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## Feynman graphA ## Additional recommended knowledge(The most common use is when each field has quanta (particles) associated with it - as the quantum of the electromagnetic field is a photon. For simplicity, we will discuss them in terms of that case. Those who would apply Feynman diagrams in other subjects or categories should translate the explanation with an appropriate isomorphism.) A Feynman graph is a finite, partially directed, colored pseudograph which satisfies certain conditions.(" Thus, there is a set of Both edges and vertices are colored. Internal vertices are colored with the interaction label corresponding to the interaction they represent; similarly, edges are colored with field labels. The edges with orientable field labels (and only those) are directed; give the label the same direction as its edge. An external vertex is colored with the field label of its incident edge (and if the field label is orientable, with a head or a tail). Each type of interaction must involve the right number of the right type of particles. To represent this, each interaction label has a An internal vertex, which is colored with an interaction label, A pseudograph is a Feynman graph if it is finite, partially directed, and colored, such that: Every edge is colored with a field label; an edge is directed if and only if its color is orientable. There are no isolated vertices; each vertex of degree 1 is colored with the field label (and orientation, if any) of its incident edge; every vertex of degree more than one is colored with an interaction label and satisfies the corresponding matching condition. An automorphism of a Feynman graph is a map from the graph onto itself which preserves the coloring and the graph structure (both the orientation of edges and their incidence with vertices) and leaves the external vertices untouched. The size of the automorphism group is called the
A Feynman graph decomposes uniquely into a union of connected components. A
We analyse the connected components of a Feynman graph as follows: Define a relation The following lemmata are then obvious, at least in the sense of Laplace: - Weak connection is an equivalence relation.
- An edge connects different equivalence classes if and only if it is a bridge.
- There is at most one bridge connecting any two different equivalence classes.
- Each external vertex is the only element of its equivalence class.
A - A Feynman graph is the edge-disjoint union of its external vertices, one particle irreducible subgraphs, and bridges.
The - The reduced graph of a Feynman graph is a forest. It is a tree if and only if the Feynman graph is connected.
Isolated vertices of the reduced graph are bubbles. Any other tree in the forest can be - if any non-external vertex has degree 1 (and is not directly adjacent to an external vertex), removing it and its incident edge; or
- if any vertex has degree 2, replacing it and its incident edges by a single edge (which joins the vertices adjacent to the original vertex);
and repeating as often as possible. Neither kind of simplication can change the external vertices. Eventually a tree will be left with no vertices of either kind. The resulting ## Numerical evaluationThe model will assign operators: one to each interaction label (called Usually, in quantum field theory and statistical mechanics, each of the operators is just a multiplication by a complex constant, so the value is their product, effectively also a complex number. In other applications, the value will be more general. The Because every such graph can be reduced uniquely into a forest of reduced trees, we can use a two step procedure to compute the correlation function. - 1. sum over 1PI graphs to get the one particle irreducible correlation functions.
- 2. compute the tadpole correlation functions and 2-point connected correlation functions.
- 3. using the intermediate values obtained from steps 1 and 2, sum over the reduced trees to get the n-point connected correlation functions
- 4. look at the forests and compute the correlation function from the connected correlation functions
Because the sum is not convergent in general, much less absolutely convergent, there might be some problems with the rearrangement. In the usual derivations of the feynman rules using perturbation theory, the infinite series is summed in the order of the power of the coupling constants (in other words, according to the number of vertices) while the 1PI method performs the summation in a different order. This has led to some occasional subtleties. See resummation. The previous algorithm used 1PIs as an intermediate step. This isn't the only possible algorithm. For instance, we might have used two particle irreducible subgraphs in an intermediate step instead. |

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Feynman_graph". A list of authors is available in Wikipedia. |