Intuition Beginner

Wick's theorem is the rule that turns a long product of free quantum fields into a controlled list of pairings. In ordinary arithmetic, multiplying many terms can create a mess. In free quantum field theory, the fields are built from creation and annihilation operators, and the vacuum state kills many terms. Wick's theorem says exactly which pairings survive and how to write the rest in a tidy ordered form.

The idea is close to the bell-curve rule in probability. For a centered Gaussian random variable, all odd moments vanish, and every even moment is made from pairings of two-point moments. Wick's theorem is the operator version of that same fact. The two-point building block is called a contraction. In physics language, it becomes a propagator.

This theorem is the bridge from operator products to Feynman diagrams. Each pairing can be drawn as a line connecting two fields. A product with four fields has three possible pairings. A product with six fields has fifteen. Perturbation theory in quantum field theory works because Wick's theorem converts complicated operator products into pairings that can be counted, drawn, and assigned numerical factors.

Visual Beginner

Picture four field operators sitting in a row. Wick's theorem asks how they can be paired off. One pairing links the first with the second and the third with the fourth. Another links the first with the third and the second with the fourth. A third links the first with the fourth and the second with the third.

The unpaired part is written in normal order, meaning all creation moves are placed before annihilation moves. The paired parts are contractions, meaning two-point vacuum values. For a free field, all vacuum averages are built from this pairing pattern.

Worked example Beginner

Suppose a free field has four appearances, labelled A, B, C, and D. Wick's theorem says that the vacuum average is the sum of the three ways to pair them.

Step 1. Pair neighbors: A with B, and C with D. This contributes the two-point value for A-B times the two-point value for C-D.

Step 2. Cross once: A with C, and B with D. This contributes the two-point value for A-C times the two-point value for B-D.

Step 3. Cross the other way: A with D, and B with C. This contributes the two-point value for A-D times the two-point value for B-C.

Step 4. Add the three contributions. If every two-point value were equal to 2 in this toy model, the total four-field average would be 2 times 2, plus 2 times 2, plus 2 times 2, giving 12.

What this tells us: the hard part is not multiplying every operator by hand. The hard part is listing the allowed pairings and assigning the correct two-point value to each pair.

Check your understanding Beginner

Formal definition Intermediate+

Let ${\mathcal{F}}_s(\mathcal{H})$ be bosonic Fock space over a one-particle Hilbert space $\mathcal{H}$ with vacuum vector $\Omega$. Write $a^{\ast }(f)$ and $a(f)$ for creation and annihilation operators on the finite-particle domain, satisfying $$ [a(f),a^(g)] = \langle f,g\rangle I,\qquad [a(f),a(g)] = [a^(f),a^*(g)] = 0. $$

For a finite word $A_1\cdots A_n$ in creation and annihilation operators, its normal ordering $:A_1\cdots A_n:$ is obtained by moving every creation operator to the left of every annihilation operator, preserving the internal order among creation operators and among annihilation operators. For free scalar fields, the same notation extends by linearity to field operators $\varphi (f)=a^{\ast }(Kf)+a(Kf)$ on a suitable test-function space.

The contraction of two free fields is the scalar $$ \contraction{}{\phi}{(f)}{\phi}\phi(f)\phi(g) := \langle \Omega,\phi(f)\phi(g)\Omega\rangle. $$ For time-ordered products, the contraction is instead $$ \contraction{}{\phi}{(x)}{\phi}\phi(x)\phi(y) := \langle \Omega,T{\phi(x)\phi(y)}\Omega\rangle = \Delta_F(x-y), $$ the Feynman propagator, interpreted as a distribution ^{[Peskin 1995]}.

Key theorem with proof Intermediate+

Theorem (Wick's theorem for bosonic free fields). Let ${\varphi }_1,\dots ,{\varphi }_n$ be linear bosonic free-field operators smeared against test functions. Then $$ \phi_1\cdots\phi_n

:\phi_1\cdots\phi_n: + \sum_{\text{one contraction}} :\text{remaining fields}: + \sum_{\text{two disjoint contractions}} :\text{remaining fields}: +\cdots, $$ where the final sum ranges over all collections of pairwise disjoint contractions. Taking the vacuum expectation keeps only complete pairings: $$ \langle \Omega,\phi_1\cdots\phi_n\Omega\rangle

\sum_{\text{pairings }P} \prod_{{i,j}\in P} \langle \Omega,\phi_i\phi_j\Omega\rangle. $$ If $n$ is odd, the pairing sum is empty and the vacuum expectation is zero.

Proof. Write each free field as ${\varphi }_i=C_i+A_i$, where $C_i$ is the creation part and $A_i$ is the annihilation part. Normal ordering moves all $C$ terms to the left of all $A$ terms. Whenever an annihilation part $A_i$ passes a creation part $C_j$, the canonical commutation relation contributes the scalar commutator $[A_i,C_j]$, which is exactly the contraction of the two corresponding field factors.

