Hello! In our last lesson, we constructed the Dyson series, which gives us a formal expression for the time-evolution operator in an interacting theory like $\lambda\phi^4$. This was a major step, allowing us to write down the S-matrix, $S = U(+\infty, -\infty)$, which connects initial and final states in a scattering process.
However, we were left with a practical problem. The Dyson series is a time-ordered exponential, and its expansion involves calculating vacuum expectation values (VEVs) of time-ordered products of many field operators, like $\langle 0 | T(\phi(x_1)\phi(x_2)\dots\phi(x_n)) | 0 \rangle$. Evaluating these directly seems intractable.
Today, we will learn the master tool for solving this problem: Wick's Theorem. This remarkable theorem provides a simple, combinatorial algorithm for breaking down any time-ordered product of fields into a sum of more manageable terms. It is the crucial bridge between the abstract formalism of the Dyson series and the practical, graphical method of Feynman diagrams.
Our learning outcome is to: Apply Wick's theorem to express time-ordered products of interacting fields in terms of normal-ordered products and propagators.
1. Normal Ordering and Contractions
The two key ingredients of Wick's theorem are normal ordering and contractions.
First, recall that we can split our field operator $\phi(x)$ into a part that contains only annihilation operators, which we'll call $\phi^+(x)$, and a part with only creation operators, $\phi^-(x)$:
where
(Note: This +/- convention is common, but some authors, like Peskin & Schroeder, use the opposite. The physics is the same.)
Normal Ordering, denoted by $N(\dots)$ or $:\dots:$, is an instruction to rearrange a product of field operators so that all creation operators ($\phi^-$) are placed to the left of all annihilation operators ($\phi^+$).
The most important property of a normal-ordered product is that its vacuum expectation value is always zero. This is because any term will either have an annihilation operator acting on the ket vacuum $\lvert 0 \rangle$ or a creation operator acting on the bra vacuum $\langle 0 \rvert$:
Now, let's compare the time-ordered product $T(\phi(x_1)\phi(x_2))$ with the normal-ordered product $N(\phi(x_1)\phi(x_2))$. They are clearly not the same. Their difference defines the concept of a contraction.
Quantum Field Theory I, Chapter 8
Let's formalize these ideas. The following reading from the ETH Zurich QFT I notes provides a concise and rigorous development of normal ordering and shows how the difference between a time-ordered and a normal-ordered product of two fields gives rise to a familiar object.
Please read Section 8.3, 'Wick’s Theorem', starting from the definition of normal ordering down to equation (8.24). Focus on how the difference between a time-ordered product and a normal-ordered product of two fields is calculated.
As you saw in the reading, the difference between the time-ordered and normal-ordered product of two fields is not an operator, but a c-number function:
This is precisely the Feynman propagator for the scalar field! This difference is called the contraction of the two fields, often denoted with a line connecting them:
2. Wick's Theorem: The General Statement
Wick's theorem generalizes this relationship to a product of any number of field operators. It states that:
The time-ordered product of a set of field operators is equal to the sum of all their possible normal-ordered products with all possible pairings of contractions.
Let's unpack this with the examples from the notes you just read.
Quantum Field Theory I, Chapter 8
Now we'll see the general statement of the theorem. Continue reading the same section to see the formal statement and examples for 3 and 4 fields.
Continue reading from just after equation (8.24) to the end of equation (8.26). Pay close attention to the notation for contractions and how the theorem expresses a T-product as a sum over normal-ordered terms with zero, one, or more contractions.
To summarize the structure of the theorem:
Each term in this sum consists of a normal-ordered product of the uncontracted fields, multiplied by the propagators corresponding to the contracted fields.
3. The Power of Wick's Theorem in Calculations
The true utility of Wick's theorem emerges when we need to calculate a vacuum expectation value.
Quantum Field Theory I, Chapter 8
The real power of Wick's theorem becomes apparent when we calculate VEVs. Read the next short subsection to see why this provides a massive simplification.
Read the subsection 'Time-Ordered Correlators' (up to equation 8.27). The key insight is that only fully contracted terms survive when taking the VEV.
Since $\langle 0 | N(\dots) | 0 \rangle = 0$ for any product of operators, the only terms from the Wick expansion that survive are the ones where all fields are contracted. Each contraction is a c-number (a propagator), so it can be pulled out of the expectation value, leaving only $\langle 0 | 0 \rangle = 1$.
Therefore, the VEV of a time-ordered product of fields is simply the sum of all possible ways to pair up the fields into propagators.
