The Generating Functional for Correlation Functions

Greetings. In our last lesson, we derived the Ginzburg criterion, which revealed a crucial limitation of mean-field theory. We found that for spatial dimensions $d < 4$ , fluctuations inevitably overwhelm the mean-field behavior as a system approaches its critical point. This means that to accurately describe critical phenomena, we must move beyond the simple Gaussian approximation and confront the interacting nature of the system, encapsulated by the $ϕ^{4}$ term in the Landau-Ginzburg functional.

To do this systematically, we need a more powerful and general method for calculating correlation functions. Today, we will develop this essential tool: the generating functional. The core idea is a clever mathematical maneuver. Instead of computing each $n$ -point correlation function through a separate, challenging functional integral, we will construct a single object, $Z [J]$ , from which all correlation functions can be generated by the comparatively simple act of taking derivatives. This approach is central to modern statistical and quantum field theory.

The Source of All Correlations

The concept of a generating function is likely familiar to you from your studies in mathematics and probability theory, where moment-generating functions are used to find moments of a distribution. We will apply the exact same logic, but elevated from a single variable to a field.

The trick is to introduce an auxiliary "source" field, $J (x)$ , that couples linearly to our order parameter field, $ϕ (x)$ . We then modify the partition function to depend on this source.

To see how this works, let's start with the basic idea as presented in a lecture from MIT OpenCourseWare. The professor first introduces the concept for an ordinary one-dimensional integral before generalizing it.

Lecture 9: Path Integral Formalism for QFT; Computation of Time-Ordered Correlation Functions

This video segment by Prof. Hong Liu introduces the motivation for using a generating function to calculate moments of a distribution, which in our case will be correlation functions.

Please watch the clip from this point. The key idea is that by adding a term like iax to the exponent of an integral, the coefficients of the Taylor series expansion of the resulting integral in powers of a are precisely the moments \langle x^n \rangle we wish to calculate. Alternatively, these moments can be found by taking derivatives with respect to a.

The Generating Functional in Statistical Field Theory

Now, we elevate this idea to the continuum. The integration variable $x$ becomes the field $ϕ (x)$ , the source $a$ becomes a source field $J (x)$ , and sums over indices become integrals over space. The generating functional, $Z [J]$ , is defined as the partition function in the presence of this external source field:

Z [J] = \int D ϕ exp (- β F [ϕ] + \int d^{d} x J (x) ϕ (x))

Here, $F [ϕ]$ is our familiar Landau-Ginzburg free energy functional. Notice that $Z [J]$ is a functional—it takes a function, $J (x)$ , as its input and produces a single number as its output.

The magic happens when we take functional derivatives of $Z [J]$ with respect to the source field $J (x)$ . Since the functional derivative $δ / δ J (y)$ acts like a regular derivative, its only effect on the term $\int d^{d} x J (x) ϕ (x)$ is to "pull down" a factor of $ϕ (y)$ .

Let's see this in action. The thermal average of any quantity $A [ϕ]$ in the presence of the source is:

⟨ A [ϕ] ⟩_{J} = \frac{1}{Z [ J ]} \int D ϕ A [ϕ] exp (- β F [ϕ] + \int d^{d} x J (x) ϕ (x))

If we take a functional derivative of $Z [J]$ itself, we get:

\frac{δ Z [ J ]}{δ J ( y )} = \int D ϕ ϕ (y) exp (- β F [ϕ] + \int d^{d} x J (x) ϕ (x)) = Z [J] ⟨ ϕ (y) ⟩_{J}

By repeating this process, we can generate any $n$ -point correlation function. We first take $n$ derivatives and then set the source $J$ to zero to recover the correlations of the original, unperturbed system. The general formula is:

⟨ 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}

Here $Z [0]$ is just our original partition function. The following video segment formally lays out this generalization from quantum mechanics to field theory. The context is quantum field theory, so time ordering is mentioned, but for our statistical mechanics problems in Euclidean space, this is not a concern as fields at different points commute.

Lecture 10: Time-Ordered Correlation Functions in Field Theory

In this video, Prof. Hong Liu formalizes the definition of the generating functional for a field theory and shows how it generates n-point functions.

Watch from this section, where the generating functional Z[J] is defined and the rule for obtaining n-point correlation functions via functional differentiation is derived. Then, watch from the expansion of Z[J] in powers of J, which shows that the correlation functions appear as the coefficients. Finally, see how this generalizes to field theory from this point.

Connected Correlation Functions and $W [J]$

A closely related and often more useful object is the generator of connected correlation functions, defined as $W [J] = ln Z [J]$ . In statistical terms, taking derivatives of $ln Z$ generates cumulants, rather than raw moments.

