In this Feynman diagram, an electron and positron annihilate and become a quark-antiquark pair. Then one radiates a gluon. (Time goes left to right.)

A Feynman diagram is a method for performing calculations in quantum field theory, invented by American physicist Richard Feynman. They are also (rarely) referred to as Stückelberg diagrams or (for a subset of special cases) penguin diagrams. Image File history File links Radiate_gluon. ... Image File history File links Radiate_gluon. ... Properties The electron is a lightweight fundamental subatomic particle that carries a negative electric charge. ... The first detection of the positron in 1932 by Carl D. Anderson The positron is the antiparticle or the antimatter counterpart of the electron. ... Annihilation occurs when a particle collides with an antiparticle. ... Quarks are one of the two basic constituents of matter in the Standard Model of particle physics. ... This article is being considered for deletion in accordance with Wikipedias deletion policy. ... In particle physics, gluons are vector gauge bosons that mediate strong color charge interactions of quarks in quantum chromodynamics (QCD). ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ... A physicist is a scientist trained in physics. ... Richard Phillips Feynman (May 11, 1918 – February 15, 1988) (surname pronounced FINE-man; in IPA) was an influential American physicist known for expanding greatly on the theory of quantum electrodynamics, quark theory, and the physics of the superfluidity of supercooled liquid helium. ...

The interaction between two particles is quantified by the cross section corresponding to their collision. If this interaction is not too large, i.e. if it can be tackled via perturbation theory, this cross section (or more precisely the corresponding time evolution operator, propagator or S matrix) can be expressed as a sum of terms (the Dyson series) which can be described as a short story in time sounding like: Interaction is a kind of action which occurs as two or more objects have an effect upon one another. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem. ... Please wikify (format) this article as suggested in the Guide to layout and the Manual of Style. ... In quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. ... The S-matrix is the matrix in quantum mechanics or quantum field theory that relates the final state in the infinite future and the initial state in the infinite past. ... It has been suggested that this article or section be merged with Dyson operator. ...

• (once upon a time) two particles were moving freely with some relative speed (one draws two lines --edges -- going upwards),
• they met each other (the two lines meet at a first point -- vertex),
• took a stroll together on a common path (the lines merge in one vertical line)
• and, then separated again (second vertex)
• but they realized their speed had changed and they were not really the same anymore (two lines are drawn upwards coming from the last vertex -- maybe in a different style for symbolizing the change experienced by the particles).

And this nice story can be drawn as a diagram (where the evolving time is the upwards direction) which is much easier to remember than the corresponding mathematical formula in the Dyson series. These diagrams are called Feynman diagrams. They are of course meaningful only if the Dyson series converges fast. Their easy story telling character and the similarity with the early bubble chamber experiments have made the Feynman diagrams very popular. A labeled graph with 6 vertices and 7 edges. ... This article just presents the basic definitions. ... It has been suggested that this article or section be merged with Dyson operator. ... It has been suggested that this article or section be merged with Dyson operator. ... A bubble chamber A bubble chamber is a vessel filled with a superheated transparent liquid used to detect electrically charged particles moving through it. ...

Motivation and history

The problem of calculating scattering cross sections in particle physics reduces to summing over the amplitudes of all possible intermediate states (each corresponding to one term in the perturbation expansion which is known as the Dyson series). These states can be represented by Feynman diagrams, which are much easier to keep track of than frequently tortuous calculations. Feynman showed how to calculate diagram amplitudes using so-called Feynman rules, which can be derived from the system's underlying Lagrangian. Each internal line corresponds to a factor of the corresponding 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 provide constraints on energy, momentum and spin. A Feynman diagram is therefore a symbolic notation for the factors appearing in each term of the Dyson series. In particle physics, scattering is a class of phenomena by which particles are deflected by collisions with other particles. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Particles explode from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. ... It has been suggested that this article or section be merged with Dyson operator. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... In physics, a virtual particle is a particle-like abstraction used in some models of quantum field theory. ... In quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. ... In classical mechanics, momentum (pl. ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is generated by the motion of its center of mass about an external point. ... It has been suggested that this article or section be merged with Dyson operator. ...

However, being a perturbative expansion, nonperturbative effects do not show up in Feynman diagrams. In quantum mechanics, perturbation theory is a set of approximation schemes for describing a complicated quantum system in terms of a simpler one. ...

