A Feynman diagram is a bookkeeping device for performing calculations in quantum field theory, invented by American physicist Richard Feynman. They are sometimes also referred to as Stückelberg diagrams or (for a subset of special cases) penguin diagrams.
Motivation and History
The problem of calculating scattering cross sections in particle physics reduces to summing over the amplitudes of all possible intermediate states, in what is known as a perturbation expansion. 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.
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 does not violate relativity for deep reasons; in fact, it helps preserve causality in a relativistic spacetime.) The probability of each outcome 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 renormalization has been carried out, Feynman diagram calculations 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 #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 interpretation. It's only the infinite sum over all possible (without vacuum bubbles) feynman diagrams which gives physical results. Unfortunately, this infinite sum is only asymptotically convergent.
Mathematical details
A Feynman graph is a finite, directed colored graph (a pseudograph in general) where each vertex is either an external vertex or an internal vertex but not both. Each internal vertex is colored with an element of the set of interaction labels. Each edge is colored with an element of the set of field labels. Some field labels are unoriented, meaning the orientation of the edges colored by it is irrelevant. Otherwise, the field labels are oriented. Each external vertex has a degree of 1 and is colored with the same color as the edge incident to it and is also labeled with head/tail depending on whether or not it is the head or tail of that edge if the field label is oriented. (So that external vertices, unlike internal vertices are colored by field labels instead of interaction labels). In addition, to be a Feynman graph, every vertex must satisfy the following matching conditions :
- Each interaction label comes together with a positive integer n and with an assignment for each integer from 1 to n of a field label and if that field label is oriented, a direction (head/tail).
- Each internal vertex must have the same degree as the positive integer assigned to its color and there must be an ordering of all the edges incident to it (with loops being double counted, one for the head and the other for the tail) such that the ith edge is colored with the same color as the ith assignment of the interaction label and also, if the edge is oriented, whether or not the vertex in question is the head or tail of that edge matches the head/tail assignment.
The number of external vertices is called the number of legs.
An automorphism of a Feynman graph is basically a map from the graph to itself which preserves the graph structure and coloring. More precisely, an automorphism here is a permutation of the vertices and a permutation of the edges which preserves the incidence relation (this includes the orientation for the oriented edges) and the color assignments. The size of the automorphism group is called the symmetry factor.
A Feynman graph decomposes uniquely into a union of connected components. A connected component without any external vertices is called a vacuum bubble. A Feynman graph without any vacuum bubbles is called a bubbleless graph.
A connected component with only one external vertex is called a tadpole.
Below is a description of each connected component. We can define relation weakly connected between the vertices such that two vertices are weakly connected if and only if cutting any edge away will not separate both vertices.
Lemma: Weak connectivity is an equivalence relation.
Lemma: An edge connects different equivalence classes if and only if it is a bridge.
Lemma: There is at most one bridge connecting any two different equivalence classes.
Lemma: Each external vertex is the only element of its equivalence class.
Associated with each equivalence class which does not contain an external vertex is the subgraph of vertices in that class together with all the edges with both their heads and tails in that class called the one particle irreducible (1PI) subgraph.
Lemma: The feynman graph is the edge-disjoint union of one particle irreducible subgraphs and bridges.
We can define a (unique) new graph associated with the Feynman graph whose vertices are the external vertices and one particle irreducible subgraphs (each of them as a single vertex) and edges are the bridges called the reduced graph.
Lemma: The reduced graph associated with a connected feynman graph is a tree. Otherwise, it is a forest.
We can further reduce this tree by removing every vertex of degree two and replacing the two edges incident to that vertex with a single edge connecting the two vertices adjacent to that vertex and also removing every nonexternal vertex of degree 1 (called a tadpole vertex) and the edge incident to it (UNLESS its adjacent vertex is an external vertex in which case we do nothing or the adjacent vertex is also another tadpole vertex in which case we replace both vertices with a single vertex called the vacuum bubble vertex and remove the connecting edge)(and proceeding iteratively until all vertices of degree two and all tadpole vertices not adjacent to an external vertex or another tadpole vertex are gone). The (unique) resulting tree will not contain any vertex of degree 2 and if the number of legs is 2 or more, the only vertices of degree 1 will be the external vertices. If the number of legs is 1, we have a tadpole graph with only one external vertex and one nonexternal vertex which happens to be a tadpole vertex and only one edge connecting the two. If the number of legs is zero, we have only one vertex (the vacuum bubble vertex) and no edges.
Numerical assignment
Basically, operators are assigned to each interaction label (called coupling constants) and each field label (called bare propagators ) (these are fixed by the model)and a position/momentum is assigned to each external vertex. A contraction process is done resulting in the assignment of a value (which is usually a complex number in quantum field theory/statistical mechanics but there are other applications where the value happens to be more general) to each graph. This value often needs some regularization to be computed.
The correlation function is the sum over all bubbleless feynman graphs with fixed external vertices and fixed positions/momenta of the value computed for each graph 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.
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.
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 against or backwards in time. But using such notation only means that that fermion is not the particle travelling backwards in time but its antiparticle travelling forwards in time. Hence the particle labelled neutrino is, in fact, an antineutrino. This method works very well for all particles and antiparticles.
Numerical Assignment
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".
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.
See also
External links
Last updated: 06-01-2005 22:27:39
Last updated: 10-29-2005 02:13:46
