Wick Theorem and n-Point Functions

The Feynman propagator is the two-point function of a free quantum field. The next question is what happens when more than two fields are inserted. In perturbation theory this is not a small technical issue: the Dyson expansion produces time-ordered products of many fields, and every term must be reduced to objects we know how to compute.

For a free field, the answer is exceptionally rigid. Once the two-point function is known, every vacuum time-ordered $n$-point function is fixed. Odd-point functions vanish, and even-point functions are sums over all possible pairings of insertions. This is Wick theorem. It is the algebraic reason Feynman diagrams exist: a line is a two-point function, and a diagram is a bookkeeping device for a pattern of pairings.

The theorem is sometimes presented as a combinatorial trick. A better way to view it is this: a free field is a Gaussian quantum variable. Gaussian variables have no independent higher cumulants. Their higher moments are built entirely out of their two-point covariance. Wick theorem is the operator version of that Gaussian fact, with time ordering and the Feynman boundary condition built in.

The rule in one line

Free Gaussian vacuum

Wick theorem on this page is an exact statement about the free Gaussian vacuum. In an interacting theory, exact correlators do not generally factor into two-point functions. Perturbation theory works by expanding the interaction and evaluating each term in the free theory, where Wick theorem is valid. This is why the theorem is both limited and powerful.

Scalar two-point-function notation

In this page

$$G_F(x-y)=\langle0|\mathcal T\phi(x)\phi(y)|0\rangle$$

denotes the same scalar Feynman two-point function called $D_F$ in the qft.org convention page. No extra factor of $i$ is extracted. It obeys

$$(\Box_x+m^2)G_F(x-y)=-i\delta^{(4)}(x-y).$$

For the oscillator warm-up, write

$$q(t)=\frac{1}{\sqrt{2\omega}} \left(ae^{-i\omega t}+a^\dagger e^{i\omega t}\right), \qquad [a,a^\dagger]=1.$$

For a free real scalar field, Wick theorem says

$$\boxed{ \langle0|\mathcal T\phi_1\phi_2\cdots\phi_{2N}|0\rangle =\sum_{\text{complete pairings}} \prod_{(i,j)}G_F(x_i-x_j), }$$

while all odd vacuum correlators vanish. Here $\phi_i$ means $\phi(x_i)$, and a “complete pairing” means that every insertion appears in exactly one pair.

This boxed formula is the destination of the page. The rest of the derivation explains why the contraction is the Feynman propagator, why no other higher connected data appear in a free theory, and where the contact terms come from when the equation of motion acts on a time-ordered product.

Time ordering beyond two insertions

For two fields,

$$\langle0|\mathcal T\phi(t_1)\phi(t_2)|0\rangle=G_F(t_1,t_2).$$

For three or more operators the definition is the same, but the notation becomes heavier. The time-ordering symbol rearranges operators so that later times stand to the left:

$$\mathcal T\{A(t_1)B(t_2)C(t_3)\} = A(t_1)B(t_2)C(t_3) \qquad \text{if } t_1>t_2>t_3,$$

with analogous formulas in the other time-ordering regions. For bosonic operators there is no sign from the rearrangement. For fermionic operators each exchange contributes a minus sign; that will become important later.

The time-ordered vacuum expectation value is not merely a real-time object. The same expression can be understood by slightly complexifying the times. If

$$A(t)=e^{iHt}Ae^{-iHt}$$

and $t_1$ is the latest operator in

$$\langle0|A(t_1)\cdots|0\rangle,$$

then inserting energy eigenstates immediately to the right of $A(t_1)$ gives terms of the form

$$\sum_n \langle0|A|n\rangle\langle n|\cdots|0\rangle e^{-i(E_n-E_0)t_1}.$$

These are positive-frequency contributions. To damp them at large positive time one sends the latest time slightly below the real axis:

$$\operatorname{Im}t_{\max}<0.$$

Similarly, if $t_1$ is the earliest time, then

$$\langle0|\cdots A(t_1)|0\rangle =\sum_n \langle0|\cdots|n\rangle\langle n|A|0\rangle e^{+i(E_n-E_0)t_1},$$

which is a negative-frequency contribution. It is damped by placing the earliest time slightly above the real axis:

$$\operatorname{Im}t_{\min}>0.$$

This is the same Feynman boundary condition encoded by the $i\epsilon$ prescription. In words: negative-frequency pieces are selected toward the past, while positive-frequency pieces are selected toward the future.