In addition to their value as a mathematical technology, Feynman diagrams provide deep physical insight to the nature of particle interactions. Particles interact in every way available; in fact, intermediate virtual particles are allowed to propagate faster than light. (This is due to the Heisenberg Uncertainty Principle and does not violate relativity for deep reasons; in fact, it helps preserve causality in a relativistic spacetime.) 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 Heisenberg uncertainty principle or just uncertainty principle (sometimes also the Heisenberg indeterminacy principle - a name given to it by N. Bohr) is one of the cornerstones of quantum mechanics. ... For a non-technical introduction to the topic, please see Introduction to Special relativity. ... In physics, spacetime is a model that combines three-dimensional space and one-dimensional time into a single construct called the space-time continuum, in which time plays the role of the 4th dimension. ... In physics, functional integration is integration over certain infinite-dimensional spaces. ... For a non-technical introduction to the topic, please see Introduction to Quantum mechanics. ... This article or section is in need of attention from an expert on the subject. ...

The naïve application of such calculations often produces diagrams whose amplitudes are infinite, which is undesirable in a physical theory. The problem is that particle self-interactions are erroneously ignored. The technique of renormalization, pioneered by Feynman, Schwinger, and Tomonaga compensates for this effect and eliminates the troublesome infinite terms. After such renormalization, calculations using Feynman diagrams often match experimental results with very good accuracy. The word infinity comes from the Latin infinitas or unboundedness. It refers to several distinct concepts which arise in theology, philosophy, mathematics and everyday life. ... Figure 1. ... Julian Seymour Schwinger (February 12, 1918 -- July 16, 1994) was an American theoretical physicist. ... Sin-Itiro Tomonaga Sin-Itiro Tomonaga or Shinichirō Tomonaga (朝永 振一郎 Tomonaga Shinichirō, March 31, 1906–July 8, 1979) was a Japanese physicist, influential in the development of quantum electrodynamics, work for which he was jointly awarded the Nobel Prize in Physics in 1965 along with Richard Feynman and Julian Schwinger. ...

Feynman diagram and path integral methods are also used in statistical mechanics. Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ...

Alternative names

Example of a penguin diagram

Murray Gell-Mann always referred to Feynman diagrams as Stückelberg diagrams, after a Swiss physicist, Ernst Stückelberg, who devised a similar notation[1]. Image File history File links Pinguindiagramm. ... Image File history File links Pinguindiagramm. ... Murray Gell-Mann at Harvard University Murray Gell-Mann (born September 15, 1929) is an American physicist who received the 1969 Nobel Prize in physics for his work on the theory of elementary particles. ... Ernst Carl Gerlach Stueckelberg (February 1, 1905, Basel - September 4, 1984, Basel) was a Swiss mathematician and physicist. ...

John Ellis was the first to refer to a certain class of Feynman diagrams as penguin diagrams, due in part to their shape, and in part to a legendary bar-room bet with Melissa Franklin (the loser reportedly had to incorporate the term "penguin" into their next research paper). Thorsten Ohl's paper on generating Feynman diagrams with LaTeX (see the external links) illustrates their penguin-like shape. John Ellis is a British theoretical physicist born in 1946. ... Melissa Franklin is an experimental particle physicist and professor at Harvard University. ...

Historically they were also called Feynman-Dyson diagrams.

Interpretation

Feynman diagrams are really a graphical way of keeping track of deWitt indices, much like Penrose's graphical notation for indices in multilinear algebra. There are several different types for the indices, one for each field (this depends on how the fields are grouped; for instance, if the up quark field and down quark field are treated as different fields, then there would be different type assigned to both of them but if they are treated as a single multicomponent field with "flavors", then there would only be one type). The edges, (i.e. propagators) are tensors of rank (2,0) in deWitt's notation (i.e. with two contravariant indices and no covariant indices), while the vertices of degree n are rank n covariant tensors which are totally symmetric among all bosonic indices of the same type and totally antisymmetric among all fermionic indices of the same type and the contraction of a propagator with a rank n covariant tensor is indicated by an edge incident to a vertex (there is no ambiguity in which "slot" to contract with because the vertices correspond to totally symmetric tensors). The external vertices correspond to the uncontracted contravariant indices. In physics, we often deal with classical models where the dynamical variables are a collection of functions {φα}α over a d-dimensional space/spacetime manifold M where α is the flavor index. ... Abstract index notation is a mathematical notation for tensors, and more generally spinors, which uses indices to indicate their type. ... In mathematics, multilinear algebra extends the methods of linear algebra. ... Type has historically had the following uses: In biology, a type is the specimen or specimens upon which an original species description is based. ... In quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... Note: This is a fairly abstract mathematical approach to tensors. ... Contravariant is a mathematical term with a precise definition in tensor analysis. ... In category theory, see covariant functor. ... In multilinear algebra, a tensor contraction is a sum of products of scalar components of one or more tensors caused by applying the summation convention to a pair of dummy indices which are bound to each other in an expression. ... Contravariant is a mathematical term with a precise definition in tensor analysis. ...