Let's see what the first two derivatives of $W [J]$ give:

First Derivative:
$\frac{δ W [ J ]}{δ J ( x )} = \frac{1}{Z [ J ]} \frac{δ Z [ J ]}{δ J ( x )} = ⟨ ϕ (x) ⟩_{J}$
Setting $J = 0$ , this gives the average value of the order parameter, $⟨ ϕ (x)⟩$ .
Second Derivative:
\begin{align*}
\frac{\delta^2 W[J]}{\delta J(x) \delta J(y)} &= \frac{\delta}{\delta J(x)} \left( \frac{1}{Z[J]} \frac{\delta Z[J]}{\delta J(y)} \right) \
&= -\frac{1}{Z[J]^2} \frac{\delta Z[J]}{\delta J(x)} \frac{\delta Z[J]}{\delta J(y)} + \frac{1}{Z[J]} \frac{\delta^2 Z[J]}{\delta J(x) \delta J(y)} \
&= \langle \phi(x)\phi(y) \rangle_J - \langle \phi(x) \rangle_J \langle \phi(y) \rangle_J
\end{align*}
Setting $J = 0$ , this gives precisely the connected two-point correlation function, $⟨ ϕ (x) ϕ (y) ⟩_{c}$ , which measures the correlated fluctuations around the mean.

This is a general and powerful result. Kevin Zhou's excellent notes on Statistical Field Theory provide a very clear and compact summary of this idea.

Statistical Field Theory

These notes introduce the generating functional by analogy with how differentiating the log of the standard partition function with respect to inverse temperature \beta yields moments of the energy.

Please read the first page of the notes, up to the point where the two-point function is recovered. Focus on the section beginning here, which defines the generating functional in the presence of a source B(x) (equivalent to our J(x)) and shows that its second derivative generates the connected correlation function, which he refers to as the joint cumulant.

Example: The Gaussian Theory

To make this machinery concrete, let's calculate $Z [J]$ for the free, or Gaussian, theory, which has been our workhorse so far. The Landau-Ginzburg functional is purely quadratic:

β F [ϕ] = \frac{1}{2} \int d^{d} x [(\nabla ϕ)^{2} + r ϕ^{2}] = \frac{1}{2} \int d^{d} x ϕ (x) (- \nabla^{2} + r) ϕ (x)

The exponent in our generating functional is therefore:

- \frac{1}{2} \int d^{d} x ϕ (x) (- \nabla^{2} + r) ϕ (x) + \int d^{d} x J (x) ϕ (x)

This is a Gaussian integral with a term linear in the integration variable $ϕ$ . You know from your experience with path integrals that such integrals can be solved by "completing the square". The procedure is analogous to solving the ordinary integral $\int e^{- a x^{2} /2 + J x} d x$ .

The following video calculates exactly this type of integral for the path integral of a simple harmonic oscillator, which is mathematically identical to our Gaussian field theory.

Lecture 10: Time-Ordered Correlation Functions in Field Theory

This part of the lecture explicitly calculates the generating functional for a free theory (a harmonic oscillator).

Please watch the detailed calculation from this segment. The key steps are: The action is written in a quadratic form, \frac{1}{2} \int x K x, where K is a differential operator. The source term Jx is added. The integral is solved by completing the square. The result for the generating functional Z[J] is shown to be Z[0] \exp\left(-\frac{1}{2} \int J K^{-1} J\right).

The result of this calculation is profound. For a Gaussian theory, the generating functional is:

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 inverse of the operator $K = - \nabla^{2} + r$ . This is exactly the Green's function for that operator, which we have already identified as the Ornstein-Zernike two-point correlation function we derived in a previous lesson!

This tells us that for a free theory, $Z [J]$ is itself a Gaussian functional of the source $J$ . The covariance of this Gaussian is none other than the two-point propagator of the field $ϕ$ .

Conclusion

Today we have constructed one of the most powerful formal tools in statistical field theory. While the concept may seem abstract, it provides a unified and systematic path to understanding correlations in complex systems.

Key takeaways from this lesson are:

The generating functional $Z [J]$ is defined by adding a source term $\int d^{d} x J (x) ϕ (x)$ to the exponent of the functional integral for the partition function.
Functional derivatives of $Z [J]$ with respect to $J (x)$ systematically generate all $n$ -point correlation functions of the field $ϕ$ .
Derivatives of $W [J] = ln Z [J]$ generate the connected correlation functions (or cumulants), which describe the intrinsic correlations of the system.
For a Gaussian (free) theory, $Z [J]$ can be calculated exactly and takes the form of a Gaussian in the source field $J$ . The covariance of this Gaussian is precisely the two-point correlation function (propagator) $G (x - y)$ .

We now have the machinery in place. We have defined the object that "knows" about all correlation functions, and we have solved for it in the simple case of a non-interacting theory. In our next lesson, we will put this result to work. We will use the explicit form of $Z [J]$ for the Gaussian theory to derive Wick's theorem, a combinatorial rule that allows us to calculate any $n$ -point correlation function as a sum of products of two-point functions. This will complete our understanding of free theories and lay the final piece of groundwork for tackling interacting systems with perturbation theory.

Recommended resources

The Source of All Correlations

The Generating Functional in Statistical Field Theory

Connected Correlation Functions and W[J]

Example: The Gaussian Theory

Conclusion

Connected Correlation Functions and $W [J]$