In a time-ordered vacuum matrix element, the latest insertion contributes phases $e^{-i(E_n-E_0)t_{\max}}$, while the earliest insertion contributes phases $e^{+i(E_n-E_0)t_{\min}}$. This is the time-domain form of the Feynman boundary condition.

Wick theorem now asks how this two-point object controls all higher products in a free theory.

Normal ordering and contractions

Split the oscillator into annihilation and creation parts,

$$q(t)=q_-(t)+q_+(t), \qquad q_-(t)=\frac{a}{\sqrt{2\omega}}e^{-i\omega t}, \qquad q_+(t)=\frac{a^\dagger}{\sqrt{2\omega}}e^{i\omega t}.$$

Here the subscripts are bookkeeping for annihilation and creation parts. The $e^{-i\omega t}$ term lowers the oscillator occupation number, while the $e^{+i\omega t}$ term raises it. They are not charge labels and should not be confused with the charge labels used later for complex fields.

The notation is chosen so that $q_-$ annihilates the vacuum on the right and $q_+$ annihilates the vacuum on the left:

$$q_-|0\rangle=0, \qquad \langle0|q_+=0.$$

A normal-ordered product, written with colons, places all creation operators to the left of all annihilation operators:

$$:q_-(t_1)q_+(t_2):=q_+(t_2)q_-(t_1).$$

Consequently, every vacuum expectation value of a nontrivial normal-ordered product vanishes:

$$\langle0|:q(t_1)\cdots q(t_n):|0\rangle=0 \qquad n>0.$$

The contraction of two fields is the difference between their time-ordered product and their normal-ordered product:

$$\operatorname{Contr}(q_iq_j) \equiv \mathcal T\{q(t_i)q(t_j)\}-:q(t_i)q(t_j):.$$

For a free oscillator this contraction is a c-number and equals the Feynman two-point function:

$$\operatorname{Contr}(q_iq_j) =\langle0|\mathcal T q(t_i)q(t_j)|0\rangle \equiv G_{ij}.$$

Indeed, if $t_i>t_j$, then

$$\begin{aligned} \mathcal T\{q_iq_j\} &=q_iq_j \\ &=:q_iq_j:+[q_{-,i},q_{+,j}] \\ &=:q_iq_j:+\frac{1}{2\omega}e^{-i\omega(t_i-t_j)}. \end{aligned}$$

If $t_j>t_i$, the same argument gives the other time ordering. Together,

$$G_{ij}=\frac{1}{2\omega}e^{-i\omega|t_i-t_j|}.$$

This is the same function obtained from the pole prescription. The point is that the contraction is not a new object; it is the Feynman propagator in operator clothing.

Two cautions are worth keeping in mind. First, normal ordering is defined relative to the chosen free vacuum and free oscillator split; changing the vacuum changes what “normal” means. Second, fields at the same spacetime point are operator-valued distributions, not ordinary functions. Wick theorem is safest when the insertions are kept at distinct points or smeared with test functions; coincident limits are where ultraviolet divergences enter.

Wick theorem

Wick theorem says that a time-ordered product of free fields equals the sum of all normal-ordered products obtained by contracting zero, one, two, and more disjoint pairs of fields:

$$\boxed{ \mathcal T\{q_1q_2\cdots q_n\} = :q_1q_2\cdots q_n: +\sum_{\text{one pair}}G_{ij}: \!\prod_{k\ne i,j}q_k\!: +\sum_{\text{two pairs}}G_{ij}G_{\ell m}: \!\prod_{k\ne i,j,\ell,m}q_k\!: +\cdots }$$

where $q_i$ abbreviates $q(t_i)$ and no field can be used in more than one contraction.