A derivation of the Feynman rules using Gaussian functional integrals is given in the functional integral article. In physics, functional integration is integration over certain infinite-dimensional spaces. ... In physics, functional integration is integration over certain infinite-dimensional spaces. ...

Each Feynman diagram on its own does not have a physical significance. It's only the infinite sum over all possible (bubble-free) Feynman diagrams which gives physical results. Unfortunately, this infinite sum is only asymptotically convergent. In mathematics an asymptotic expansion, asymptotic series or Poincaré expansion is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point. ...

Mathematical details

See main article: Feynman graph A Feynman graph is a graph suitable to be a Feynman diagram in a particular application of quantum field theory. ...

A Feynman diagram can be considered as a graph. When considering a field composed of particles, the edges will represent (sections) of particle world lines; the vertices represent virtual interactions. Since only certain interactions are permitted, the graph is constrained to have only certain types of vertices. The type of field of an edge is its field label; the permitted types of interaction are interaction labels. This article just presents the basic definitions. ... World line of the orbit of the Earth depicted in two spatial dimensions X and Y (the plane of the Earth orbit) and a time dimension, usually put as the vertical axis. ... This article just presents the basic definitions. ... Interaction is a kind of action which occurs as two or more objects have an effect upon one another. ...

The value of a given diagram can be derived from the graph; the value of the interaction as a whole is obtained by summing over all diagrams.

Examples

beta decay File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...

Beta decay

To the right is the Feynman diagram for beta decay. The straight lines in the diagrams represent fermions, while the wavy line represents virtual bosons. In this particular case, the diagram is set in the manifold spacetime, where the y-coordinate is time and the x-coordinate is space; the x-coordinate also represents the "location" for some interaction (think collision) of particles. As time runs along the y-coordinate of the diagram, the neutrino looks as if it is moving backwards in time; however, that fermion is normally interpreted not as the particle travelling backwards, but its antiparticle travelling forwards in time. (There is no mathematical difference between the two concepts.) Hence the particle labelled neutrino is, in fact, an antineutrino. This applies to all particles and antiparticles. In nuclear physics, beta decay (sometimes called neutron decay) is a type of radioactive decay in which a beta particle (an electron or a positron) is emitted. ... In particle physics, fermions, (named after Enrico Fermi), are particles with semi-integer spin. ... In physics, bosons, named after Satyendra Nath Bose, are particles with integer spin. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In physics, spacetime is a model that combines three-dimensional space and one-dimensional time into a single construct called the space-time continuum, in which time plays the role of the 4th dimension. ... The neutrino is an elementary particle. ... Please wikify (format) this article as suggested in the Guide to layout and the Manual of Style. ...

Quantum electrodynamics

In QED, there are two field labels, called "electron" and "photon". "Electron" is oriented while "photon" is unoriented. There is only one interaction label with degree 3 called "γ" to which is assigned a "photon", an "electron" "head" and an "electron" "tail". Quantum electrodynamics (QED) is a relativistic quantum field theory of electromagnetism. ... Properties The electron is a lightweight fundamental subatomic particle that carries a negative electric charge. ... In quantum physics, the photon (from Greek φως, phōs, meaning light) is the quantum of the electromagnetic field (light). ...

Real φ⁴

In (real) φ⁴, there is only one field label, called "φ" which is unoriented. There is also only one interaction label with degree 4 called "λ" to which is assigned four "φ"'s. In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... This article is in need of attention from an expert on the subject. ...

the feynman rules for phi to the fourth I, the creator of this image, hereby release it into the public domain. ...

See also

• Stückelberg-Feynman interpretation

The Stückelberg-Feynman interpretation, named for Ernst Stueckelberg and Richard Feynman, of antimatter asserts that antiparticles can be treated to be normal particles traveling backwards in time. ...

References

• Gerardus 't Hooft, Martinus Veltman, Diagrammar, CERN Yellow Report 1973, online
• David Kaiser, Drawing Theories Apart: The Dispersion of Feynman Diagrams in Postwar Physics, Chicago: University of Chicago Press, 2005. ISBN 0-226-42266-6
• Martinus Veltman, Diagrammatica: The Path to Feynman Diagrams, Cambridge Lecture Notes in Physics, ISBN 0521456924 (expanded, updated version of above)

External links

• Feynman diagram page at SLAC
• AMS article: "What's New in Mathematics: Finite-dimensional Feynman Diagrams"
• WikiTeX supports editing Feynman diagrams directly in Wiki articles.
• Drawing Feynman diagrams with LaTeX and METAFONT, from a CERN site
• Drawing Feynman diagrams with FeynDiagram C++ library that produces PostScript output.
• Feynman Rules.it a kind and detailed introduction for Italian students