Editing Feynman diagram (section)

==== Interaction ====
Interactions are represented by higher order contributions, since quadratic contributions are always Gaussian. The simplest interaction is the quartic self-interaction, with an action:

:<math> S = \int \partial^\mu \phi \partial_\mu\phi +\frac {\lambda}{ 4!} \phi^4. </math>

The reason for the combinatorial factor 4! will be clear soon. Writing the action in terms of the lattice (or continuum) Fourier modes:

:<math> S = \int_k k^2 \left|\phi(k)\right|^2 + \frac{\lambda}{4!}\int_{k_1k_2k_3k_4} \phi(k_1) \phi(k_2) \phi(k_3)\phi(k_4) \delta(k_1+k_2+k_3 + k_4) = S_F + X. </math>

Where {{mvar|S<sub>F</sub>}} is the free action, whose correlation functions are given by Wick's theorem. The exponential of {{mvar|S}} in the path integral can be expanded in powers of {{mvar|λ}}, giving a series of corrections to the free action.

:<math> e^{-S} = e^{-S_F} \left( 1 + X + \frac{1}{2!} X X + \frac{1}{3!} X X X + \cdots \right) </math>

The path integral for the interacting action is then a [[power series]] of corrections to the free action. The term represented by {{mvar|X}} should be thought of as four half-lines, one for each factor of {{math|''φ''(''k'')}}. The half-lines meet at a vertex, which contributes a delta-function that ensures that the sum of the momenta are all equal.

To compute a correlation function in the interacting theory, there is a contribution from the {{mvar|X}} terms now. For example, the path-integral for the four-field correlator:

:<math>\left\langle \phi(k_1) \phi(k_2) \phi(k_3) \phi(k_4) \right\rangle = \frac{\displaystyle\int e^{-S} \phi(k_1)\phi(k_2)\phi(k_3)\phi(k_4) D\phi }{ Z}</math>

which in the free field was only nonzero when the momenta {{mvar|k}} were equal in pairs, is now nonzero for all values of {{mvar|k}}. The momenta of the insertions {{math|''φ''(''k<sub>i</sub>'')}} can now match up with the momenta of the {{mvar|X}}s in the expansion. The insertions should also be thought of as half-lines, four in this case, which carry a momentum {{mvar|k}}, but one that is not integrated.

The lowest-order contribution comes from the first nontrivial term {{math|''e''<sup>−''S<sub>F</sub>''</sup>''X''}} in the Taylor expansion of the action. Wick's theorem requires that the momenta in the {{mvar|X}} half-lines, the {{math|''φ''(''k'')}} factors in {{math|X}}, should match up with the momenta of the external half-lines in pairs. The new contribution is equal to:

:<math> \lambda \frac{1}{ k_1^2} \frac{1}{ k_2^2} \frac{1}{ k_3^2} \frac{1}{ k_4^2}\,. </math>

The 4! inside {{math|X}} is canceled because there are exactly 4! ways to match the half-lines in {{mvar|X}} to the external half-lines. Each of these different ways of matching the half-lines together in pairs contributes exactly once, regardless of the values of {{math|''k''<sub>1,2,3,4</sub>}}, by Wick's theorem.