Feynman diagram

Feynman diagrams encode the contributions of virtual particles to a particle interaction as a series expansion in increasing powers of the coupling constant.¹ In a diagram, time usually flows upward or left to right, with straight lines representing fermions, wavy lines for photons, and loops for virtual particle–antiparticle pairs. Each diagram corresponds to a specific integral (the amplitude), and the squared amplitude yields the probability of the process. The sum over all topologically distinct diagrams of a given order gives the perturbative prediction for a scattering cross section or decay rate.

Feynman developed the diagrams while working on a space–time approach to QED at Cornell University, partly to avoid the complicated algebra of earlier methods.¹ Freeman Dyson later showed that Feynman's diagrams were equivalent to the covariant perturbation theory of Julian Schwinger and Sin-Itiro Tomonaga, and he derived the rules (the Feynman rules) for converting diagrams into integrals.² The diagrams quickly became the standard tool in high-energy physics, especially after the stunning success of QED in predicting the electron's anomalous magnetic moment.

Each line in a diagram carries a momentum, and energy–momentum conservation holds at every vertex. The Feynman rules assign a specific factor for each internal line (propagator), vertex, and external leg, as well as a symmetry factor for identical particles.³ For example, the electron propagator is i/(p̸ − m + iε), and the photon propagator is −ig_μν/(k² + iε). A loop integral requires a regularization procedure (e.g., dimensional regularization) to handle divergences. The diagrams thus provide an intuitive yet rigorous way to compute scattering amplitudes in quantum field theory.

Beyond QED, Feynman diagrams are used in quantum chromodynamics (QCD), the electroweak theory, and essentially all perturbative calculations in the Standard Model of particle physics.⁴ They also appear in condensed matter physics (e.g., many-body perturbation theory, diagrammatic expansions in the Hubbard model) and in statistical mechanics (e.g., cluster expansions).⁵ Modern computational tools, such as FeynArts and FormCalc, automate the generation and evaluation of large numbers of diagrams, enabling predictions for experiments at the LHC.

Feynman himself initially considered the diagrams a mnemonic device and did not publish them until 1949.¹ The diagrams can be reinterpreted in terms of worldlines in the Feynman–Schwinger formalism, and they have a deep connection with the Wick rotation and Euclidean path integrals. In the 1970s, Gerard 't Hooft and others used diagrams to study gauge theories and renormalization, leading to the discovery of the beta function of QCD. A lesser-known variant is the cut diagram used in finite-temperature field theory (the real-time formalism).⁵ Feynman diagrams also appear in pure mathematics, such as knot invariants and the construction of Kashiwara crystals, where they represent terms in certain combinatorial expansions.

Vertex

A point in a Feynman diagram where three or more particle lines meet, representing a fundamental interaction (e.g., electron–photon coupling).

Propagator

The function (or line) that describes the probability amplitude for a particle to travel from one spacetime point to another.

Virtual particle

A particle that exists only within a Feynman diagram, not directly observable, but whose exchange mediates the force between real particles.

Perturbation theory

An approximation method in quantum field theory that expands the scattering amplitude as a power series in the coupling constant, each term represented by a Feynman diagram.

Loop

A closed path in a Feynman diagram, representing virtual particle–antiparticle pairs and giving rise to quantum corrections and infinities that require renormalization.