Reading Feynman Diagrams

In 1948, at a cramped conference in the Pocono Mountains, Richard Feynman stood at the blackboard and drew a few squiggles and lines, and claimed they were the answer to a calculation that had been torturing the best physicists in the world. Most of the audience was baffled. Within a decade his little cartoons had become the single most-used tool in particle physics — the pictures you now see on T-shirts, book covers, and the walls of every physics department. This page is about learning to read them.

Here is the one idea, and everything else hangs from it: a Feynman diagram is not a picture of particles flying through space — it is a bookkeeping cartoon of a term in an amplitude, read vertex by vertex. Quantum electrodynamics (QED) computes the probability of a process as a sum over infinitely many contributions, each smaller than the last. Each contribution is a many-dimensional integral that would take a page to write out — so Feynman invented a picture that stands in for it, and a set of rules that turn the picture straight back into the maths. Learn to read the picture and you can read off the physics: which particles, how they couple, and how big the contribution is — all without writing the integral at all.

The stage: one axis of time, one of space

Every Feynman diagram lives on a little spacetime grid. In this page we let time run to the right and space run up the page (some textbooks flip these — always check the caption). A particle that exists before the interaction enters from the left; one that exists afterwards leaves to the right. But do not read the wiggles and diagonals as trajectories: the exact angle and length of every line is arbitrary. Only the connectivity — what joins to what — carries meaning.

Particles are drawn as lines, and the kind of line names the particle:

Straight lines with an arrow are spin-\tfrac12 fermions (electrons, quarks, neutrinos).
Wavy (or sometimes coiled/dashed) lines are the force carriers — for QED that is the photon, \gamma.

The arrow on a fermion line is the crucial bit of grammar. For a particle the arrow points along the direction of time (here, to the right). For an antiparticle the arrow points against time (to the left). So a positron — the electron's antiparticle — is drawn as an electron line whose arrow runs backward. This is not just a convention: Feynman and Stückelberg showed that the mathematics of an antiparticle moving forward in time is identical to that of a particle moving backward, so one arrow does the job of both.

External lines (with a loose end) are real, observable particles that go in or come out — they satisfy E^2 = (pc)^2 + (mc^2)^2 (they are on mass-shell).
Arrow with time = particle; arrow against time = antiparticle. Charge, lepton number and the like flow with the arrow.
The photon is its own antiparticle, so its wavy line carries no arrow.

The vertex: where the physics — and the coupling — lives

Look for the points where three lines meet. Each such junction is a vertex, and it is where an interaction actually happens. The fundamental QED vertex is always the same shape: two fermion lines and one photon line meeting at a point — an electron absorbs or emits a photon. Nothing else is allowed in pure QED; every diagram, no matter how baroque, is built by gluing copies of this one vertex together.

Here is the payoff. When you translate the diagram back into maths, each vertex contributes a factor of the coupling constant e (the electron's charge) to the amplitude. This is the whole reason the pictures are useful for estimating sizes: you do not need the integral to know how strongly a process is suppressed — you just count the vertices. The coupling is small,

\alpha = \frac{e^2}{4\pi\varepsilon_0\hbar c} \approx \frac{1}{137},

so every extra vertex you add multiplies the amplitude by another small factor of e (roughly \sqrt{\alpha}). A diagram with V vertices therefore has an amplitude

\mathcal{M} \;\propto\; e^{\,V},

and — since a probability is the amplitude squared — the rate of the process scales as

|\mathcal{M}|^2 \;\propto\; e^{\,2V} \;=\; \alpha^{\,V}.

The power of the coupling is the order of the process. The fewest-vertex diagram is the leading order; every diagram with two more vertices is a small correction on top (a "higher-order" or "loop" term), suppressed by another factor of \alpha \approx 1/137. That is why QED can be computed to astonishing precision: the series converges fast because \alpha is tiny.

Internal lines: propagators, and the ghostly virtual particle

A line joining two vertices — with no loose end — is an internal line, and it represents a propagator: a virtual particle that is created at one vertex and destroyed at the other, existing only for the fleeting instant in between. Virtual particles are the messengers of force. In QED the electromagnetic repulsion between two electrons is the exchange of a virtual photon.

