Feynman Diagrams: A Visual Language for Quantum Electrodynamics

A diagram that is also an equation

A Feynman diagram, introduced by Richard Feynman in 1948, is a picture that stands in directly for a term in the mathematical expansion of a quantum-mechanical scattering amplitude, with time usually running left to right. In quantum electrodynamics (QED), straight lines with arrows represent electrons or positrons, wavy lines represent photons, and the entire theory is built from copies of a single building block: the vertex where one electron line, one photon line, and one more electron line all meet.

Every diagram is copies of one vertex

Each vertex contributes one factor of the elementary electric charge e to the amplitude, so a diagram with V vertices is suppressed by roughly the fine-structure constant α = e²/4π ≈ 1/137 raised to a power related to V — which is why diagrams with more vertices contribute exponentially less and a perturbative expansion in increasing numbers of vertices actually converges usefully.

vertex: electron in --- * --- electron out
                        |
                     photon (emitted or absorbed)
each vertex costs one factor of e  (probability weight ∝ α)

Reading the arrows: particles versus antiparticles

An arrow pointing forward along the direction of increasing time represents a particle (an electron); an arrow pointing backward in time represents its antiparticle (a positron) — the Feynman–Stueckelberg interpretation, which lets a single mathematical propagator describe both an electron moving forward in time and a positron moving backward in time, so pair production and pair annihilation diagrams use exactly the same propagator drawn with the arrow reversed rather than needing separate machinery.

The processes this site animates

Compton scattering (an electron and a photon in, an electron and a photon out) is the simplest tree diagram, using one electron propagator between the two vertices. Møller scattering (electron-electron to electron-electron) and Bhabha scattering (electron-positron to electron-positron) each need two diagrams rather than one, because the particles involved can reach the same final state through more than one distinct vertex arrangement. Pair production converts a photon into an electron-positron pair near a nucleus that absorbs the recoil momentum, annihilation is the exact reverse process turning an electron-positron pair back into photons, and a self-energy loop is a single electron emitting a virtual photon and then reabsorbing it a moment later — a one-loop quantum correction to the electron's own propagator.

Why loops make the sum hard

Every extra vertex adds a factor of α and generally makes a diagram's contribution smaller, but loop diagrams also introduce an integral over the undetermined momentum circulating inside the loop, and that integral often diverges. The resolution, renormalisation, systematically absorbs those infinities into a redefinition of the electron's measured mass and charge, leaving finite, testable predictions behind. The reward for doing this correctly is one of the most precisely verified results in all of physics: QED's prediction for the electron's magnetic moment agrees with experiment to roughly twelve significant digits.

Frequently asked questions

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