A Feynman diagram is a tool used by physicists to calculate probabilities of reactions between elementary particles.
In theoretical quantum physics, a physical quantity—like a scattering cross section—is often calculated using the so-called perturbation theory, where the quantity is represented as an infinite series, that is, a sum of ever decreasing terms. A Feynman diagram is an intuitive graphical representation of a term in this series.
More precisely, a Feynman diagram is a graphical representation of a perturbative contribution to the transition amplitude or correlation function of a quantum mechanical or statistical field theory.
Within the canonical formulation of quantum field theory a Feynman diagram represents a term in the Wick's expansion of the perturbative S-matrix.
Alternatively, the path integral formulation of quantum field theory represents the transition amplitude as a weighted sum of all possible histories of the system from the initial to the final state, in terms of either particles or fields. A Feynman diagram is then a contribution of a particular class of particle paths, which join and split as described by the diagram.
The transition amplitude is then given as the matrix element of the S-matrix between the initial and the final states of the quantum system.
Feynman diagrams were developed by Richard Feynman, and are named after him. There are many applications, primarily in quantum field theory, but also in other fields, e.g. in solid-state theory.
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