Editing Feynman diagram (section)

==== Feynman diagrams ====
The expansion of the action in powers of {{mvar|X}} gives a series of terms with progressively higher number of {{mvar|X}}s. The contribution from the term with exactly {{mvar|n}} {{mvar|X}}s is called {{mvar|n}}th order.

The {{mvar|n}}th order terms has:
# {{math|4''n''}} internal half-lines, which are the factors of {{math|''φ''(''k'')}} from the {{mvar|X}}s. These all end on a vertex, and are integrated over all possible {{mvar|k}}.
# external half-lines, which are the come from the {{math|''φ''(''k'')}} insertions in the integral.

By Wick's theorem, each pair of half-lines must be paired together to make a ''line'', and this line gives a factor of

:<math> \frac{\delta(k_1 + k_2)}{k_1^2} </math>

which multiplies the contribution. This means that the two half-lines that make a line are forced to have equal and opposite momentum. The line itself should be labelled by an arrow, drawn parallel to the line, and labeled by the momentum in the line {{mvar|k}}. The half-line at the tail end of the arrow carries momentum {{mvar|k}}, while the half-line at the head-end carries momentum {{mvar|−''k''}}. If one of the two half-lines is external, this kills the integral over the internal {{mvar|k}}, since it forces the internal {{mvar|k}} to be equal to the external {{mvar|k}}. If both are internal, the integral over {{mvar|k}} remains.

The diagrams that are formed by linking the half-lines in the {{mvar|X}}s with the external half-lines, representing insertions, are the Feynman diagrams of this theory. Each line carries a factor of {{math|{{sfrac|1|''k''<sup>2</sup>}}}}, the propagator, and either goes from vertex to vertex, or ends at an insertion. If it is internal, it is integrated over. At each vertex, the total incoming {{mvar|k}} is equal to the total outgoing {{mvar|k}}.

The number of ways of making a diagram by joining half-lines into lines almost completely cancels the factorial factors coming from the Taylor series of the exponential and the 4! at each vertex.