Feynman diagrams

How can a cartoon of lines and vertices stand in for an integral nobody can picture?

A Feynman diagram looks like a story. Two electrons come in, exchange a photon, go out again; the axes suggest space and time; the lines suggest paths. Every popular account reads it that way, and physicists draw them on napkins as if narrating an event.

Yet the theory the diagrams belong to does not contain trajectories at all. So either the picture is a helpful lie, or it is not a picture in the sense we assumed. Before deciding, it is worth asking a more basic question: what does a good notation actually do, and what would we have to check to know whether these squiggles are doing it?

Take the problem the diagrams were invented to solve. In quantum electrodynamics you want the amplitude for some process — say two electrons scattering. There is no way to write that amplitude in closed form. What you can do is expand it as a series in the strength of the electromagnetic interaction, whose coupling is small: the fine-structure constant is about 1/137. A small parameter means the first few terms may be enough, and each successive term is a more elaborate integral than the last.

Now the difficulty is bookkeeping. At each order there are many distinct contributions, each a many-dimensional integral, each with signs and factors that are easy to get wrong. Feynman's move, presented at the Pocono conference in 1948 and published the following year, was to give each contribution a graph, and to give a rule translating every feature of the graph into a factor in the integral.

That is the whole trick, and notice its shape. An external line means an incoming or outgoing particle. An internal line means a propagator — a specific algebraic expression built from the particle's mass and the momentum flowing along it. Each vertex where lines meet contributes a factor of the coupling, and enforces conservation of energy and momentum at that point. A closed loop means an integral over a momentum that the external conditions leave undetermined. Draw every topologically distinct graph with the right external lines and the allowed number of vertices, translate each one, and add them. Freeman Dyson showed in 1949 that this recipe reproduces exactly the terms of the conventional perturbation expansion — which is what turned the diagrams from one man's private shorthand into the field's standard method.

So ask the representation question properly: what does the notation preserve, and what does it discard? It preserves connectivity — which line joins which vertex — and through that, the structure of the integral. It preserves the counting: the number of vertices fixes the power of the coupling, so a diagram's order is legible at a glance. What it discards is everything geometric. Line lengths mean nothing. Angles mean nothing. The distance between two vertices on the page is not a distance, and the vertical position is not a time at which anything occurred. The diagram is a graph in the mathematician's sense, and two drawings that look quite different are the same diagram if the connections match.

Once that is clear, the awkward feature stops being awkward. Internal lines are called virtual particles, and they are not particles that were there. They do not satisfy the relation between energy and momentum that a real particle must; they are terms, and a term is not an event. The strongest evidence is that no single diagram is measurable. Only the sum is. You cannot ask which diagram the electrons "actually did", any more than you can ask which term of a series a number actually is.

Two further facts sharpen how seriously to take the whole scheme. First, the number of diagrams explodes with order: for the electron's anomalous magnetic moment there is one diagram at first order in the coupling, seven at second, seventy-two at third, and in the tens of thousands by the fifth — which is why this calculation, agreeing with experiment to around ten significant figures, is among the most laborious in physics as well as the most precise.

Second, and more surprising: the series does not converge. Dyson gave an argument in 1952 that the QED perturbation series has zero radius of convergence, so it is asymptotic — adding terms improves the answer for a while and would eventually make it worse. In practice the turnaround lies far beyond any order anyone can compute, so the method works superbly. But it means the diagrams are an extraordinarily good approximation scheme rather than a definition of the theory, and that is not a quibble.

Three refinements. First, calling the diagrams notation is not calling them arbitrary. A good notation makes a problem's structure visible, and these do: symmetries, conservation laws and the order-counting are readable straight off the topology, and physicists genuinely reason by manipulating the pictures — operations on the symbols mirroring operations on the mathematics.

Second, the space-time reading is not simply an error. Feynman arrived at the diagrams through a space-time way of thinking, and the graph does record an ordering of interactions. What is illegitimate is treating a single diagram as a definite history, or its internal lines as observed particles.

Third, the misconception worth naming: diagrams do not explain where their own rules come from. They are downstream of a Lagrangian, and change the theory and the vertices change with it. The method also depends entirely on the expansion parameter being small — which is why a diagram of the strong interaction at low energy, where the coupling is not small, cannot be trusted the way a QED diagram can.

• Renormalisation: why loop integrals diverge, and what is actually done about it.
• Why the same expansion fails for the strong force at low energy, and what replaces it.
• Asymptotic series in general — when adding another term stops helping.

Every term the collection defines is gathered in the glossary.