An immediate consequence is that the VEV of an odd number of fields is always zero, because it's impossible to pair them all up.
Test your understanding!
Using Wick's theorem, write out the expression for the four-point correlation function, $\langle 0 | T(\phi(x_1)\phi(x_2)\phi(x_3)\phi(x_4)) | 0 \rangle$, in terms of Feynman propagators $D_F(x_i - x_j)$.
Show answer
We need to find all possible ways to fully contract the four fields in pairs.
- Contract (1,2) and (3,4).
- Contract (1,3) and (2,4).
- Contract (1,4) and (2,3).
There are three distinct ways to do this. Each term is a product of two propagators.
4. Application: A First Look at Perturbation Theory
We are now ready to tackle a real calculation. Let's compute the first-order correction (in $\lambda$) to the two-point correlation function $\langle 0 | T(\phi(x_1)\phi(x_2)) | 0 \rangle$ in $\phi^4$ theory.
From the Dyson series, the S-matrix is $S = T \exp(-i \int d^4z \mathcal{L}_I(z))$, where $\mathcal{L}_I = \frac{\lambda}{4!}\phi^4$. The full interacting two-point function is given by:
where $|\Omega\rangle$ is the true interacting vacuum. Let's focus on the numerator and expand $S$ to first order in $\lambda$:
The zeroth-order term is just the free propagator $D_F(x_1-x_2)$. The first-order term requires us to calculate the VEV of a time-ordered product of six fields. This is a perfect job for Wick's theorem.
Quantum Field Theory I Lecture 6: Normal ordering and Wick's thm. Feynman diagrams and rules.
We now have all the tools to tackle a term from the Dyson series expansion. The following video segment demonstrates how to calculate the first-order correction to the two-point correlation function in $\phi^4$ theory. This is a crucial example that ties everything together and provides a first glimpse of Feynman diagrams.
Watch from 57:47 to 1:20:00. This is a detailed calculation. Follow how Wick's theorem is applied to the time-ordered product of six fields, how the terms are grouped by their contraction structure, and how this structure is represented by simple diagrams. Don't worry about memorizing the symmetry factors yet; focus on the process.
This calculation is the heart of perturbative QFT. We took an abstract term from the Dyson series and, using Wick's theorem, turned it into a sum of concrete objects—propagators—which we can represent visually.
For the term involving six fields $\phi(x_1)\phi(x_2)\phi(z)\phi(z)\phi(z)\phi(z)$, you saw two main types of full contractions emerge:
- Disconnected diagrams: $x_1$ is contracted with $x_2$, and the four fields at $z$ are contracted among themselves. This gives a term proportional to $D_F(x_1-x_2) \times (\text{something at z})$.
- Connected diagrams: $x_1$ contracts with a $\phi(z)$, $x_2$ contracts with another $\phi(z)$, and the remaining two $\phi(z)$'s contract with each other. This gives a term proportional to $\int d^4z D_F(x_1-z)D_F(x_2-z)D_F(z-z)$.
The visual diagrams introduced in the video are not just helpful pictures; they are Feynman diagrams, and each line and vertex corresponds precisely to a term in the Wick expansion.
Conclusion
Today, we've unlocked the primary computational engine of perturbative QFT. We've seen that the intimidating time-ordered products in the Dyson series can be systematically evaluated using a simple set of combinatorial rules.
Key Takeaways:
- Normal Ordering ($N(\dots)$) arranges operators with creation operators to the left of annihilation operators, ensuring the VEV is zero.
- A Contraction is the difference between a time-ordered and a normal-ordered product of two fields, and it is equal to the Feynman propagator $D_F(x-y)$.
- Wick's Theorem states that any time-ordered product of fields equals the sum of their normal-ordered products with all possible contractions.
- In VEV calculations, this simplifies dramatically: the result is just the sum of all possible full contractions (pairings) of the fields, where each pair becomes a propagator.
- This process directly generates the terms that are represented by Feynman diagrams.
We now have a method to calculate the fundamental building blocks of scattering processes. The next step is to formalize how these calculated quantities, the correlation functions, are related to the physical observables we measure in experiments, such as scattering cross-sections.
Preview of the Next Lesson:
In the next lesson, we will introduce the Lehmann-Symanzik-Zimmermann (LSZ) reduction formula. This crucial formula provides the exact relationship between the S-matrix elements (which we calculate from correlation functions using Wick's theorem) and the cross-sections and decay rates measured in particle physics experiments. It is the final link between our theoretical calculations and the real world.