Perturbative Expansion and Feynman Rules

Feynman rules are a compact representation of a perturbative expansion, not independent postulates. This chapter derives graphs from time-ordered evolution and source differentiation, explaining their combinatorial weights, fermion signs, contact terms, and gauge-consistency checks.

Enter this chapter

Begin with The Interaction Picture and Dyson Series to identify the expansion and its state prescription. The functional derivation later provides an independent check of the same coefficients.

Topics

1. The Interaction Picture and Dyson Series — Constructs time-ordered evolution around a solvable reference system.
2. Wick Expansion for Interacting Fields — Applies free-field contractions to each perturbative interaction insertion.
3. Diagrammatics and Symmetry Factors — Counts relabelings without overcounting a graph.
4. Momentum-Space Feynman Rules — Translates the declared quadratic kernel and interactions into momentum rules.
5. Fermion Signs and Closed Loops — Derives reordering signs and the closed-fermion-loop factor.
6. Derivative Interactions and Contact Terms — Tracks momentum factors and local terms from derivative couplings.
7. Connected, Disconnected, and Vacuum Diagrams — Separates connected external contributions from exponentiated vacuum factors.
8. Functional Derivation of Perturbation Theory — Reproduces the expansion using derivatives of a Gaussian functional.
9. Gauge-Fixed Perturbation Rules, Ghost Diagrams, and Identity Checks — Derives gauge-fixed vertices and tests identities on complete quantities.

How the ideas fit together

Wick’s theorem is applied to the free interaction-picture fields inside each Dyson coefficient. It does not assert that the full interacting state is Gaussian. Vacuum bubbles cancel from normalized vacuum correlators, while taking the logarithm organizes connected source-dependent graphs. See Tong 2006–2007, §§ 3.3 and 3.7.

The action’s factorials, expansion factorials, and contraction counts must all describe the same labeled problem. A graph’s appearance alone cannot fix its weight. Derivative interactions and gauge fixing add further terms that must be retained before checking a physical amplitude.

Review the chapter

For $\mathcal L_{\mathrm{int}}=-\lambda\phi^4/4!$, why is the single four-scalar vertex proportional to $-i\lambda$ rather than $-i\lambda/4!$?

Solution

The first-order expansion supplies $-i\lambda/4!$, and contracting four labeled external legs with the four vertex fields gives $4!$ assignments. They cancel. Identical-graph automorphisms in other processes require a fresh count; this cancellation is not a universal symmetry factor.

Where to go next

Continue to Asymptotic States, LSZ, and Scattering Observables to identify which amputated correlators become physical amplitudes.

References

• David Tong, Quantum Field Theory, University of Cambridge Part III lecture notes, 2006–2007, Chapter 3. Author’s online text.