Taking the vacuum expectation value kills every remaining normal-ordered factor. Therefore the free time-ordered $n$-point function is especially simple. Define

$$G_n(t_1,\ldots,t_n) \equiv \langle0|\mathcal Tq(t_1)\cdots q(t_n)|0\rangle.$$

Then

$$\boxed{ G_{2N}(t_1,\ldots,t_{2N}) =\sum_{\text{pairings }P}\prod_{(i,j)\in P}G_{ij}, \qquad G_{2N+1}=0. }$$

A pairing $P$ is a partition of the labels $\{1,\ldots,2N\}$ into $N$ unordered pairs. The labels stay attached to the insertions throughout the calculation; Wick theorem is not saying that the points are indistinguishable. Time ordering is already contained inside each $G_{ij}$, so the pairing sum is over labels, not over time-ordering regions.

The number of pairings is

$$(2N-1)!!=\frac{(2N)!}{2^N N!}.$$

The denominator has a clear meaning: $2^N$ removes the order inside each pair, and $N!$ removes the order of the pairs themselves.

For actual calculations, it helps to separate the algorithm from the formula:

1. label the fields before doing any algebra;
2. list complete pairings of the labels;
3. replace each pair $(i,j)$ by $G_F(x_i-x_j)$ or $G_{ij}$;
4. multiply within each pairing and sum over pairings.

Do not time-order the labels by hand after this step. The propagator attached to a pair is already the time-ordered two-point function.

The first nontrivial example is the four-point function:

$$\boxed{ G_4(t_1,t_2,t_3,t_4) =G_{12}G_{34}+G_{13}G_{24}+G_{14}G_{23}. }$$

The free four-point function has three Wick pairings. The labels are not assumed to be time ordered; the time ordering is already contained in each propagator $G_{ij}=\langle0|\mathcal Tq_iq_j|0\rangle$.

For six fields there are fifteen terms:

$$G_6(t_1,\ldots,t_6) =G_{12}G_{34}G_{56}+G_{12}G_{35}G_{46}+\cdots,$$

where the dots stand for all distinct pairings of the six labels. Listing them by hand is already unpleasant; drawing lines is better. This is the first taste of why diagrams are so efficient.

Proof by induction

The proof is almost the same as the proof of Gaussian moment factorization. For clarity, assume first that the times are ordered,

$$t_1>t_2>\cdots>t_n.$$

Then the vacuum correlator is

$$G_n(t_1,\ldots,t_n) =\langle0|q_1q_2\cdots q_n|0\rangle.$$

Write $q_1=q_{1,+}+q_{1,-}$. Because $\langle0|q_{1,+}=0$, only the annihilation part contributes. Move $q_{1,-}$ to the right through $q_2,\ldots,q_n$. Each time it passes a field $q_j$, it produces the c-number commutator

$$[q_{1,-},q_{j,+}]=G_{1j} \qquad (t_1>t_j).$$

After the last commutation, the term with $q_{1,-}$ all the way on the right vanishes because $q_{1,-}|0\rangle=0$. What remains is the exact recursion

$$\boxed{ G_n(t_1,\ldots,t_n) =\sum_{j=2}^n G_{1j}\, G_{n-2}(t_2,\ldots,\widehat{t_j},\ldots,t_n), }$$

where $G_0=1$ and the hat means that the argument is omitted. The recursion pairs label $1$ with exactly one of the other labels, then applies the same step to the remaining $n-2$ insertions. It therefore generates every complete pairing exactly once. If $n$ is odd, the recursion eventually reaches $G_1=\langle0|q(t)|0\rangle=0$.

This proves the vacuum-correlator form of Wick theorem. The full operator identity stated above follows by the same commutation procedure before taking the vacuum expectation value: one must retain both the terms in which $q_1$ is contracted and all normal-ordered terms in which it remains uncontracted.

