Editing Feynman diagram (section)

== Motivation and history ==
[[File:RichardFeynman-PaineMansionWoods1984 copyrightTamikoThiel bw.jpg|thumb|[[Richard Feynman]] in 1984]]
[[File:Kaon-Decay.svg|class=skin-invert-image|thumb|upright=1.1|In this diagram, a [[kaon]], made of an [[up quark|up quark]] and [[strange quark|strange antiquark]], decays both [[Weak interaction|weakly]] and [[strong interaction|strongly]] into three [[pion]]s, with intermediate steps involving a [[W and Z bosons|W boson]] and a [[gluon]], represented by the blue sine wave and green spiral, respectively.]]

When calculating [[scattering cross-section]]s in [[particle physics]], the interaction between particles can be described by starting from a [[free field]] that describes the incoming and outgoing particles, and including an interaction [[Hamiltonian (quantum mechanics)|Hamiltonian]] to describe how the particles deflect one another. The amplitude for scattering is the sum of each possible interaction history over all possible intermediate particle states. The number of times the interaction Hamiltonian acts is the order of the [[perturbation theory (quantum mechanics)|perturbation expansion]], and the time-dependent perturbation theory for fields is known as the [[Dyson series]]. When the intermediate states at intermediate times are energy [[Eigenvalues and eigenvectors|eigenstates]] (collections of particles with a definite momentum) the series is called [[Perturbation theory (quantum mechanics)#Time-dependent perturbation theory|old-fashioned perturbation theory]] (or time-dependent/time-ordered perturbation theory).

The Dyson series can be alternatively rewritten as a sum over Feynman diagrams, where at each vertex both the [[energy]] and [[momentum]] are [[conservation law|conserved]], but where the length of the [[four-momentum|energy-momentum four-vector]] is not necessarily equal to the mass, i.e. the intermediate particles are so-called [[On shell and off shell|off-shell]]. The Feynman diagrams are much easier to keep track of than "old-fashioned" terms, because the old-fashioned way treats the particle and antiparticle contributions as separate. Each Feynman diagram is the sum of exponentially many old-fashioned terms, because each internal line can separately represent either a particle or an antiparticle. In a non-relativistic theory, there are no antiparticles and there is no doubling, so each Feynman diagram includes only one term.

Feynman gave a prescription for calculating the amplitude ([[#Feynman rules|the Feynman rules, below]]) for any given diagram from a [[Lagrangian (field theory)|field theory Lagrangian]]. Each internal line corresponds to a factor of the [[virtual particle]]'s [[propagator]]; each vertex where lines meet gives a factor derived from an interaction term in the Lagrangian, and incoming and outgoing lines carry an energy, momentum, and [[Spin (physics)|spin]].

In addition to their value as a mathematical tool, Feynman diagrams provide deep physical insight into the nature of particle interactions. Particles interact in every way available; in fact, intermediate virtual particles are allowed to propagate faster than light. The probability of each final state is then obtained by summing over all such possibilities. This is closely tied to the [[functional integral]] formulation of [[quantum mechanics]], also invented by Feynman—see [[path integral formulation]].

The naïve application of such calculations often produces diagrams whose amplitudes are [[infinity|infinite]], because the short-distance particle interactions require a careful limiting procedure, to include particle [[self-interaction]]s. The technique of [[renormalization]], suggested by [[Ernst Stueckelberg]] and [[Hans Bethe]] and implemented by [[Freeman Dyson|Dyson]], Feynman, [[Julian Schwinger|Schwinger]], and [[Sin-Itiro Tomonaga|Tomonaga]] compensates for this effect and eliminates the troublesome infinities. After renormalization, calculations using Feynman diagrams match experimental results with very high accuracy.

Feynman diagram and path integral methods are also used in [[statistical mechanics]] and can even be applied to [[classical mechanics]].<ref>{{Cite journal |first1=R. |last1=Penco |first2=D. |last2=Mauro |arxiv=hep-th/0605061 |title=Perturbation theory via Feynman diagrams in classical mechanics |journal=European Journal of Physics |volume=27 |issue=5 |pages=1241–1250 |year=2006 |doi=10.1088/0143-0807/27/5/023 |bibcode=2006EJPh...27.1241P |s2cid=2895311 }}</ref>

=== Alternate names ===
[[Murray Gell-Mann]] always referred to Feynman diagrams as '''Stueckelberg diagrams''', after Swiss physicist [[Ernst Stueckelberg]], who devised a similar notation many years earlier. Stueckelberg was motivated by the need for a manifestly covariant formalism for quantum field theory, but did not provide as automated a way to handle symmetry factors and loops, although he was first to find the correct physical interpretation in terms of forward and backward in time particle paths, all without the path-integral.<ref>{{cite news |author=George Johnson |title=The Jaguar and the Fox |url=https://www.theatlantic.com/issues/2000/07/johnson.htm |work=The Atlantic |date=July 2000 |access-date=February 26, 2013}}</ref>

Historically, as a book-keeping device of covariant perturbation theory, the graphs were called '''Feynman–Dyson diagrams''' or '''Dyson graphs''',<ref>{{cite book |last1=Gribbin |first1=John |last2=Gribbin |first2=Mary |title=Richard Feynman: A Life in Science |publisher=Penguin-Putnam |year=1997 |chapter=5}}</ref> because the path integral was unfamiliar when they were introduced, and [[Freeman Dyson]]'s derivation from old-fashioned perturbation theory borrowed from the perturbative expansions in statistical mechanics was easier to follow for physicists trained in earlier methods.<ref group=lower-alpha>"It was Dyson's contribution to indicate how Feynman's visual insights could be used [...] He realized that Feynman diagrams [...] can also be viewed as a representation of the logical content of field theories (as stated in their perturbative expansions)". Schweber, op.cit (1994)</ref> Feynman had to lobby hard for the diagrams, which confused physicists trained in equations and graphs.<ref>{{cite book |first=Leonard |last=Mlodinow |title=Feynman's Rainbow |publisher=Vintage |year=2011 |page=29}}</ref>