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Feynman diagrams are a systematic graphical representation of terms in the perturbative expansion of scattering amplitudes. Each diagram encodes a precise mathematical expression: external lines represent incoming/outgoing particles, internal lines are propagators, and vertices carry coupling constants. The Feynman rules translate any diagram into an integral.
Feynman diagrams are not merely illustrations -- they are a precise computational tool. Each diagram corresponds to a specific term in the perturbative expansion of a scattering amplitude, and the Feynman rules translate the diagram into a mathematical expression that can be evaluated. The rules are derived rigorously from Wick's theorem and the interaction Lagrangian, but once derived, they can be applied mechanically without re-deriving them each time.
The rules for any theory are: (1) draw all topologically distinct diagrams with the correct external particles at the desired order in the coupling constant; (2) for each external line, write the appropriate wave function factor (spinor, polarization vector, or 1 for scalars); (3) for each internal line, write the propagator for that field type; (4) for each vertex, write the vertex factor derived from the interaction Lagrangian; (5) impose four-momentum conservation at each vertex; (6) integrate over each undetermined internal momentum with d4p/(2pi)4; (7) include a factor of (-1) for each closed fermion loop; (8) divide by the symmetry factor of the diagram.
The symmetry factor accounts for the fact that different contractions in Wick's theorem can produce the same diagram. If a diagram has S internal symmetries (permutations of internal lines and vertices that leave the topology unchanged), the amplitude must be divided by S to avoid overcounting. For simple diagrams the symmetry factor is 1, but loops with identical propagators or vertices with multiple identical fields can have larger symmetry factors.
The organizing principle is the coupling constant. In QED, each vertex contributes a factor of e (the electron charge), and each loop introduces an additional power of alpha = e2/(4pi) approximately 1/137. Tree-level diagrams (no loops) give the leading contribution. One-loop diagrams are suppressed by alpha, two-loop diagrams by alpha2, and so on. This is why perturbation theory converges rapidly for QED -- higher-order corrections are systematically smaller. The same structure applies to any weakly coupled theory, though for strongly coupled theories (like QCD at low energies), the perturbative expansion breaks down and non-perturbative methods are needed.