A recursive view of Wick theorem. Pair the first insertion with one of the remaining fields, remove both, and repeat. This generates each pairing exactly once.

For arbitrary real times, first divide the product into time-ordering regions and apply the ordered proof in each region. The result can be written uniformly because $G_{ij}$ itself already knows which of $t_i$ and $t_j$ is later.

The same proof works for a free scalar field because each momentum mode is an oscillator. The only replacement is

$$G_{ij}\longrightarrow G_F(x_i-x_j).$$

Therefore

$$\boxed{ \langle0|\mathcal T\phi(x_1)\cdots\phi(x_{2N})|0\rangle = \sum_{\text{pairings }P}\prod_{(i,j)\in P}G_F(x_i-x_j), }$$

and all odd correlators vanish.

Contact-term recursion

This is the point where many readers accidentally drop the most important term: a field satisfies the free equation only away from coincident insertions. At coincident insertions, differentiating the step functions inside $\mathcal T$ produces delta functions.

There is another way to recognize Wick factorization: apply the free equation of motion to one insertion. Since

$$(\Box_{x_1}+m^2)G_F(x_1-x_j)=-i\delta^{(4)}(x_1-x_j),$$

the Wick expansion immediately gives

$$\boxed{ (\Box_{x_1}+m^2) G_n(x_1,\ldots,x_n) =-i\sum_{j=2}^n\delta^{(4)}(x_1-x_j) G_{n-2}(x_2,\ldots,\widehat{x_j},\ldots,x_n). }$$

For the oscillator this becomes

$$\left(\frac{d^2}{dt_1^2}+\omega^2\right) G_n(t_1,\ldots,t_n) =-i\sum_{j=2}^n\delta(t_1-t_j) G_{n-2}(t_2,\ldots,\widehat t_j,\ldots,t_n).$$

This equation says that a free field satisfies its classical equation of motion inside correlators except when it collides with another field insertion. The collision produces a delta-function contact term. That is exactly what one should expect from differentiating a time-ordered product: the derivative can hit the step functions that define the ordering.

Acting with the free wave operator on one insertion collapses the propagator connecting it to another insertion. The result is a contact term and a lower-point correlator.

For example, in the four-point function,

$$G_4=G_{12}G_{34}+G_{13}G_{24}+G_{14}G_{23},$$

we get

$$\begin{aligned} (\Box_{x_1}+m^2)G_4 &=-i\delta^{(4)}(x_1-x_2)G_{34} -i\delta^{(4)}(x_1-x_3)G_{24} -i\delta^{(4)}(x_1-x_4)G_{23}. \end{aligned}$$

The contact terms are not optional decorations. They are what make the time-ordered product the inverse of the quadratic kinetic operator.

From oscillator modes to fields

The oscillator proof is not a toy calculation that later gets discarded. A free field is a continuum of independent oscillators, one for each momentum mode. Wick theorem for the field follows because the vacuum is Gaussian in all those oscillator coordinates simultaneously.

A real scalar field can be expanded in the covariantly normalized form

$$\phi(x)=\int\frac{d^3\mathbf p}{(2\pi)^3 2E_{\mathbf p}} \left[a(\mathbf p)e^{-ip\cdot x}+a^\dagger(\mathbf p)e^{ip\cdot x}\right], \qquad p^0=E_{\mathbf p},$$

where $[a(\mathbf p),a^\dagger(\mathbf q)]=(2\pi)^3 2E_{\mathbf p}\delta^{(3)}(\mathbf p-\mathbf q)$. The field is therefore an infinite collection of oscillators. Wick theorem follows mode by mode, while the contraction becomes the spacetime propagator

$$G_F(x-y)=\int\frac{d^4p}{(2\pi)^4} \frac{i e^{-ip\cdot(x-y)}}{p^2-m^2+i\epsilon}.$$

The theorem is powerful because it separates two problems. The free theory supplies propagators. The interaction supplies insertions from the Dyson expansion. Perturbation theory is then the systematic pairing of free fields produced by those insertions.

