Interacting Fields & Perturbation Theory

Free fields are exactly solvable but physically trivial — real physics requires interactions. Adding a λφ⁴/4! coupling to the scalar Lagrangian makes exact solutions impossible, but perturbation theory in small λ provides a systematic expansion. Wick's theorem converts time-ordered products of field operators into sums over all possible propagator pairings, turning the perturbation series into a precise set of Feynman diagram rules.

Key Concepts

ℒ = ½(∂φ)² − ½m²φ² − λ/4! φ⁴, with coupling λ ≪ 1 controlling the expansion
Interaction picture: fields evolve freely, states evolve under H_int
S-matrix: S = T exp(−i∫d⁴x H_int) in the interaction picture
Wick's theorem: T{φ(x₁)…φ(xₙ)} = :φ₁…φₙ: + all contractions, where contraction = ΔF
Vacuum bubbles factor out and cancel between numerator and denominator of ⟨Ω|…|Ω⟩
LSZ formula: amputate external legs and go on-shell to extract the S-matrix from Green's functions

Key Equations

Interacting Lagrangian

\mathcal{L}=\tfrac{1}{2}(\partial_\mu\phi)^2-\tfrac{1}{2}m^2\phi^2-\frac{\lambda}{4!}\phi^4

Dyson series

S=\mathcal{T}\exp\!\Bigl(-i\!\int\!d^4x\,\mathcal{H}_{\rm int}\Bigr)

Wick contraction

\langle 0|\mathcal{T}\{\phi(x)\phi(y)\}|0\rangle = \Delta_F(x-y)

LSZ formula

\langle p_1 p_2|S|k_1 k_2\rangle = \prod\frac{i(p^2-m^2)}{\sqrt{Z}}\cdot\tilde{G}^{(4)}(p_i,k_j)\Big|_{\rm on-shell}

Worked Example

Example Problem

Problem

At order λ in λφ⁴ theory, the connected 4-point function has one vertex. How many Wick contractions give a distinct connected diagram, and what is its symmetry factor?

Solution

The vertex −iλ/4!∫d⁴x φ(x)⁴ has four fields at x. Contracting each with one external momentum gives 4! = 24 contractions, canceling the 1/4! denominator. The symmetry factor is S=1. The single Feynman diagram is four external lines meeting at one vertex with amplitude −iλ (momentum conservation imposed).

Key Takeaways

Coupling terms in the Lagrangian generate interactions; the coupling constant λ controls the order of the perturbation expansion
Wick's theorem reduces time-ordered products to propagator pairings — the combinatorial engine behind Feynman diagrams
Vacuum bubble diagrams exponentiate and cancel in connected S-matrix elements — only connected diagrams contribute to physical amplitudes
The LSZ formula bridges Green's functions (computed from Feynman diagrams) and S-matrix elements (measured in experiments)