Wick's Theorem and Contractions

The Dyson series expresses the S-matrix as a sum of vacuum and in/out matrix elements of time-ordered products of free fields. Evaluating those requires converting time-ordered products into a form whose vacuum expectation values are read off immediately. Wick's theorem is the algorithm that does this: it reduces any time-ordered product to a sum of normal-ordered terms times contractions (propagators). This is the combinatorial engine behind Feynman rules.

Conventions: $\hslash =c=1$; free fields as built in canonical quantization.

Normal ordering and contractions

Recall two constructions from the scalar field:

• Normal ordering $:\cdots :$ places all annihilation operators to the right, so that $⟨0|:\cdots :|0⟩=0$ (unless the product is empty).
• Splitting the field into positive- and negative-frequency parts $\varphi ={\varphi }^{+}+{\varphi }^{-}$, where ${\varphi }^{+}\sim \int a_{\mathbf{p}}{e}^{-ip\cdot x}$ (annihilation) and ${\varphi }^{-}\sim \int a_{\mathbf{p}}^{†}{e}^{+ip\cdot x}$ (creation).

The contraction of two fields — written with an overline, $\overline{\varphi (x)\varphi (y)}$ — is the difference between their time-ordered and normal-ordered products:

$$\overline{\varphi (x)\varphi (y)}\equiv T\{\varphi (x)\varphi (y)\}-:\varphi (x)\varphi (y):.$$

A short computation shows this is a c-number equal to the Feynman propagator:

$$\boxed{\overline{\varphi (x)\varphi (y)}=⟨0|T\{\varphi (x)\varphi (y)\}|0⟩=D_F(x-y).}$$

(The vacuum expectation value of the normal-ordered term vanishes, so the contraction is the propagator.) For fermions the contraction is the Dirac propagator $S_F$, and each contraction that requires reordering anticommuting fields carries a minus sign.

Statement of Wick's theorem (Theorem)

Wick's theorem. A time-ordered product of free fields equals the sum of all possible ways of contracting the fields in pairs, each accompanied by the normal-ordered product of the remaining (uncontracted) fields:

$$\boxed{T\{{\varphi }_1{\varphi }_2\cdots {\varphi }_n\}=:{\varphi }_1{\varphi }_2\cdots {\varphi }_n:+\sum _{\text{single contractions}}:(\cdots ):+\sum _{\text{double}}:(\cdots ):+\cdots }$$

with the sum running over all pairings, including the fully contracted term when $n$ is even. Schematically, for four fields (writing $\overline{{\varphi }_i{\varphi }_j}$ for the contraction of ${\varphi }_i$ with ${\varphi }_j$):

$$T\{{\varphi }_1{\varphi }_2{\varphi }_3{\varphi }_4\}=:{\varphi }_1{\varphi }_2{\varphi }_3{\varphi }_4:+[\overline{{\varphi }_1{\varphi }_2}:{\varphi }_3{\varphi }_4:+\text{(5 more single)}]+[\overline{{\varphi }_1{\varphi }_2}\overline{{\varphi }_3{\varphi }_4}+\overline{{\varphi }_1{\varphi }_3}\overline{{\varphi }_2{\varphi }_4}+\overline{{\varphi }_1{\varphi }_4}\overline{{\varphi }_2{\varphi }_3}].$$

Proof sketch. By induction, moving each ${\varphi }^{+}$ through the normal-ordered product to the right; every time it passes a ${\varphi }^{-}$ it generates a commutator (for bosons) or anticommutator (for fermions), which is precisely a contraction. The base case $n=2$ is the definition above. $■$

Why it matters: vacuum expectation values are trivial

The point is what happens between vacuum states. Since $⟨0|:\cdots :|0⟩=0$ for any non-empty normal-ordered product, only the fully contracted terms survive in a vacuum expectation value:

$$⟨0|T\{{\varphi }_1\cdots {\varphi }_n\}|0⟩=\sum _{\text{complete pairings}}\prod \overline{{\varphi }_i{\varphi }_j}.$$

This is the combinatorial statement of the Feynman rules: the $n$-point time-ordered correlator is the sum over all ways of pairing the $n$ points with propagators. For $n=4$:

$$⟨0|T\{{\varphi }_1{\varphi }_2{\varphi }_3{\varphi }_4\}|0⟩=D_F(x_1-x_2)D_F(x_3-x_4)+D_F(x_1-x_3)D_F(x_2-x_4)+D_F(x_1-x_4)D_F(x_2-x_3),$$

the three ways to connect four external points pairwise — the three "diagrams" of the free four-point function.

Contractions with external states

In an S-matrix element the time-ordered product is sandwiched between in/out Fock states rather than vacua. An external particle of momentum $\mathbf{p}$ in the initial state is annihilated by a ${\varphi }^{+}$ from one of the fields, contributing a factor $e^{-ip\cdot x}$; an external particle in the final state is created by a ${\varphi }^{-}$, contributing $e^{+ip\cdot x}$. These external-leg contractions are what attach the internal propagator network to the physical particles, and become the external-line factors of the Feynman rules. (The LSZ formula makes this attachment precise and shows the external legs are amputated.)

Worked example: ${\varphi }^4$ at first order

Take ${\mathcal{L}}_{\text{int}}=-\frac{\lambda }{4!}{\varphi }^4$ and the process $\varphi \varphi \to \varphi \varphi$. The first-order Dyson term is

$$S^{(1)}=i\int d^{4}x{\mathcal{L}}_{\text{int}}(x)=-\frac{i\lambda }{4!}\int d^{4}x:{\varphi }^4(x):+(\text{contractions}).$$

For $2\to 2$ scattering all four external particles must contract with the four fields at the single vertex $x$ — no internal propagators at this order. The $4!$ ways of assigning the four external legs to the four $\varphi$'s exactly cancel the $1/4!$, leaving the amplitude

$$i\mathcal{M}=-i\lambda .$$

This is the simplest Feynman rule: the ${\varphi }^4$ vertex contributes $-i\lambda$, and the vertex $1/4!$ is the symmetry factor that combinatorics removes for a fully-external diagram.

Summary

• Contraction = time-ordered minus normal-ordered = the Feynman propagator (with a sign for fermions).
• Wick's theorem rewrites any $T$-product as a sum over all pairings.
• In the vacuum, only fully contracted terms survive — the $n$-point function is a sum of propagator networks.
• External particles contribute leg factors from their contractions with the fields.

Where this leads

• Diagrammatic bookkeeping of the pairings, and the full rule set: Feynman diagrams and Feynman rules.
• Amputating external legs and converting correlators to S-matrix elements: the LSZ reduction formula.

References

• Peskin & Schroeder, An Introduction to Quantum Field Theory, Ch. 4.3.
• Weinberg, The Quantum Theory of Fields, Vol. 1, Ch. 6.1.
• Srednicki, Quantum Field Theory, Ch. 8–9.