Everything about Feynman Diagram totally explained
A
Feynman diagram is a tool invented by American
physicist Richard Feynman for performing
scattering 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 an interaction vertex, or vertex for short. Lines fall into three categories: internal lines (which connect two vertices), incoming lines (which extend from "the past" to a vertex and represent the initial noninteracting state) and outgoing lines (which extend from a vertex to "the future" and represent the final noninteracting state). Most commonly the bottom of the diagram represents the past and the top of the diagram represents the future.
Feynman diagrams are a pictorial representation of a
term in a
perturbative expansion of the
scattering amplitude for the experiment defined by the incoming and outgoing lines. In some quantum field theories (notably
quantum electrodynamics), one can obtain an excellent approximation of the scattering amplitude from a few terms of the perturbative expansion, corresponding to a few simple Feynman diagrams with the same incoming and outgoing lines connected by different vertices and internal lines.
The method, although originally invented for particle physics, has somewhat informally been adopted in solid-state physics, where the behavior of
phonons may be expressed in analogy to that of
photons, for example in the theory of superconductivity.
Feynman diagrams are frequently confused with
spacetime diagrams and
bubble chamber images because of their visual similarity, but the connection is weak. Feynman diagrams are merely
graphs; there's no concept of position or space in a Feynman diagram, and there's no concept of time aside from the distinction between incoming and outgoing lines. Additionally, only a
collection of Feynman diagrams can be said to represent any given particle interaction; particles don't choose a particular diagram each time they interact.
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 don't show up in Feynman diagrams.
In addition to their value as a mathematical tool, 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. 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.
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:
Thorsten Ohl's paper on generating Feynman diagrams with
LaTeX (see the
external links) illustrates their penguin-like shape.
In 1991 and 1994, the CLEO collaboration provided the first experimental evidence for these processes.
Alternative names
Murray Gell-Mann always referred to Feynman diagrams as
Stückelberg diagrams, after a Swiss physicist,
Ernst Stückelberg, who devised a similar notation.
Historically they were also called
Feynman-Dyson diagrams or
Dyson graphs.
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're treated as a single multicomponent field with "flavors", then there would only be one type). The edges, (for example,
propagators) are
tensors of
rank (2,0) in deWitt's notation (for example, 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 doesn't 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
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.
Further Information
Get more info on 'Feynman Diagram'.
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