Feynman graph

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

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(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.("Pseudograph" means the graph can have loops and multiple edges; "partially directed" means, in general, some edges are directed and others are not). Each edge represents a segment of the world line of a particle. There are two kinds of vertex: External vertices represent single particles entering or exiting the reaction as a whole, and so they have degree 1. The color of an edge or an exterior vertex will represent the type of particle. Internal vertices represent interactions, involving more than one particle; and so have degree more than one. The color of an internal vertex will represent the type of interaction.

Thus, there is a set of field labels, one for each type of field; also a set of interaction labels, one for each permitted type of interaction. Depending on the type of field, field labels may or may not be orientable. These labels are the colors of the graph.

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 matching condition: an ordered set of field labels. (The two directions of orientable labels are counted as distinct.)

An internal vertex, which is colored with an interaction label, satisfies the matching condition of its color if the vertex has degree equal to the size of the matching condition (as a set); and the incident edges can be ordered so that their labels are equal to the matching condition (as an ordered set). Loops are counted twice; once with each orientation.

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 symmetry factor. For the purposes of computing the symmetry factor, we have to assume that the head and the tail of even the unoriented edges are distinct.

A Feynman graph decomposes uniquely into a union of connected components. A vacuum bubble is a connected component without any external vertices . (A Feynman graph without any vacuum bubbles is a bubbleless graph.) A tadpole is a connected component with only one external vertex. The external vertices (each with its single incident edge) are legs.

We analyse the connected components of a Feynman graph as follows: Define a relation weakly connected between the vertices: two vertices are weakly connected if and only if no edge can be cut so as separate the vertices. (In graph theory, this is weakly 2-line-connected.)

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 one particle irreducible (1PI) subgraph of a Feynman graph is the subgraph induced by an equivalence class which is not an external vertex.

• A Feynman graph is the edge-disjoint union of its external vertices, one particle irreducible subgraphs, and bridges.

The reduced graph of a Feynman graph is the graph produced by identifying all equivalent vertices (that is, each vertex with all other vertices weakly connected to it). The points of the reduced graph are the external vertices and the one particle irreducible subgraphs of the Feynman graph; its edges are the bridges.

• 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 simplified by:

• 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 simplified tree is unique; it does not depend on the order of simplification. It will have the same number of legs as the corresponding component of the original Feynman graph. If this component had no legs, it was a bubble; the simplified tree will be a single vertex. If it had one leg, it was a tadpole; the simplified tree will have exactly one edge connecting two vertices (the graph K₂). If it had at least 2 legs, every other vertex of the simplified tree will have degree at least three.

Numerical evaluation

The model will assign operators: one to each interaction label (called coupling constants), and one to each field label (called bare propagators). The initial conditions provide a position/momentum for each external vertex. Contraction of these will result in a value for each graph; computation of this value often requires some regularization.

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 correlation function is the sum over all bubbleless Feynman graphs (with the given external vertices and positions/momenta) of the values computed for each graph, each divided by its symmetry factor. There are almost always infinitely many such graphs and, usually, this sum does not converge, but instead gives an asymptotic series in the coupling constants.

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.