The word "virtual" is doing real work. An internal particle is off mass-shell: it does not obey E^2 = (pc)^2 + (mc^2)^2. A virtual photon can carry an "impossible" combination of energy and momentum — it can even have an effective mass — because the energy–time uncertainty principle lets it borrow energy \Delta E for a time \Delta t \sim \hbar/\Delta E and pay it back before anyone can catch it. You can never detect a virtual particle directly; it is an internal line, a bookkeeping device inside the integral, not something that reaches a detector.

Reading a whole diagram: electron–electron scattering

Time to put it together. The figure below builds up the leading-order diagram for two electrons scattering off each other, e^-e^- \to e^-e^- (called Møller scattering). Reveal it step by step and read it like a sentence, left to right in time.

Now read off the physics without touching an integral:

Four external lines — two electrons in, two electrons out. All four are real, on-shell, and (in principle) detectable.
Two vertices — so the amplitude carries a factor e \cdot e = e^2, i.e. \mathcal{M} \propto \alpha. This is the smallest number of vertices any QED scattering can have, so this is the leading-order diagram.
One internal line — the wavy photon between the vertices. It is a virtual photon: the force carrier, off mass-shell, never observed on its own.

Squaring the amplitude, the probability (the measurable cross-section) scales as |\mathcal{M}|^2 \propto e^4 = \alpha^2 \approx (1/137)^2. The next-order correction has four vertices, so it is smaller by roughly another factor of \alpha — which is exactly why the leading diagram is usually all you need.

Three cousins, same alphabet

Once you can read one QED diagram you can read the whole family — they differ only in how the same two-fermion-one-photon vertex is wired up:

Møller scattering e^-e^- \to e^-e^- — the diagram above: two electrons trade a virtual photon.
Bhabha scattering e^-e^+ \to e^-e^+ — an electron and a positron scatter. Two diagrams contribute at leading order: they can exchange a virtual photon (as in Møller), or annihilate into a virtual photon that then re-materialises as an e^-e^+ pair. Note the incoming positron: an electron line with its arrow pointing backward in time.
Pair annihilation e^-e^+ \to \gamma\gamma — an electron and a positron meet and vanish, their energy emerging as two real photons. Here the photons are external lines, not internal ones, so they are on-shell and fly off to a detector. Two vertices again, so again \mathcal{M} \propto \alpha.

Every one of these is the same handful of vertices, rearranged — and in every one you find the order of the process just by counting the dots.

The coupling \alpha \approx 1/137.036 is the fine-structure constant, and it is one of the deepest puzzles in physics. It is dimensionless — it has no units, so it means the same thing to any alien in any unit system — and it sets the strength of the entire electromagnetic force. Nobody knows why it takes this particular value. Richard Feynman called it "one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man." Wolfgang Pauli was so obsessed with 137 that when he died in hospital room number 137, colleagues found it eerie. The smallness of \alpha is a gift, though: it is precisely what makes the vertex-counting expansion converge, and what let QED predict the electron's magnetic moment to twelve decimal places — the most accurately tested prediction in all of science.

Two traps snare almost everyone the first time.

Trap 1 — "the wavy internal line is a real photon I could catch." No. An internal line is a virtual particle: it is off mass-shell, exists only between the two vertices, and can never reach a detector. It is a term inside an integral, not a flash of light. A photon you could actually detect must be an external line (a loose end), like the two photons in e^-e^+ \to \gamma\gamma. If the line joins two vertices, it is virtual — full stop.

Trap 2 — "the diagram shows the path the particles take through space." Also no. The angles, lengths and curviness of the lines are meaningless doodling — only the connectivity matters. A Feynman diagram is a cartoon standing in for one term in a quantum amplitude; the particle does not "really" travel along that squiggle any more than a subway map is a photograph of the tunnels. And the backward-pointing arrow on an antiparticle does not mean it literally moves back in time — it is a compact way of writing the antiparticle's maths. Read connectivity and vertices, never trajectories.