Wick's Theorem for Multi-Point Correlation Functions

Hello. In our previous lesson, we constructed the generating functional $Z [J]$ and discovered a remarkable result for the non-interacting Gaussian theory: the functional itself is a Gaussian in the source field $J$ . Specifically, we found that

Z [J] = Z [0] exp (\frac{1}{2} \int d^{d} x \int d^{d} y J (x) G (x - y) J (y))

where $G (x - y)$ is the Ornstein-Zernike two-point correlation function, or propagator.

This explicit form is more than just an elegant solution; it is a computational powerhouse. Since all correlation functions can be found by taking functional derivatives of $Z [J]$ , and we now have a closed-form expression for $Z [J]$ , we can, in principle, calculate any correlation function for the free theory. Today, we will do exactly that. We will see that this process reveals a profound combinatorial structure known as Wick's theorem. This theorem provides a simple, powerful rule for expressing any multi-point correlation function in a Gaussian theory as a sum of products of its fundamental building block: the two-point propagator. This is the final piece of theoretical machinery we need before we can begin to analyze interacting systems.

Unpacking the Generating Functional

Let's begin by systematically taking derivatives of our Gaussian $Z [J]$ to see what emerges. Recall the central formula from our last lesson:

⟨ ϕ (x_{1}) ϕ (x_{2}) \dots ϕ (x_{n})⟩ = \frac{1}{Z [ 0 ]} [\frac{δ ^{n} Z [ J ]}{δ J ( x _{1} ) δ J ( x _{2} ) \dots δ J ( x _{n} )}]_{J = 0}

The key insight is that since $Z [J]$ is exponential in a term quadratic in $J$ , taking derivatives and then setting $J = 0$ will only yield a non-zero result if we differentiate an even number of times, completely exhausting the factors of $J$ .

Let's watch this happen explicitly. The following video segment from MIT OpenCourseWare carries out this procedure for the quantum mechanical harmonic oscillator, which is mathematically identical to our statistical field theory.

Lecture 10: Time-Ordered Correlation Functions in Field Theory

This video derives Wick's theorem by showing that the quadratic nature of the generating functional for a free theory leads to a simple combinatorial rule for all n-point functions.

Please watch from this section. The lecturer explains why all odd n-point functions must be zero. He then shows that all even n-point functions can be expressed as a sum of all possible pairings (contractions) of the fields, where each pairing corresponds to a two-point function (a Feynman propagator).

Let's write out the first few cases to solidify our understanding. Let $E [J] = \frac{1}{2} \iint J G J$ . Then $Z [J] = Z [0] e^{E [J]}$ .

1-point function ( $n = 1$ ):
$\frac{δ Z [ J ]}{δ J ( x _{1} )} = Z [0] (\frac{δ E [ J ]}{δ J ( x _{1} )}) e^{E [J]} = Z [0] (\int d y G (x_{1} - y) J (y)) e^{E [J]}$
When we set $J = 0$ , the term in the parenthesis vanishes. Thus, $⟨ ϕ (x_{1})⟩ = 0$ . This confirms that in the symmetric phase ( $r > 0$ ), the average order parameter is zero.
2-point function ( $n = 2$ ):
$\frac{δ ^{2} Z [ J ]}{δ J ( x _{2} ) δ J ( x _{1} )} = Z [0] [(\int d y G (x_{1} - y) J (y)) (\int d z G (x_{2} - z) J (z)) + G (x_{1} - x_{2})] e^{E [J]}$
When we set $J = 0$ , the first term containing products of $J$ vanishes, but the second term, $G (x_{1} - x_{2})$ , survives. Thus, we recover our propagator: $⟨ ϕ (x_{1}) ϕ (x_{2})⟩ = G (x_{1} - x_{2})$ .
3-point function ( $n = 3$ ):
Every term in the third derivative will contain at least one factor of $J$ , so setting $J = 0$ gives $⟨ ϕ (x_{1}) ϕ (x_{2}) ϕ (x_{3})⟩ = 0$ .
4-point function ( $n = 4$ ):
This is where the combinatorial nature becomes clear. After taking four derivatives, the only terms that survive when we set $J = 0$ are those where all factors of $J$ from the exponent have been "used up" by the derivatives. This happens in three distinct ways:
$⟨ ϕ (x_{1}) ϕ (x_{2}) ϕ (x_{3}) ϕ (x_{4})⟩ = G (x_{1}, x_{2}) G (x_{3}, x_{4}) + G (x_{1}, x_{3}) G (x_{2}, x_{4}) + G (x_{1}, x_{4}) G (x_{2}, x_{3})$
Each term corresponds to a different way of pairing up the four points.

For a detailed, step-by-step derivation in the language of field theory, the following notes are exceptionally clear.

Path Integrals in Quantum Field Theory - Sanjeev Seahra

These notes explicitly calculate the generating functional Z[J] for a free scalar field and then obtain the 1-point, 2-point, 3-point, and 4-point correlation functions by taking derivatives and setting J=0. The emerging pattern is then identified as Wick's theorem. Note that the notes use i\Delta_F for the propagator, which is equivalent to our G.

Please read the section titled "The free scalar field" starting on page 19. Focus on the explicit calculations of the derivatives of Z[J] in the subsection here, from equation (85) to (90), which clearly demonstrate how the correlation functions are built from the propagator.

Wick's Theorem: A Combinatorial Rule

The pattern we have uncovered is general. For any Gaussian theory, the calculation of an $n$ -point correlation function reduces to a combinatorial problem of pairing up fields. This is the essence of Wick's theorem.

