Wick's Theorem in Free CFTs

Hello! Welcome to your next lesson on computational methods in 2D Conformal Field Theory.

In our last session, we performed a complete calculation of the partition function for a free boson on a torus. We saw how the contributions from the zero-modes and the oscillator modes must combine in a highly specific way—governed by the Dedekind eta function—to ensure the theory is modular invariant. This gave us a picture of the full spectrum of the theory.

Today, we shift our focus from the spectrum to another set of fundamental observables: correlation functions. These functions, which represent the vacuum expectation values of products of fields, encode the dynamics and interactions within the theory. Our goal is to master the primary computational tool for calculating correlators in free theories: Wick's theorem. Since you have a background in QFT, you are likely familiar with this theorem in the context of time-ordered products. Here, we will adapt it to the CFT framework of radial ordering and operator product expansions.

Your learning objective for this lesson is to apply Wick's theorem to compute correlation functions in free bosonic and fermionic CFTs.

1. Wick's Theorem in Conformal Field Theory

In standard QFT, Wick's theorem relates time-ordered products of fields to normal-ordered products. In CFT, we work with radially-ordered products, but the principle is identical. The theorem provides a systematic algorithm for expressing a complicated product of operators in terms of simpler, known quantities.

The three key ingredients are:

Radial Ordering $R {...}$ : Operators are ordered by the magnitude of their radial coordinate, with operators at larger $∣ z ∣$ placed to the left. As we've seen, this corresponds to chronological ordering on the cylinder.
Normal Ordering $: ... :$ : Within any operator, creation modes are placed to the left of annihilation modes. This construction guarantees that the vacuum expectation value (VEV) of any normal-ordered product of operators is zero: $⟨ 0∣ : A (z) B (w) ... : ∣0 ⟩ = 0$ .
Contraction: The "contraction" of two fields, let's denote it $A (z) B (w)$ , is defined as the singular part of their Operator Product Expansion (OPE). For the free fields we are considering, this is simply their two-point function: $A (z) B (w) \equiv R {A (z) B (w)} - : A (z) B (w) := ⟨ A (z) B (w)⟩$

Wick's Theorem states that a radially-ordered product of fields is equal to the sum of the normal-ordered product of those fields with all possible contractions applied.
For example, for four fields:

R {A_{1} A_{2} A_{3} A_{4}} =: A_{1} A_{2} A_{3} A_{4} : + 1 contraction \sum : A_{i} A_{j} A_{k} A_{l} : + 2 contractions \sum : A_{i} A_{j} A_{k} A_{l} :

The power of this theorem becomes apparent when we take the vacuum expectation value. Since the VEV of any normal-ordered term is zero, the only terms that survive are the ones where all fields have been contracted.

The VEV of a radially-ordered product of fields is the sum of all possible full contractions.

Let's see this in action.

The Boson-Fermion Correspondence and its Applications

The following resource provides an excellent, self-contained introduction to Wick's theorem in the context of 2D CFT. It formalizes the concepts of normal ordering, radial ordering, and contraction.

Please read Section 4.3 up to and including the paragraph that starts with 'Wick’s theorem...'. Then, skim the first few paragraphs of Section 4.4 ('Correlation Functions') to see how these ideas are applied to computing VEVs, specifically noting how the VEV of a normal-ordered product vanishes.

2. Application: The Free Boson

Let's compute the four-point function of the chiral free boson field, $\partial ϕ (z)$ . In our previous lessons, we established the OPE, which gives us our fundamental contraction (setting normalization constants to 1 for simplicity):

\partial ϕ (z_{i}) \partial ϕ (z_{j}) = ⟨ \partial ϕ (z_{i}) \partial ϕ (z_{j})⟩ = \frac{1}{( z _{i} - z _{j} ) ^{2}}

Now we want to compute $⟨ \partial ϕ (z_{1}) \partial ϕ (z_{2}) \partial ϕ (z_{3}) \partial ϕ (z_{4})⟩$ . According to Wick's theorem, we just need to sum over all ways to fully pair up the four fields. There are three such ways:

Pairing (1,2) and (3,4): $⟨ \partial ϕ (z_{1}) \partial ϕ (z_{2}) \partial ϕ (z_{3}) \partial ϕ (z_{4}) ⟩ = \frac{1}{( z _{1} - z _{2} ) ^{2}} \frac{1}{( z _{3} - z _{4} ) ^{2}}$
Pairing (1,3) and (2,4): $⟨ \partial ϕ (z_{1}) \partial ϕ (z_{3}) \partial ϕ (z_{2}) \partial ϕ (z_{4}) ⟩ = \frac{1}{( z _{1} - z _{3} ) ^{2}} \frac{1}{( z _{2} - z _{4} ) ^{2}}$
Pairing (1,4) and (2,3): $⟨ \partial ϕ (z_{1}) \partial ϕ (z_{4}) \partial ϕ (z_{2}) \partial ϕ (z_{3}) ⟩ = \frac{1}{( z _{1} - z _{4} ) ^{2}} \frac{1}{( z _{2} - z _{3} ) ^{2}}$

The full four-point function is the sum of these three terms:

⟨ \partial ϕ (z_{1}) \partial ϕ (z_{2}) \partial ϕ (z_{3}) \partial ϕ (z_{4})⟩ = \frac{1}{( z _{1} - z _{2} ) ^{2} ( z _{3} - z _{4} ) ^{2}} + \frac{1}{( z _{1} - z _{3} ) ^{2} ( z _{2} - z _{4} ) ^{2}} + \frac{1}{( z _{1} - z _{4} ) ^{2} ( z _{2} - z _{3} ) ^{2}}

The resource you just read, specifically equation (4.25), works through this exact example, showing how the fully contracted terms are just one part of the full OPE of four fields.