This explains why the next step in the course is diagrammatic. A Wick pairing is a line. A product of pairings is a graph. Symmetry factors count how many algebraic pairings produce the same graph.

Summary

Wick theorem is the central computational statement about free quantum fields. It says that every free vacuum time-ordered correlator is determined by the two-point function. Odd correlators vanish. Even correlators are sums over all pairings,

$$G_{2N}=\sum_{\text{pairings}}\prod G_F.$$

The theorem is the operator version of Gaussian moment factorization. Its physical content is that a free theory has no independent connected interactions: all higher correlations are built by propagating pairs of quanta between insertions.

The contact-term equation

$$(\Box_{x_1}+m^2)G_n =-i\sum_{j=2}^n\delta^{(4)}(x_1-x_j)G_{n-2}$$

is the differential form of the same statement. Away from coincident insertions, each field satisfies the free equation of motion. At coincident insertions, time ordering produces delta-function sources. This is the bridge from Green functions to Feynman diagrams.

Common pitfalls

The contraction is not an arbitrary line drawn between two fields. For the vacuum time-ordered theory it is exactly the Feynman propagator. Changing the boundary condition changes the contraction.

Normal ordering is not the same as time ordering. Normal ordering rearranges creation and annihilation operators; time ordering rearranges operators according to time. Wick theorem is useful precisely because it relates the two.

Do not forget the disconnected terms. In a free theory the four-point function is not zero, but it is disconnected:

$$G_4=G_{12}G_{34}+G_{13}G_{24}+G_{14}G_{23}.$$

A zero connected four-point function does not mean a zero four-point function.

For fermions, the same theorem holds with signs. Every interchange of fermionic fields contributes a minus sign. The scalar formulas on this page are the bosonic version.

A final overcounting check: for labeled insertions there is no additional numerical factor multiplying a Wick pairing. The coefficient is $1$ for each pairing. Symmetry factors appear later only after many labeled contractions are represented by the same unlabeled Feynman diagram.

Exercises

Exercise 1: Counting pairings

Show that the number of pairings of $2N$ labeled points is

$$(2N-1)!!=\frac{(2N)!}{2^N N!}.$$

Solution

Start with $2N$ labeled points. If we order all labels in a sequence, there are $(2N)!$ possible sequences. Turn a sequence into pairs by grouping adjacent entries:

$$(i_1,i_2),(i_3,i_4),\ldots,(i_{2N-1},i_{2N}).$$

This overcounts each pairing. First, each pair can be written in two orders, giving a factor $2^N$. Second, the $N$ pairs themselves can be permuted, giving a factor $N!$. Hence

$$\#\text{ pairings}=\frac{(2N)!}{2^N N!}.$$

Equivalently, choose a partner for point $1$ in $2N-1$ ways, then a partner for the smallest remaining label in $2N-3$ ways, and so on:

$$(2N-1)(2N-3)\cdots 3\cdot1=(2N-1)!!.$$

Exercise 2: Six-point Wick expansion

Use Wick theorem to compute the free six-point function

$$G_6(t_1,t_2,t_3,t_4,t_5,t_6)$$

in the partially organized form where $t_1$ is paired first.

Solution

Pair $t_1$ with one of the other five times. For each choice, Wick theorem gives the propagator for that pair times the four-point function of the remaining fields:

$$\begin{aligned} G_6(1,2,3,4,5,6) ={}&G_{12}\bigl(G_{34}G_{56}+G_{35}G_{46}+G_{36}G_{45}\bigr)\\ &+G_{13}\bigl(G_{24}G_{56}+G_{25}G_{46}+G_{26}G_{45}\bigr)\\ &+G_{14}\bigl(G_{23}G_{56}+G_{25}G_{36}+G_{26}G_{35}\bigr)\\ &+G_{15}\bigl(G_{23}G_{46}+G_{24}G_{36}+G_{26}G_{34}\bigr)\\ &+G_{16}\bigl(G_{23}G_{45}+G_{24}G_{35}+G_{25}G_{34}\bigr). \end{aligned}$$

