Operator Product Expansion and Operator Algebra

Hello! Welcome to the next lesson in our course on Conformal Field Theory.

In our previous lesson, we established that while 2- and 3-point functions are fixed by conformal symmetry, the 4-point function introduces a theory-dependent function $G (u, v)$ of the conformal cross-ratios. This function encodes the dynamics of the theory. The natural question is: how can we constrain or determine this function?

The answer lies in a concept that is arguably the most important structural element of any quantum field theory, and which becomes extraordinarily powerful in CFT: the Operator Product Expansion (OPE). The OPE provides a precise way to express what happens when two operators approach each other. It effectively defines the "multiplication" in the algebra of local operators and, as we'll see, forges a deep connection between correlation functions of different numbers of points.

Today, we will formally define the OPE and explore the deep reasons for its power in CFT, relating it directly to the algebraic structure of the theory.

The Operator Product Expansion (OPE)

At its core, the OPE is the statement that the product of two local operators, $O_{1} (x_{1})$ and $O_{2} (x_{2})$ , at nearby points can be replaced by an infinite sum of single local operators at a point midway between them.

Operator Product Expansion (OPE) Illustration — This image illustrates the central idea of the OPE. When two operators $\mathcal{O}_1$ and $\mathcal{O}_2$ are brought close together, their combined effect can be represented by a sum over a complete basis of local operators $\mathcal{O}_k$ located at their midpoint, weighted by coefficients that depend on the separation.

The general form of the OPE is written as:

O_{i} (x) O_{j} (0) x \to 0 \sim k \sum C_{ij}^{k} (x) O_{k} (0)

Here, the sum is over a complete basis of local operators in the theory, ${O_{k}}$ . The functions $C_{ij}^{k} (x)$ depend on the separation $x$ and encode how much of each operator $O_{k}$ is contained in the product $O_{i} O_{j}$ .

In a general QFT, this is an asymptotic expansion, valid only in the short-distance limit $x \to 0$ and only when inserted into correlation functions. However, a remarkable feature of CFTs is that the OPE is an exact, convergent series with a finite radius of convergence. To understand why this is the case, we need to introduce the powerful framework of radial quantization.

A Rigorous Foundation: Radial Quantization and the State-Operator Correspondence

Your background in QFT is likely centered on canonical quantization, where time is a special direction, and states evolve via the Hamiltonian $H = P^{0}$ . In CFT, it is more natural to treat space and time on an equal footing and choose a different "time" direction.

Radial quantization foliates Euclidean space $R^{d}$ with concentric spheres $S^{d - 1}$ of radius $r$ . The evolution from one sphere to the next is generated not by the Hamiltonian, but by the dilatation operator $D$ . The "time" coordinate is $τ = lo g r$ .

This framework leads to one of the most profound properties of CFTs: the state-operator correspondence.

In this picture, a local operator $O (0)$ inserted at the origin of $R^{d}$ creates a state $∣ O ⟩ = O (0) ∣0 ⟩$ on the sphere at $r = 0$ .
This state $∣ O ⟩$ is an eigenstate of the radial Hamiltonian $D$ , with an eigenvalue equal to its scaling dimension $Δ$ : $D ∣ O ⟩ = i Δ∣ O ⟩$ .
Crucially, this mapping is an isomorphism: there is a one-to-one correspondence between the set of all local operators in the theory and the Hilbert space of states defined on a sphere.

This correspondence is the reason for the OPE's exactness.

EPFL Lectures on Conformal Field Theory in D ⩾ 3

The following resource provides a brilliant derivation of the OPE from the state-operator correspondence. It elevates the OPE from a useful trick to a fundamental, convergent property of the theory.

Please read section 3.3, "Operator Product Expansion (OPE)" (pages 47-49). Focus on the derivation starting from Figure 3.7. The argument is that the state created by two operators inside a sphere, |Ψ⟩ = φ₁(x)φ₂(0)|0⟩, can be expanded in the complete basis of energy eigenstates |Eₙ⟩. The state-operator map then turns this expansion of states into an expansion of operators—the OPE. Also, note the argument that shows conformal symmetry fixes the functional form of the coefficients.

As the reading explains, the state created by the operator product $O_{i} (x) O_{j} (0)$ can be expanded in the complete basis of states $∣ O_{k} ⟩$ :

O_{i} (x) O_{j} (0) ∣0 ⟩ = k \sum C_{ij}^{k} (x) ∣ O_{k} ⟩ = k \sum C_{ij}^{k} (x) O_{k} (0) ∣0 ⟩

Because this holds for any state created by any configuration of other operators outside the sphere containing $x$ and $0$ , it promotes to a general operator identity. The radius of convergence of this series is the distance to the nearest other operator insertion.

The OPE as Operator Algebra

The OPE defines a product on the space of local operators, turning it into an infinite-dimensional associative algebra. The "structure constants" of this algebra are the OPE coefficients.

Conformal symmetry heavily constrains the functions $C_{ij}^{k} (x)$ . As shown in the reading, their dependence on $x$ is completely fixed by the scaling dimensions and spins of the operators involved. The overall normalization of each term in the sum is a single number, $C_{ij k}$ , which is precisely the same constant that appears in the three-point function $⟨ O_{i} O_{j} O_{k} ⟩$ .

