Wick Expansion for Interacting Fields

Perturbative interacting correlators are obtained by expanding the Dyson operator and applying Wick’s theorem to the resulting free interaction-picture fields. Each term becomes a sum of normal-ordered products and contractions; vacuum expectation removes every term with uncontracted free fields. Dividing by the vacuum-to-vacuum amplitude cancels components disconnected from all external insertions. The result is a controlled series of propagator integrals, not an assertion that exact interacting fields obey free Gaussian factorization.

Required background. The Interaction Picture and Dyson Series supplies the ordered expansion, while Wick’s Theorem and Free Gaussian Factorization supplies the free-field operator identity and its graded signs.

The normalized interacting correlator

For a nonderivative interaction, $H_I(t)=-\int\mathrm d^{d-1}x\,\mathcal L_{\mathrm{int}}(x)$ in the interaction picture. With a vacuum-selection and switching prescription understood, the in–out time-ordered correlator is

$$\boxed{ \langle\Omega|\mathrm T\{\phi_H(x_1)\cdots\phi_H(x_n)\}|\Omega\rangle = \frac{ \langle0|\mathrm T\!\left\{ \phi_I(x_1)\cdots\phi_I(x_n) e^{\,i\int\mathrm d^d z\,\mathcal L_{\mathrm{int}}(z)} \right\}|0\rangle }{ \langle0|\mathrm T\!\left\{ e^{\,i\int\mathrm d^d z\,\mathcal L_{\mathrm{int}}(z)} \right\}|0\rangle } }.$$

The numerator, denominator, and instruction to expand their exponentials are displayed together in Srednicki 2007, § 9, p. 86. The formula presupposes that the selected interacting vacuum has nonzero overlap with the regulated free vacuum. It is an in–out object; thermal, in–in, and finite-time correlators require their own contours and states.

Expand the numerator as

$$\sum_{V=0}^{\infty}\frac{i^V}{V!} \int\mathrm d^d z_1\cdots\mathrm d^d z_V\, \left\langle0\left|\mathrm T\!\left\{ \phi_I(x_1)\cdots\phi_I(x_n) \prod_{a=1}^{V}\mathcal L_{\mathrm{int}}(z_a) \right\}\right|0\right\rangle .$$

Every bracket now contains only free fields. Wick’s theorem replaces it by the sum over complete contractions. A scalar contraction is the raw ordered two-point function

$$D_F(x-y) =\langle0|\mathrm T\{\phi_I(x)\phi_I(y)\}|0\rangle =\int\frac{\mathrm d^d p}{(2\pi)^d} \frac{i\,e^{-ip\cdot(x-y)}}{p^2-m^2+i0}.$$

For fermions the same reduction holds with graded time ordering: each term carries the parity of the permutation needed to produce its declared contraction order. Wick’s original theorem treats both commuting and anticommuting fields Wick 1950, Rules B, D, and C″, pp. 269–270; Theorems 1–2, pp. 270–271.

For $2r$ bosonic fields the vacuum expectation can be written without choosing a drawing:

$$\langle0|\mathrm T\{\phi_1\cdots\phi_{2r}\}|0\rangle =\sum_{P\in\mathcal P_{2r}} \prod_{(i,j)\in P}D_F(x_i-x_j), \qquad |\mathcal P_{2r}|=(2r-1)!!,$$

where $\mathcal P_{2r}$ is the set of perfect pairings. An odd vacuum expectation vanishes in the free zero-mean vacuum. These statements depend on the Gaussian reference state: a thermal Gaussian state has a different contraction, while a non-Gaussian state requires higher connected cumulants as additional building blocks.

First order in scalar φ⁴ theory

Take

$$\mathcal L_{\mathrm{int}}=-\frac{\lambda}{4!}\phi^4.$$

Use one ultraviolet regulator consistently in the numerator and denominator, and let $D_F(0)$ below denote the resulting regulated coincidence limit. At first order, the unnormalized two-point numerator contains

$$-\frac{i\lambda}{4!}\int\mathrm d^d z\, \langle0|\mathrm T\{\phi(x)\phi(y)\phi(z)^4\}|0\rangle_0.$$

There are two contraction classes.

1. Connect $x$ to one of the four fields at $z$, connect $y$ to one of the remaining three, and pair the final two fields at $z$. The $4\times3=12$ labeled contractions give

$$-\frac{i\lambda}{2}\int\mathrm d^d z\, D_F(x-z)D_F(z-y)D_F(0).$$

2. Contract $x$ with $y$ and pair the four fields at $z$ in $(4-1)!!=3$ ways. This gives the disconnected vacuum factor

$$D_F(x-y) \left[-\frac{i\lambda}{8} \int\mathrm d^d z\,D_F(0)^2\right].$$

The bracket is exactly the first-order contribution to the denominator. Expanding the ratio cancels it, leaving only the term connected to the external insertions. The same normalized canonical and path-integral organization is derived in Schwartz 2014, §§ 7.2 and 14.3.3, pp. 84–93 and 264.

