᵘ to the weak interaction’s vertex with the chirality projector built in, but the apparatus is identical. Once you can draw a vertex and a propagator, you can compute the rest of physics.