There are $5\times3=15$ terms, as expected from

$$(6-1)!!=15.$$

Exercise 3: Contact terms in a four-point function

Let

$$\begin{aligned} G_4(x_1,x_2,x_3,x_4) ={}&G_F(x_1-x_2)G_F(x_3-x_4) +G_F(x_1-x_3)G_F(x_2-x_4)\\ &+G_F(x_1-x_4)G_F(x_2-x_3). \end{aligned}$$

Using $(\Box+m^2)G_F=-i\delta^{(4)}$, prove the contact-term identity for $G_4$.

Solution

Act with $\Box_{x_1}+m^2$. Only propagators containing $x_1$ are differentiated. Therefore

$$\begin{aligned} (\Box_{x_1}+m^2)G_4 ={}&\bigl[(\Box_{x_1}+m^2)G_F(x_1-x_2)\bigr]G_F(x_3-x_4)\\ &+\bigl[(\Box_{x_1}+m^2)G_F(x_1-x_3)\bigr]G_F(x_2-x_4)\\ &+\bigl[(\Box_{x_1}+m^2)G_F(x_1-x_4)\bigr]G_F(x_2-x_3). \end{aligned}$$

Using

$$(\Box_{x_1}+m^2)G_F(x_1-x_j)=-i\delta^{(4)}(x_1-x_j),$$

we obtain

$$\begin{aligned} (\Box_{x_1}+m^2)G_4 ={}&-i\delta^{(4)}(x_1-x_2)G_F(x_3-x_4)\\ &-i\delta^{(4)}(x_1-x_3)G_F(x_2-x_4)\\ &-i\delta^{(4)}(x_1-x_4)G_F(x_2-x_3), \end{aligned}$$

which is the $n=4$ case of the general recursion formula.

Exercise 4: Complex scalar pairings

For a complex free scalar field with nonzero contractions

$$\langle0|\mathcal T\phi(x)\phi^\dagger(y)|0\rangle=G_F(x-y), \qquad \langle0|\mathcal T\phi(x)\phi(y)|0\rangle=0,$$

compute

$$\langle0|\mathcal T\phi(x_1)\phi^\dagger(y_1)\phi(x_2)\phi^\dagger(y_2)|0\rangle.$$

Solution

Only contractions between a $\phi$ and a $\phi^\dagger$ are nonzero. Therefore there are two pairings:

$$\phi(x_1)\leftrightarrow\phi^\dagger(y_1), \qquad \phi(x_2)\leftrightarrow\phi^\dagger(y_2),$$

and

$$\phi(x_1)\leftrightarrow\phi^\dagger(y_2), \qquad \phi(x_2)\leftrightarrow\phi^\dagger(y_1).$$

Thus

$$\boxed{ \langle0|\mathcal T\phi(x_1)\phi^\dagger(y_1)\phi(x_2)\phi^\dagger(y_2)|0\rangle =G_F(x_1-y_1)G_F(x_2-y_2)+G_F(x_1-y_2)G_F(x_2-y_1). }$$

This is the complex-field version of Wick factorization, with charge conservation built into the allowed contractions.

References and further reading

• Sidney Coleman, Lectures of Sidney Coleman on Quantum Field Theory, Chapter 8, for Wick theorem and the transition from Wick diagrams to perturbation theory.
• Mark Srednicki, Quantum Field Theory, Sections 8–10, for the path-integral derivation of free-field correlators and Feynman rules.
• Steven Weinberg, The Quantum Theory of Fields, Volume I, Sections 6.1–6.2, for Wick pairings, coordinate-space rules, and propagators in perturbative QFT.
• Michael E. Peskin and Daniel V. Schroeder, An Introduction to Quantum Field Theory, Chapter 4, for the standard perturbative use of Wick theorem in scalar field theory.