Feynman Diagrams and Feynman Rules

Wick's theorem turns each order of the Dyson series into a sum of products of propagators and external-leg factors. Feynman diagrams are the pictorial bookkeeping of those terms, and the Feynman rules are the dictionary translating a diagram directly into the corresponding contribution to the amplitude $i\mathcal{M}$ — bypassing the Wick algebra entirely once the rules are known. This page derives the rules, states them for the standard theories, and explains symmetry factors and diagram topology. The $\mathcal{M}$ produced here is the input to the cross-section and decay-rate master formulas.

Conventions: $\hslash =c=1$, ${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$.

From Wick contractions to diagrams

Each term in the Wick expansion of an S-matrix element corresponds to a unique diagram:

• an internal line = a contraction between two interaction fields = a propagator $D_F$;
• an external line = a contraction of an interaction field with an in/out particle = a leg factor;
• a vertex = one factor of ${\mathcal{L}}_{\text{int}}$ (one point $x$ integrated over spacetime).

The spacetime integrals $\int d^4{x}_i$ at each vertex enforce energy–momentum conservation at every vertex (each produces a $(2\pi {)}^4{\delta }^4(\sum p)$ after Fourier transform), and the overall ${\delta }^4(p_i-p_f)$ factors out into the S-matrix definition $S=1+iT$. Working directly in momentum space is almost always simplest.

Momentum-space Feynman rules for a scalar theory

For $\mathcal{L}=\frac{1}{2}(\partial \varphi {)}^2-\frac{1}{2}{m}^2{\varphi }^2-\frac{\lambda }{4!}{\varphi }^4$, the rules for computing $i\mathcal{M}$ (all momenta flowing in; amputated external legs, as justified by LSZ):

Recipe. Draw all topologically distinct diagrams with the required external legs at the desired order in $\lambda$; for each, write the product of propagators and vertices, integrate over undetermined loop momenta, divide by the symmetry factor, and sum. The result is $i\mathcal{M}$.

QED Feynman rules

For ${\mathcal{L}}_{\text{int}}=-e\bar{\psi }{\gamma }^{\mu }\psi A_{\mu }$ (coupling the Dirac field to the photon), the rules involve spinor and Lorentz structure. The core set (Feynman gauge for the photon):

The antifermion, loop-sign, and trace rules are direct consequences of the anticommuting quantization. These are exactly the rules invoked in the Compton calculation and the QED derivation.

Connected, amputated, and 1PI diagrams

Three topological classifications organize which diagrams to compute:

• Connected — every part is linked; disconnected pieces correspond to independent sub-processes and factor out. Only connected diagrams contribute to a given scattering amplitude (linked-cluster theorem), and the disconnected vacuum bubbles cancel against the normalization denominator $⟨0|S|0⟩$.
• Amputated — external-leg propagators (and their self-energy corrections) are stripped off. The LSZ formula shows the S-matrix uses amputated correlators, with external legs replaced by on-shell wavefunction factors.
• One-particle irreducible (1PI) — cannot be split into two by cutting a single internal line. 1PI diagrams are the building blocks of the effective action and the natural objects of renormalization: the self-energy $\Sigma$ and the vertex function $\Gamma$ are sums of 1PI graphs.

Symmetry factors

The vertex normalization $1/4!$ (or $1/2$ for a cubic vertex) is chosen so that most diagrams' combinatorial factors cancel. What remains is the symmetry factor $S$ — the order of the diagram's automorphism group (the number of ways to permute internal lines and vertices leaving the diagram unchanged). Examples in ${\varphi }^4$:

Getting $S$ right is the most error-prone step in hand calculations; it is one reason the path-integral derivation of the rules — where symmetry factors emerge automatically from functional differentiation — is often preferred.

Tree level vs. loops

• Tree diagrams (no closed loops) give the leading, classical approximation to a process; all internal momenta are fixed by the external ones, no integration remains, and the result is finite. These are the subject of tree-level worked examples.
• Loop diagrams carry undetermined loop momenta $\int d^4\ell$; they encode quantum corrections and are generically divergent, launching the program of regularization and renormalization.

The number of loops $L$ counts powers of $\hslash$ (restored momentarily): the loop expansion is the semiclassical expansion.

Summary

• A Feynman diagram encodes one Wick term: lines = propagators/legs, vertices = ${\mathcal{L}}_{\text{int}}$.
• The Feynman rules read $i\mathcal{M}$ off a diagram directly.
• Connected/amputated/1PI classify which diagrams enter amplitudes, and organize renormalization.
• Symmetry factors correct for over/undercounting from vertex normalizations.

Where this leads

• Explicit tree amplitudes: tree-level worked examples.
• From amplitude to observable: the S-matrix, cross sections and decay rates, then observables.
• Loops and their divergences: loop integrals.
• The functional derivation of these same rules: the generating functional.
• The contour technique for evaluating loop integrals (residues, the $i\epsilon$ prescription): math/complex-analysis: evaluating integrals by residues.

References

• Peskin & Schroeder, An Introduction to Quantum Field Theory, Ch. 4.4–4.7.
• Weinberg, The Quantum Theory of Fields, Vol. 1, Ch. 6.
• Srednicki, Quantum Field Theory, Ch. 9–10.