Functional Derivatives and Correlation Functions

The Master Formula and the Functional Derivative

The central formula for this lesson connects the generating functional to the vacuum expectation values of time-ordered products of field operators, also known as Green's functions or correlation functions. It is given by:

This formula tells us that to get the n-point function, we simply differentiate $Z_0[J]$ with respect to the source $J(x)$ at each of the n spacetime points and then set the source to zero. The heart of this operation is the functional derivative.

Given your strong mathematical background, you can think of the functional derivative $\delta F[J]/\delta J(x)$ as a generalization of the partial derivative for functions. It measures how a functional $F[J]$ (a "function of a function") changes when the function $J(y)$ is perturbed by a delta function at point $x$. For our purposes, the most important rule is the analogue of $\partial x_i/\partial x_j={\delta }_{ij}$:

This rule, combined with the standard rules of differentiation like the product and chain rules, is all we need.

To ensure we're on solid ground with this concept, the following resource provides a clear and practical introduction to functional derivatives.

Computing the Two-Point Function

Let's start by applying the master formula for the simplest non-trivial case: the two-point function ($n=2$). This should recover the Feynman propagator, confirming that our formalism is consistent.

We need to calculate:

Let's perform the differentiation step-by-step. For clarity, let the exponent in $Z_0[J]$ be $K[J]=-\frac{i}{2}\iint d^{4}y{d}^{4}zJ(y){\Delta }_F(y-z)J(z)$.

1. First Derivative: Using the chain rule, $\delta (e^K)/\delta J=e^K\cdot \delta K/\delta J$.

   $$\frac{\delta Z_0[J]}{\delta J(x_2)}=Z_0[0]e^{K[J]}\cdot \frac{\delta K[J]}{\delta J(x_2)}=Z_0[J]\left(-i\int d^{4}y{\Delta }_F(x_2-y)J(y)\right)$$

2. Second Derivative: Now we differentiate with respect to $J(x_1)$ using the product rule.

   $$\frac{{\delta }^2{Z}_0[J]}{\delta J(x_1)\delta J(x_2)}=\underset{\text{Term 1}}{\underbrace{\frac{\delta Z_0[J]}{\delta J(x_1)}\left(-i\int d^{4}y{\Delta }_F(x_2-y)J(y)\right)}}+\underset{\text{Term 2}}{\underbrace{Z_0[J]\left(-i\int d^{4}y{\Delta }_F(x_2-y)\frac{\delta J(y)}{\delta J(x_1)}\right)}}$$

3. Evaluate at J=0:

   • Term 1 contains a factor of $J(y)$, so when we set $J=0$, this entire term vanishes.
   • Term 2 simplifies using $\delta J(y)/\delta J(x_1)={\delta }^{(4)}(x_1-y)$. The integral collapses: $$Z_0[J]\left(-i\int d^{4}y{\Delta }_F(x_2-y){\delta }^{(4)}(x_1-y)\right)=Z_0[J](-i{\Delta }_F(x_2-x_1))$$

   Setting $J=0$ gives $Z_0[0](-i{\Delta }_F(x_1-x_2))$, since the propagator is symmetric.

4. Final Result: Plugging this back into our master formula:

   $$⟨\Omega |T\{\varphi (x_1)\varphi (x_2)\}|\Omega ⟩=\frac{-1}{Z_0[0]}\left[Z_0[0](-i{\Delta }_F(x_1-x_2))\right]=i{\Delta }_F(x_1-x_2)$$

This is exactly the result we expect from canonical quantization! It confirms that the two-point correlation function is, up to a factor of $i$, the Feynman propagator.

The following short video provides a dynamic walkthrough of this very calculation.

Computing the Four-Point Function and Wick's Theorem

Now for the main event: the four-point function ($n=4$). This is where the true power and structure of this method become apparent. The calculation is a straightforward, if lengthy, application of the same rules. We need to apply $\frac{1}{i}\frac{\delta }{\delta J}$ four times.

Let's think about the structure. Each time we differentiate, we "bring down" a factor of $\int {\Delta }_{F}J$. When we differentiate that factor, the derivative can either hit the $J$ inside the integral (bringing down a ${\Delta }_F$) or it can hit the $Z_0[J]$ prefactor (bringing down another $\int {\Delta }_{F}J$).

The final result is obtained by setting $J=0$. This means any term with a remaining factor of $J$ will vanish. A non-zero contribution only occurs when every derivative $\delta /\delta J(x_i)$ has been used to eliminate a $J$ from one of the $\int {\Delta }_{F}J$ terms.

Since each propagator term ${\Delta }_F(y-z)$ involves two sources, $J(y)$ and $J(z)$, and we have four derivatives, the only way for everything to be non-zero is to pair up the derivatives. For $⟨T{\varphi }_1{\varphi }_2{\varphi }_3{\varphi }_4⟩$, the pairings are (1,2)-(3,4), (1,3)-(2,4), and (1,4)-(2,3). Each pair corresponds to a product of two propagators.

The result of this calculation is:
\begin{align*}
\langle \Omega | T{\phi(x_1)\phi(x_2)\phi(x_3)\phi(x_4)} | \Omega \rangle = & (i\Delta_F(x_1-x_2))(i\Delta_F(x_3-x_4)) \
& + (i\Delta_F(x_1-x_3))(i\Delta_F(x_2-x_4)) \
& + (i\Delta_F(x_1-x_4))(i\Delta_F(x_2-x_3))
\end{align*}

This result—that the n-point function in a free theory is the sum over all possible pairings of the fields, with each pair replaced by a propagator—is Wick's Theorem. In the path integral formalism, it is simply a consequence of repeatedly differentiating a Gaussian function!

The following resource explicitly states this result and provides a helpful graphical way to think about the calculation.

Conclusion

In this lesson, we have seen how the generating functional serves as a true "generator" for all correlation functions in a quantum field theory. By mastering the simple mechanics of functional differentiation, we can systematically compute any n-point Green's function.

Key Takeaways:

• Time-ordered correlation functions are computed by taking functional derivatives of the generating functional $Z[J]$ with respect to the source $J(x)$ at the desired spacetime points.
• For the free scalar theory, the generating functional $Z_0[J]$ is Gaussian. This has a profound consequence: all odd-point correlation functions are zero.
• The n-point correlation function for an even $n$ is the sum of all possible ways to pair up the $n$ points, where each pair contributes a factor of the Feynman propagator, $i{\Delta }_F$. This result is Wick's Theorem.

Preview of the Next Lesson:
Our entire discussion so far has been about scalar fields, which describe bosons. What about fermions, like electrons? To build a path integral for fermionic fields, we need a new kind of number system where the variables anti-commute, just like fermion creation operators do. In the next lesson, we will introduce Grassmann numbers and the rules for integrating them, known as Berezin integration. This will be the final mathematical tool we need to construct path integrals for Dirac fields and, ultimately, for QED.