In 2023 a team weighed the magnetism of a single electron and got a number that agrees with theory to twelve significant figures. Twelve. That is like predicting the distance from New York to Los Angeles and being right to the width of a human hair. No other theory in all of science has ever been checked so hard, from so many angles, and simply refused to break. That theory is Quantum Electrodynamics — QED — the quantum field theory of the electromagnetic force, and it is the most precisely tested description of nature we have.

What makes QED almost miraculous is how little it contains. The entire theory is built from two kinds of thing and one kind of event: electrons, photons, and a single junction where an electron emits or absorbs a photon. Everything electromagnetic — light, chemistry, the stiffness of a table, the colour of gold, the magnetism above — is that one junction, drawn over and over and added up. This whole page rests on a single sentence, worth reading twice: QED is electrons plus photons plus one vertex, expanded as a power series in a small number called \alpha, about 1/137.

The whole theory on one napkin

A quantum field theory is a recipe for what can happen and how likely each thing is. QED's recipe is astonishingly short. There is a field for the electron (its ripples are electrons and positrons) and a field for the photon (its ripples are particles of light). Left alone, each field just propagates. The physics — the interaction — enters through exactly one rule, the fundamental QED vertex:

The photon is the gauge boson of QED — the messenger that carries the electromagnetic force. It emerges from a symmetry called U(1) (roughly, the freedom to rotate the phase of the electron's quantum field independently at every point in space). Demanding that this local freedom cost nothing forces a massless spin-1 field into existence, and that field is the photon. Because the photon is massless, the electromagnetic force has infinite range: it falls off as 1/r^2 but never switches off, which is why you can see stars billions of light-years away.

The magic number: \alpha \approx 1/137

How strongly do electrons and photons actually couple? The answer is a single pure number, the fine-structure constant \alpha. Conceptually it packages the electron charge together with the fundamental constants of relativity and quantum mechanics:

Notice what is not there: no metres, no seconds, no kilograms. The units all cancel, leaving a bare number. That is what makes \alpha so special — it is a dimensionless measure of how hard electricity pulls, the same for a physicist on Earth or an alien in another galaxy. And it is small: about 0.73\%. That smallness is the secret of QED's success, because it means the interaction is a gentle nudge, not a violent shove — and gentle nudges can be added up one at a time.

Adding up the pictures: a power series in \alpha

Here is the master trick. To compute how likely some electromagnetic process is, you draw every Feynman diagram that connects the start to the finish, turn each into a number, and add them up. There are infinitely many diagrams — but they are not all equally important. The simplest diagram (fewest vertices) is the biggest; every extra loop you add pins on two more vertices, and so multiplies the contribution by another factor of \alpha \approx 1/137. The sum organises itself into a power series:

Because each successive term is smaller than the last by a factor of about 1/137, you can get a spectacular answer by keeping only the first couple of terms — the tree diagram plus its one-loop correction — and throwing the rest away. The graph below shows why the trick works. It plots the size of the order-n term, \alpha^n, against the order n, for QED's tiny \alpha = 1/137 next to a hypothetical strongly coupled world with \alpha = 1/2.

The QED curve falls off a cliff: after the first order the terms are essentially zero on this scale. The strong-coupling curve barely shrinks at all — which is precisely the situation for the strong nuclear force, whose coupling is close to 1, and why you cannot compute it by adding a handful of diagrams. QED is easy for one reason and one reason only: \alpha is small.

Worked examples

Example 1 — how small is \alpha, really? Convert the coupling to a percentage:

So each extra pair of vertices in a diagram knocks the contribution down to under 1\% of what came before. That is the whole reason a truncated series works.

Example 2 — the terms shrink like a stone. Set the leading (tree) contribution to 1 for comparison. Then the successive corrections go as powers of \alpha:

• tree (order 0): \alpha^0 = 1;
• one loop (order 1): \alpha^1 = 1/137 \approx 0.0073 — a 0.73\% tweak;
• two loops (order 2): \alpha^2 \approx 5.3\times 10^{-5} — five parts in a hundred thousand;
• three loops (order 3): \alpha^3 \approx 3.9\times 10^{-7}.

To match an experiment good to a part in a million you would need to compute out to about three loops — hard, but finite. To match a part in a trillion (as for g-2) you push to four and five loops, tens of thousands of diagrams, and it still works.

Example 3 — the electron's magnetism, from one loop. A classical spinning charge, and Dirac's 1928 equation, both predict the electron's magnetic strength factor to be exactly g = 2. QED says: not quite — the electron is constantly emitting and reabsorbing virtual photons, and each such loop nudges g upward. The leading correction, first computed by Schwinger in 1948 (it is carved on his tombstone), is beautifully simple:

That one-loop estimate already nails the measured value to four decimal places. Adding the higher-order terms in the \alpha series drags the agreement out to twelve. The tiny anomaly a — the amount by which g exceeds Dirac's clean 2 — is the fingerprint of the quantum vacuum, and measuring it is how we test QED to the hilt.

Precision, and a coupling that won't sit still

The g-2 experiments are the crown jewels. Because both theory and measurement of the electron's anomaly reach twelve significant figures and agree, QED earns its title as the most precisely tested theory in science. The muon — the electron's heavier cousin — gets the same treatment, and its g-2 is watched even more nervously, because a heavier particle is more sensitive to any new physics lurking in the loops. Any lasting mismatch there would be a doorway beyond QED.

One last twist. The number \alpha \approx 1/137 is the value measured at low energies. Probe an electron at very high energy — smash particles together hard — and you find the coupling has grown: at the energies of the Large Hadron Collider, \alpha is closer to 1/127. This is the running of the coupling: an electron is surrounded by a fizz of virtual electron–positron pairs that partly screen its charge, like a haze around a lamp. Get closer (higher energy) and you peer inside the haze, seeing more of the bare charge, so the effective coupling rises. The "constant" is not constant.