Expand the word ${\varphi }_1\cdots {\varphi }_n$ into a sum of $2^n$ words in $C_i$ and $A_i$. For each word, reorder it to normal form by repeatedly commuting an $A$ past a later $C$. Each commutation has two outcomes: the reordered operator product and the scalar contraction term. Choosing no scalar terms gives the fully normal-ordered product. Choosing one scalar term gives a word with one contraction and the remaining factors normal ordered. Choosing several scalar terms gives a collection of disjoint contractions, because a field factor disappears after it is contracted.

Summing over all expanded words gives the displayed operator identity. For the vacuum expectation, every normal-ordered word with at least one remaining operator has zero vacuum expectation: annihilation operators kill $\Omega$ on the right, while creation operators leave vectors orthogonal to $\Omega$ on the left. Therefore only the terms in which all factors are contracted survive. These are exactly the complete pairings. The odd case has no complete pairing.

Bridge. This theorem builds toward the Feynman-rule machinery in 08.10.03 and appears again in the propagator unit 08.10.05. The operator computation is also the same pairing count that the Gaussian path-integral formulation packages in 08.06.01 and 08.07.01.

Exercises Intermediate+

Advanced results Master

The operator theorem has a Gaussian-measure counterpart. Let ${\mu }_C$ be a centered Gaussian measure with covariance $C$ on a finite-dimensional vector space or on a nuclear test-function dual after the usual analytic construction. For linear observables $\Phi (f)$, the moment formula is $$ \int \Phi(f_1)\cdots \Phi(f_n),d\mu_C(\Phi)

\sum_{\text{pairings }P} \prod_{{i,j}\in P} C(f_i,f_j). $$ This is Isserlis-Wick combinatorics in probability and Wick's theorem in Euclidean field theory ^{[Glimm 1987]}.

Wick products remove the self-contractions. In Gaussian language, $:\Phi (f{)}^k:$ is the degree-$k$ Hermite polynomial in the random variable $\Phi (f)$, scaled by its variance. In field theory, this is not cosmetic: products such as $\varphi (x{)}^4$ are distributional and require renormalised definitions. Normal ordering in $P(\varphi {)}_2$ theory replaces the raw monomial by its Wick product, making the interaction density a well-defined random distribution after ultraviolet regularisation and limit passage ^{[Simon 1974]}.

The time-ordered version introduces causal or Euclidean propagators. In Minkowski perturbation theory, $$ \langle\Omega,T{\phi(x_1)\cdots\phi(x_n)}\Omega\rangle

\sum_P \prod_{{i,j}\in P}\Delta_F(x_i-x_j), $$ while the Euclidean Schwinger-function version uses the covariance of the Euclidean free field. Wick rotation connects the two under the hypotheses in which analytic continuation of correlation functions is available ^{[Streater 2000]}.

Synthesis. Wick's theorem builds toward perturbative renormalisation because it turns each interaction monomial into a finite combinatorial problem at fixed order, and it appears again in constructive field theory as the definition of Wick powers. The same pairing identity is simultaneously an operator theorem, a Gaussian moment theorem, and the bookkeeping law behind Feynman diagrams.

Full proof set Master

Proposition (Gaussian generating-functional proof). Let $X_1,\dots ,X_n$ be centered jointly Gaussian variables with covariance matrix $G_{ij}=\mathbb{E}[X_i{X}_j]$. Then $$ \mathbb{E}[X_1\cdots X_n]

\sum_{\text{pairings }P} \prod_{{i,j}\in P}G_{ij}, $$ with zero value for odd $n$.

Proof. The moment-generating function of the vector $X=(X_1,\dots ,X_n)$ is $$ M(t)=\mathbb{E}\exp\left(\sum_{i=1}^n t_iX_i\right) =\exp\left(\frac12\sum_{i,j=1}^n t_iG_{ij}t_j\right). $$ The coefficient of $t_1\cdots t_n$ in $M(t)$ is $\mathbb{E}[X_1\cdots X_n]$, since differentiating once in each $t_i$ and evaluating at zero extracts that moment.

Expand the exponential: $$ M(t)= \sum_{m=0}^{\infty}\frac{1}{m!} \left(\frac12\sum_{i,j=1}^n t_iG_{ij}t_j\right)^m. $$ The monomial $t_1\cdots t_n$ has total degree $n$, so it can occur only when $n=2m$. If $n$ is odd, no such term exists.