This example checks three independent pieces at once: the vertex normalization $1/4!$, the contraction multiplicity $12$, and the cancellation of the source-independent vacuum bubble. The coincident distribution $D_F(0)$ is ultraviolet singular before regularization; displaying it does not define it or renormalize the theory.

From contractions to graph data

The graph translation is bookkeeping for the contraction pattern:

Algebraic object	Graphical object	Information retained
insertion $\mathcal L_{\mathrm{int}}(z_a)$	vertex at $z_a$	interaction type and integration point
contraction $D_F(z_a-z_b)$	internal line	ordered two-point distribution and its prescription
contraction $D_F(x_i-z_a)$	line ending at external insertion	which labeled insertion is attached
uncontracted normal product	no vacuum contribution	operator information remains for nonvacuum matrix elements
graded permutation parity	sign attached to the term	fermion ordering, not geometry alone

The graph does not replace the algebra: two labeled contractions can have the same unlabeled topology, and their multiplicity is what becomes a symmetry factor. That conversion is developed on Diagrammatics and Symmetry Factors.

Bosonic and fermionic ordering

For bosons, reordering fields to expose a contraction introduces no statistics sign. For odd fields, the sign is fixed before drawing by a declared order of external operators, sources, and contractions. Two reliable procedures are equivalent:

• count the parity of the graded permutation that brings each paired set together; or
• keep the original Grassmann source order and differentiate with fixed left/right derivative conventions.

An arc crossing in a drawing is only a mnemonic after those conventions are fixed. Closed-fermion-loop signs and external-line ordering receive their full treatment on Fermion Signs and Closed Loops.

What is and is not being expanded

The interaction-picture fields inside the series obey the free equations and use free contractions. The correlator reconstructed by the normalized series is interacting order by order. Consequently:

• Wick factorization is applied termwise, not to the exact interacting correlator as a whole;
• a nonzero interacting connected four-point function is expected even though each free expectation is a sum of pairings;
• the denominator removes vacuum components but not disconnected pieces built from separate groups of external insertions; and
• regularization and renormalization are additional operations. A formal contraction such as $D_F(0)$ is not made finite by drawing it.

Normal ordering the interaction changes the perturbative definition. For example, using $:\!\phi^4\!:$ excludes contractions internal to a single normal-ordered vertex, so the tadpole in the worked example is absent. That convention must be declared; it cannot be inferred from a graph drawn after the contractions were counted.

Common pitfalls

Applying Wick’s theorem directly to Heisenberg fields. The theorem used here acts on free interaction-picture fields after the Dyson expansion. Exact interacting vacuum correlators are not Gaussian in general.

Deleting every disconnected diagram. Only components disconnected from all external insertions cancel against the denominator. Products of connected correlators attached to different external subsets remain in a full correlator.

Forgetting that coincident contractions are distributions. $D_F(0)$ signals a short-distance singularity requiring a regulator and, for renormalized predictions, the machinery developed in the renormalization volume.

Check your understanding

Recount the first-order $\phi^4$ two-point term without using a diagram. Find the externally connected and vacuum-bubble multiplicities, reduce their coefficients, and show the latter cancel in the normalized ratio.

Solution

Choose the field at $z$ contracted with $\phi(x)$ in four ways and the one contracted with $\phi(y)$ in three ways; the last two contract with each other. Thus $12/4!=1/2$, giving the connected coefficient $-i\lambda/2$. For the vacuum class, contract $x$ with $y$ and pair four fields at $z$ in $(4-1)!!=3$ ways, so $3/4!=1/8$. This is the free two-point function times the first-order denominator term and therefore cancels when the ratio is expanded.

Where to continue

• Convert labeled contractions into graph weights: Diagrammatics and Symmetry Factors explains automorphisms and multiplicities.
• Separate connected and vacuum sectors globally: Connected, Disconnected, and Vacuum Diagrams derives exponentiation and the partition formula.
• Check the same expansion independently: Functional Derivation of Perturbation Theory replaces fields by source derivatives.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI.
• Srednicki, Mark. Quantum Field Theory. Cambridge: Cambridge University Press, 2007. DOI.
• Wick, Gian-Carlo. “The Evaluation of the Collision Matrix.” Physical Review 80, no. 2 (1950): 268–272. DOI.