3. Application: The Free Fermion

Calculating correlators for free fermions follows the same logic, but with a crucial twist owing to Fermi-Dirac statistics: swapping the order of any two fermionic fields introduces a minus sign.

This is reflected in a modified version of Wick's theorem. When you draw the contraction pairings, each term acquires a sign of $(- 1)^{P}$ , where $P$ is the number of times the contraction lines cross each other.

The fundamental contraction for a chiral fermion field $ψ (z)$ is:

ψ (z_{i}) ψ (z_{j}) = ⟨ ψ (z_{i}) ψ (z_{j})⟩ = \frac{1}{z _{i} - z _{j}}

Let's compute the fermionic four-point function $⟨ ψ (z_{1}) ψ (z_{2}) ψ (z_{3}) ψ (z_{4})⟩$ .

Pairing (1,2) and (3,4):
The contraction lines do not cross ( $P = 0$ ). The sign is $(- 1)^{0} = + 1$ .

A non-crossing pairing of four operators. The lines connecting (1,2) and (3,4) do not intersect.

Term: $+ \frac{1}{( z _{1} - z _{2} )} \frac{1}{( z _{3} - z _{4} )}$
Pairing (1,3) and (2,4):
The contraction lines cross once ( $P = 1$ ). The sign is $(- 1)^{1} = - 1$ .

A crossing pairing of four operators. The line connecting (1,3) intersects the line connecting (2,4).

Term: $- \frac{1}{( z _{1} - z _{3} )} \frac{1}{( z _{2} - z _{4} )}$
Pairing (1,4) and (2,3):
The contraction lines cross once ( $P = 1$ ). The sign is $(- 1)^{1} = - 1$ .
Term: $- \frac{1}{( z _{1} - z _{4} )} \frac{1}{( z _{2} - z _{3} )}$

The full fermionic four-point function is therefore:

⟨ ψ (z_{1}) ψ (z_{2}) ψ (z_{3}) ψ (z_{4})⟩ = \frac{1}{( z _{1} - z _{2} ) ( z _{3} - z _{4} )} - \frac{1}{( z _{1} - z _{3} ) ( z _{2} - z _{4} )} - \frac{1}{( z _{1} - z _{4} ) ( z _{2} - z _{3} )}

Notice the difference in signs compared to the bosonic result.

The Boson-Fermion Correspondence and its Applications

Let's return to the same resource, which also covers the fermionic case with its unique rules.

First, review Section 5.1 and 5.2 to understand the definition of fermionic normal ordering (eq. 5.13), which includes a minus sign. Then, in Section 5.2, read the paragraph beginning 'Wick’s Theorem introduced in section 4.3 should be modified...', which explicitly states the sign rule. Finally, look at Section 6.5, particularly equations (6.57) and (6.58), to see this rule applied to compute 2- and 3-point fermionic correlators.

Test your understanding!

Using the fermionic Wick's theorem, calculate the three-point function $⟨ ψ (z_{1}) ψ (z_{2}) ψ (z_{3})⟩$ . What do you find?

Show answer

To compute the VEV of an odd number of fields, we need to fully contract them. However, since contractions always involve pairs of fields, it's impossible to fully contract an odd number of them. Any term in the Wick expansion will have at least one uncontracted field left inside a normal-ordered product, e.g., $: ψ (z_{1}) ψ (z_{2}) ψ (z_{3}) :$ . When we take the vacuum expectation value, this term vanishes. Since all possible terms will have at least one uncontracted field, the entire sum is zero.

⟨ ψ (z_{1}) ψ (z_{2}) ψ (z_{3})⟩ = 0

This is a general feature: for a free fermion theory of this type, all correlation functions with an odd number of fields are zero.

Conclusion

In this lesson, we have added a crucial computational technique to our CFT toolkit. By adapting Wick's theorem from QFT to the language of radial ordering and OPEs, we have developed a systematic method for calculating correlation functions in free theories. We saw how this works for both free bosons and free fermions, with the key difference being the inclusion of a sign factor for fermions due to their anticommuting nature.

Key Takeaways:

Wick's Theorem for CFTs states that a radially-ordered product of fields can be decomposed into a sum over all possible normal-ordered contractions.
The vacuum expectation value of a product of fields is the sum of all terms where every field is fully contracted.
A contraction is the singular part of the OPE between two fields, which for free theories is just their two-point function.
For bosonic fields, all terms in the Wick expansion contribute with a positive sign.
For fermionic fields, each term acquires a sign of $(- 1)^{P}$ , where $P$ is the number of crossings in the contraction diagram.

In our next lesson, we will return to the structure of the operator spectrum. We will define Virasoro characters, which are generating functions that count the number of states at each energy level within a given representation. We will then see how the powerful constraint of modular invariance, which we explored in the context of the full partition function, can be used on these characters to constrain the possible operator content of a rational CFT.

Recommended resources

1. Wick's Theorem in Conformal Field Theory

2. Application: The Free Boson

3. Application: The Free Fermion

Conclusion