For $n=2m$, choosing $m$ quadratic factors whose indices cover $\{1,\dots ,n\}$ exactly once is the same as choosing a complete pairing. The factor $1/2^m$ cancels the two orientations of each pair, and the factor $1/m!$ cancels the order of the $m$ pairs. The remaining contribution of a pairing $P$ is ${\prod }_{\{i,j\}\in P}{G}_{ij}$. Summing all such choices gives the formula.

Proposition (normal ordering recursion). For bosonic free fields, define $\text{wick}W$ recursively by normal ordering words and recording one contraction whenever an annihilation part crosses a later creation part. Then $$ \phi_1 :\phi_2\cdots\phi_n:

:\phi_1\phi_2\cdots\phi_n: + \sum_{j=2}^n \langle\Omega,\phi_1\phi_j\Omega\rangle :\phi_2\cdots\widehat{\phi_j}\cdots\phi_n:, $$ where the hat means that the factor is omitted.

Proof. Decompose ${\varphi }_1=C_1+A_1$. The creation part $C_1$ is already on the left of the normally ordered tail, so it contributes to $:{\varphi }_1\cdots {\varphi }_n:$. The annihilation part $A_1$ must move to the right through the creation parts present inside the normally ordered tail. Each crossing contributes a scalar commutator $[A_1,C_j]$, equal to the corresponding contraction, and removes ${\varphi }_j$ from the remaining normal-ordered word. After all crossings, the residual annihilation part belongs inside the fully normal-ordered product. This gives the recursion. Iterating the recursion proves the full Wick expansion by induction on word length.

Connections Master

• 08.10.01 supplies the Fock-space operators $a^{\ast }(f)$ and $a(f)$ whose canonical commutation relation generates the contraction terms.

• 08.06.01 gives the Gaussian free-field side, where the same formula is a moment identity for a centered Gaussian measure.

• 08.07.01 uses the Gaussian path integral as the base measure around which interacting perturbation theory is expanded.

• 08.10.03 depends on Wick's theorem to convert the Dyson series for ${\varphi }^4$ theory into Feynman diagrams and symmetry factors.

• 08.09.01 explains why "Wick rotation" and "Wick's theorem" are distinct ideas that meet when Minkowski vacuum expectations are analytically continued to Euclidean Schwinger functions.

Historical & philosophical context Master

Gian-Carlo Wick introduced the theorem in 1950 to evaluate collision-matrix terms in the postwar operator formulation of quantum electrodynamics ^{[Wick 1950]}. Dyson's 1949 series had organised perturbation theory in powers of the interaction, but Wick supplied the reduction step that made the terms computable by pairings rather than by direct manipulation of long operator products.

The theorem also marks a philosophical shift in what counts as a calculation in physics. After Wick and Feynman, a diagram was not merely a picture of a process. It became a compressed algebraic instruction: choose a pairing pattern, attach a propagator to every line, integrate over internal vertices, and divide by the relevant symmetry factor. The diagrammatic syntax is justified by an operator identity.

Mathematically, Wick's theorem reveals why free fields are special. Their entire hierarchy of vacuum correlations is determined by the two-point function. Interacting fields are measured against that benchmark: departures from Wick factorisation are precisely the connected higher correlations that perturbation theory and constructive field theory try to control ^{[Chatterjee 2022]}.

Bibliography Master

@article{Wick1950CollisionMatrix,
  author = {Wick, G. C.},
  title = {The Evaluation of the Collision Matrix},
  journal = {Physical Review},
  volume = {80},
  pages = {268--272},
  year = {1950}
}

@misc{Chatterjee2022QFT,
  author = {Chatterjee, Sourav},
  title = {Introduction to Quantum Field Theory for Mathematicians},
  year = {2022},
  note = {Stanford lecture notes}
}

@book{PeskinSchroeder1995QFT,
  author = {Peskin, Michael E. and Schroeder, Daniel V.},
  title = {An Introduction to Quantum Field Theory},
  publisher = {Addison-Wesley},
  year = {1995}
}

@book{Folland2008TouristGuide,
  author = {Folland, Gerald B.},
  title = {Quantum Field Theory: A Tourist Guide for Mathematicians},
  publisher = {American Mathematical Society},
  year = {2008}
}

@book{GlimmJaffe1987QuantumPhysics,
  author = {Glimm, James and Jaffe, Arthur},
  title = {Quantum Physics: A Functional Integral Point of View},
  edition = {2},
  publisher = {Springer},
  year = {1987}
}

@book{StreaterWightman2000PCT,
  author = {Streater, R. F. and Wightman, A. S.},
  title = {PCT, Spin and Statistics, and All That},
  publisher = {Princeton University Press},
  year = {2000}
}

@book{Simon1974Pphi2,
  author = {Simon, Barry},
  title = {The P(φ)_2 Euclidean (Quantum) Field Theory},
  publisher = {Princeton University Press},
  year = {1974}
}