Feynman diagram

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File:Radiate gluon.jpg
In this Feynman diagram, electrons annihilate and become a quark-antiquark pair. Then one radiates a gluon. (Time goes left to right.)

A Feynman diagram is a bookkeeping device 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.

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:

  • (once upon a time) two particle 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.

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.

However, being a perturbative expansion. nonperturbative effects do not show up in feynman diagrams.

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 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.

Feynman diagram and path integral methods are also used in statistical mechanics.

Alternative names

Murray Gell-Mann always referred to Feynman diagrams as Stückelberg diagrams, after a Swiss physicist, Ernst Stückelberg, who devised a similar notation[1].

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.

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.

A derivation of the Feynman rules using Gaussian functional integrals is given in the functional integral article.

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.

Mathematical details

See main article: Feynman graph

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.

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

Examples

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.

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".

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.

File:Phi4 feynrules.png

See also

Literature

  • Gerardus 't Hooft, Martinus Veltman, Diagrammar, CERN Yellow Report 1973, online
  • Martinus Veltman, Diagrammatica: The Path to Feynman Diagrams, Cambridge Lecture Notes in Physics, ISBN 0521456924 (expanded, updated version of above)

External links

de:Feynman-Diagramm es:Diagrama de Feynman fr:Diagramme de Feynman hu:Feynman-gráf pl:Diagram Feynmana ru:Диаграммы Фейнмана sl:Feynmanov diagram