Wick's Theorem: For a Gaussian field theory with $⟨ ϕ ⟩ = 0$ :

The expectation value of a product of an odd number of fields is zero. $⟨ ϕ (x_{1}) ϕ (x_{2}) \dots ϕ (x_{2 n + 1})⟩ = 0$
The expectation value of a product of an even number of fields is the sum over all distinct ways of pairing them up. Each pair (a "contraction") is replaced by the two-point function connecting the fields' coordinates. $⟨ ϕ (x_{1}) ϕ (x_{2}) \dots ϕ (x_{2 n})⟩ = all pairings \sum pairs (i, j) \prod G (x_{i}, x_{j})$

The number of terms in this sum is given by the number of ways to form $n$ pairs from $2 n$ objects, which is $(2 n - 1)!! = (2 n - 1) (2 n - 3) \dots (3) (1)$ . For our 4-point function example ( $n = 2$ ), this gives $(4 - 1)!! = 3!! = 3 \times 1 = 3$ terms, which matches our result. For a 6-point function, there would be $5!! = 15$ terms.

The following resource provides a concise and formal statement of the theorem in the context of statistical field theory.

[PDF] Advanced Statistical Mechanics Lecture 4: Field Theory Primer

This lecture note from Prof. Leo Radzihovsky at the University of Colorado Boulder provides a succinct summary of Gaussian integrals and a clear statement of Wick's theorem.

Please read the section "Propagators and Wick’s theorem for scalar field theory" on page 4. Focus on the paragraph that begins here, which formally states the theorem (equation 28). Note that the notation \phi(x_i) is used for the fields and G(x_i, x_j) for the two-point function.

Application: A Toy Model Calculation

Why is this theorem so indispensable? Its primary application is in perturbation theory. When we add an interaction term like $u ϕ^{4}$ to our Hamiltonian, we can no longer solve the functional integral exactly. Instead, we expand the exponential of the interaction term in a power series:

Z = \int D ϕ e^{- β H_{0} [ϕ]} e^{- u \int ϕ^{4}} = \int D ϕ e^{- β H_{0} [ϕ]} (1 - u \int ϕ^{4} + \frac{u ^{2}}{2} (\int ϕ^{4})^{2} - \dots)

Calculating expectation values in the interacting theory then involves calculating Gaussian averages (denoted by $⟨ \dots ⟩_{0}$ ) of powers of $ϕ$ , like $⟨ ϕ^{4} ⟩_{0}$ , $⟨ ϕ^{6} ⟩_{0}$ , etc. This is precisely what Wick's theorem allows us to do.

Let's consider the simplest non-trivial term that will appear: $⟨ ϕ^{4} (x) ⟩_{0}$ . This is a 4-point function where all coordinates are the same. Applying Wick's theorem:
\begin{align*}
\langle \phi(x)\phi(x)\phi(x)\phi(x) \rangle_0 &= \langle \phi(x)\phi(x)\rangle_0 \langle \phi(x)\phi(x) \rangle_0 \
&+ \langle \phi(x)\phi(x)\rangle_0 \langle \phi(x)\phi(x) \rangle_0 \
&+ \langle \phi(x)\phi(x)\rangle_0 \langle \phi(x)\phi(x) \rangle_0 \
&= 3 G(x,x)^2
\end{align*}
The three terms correspond to pairing the 1st field with the 2nd, 3rd, and 4th, respectively. Once the first pair is chosen, the second is fixed.

More generally, for a $2 n$ -point function with all coordinates identical, Wick's theorem gives:

⟨ ϕ^{2 n} (x) ⟩_{0} = (2 n - 1)!! ⟨ ϕ^{2} (x) ⟩_{0}^{n} = (2 n - 1)!! G (x, x)^{n}

This result is foundational for perturbation theory. Let's look at a concrete example from Radzihovsky's notes, where this is worked out for a zero-dimensional "toy" model.

[PDF] Advanced Statistical Mechanics Lecture 4: Field Theory Primer

This section presents a warm-up calculation for perturbation theory using a single-mode "toy" \phi^4 model. It's a perfect illustration of Wick's theorem in action.

Please read the subsection "Perturbative expansion for a single-mode “toy” φ4 model" starting on page 16. Focus on equations (86) through (90), where the author establishes the Gaussian averages and explicitly writes down the zero-dimensional version of the theorem. This directly shows how an average like \langle\phi^6\rangle_0 is computed in terms of the propagator G_0 = 1/t.

Conclusion

In this lesson, we have unlocked the computational content of the Gaussian generating functional by deriving and applying Wick's theorem. This powerful result streamlines the calculation of correlation functions in non-interacting theories, reducing them to a simple combinatorial exercise.

Here are the key takeaways:

Wick's theorem states that for a Gaussian theory, any $n$ -point correlation function can be calculated as a sum over all possible pairings of the fields.
Each pair, or contraction, corresponds to a two-point propagator $G (x_{i}, x_{j})$ .
Correlation functions with an odd number of fields are always zero.
Correlation functions with an even number, $2 n$ , of fields are a sum of $(2 n - 1)!!$ terms, each being a product of $n$ propagators.
The primary utility of this theorem is in perturbation theory, where it allows us to evaluate the Gaussian averages that arise from expanding the interaction term of the Hamiltonian.

We are now fully equipped to move beyond the non-interacting world. In our next lesson, we will apply this machinery to the full Landau-Ginzburg model with the $ϕ^{4}$ interaction. We will perform a perturbative expansion and use Wick's theorem to calculate the first-order correction to the propagator. This will naturally lead us to the concept of Feynman diagrams, a graphical method for organizing and interpreting the terms in the expansion.

Recommended resources

Unpacking the Generating Functional

Wick's Theorem: A Combinatorial Rule

Application: A Toy Model Calculation

Conclusion