The OPE for scalar primary operators takes the schematic form:

O_{i} (x_{1}) O_{j} (x_{2}) = k \sum C_{ij k} \frac{1}{∣ x _{12} ∣ ^{Δ_{i} + Δ_{j} - Δ_{k}}} (1 + a_{1} x_{12}^{μ} \partial_{μ} + a_{2} x_{12}^{μ} x_{12}^{ν} \partial_{μ} \partial_{ν} + \dots) O_{k} (x_{2})

The term in parentheses represents the contribution of the primary operator $O_{k}$ and all its descendants. The coefficients $a_{n}$ are kinematically fixed. The only theory-specific data are the dimensions $Δ_{k}$ and the OPE coefficients $C_{ij k}$ .

Algebraic Definition of Primary Operators

The OPE provides a powerful, algebraic way to define operators and their properties. This is particularly sharp in 2D CFT, which is crucial for your interest in string theory. In 2D, the stress-energy tensor splits into holomorphic and anti-holomorphic components, $T (z)$ and $T ˉ (z ˉ)$ . The OPE of any operator with $T (z)$ encodes its transformation properties under holomorphic conformal maps.

An operator $ϕ (w, w ˉ)$ with conformal weights $(h, h ˉ)$ is primary if and only if its OPE with $T (z)$ has no poles worse than second order:

T (z) ϕ (w, w ˉ) = \frac{h}{( z - w ) ^{2}} ϕ (w, w ˉ) + \frac{1}{z - w} \partial_{w} ϕ (w, w ˉ) + (non-singular terms)

And similarly for $T ˉ (z ˉ)$ with $h ˉ$ and $\partial_{w ˉ}$ .

The $1/ (z - w)^{2}$ term governs the scaling and rotation properties (encoded in $h$ ).
The $1/ (z - w)$ term governs the translation properties (it generates the derivative $\partial_{w}$ ).
The absence of higher poles (like $1/ (z - w)^{3}$ , etc.) is the defining feature of a primary operator.

Descendant operators, like $\partial_{w} ϕ$ , will have higher-order poles in their OPE with $T (z)$ , reflecting their more complex transformation laws. This algebraic definition is equivalent to the transformation property-based definition we saw in the first module, but it is often more practical for computations.

4. Introducing Conformal Field Theory

The following text from David Tong's notes provides a very clear explanation of this algebraic viewpoint and defines primary operators through their OPE with the stress tensor.

Please read Section 4.2.1 "Operator Product Expansion" for a concise definition and its meaning within correlation functions. Then, read Section 4.2.3 "Primary Operators". This section is key: it connects the transformation properties of operators under scaling and translations to the singular terms in their OPE with the stress tensor T, culminating in the algebraic definition of a primary operator in equation (4.17) and its finite version (4.18).

Test your understanding!

Let $ϕ (w)$ be a holomorphic primary operator of weight $h$ . Its OPE with $T (z)$ is given by:

T (z) ϕ (w) = \frac{h}{( z - w ) ^{2}} ϕ (w) + \frac{1}{z - w} \partial_{w} ϕ (w) + \dots

Consider the descendant operator $ψ (w) = \partial_{w} ϕ (w)$ . By taking the derivative $\partial_{w}$ of the OPE above, find the OPE of $T (z)$ with $ψ (w)$ . Does the result have a pole higher than second order? Is $ψ (w)$ a primary operator?

Show answer

We take $\partial_{w}$ of the OPE for $T (z) ϕ (w)$ :

\partial_{w} [T (z) ϕ (w)] = T (z) (\partial_{w} ϕ (w)) = T (z) ψ (w)

Applying this to the right-hand side:
\begin{align*}
T(z) \psi(w) &= \partial_w \left( \frac{h}{(z-w)^2}\phi(w) + \frac{1}{z-w}\partial_w \phi(w) + \dots \right) \
&= \frac{2h}{(z-w)^3}\phi(w) + \frac{h}{(z-w)^2}\partial_w \phi(w) + \frac{1}{(z-w)^2}\partial_w \phi(w) + \frac{1}{z-w}\partial_w^2 \phi(w) + \dots \
&= \frac{2h}{(z-w)^3}\phi(w) + \frac{h+1}{(z-w)^2}\psi(w) + \frac{1}{z-w}\partial_w \psi(w) + \dots
\end{align*}
The resulting OPE has a $1/ (z - w)^{3}$ pole. The presence of this pole, which is higher than second order, immediately tells us that $ψ (w) = \partial_{w} ϕ (w)$ is not a primary operator. It is a descendant, as expected. This exercise demonstrates the power of the OPE in algebraically identifying operator types.

Conclusion

Today, we've introduced the Operator Product Expansion, a cornerstone of quantum and conformal field theory. We've moved beyond its use as a simple short-distance approximation and established its rigorous foundation in CFT.

Key Takeaways:

The OPE expresses a product of local operators as a sum over a complete basis of single local operators.
In CFT, due to the state-operator correspondence in radial quantization, the OPE is a convergent series, not just an asymptotic one.
Conformal symmetry fixes the entire functional form of the OPE, leaving only a set of numbers—the OPE coefficients $C_{ij k}$ —as the theory-dependent dynamical data. These are the same coefficients that determine the normalization of three-point functions.
The OPE defines the algebra of local operators. The OPE of an operator with the stress-energy tensor provides a powerful algebraic definition of what it means for an operator to be primary.

We have now seen that the OPE allows us to decompose a product of two operators into a sum over all operators in the theory. In our next lesson, we will see how to organize this sum in a more structured way. We will group the contributions from a primary operator and all of its descendants into a single object called a conformal block. This will be the final step needed to decompose the mysterious four-point function $G (u, v)$ into a sum of known functions, paving the way for the conformal bootstrap.

Recommended resources

The Operator Product Expansion (OPE)

A Rigorous Foundation: Radial Quantization and the State-Operator Correspondence

The OPE as Operator Algebra

Algebraic Definition of Primary Operators

Conclusion