Feynman diagram
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A Feynman diagram is a method invented by American physicist Richard Feynman for performing calculations in quantum field theory. Particles are represented by lines, which can be drawn in various ways depending on the type of particle being depicted. A point where lines connect to other lines is called a vertex. The axes of a diagram represent time and space. There is no fixed convention as to which axis is which. Time is sometimes depicted as increasing from left to right; alternately time can be depicted as increasing from bottom to top.
Feynman diagrams translate directly into a mathematical tool for making calculations. Each line and vertex corresponds to a mathematical term. From these terms, one can calculate the probability of the interaction depicted by the diagram.
The interaction between two particles is quantified by the cross section corresponding to their collision, essentially the probability of the interaction occurring. If the strength 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 that sounds like the following:
- two particles were moving freely with some relative speed (one draws two lines --edges -- going in some general direction),
- they met each other (the two lines meet at a first point -- vertex),
- took a stroll together on a common path (the lines merge into one)
- 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 -- sometimes in a different style to symbolize the change experienced by the particles).
This story can be drawn as a diagram which is generally easier to remember than the corresponding mathematical formula in the Dyson series. These diagrams are called Feynman diagrams. They are 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.
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[edit] 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.
[edit] Penguin diagrams
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. According to John Ellis:[1]
Mary K. [Gaillard], Dimitri [Nanopoulos] and I first got interested in what are now called penguin diagrams while we were studying CP violation in the Standard Model in 1976... The penguin name came in 1977, as follows.
In the spring of 1977, Mike Chanowitz, Mary K and I wrote a paper on GUTs predicting the b quark mass before it was found. When it was found a few weeks later, Mary K, Dimitri, Serge Rudaz and I immediately started working on its phenomenology. That summer, there was a student at CERN, Melissa Franklin who is now an experimentalist at Harvard. One evening, she, I and Serge went to a pub, and she and I started a game of darts. We made a bet that if I lost I had to put the word penguin into my next paper. She actually left the darts game before the end, and was replaced by Serge, who beat me. Nevertheless, I felt obligated to carry out the conditions of the bet.
For some time, it was not clear to me how to get the word into this b quark paper that we were writing at the time. Then, one evening, after working at CERN, I stopped on my way back to my apartment to visit some friends living in Meyrin where I smoked some illegal substance. Later, when I got back to my apartment and continued working on our paper, I had a sudden flash that the famous diagrams look like penguins. So we put the name into our paper, and the rest, as they say, is history.
Thorsten Ohl's paper on generating Feynman diagrams with LaTeX (see the external links) illustrates their penguin-like shape.
[edit] Alternative names
Murray Gell-Mann always referred to Feynman diagrams as Stückelberg diagrams, after a Swiss physicist, Ernst Stückelberg, who devised a similar notation.[2]
Historically they were also called Feynman-Dyson diagrams or Dyson graphs[3].
[edit] 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.
[edit] Mathematical details
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 obtained by summing over all diagrams.
[edit] Examples
[edit] 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 is moving forwards in time; however, that fermion could also be interpreted as its antiparticle travelling backwards in time, as there is no mathematical difference between the two concepts. This applies to all particles and antiparticles.
[edit] 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".
[edit] See also
[edit] References
- ^ http://arxiv.org/abs/hep-ph/9510397
- ^ http://www.theatlantic.com/issues/2000/07/johnson.htm
- '^ Gribbin, John and Mary. Richard Feynman: A Life in Science, Penguin-Putnam, 1997 Ch 5.
- 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 0-521-45692-4 (expanded, updated version of above)
[edit] 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 FeynDiagram C++ library that produces PostScript output.
- Feynman Diagram Examples using Thorsten Ohl's Feynmf LaTeX package.
