Editing Quantum field theory (section)

===Feynman diagram===
{{Main|Feynman diagram}}

Correlation functions in the interacting theory can be written as a perturbation series. Each term in the series is a product of Feynman propagators in the free theory and can be represented visually by a [[Feynman diagram]]. For example, the {{math|''λ''<sup>1</sup>}} term in the two-point correlation function in the {{math|''ϕ''<sup>4</sup>}} theory is
:<math>\frac{-i\lambda}{4!}\int d^4z\,\lang 0|T\{\phi(x)\phi(y)\phi(z)\phi(z)\phi(z)\phi(z)\}|0\rang.</math>
After applying Wick's theorem, one of the terms is
:<math>12\cdot \frac{-i\lambda}{4!}\int d^4z\, D_F(x-z)D_F(y-z)D_F(z-z).</math>
This term can instead be obtained from the Feynman diagram

:[[File:Phi-4 one-loop.svg|200px]].

The diagram consists of

* ''external vertices'' connected with one edge and represented by dots (here labeled <math>x</math> and <math>y</math>).
* ''internal vertices'' connected with four edges and represented by dots (here labeled <math>z</math>).
* ''edges'' connecting the vertices and represented by lines.

Every vertex corresponds to a single <math>\phi</math> field factor at the corresponding point in spacetime, while the edges correspond to the propagators between the spacetime points. The term in the perturbation series corresponding to the diagram is obtained by writing down the expression that follows from the so-called Feynman rules:

# For every internal vertex <math>z_i</math>, write down a factor <math display="inline">-i \lambda \int d^4 z_i</math>.
# For every edge that connects two vertices <math>z_i</math> and <math>z_j</math>, write down a factor <math>D_F(z_i-z_j)</math>.
# Divide by the symmetry factor of the diagram.

With the symmetry factor <math>2</math>, following these rules yields exactly the expression above. By Fourier transforming the propagator, the Feynman rules can be reformulated from position space into momentum space.{{r|peskin|page1=91–94}}

In order to compute the {{math|''n''}}-point correlation function to the {{math|''k''}}-th order, list all valid Feynman diagrams with {{math|''n''}} external points and {{math|''k''}} or fewer vertices, and then use Feynman rules to obtain the expression for each term. To be precise,
:<math>\lang\Omega|T\{\phi(x_1)\cdots\phi(x_n)\}|\Omega\rang</math>
is equal to the sum of (expressions corresponding to) all connected diagrams with {{math|''n''}} external points. (Connected diagrams are those in which every vertex is connected to an external point through lines. Components that are totally disconnected from external lines are sometimes called "vacuum bubbles".) In the {{math|''ϕ''<sup>4</sup>}} interaction theory discussed above, every vertex must have four legs.{{r|peskin|page1=98}}

In realistic applications, the scattering amplitude of a certain interaction or the [[decay rate]] of a particle can be computed from the [[S-matrix]], which itself can be found using the Feynman diagram method.{{r|peskin|page1=102–115}}

Feynman diagrams devoid of "loops" are called tree-level diagrams, which describe the lowest-order interaction processes; those containing {{math|''n''}} loops are referred to as {{math|''n''}}-loop diagrams, which describe higher-order contributions, or radiative corrections, to the interaction.{{r|zee|page1=44}} Lines whose end points are vertices can be thought of as the propagation of [[virtual particle]]s.{{r|peskin|page1=31}}