Feynman Diagrams & Perturbation Theory
Pictorial Wick contractions — turning interaction integrals into diagrams.
For an interacting theory $\mathcal L = \mathcal L_0 + \mathcal L_{\rm int}$, expand the S-matrix in powers of the coupling. The Dyson series
$$S = T\exp\left(-i\int d^4x\, \mathcal H_{\rm int}\right)$$is reorganized via Wick's theorem into a sum of diagrams. Rules:
- Internal lines = propagators $iG(p) = i/(p^2 - m^2 + i\epsilon)$.
- Vertices = interaction terms; each vertex carries a coupling factor.
- External lines = on-shell incoming/outgoing particles.
- Conserve four-momentum at each vertex; integrate over internal loop momenta.
Each diagram contributes an amplitude; the cross section is built from $|\mathcal M|^2$. Loops diverge — this is why we renormalize.