Stress-Energy Tensor OPE Calculation

Hello! Welcome to the fifth module of our course.

In our previous lessons on 2D minimal models, we saw how the abstract algebraic condition of having a null vector at a certain level translates into a concrete computational tool: the Belavin–Polyakov–Zamolodchikov (BPZ) differential equations. This allowed us to solve for correlation functions in specific theories like the critical Ising model.

Today, we shift our focus from properties of specific models to a universal feature of all 2D Conformal Field Theories. We will compute the Operator Product Expansion (OPE) of the stress-energy tensor, $T (z)$ , with itself. This is arguably one of the most fundamental calculations in CFT. The result will reveal a crucial parameter, the central charge $c$ , which characterizes the CFT and can be thought of as a measure of its degrees of freedom.

This OPE is the quantum operator statement that encodes the symmetry algebra of the theory—the Virasoro algebra. As we'll see, a fascinating outcome of this calculation is that the stress-energy tensor itself is not a primary field, a purely quantum mechanical effect with profound consequences.

The General Form of the $T (z) T (w)$ OPE

Before diving into a specific calculation, let's use the principles of conformal invariance and dimensional analysis to deduce the general structure of the $T (z) T (w)$ OPE.

Dimensionality: The stress-energy tensor $T (z)$ is the generator of transformations and is constructed from fields and their derivatives. In any 2D CFT, it has a conformal weight of $(h, h ~) = (2, 0)$ . This means its scaling dimension is $Δ = h + h ~ = 2$ . The OPE $T (z) T (w)$ involves a product of two such operators, so any operator appearing on the right-hand side must have a dimension that maintains the overall dimension of 4. The general form is $T (z) T (w) = \sum_{k} C_{k} (z - w) O_{k} (w)$ . On dimensional grounds, the coefficient functions must scale as $C_{k} (z - w) \sim (z - w)^{Δ_{k} - 4}$ .
Singularities: As $z \to w$ , the OPE can have singularities. The terms in the expansion will look like $(z - w)^{- n} O_{4 - n} (w)$ , where $O_{4 - n}$ is an operator of dimension $4 - n$ . In a unitary theory, operator dimensions must be non-negative ( $Δ \geq 0$ ). This constrains the most singular possible term to $n = 4$ , which must be multiplied by an operator of dimension 0. The only such operator is the identity, $I$ , which is simply a c-number.
Symmetry: The OPE is an identity inside a time-ordered (or radially-ordered) correlation function, so it must be symmetric under the exchange $z \leftrightarrow w$ . This forbids terms with an odd power like $(z - w)^{- 3}$ multiplying a simple local operator, but allows a term like $(z - w)^{- 1}$ if it multiplies a derivative, as we will see.

Combining these constraints leads to the most general form for the singular part of the $T (z) T (w)$ OPE:

T (z) T (w) \sim \frac{c /2}{( z - w ) ^{4}} + \frac{2 T ( w )}{( z - w ) ^{2}} + \frac{\partial _{w} T ( w )}{z - w}

Let's dissect this expression:

The terms $\frac{2 T ( w )}{( z - w ) ^{2}} + \frac{\partial _{w} T ( w )}{z - w}$ are precisely what you would expect from the general OPE of a primary field of weight $h = 2$ with $T (z)$ .
The term $\frac{c /2}{( z - w ) ^{4}}$ is new. It is the most singular term allowed and is proportional to the identity operator. Its coefficient, $c$ , is a constant called the central charge. This term signals that $T (z)$ is not a primary field (unless $c = 0$ ). This is a quantum anomaly; classically, the stress tensor transforms homogeneously.

The central charge $c$ is a fundamental, defining property of a CFT.

A Concrete Calculation: The Free Boson

To see how this OPE arises in practice and to compute a value for $c$ , we'll use the canonical example of a 2D CFT: the free massless scalar boson. Your background in QFT and path integrals will be very helpful here. The calculation relies on Wick's theorem.

1. The Theory

We start with the action for a free massless scalar field $X$ in Euclidean signature:

S = \frac{1}{4 π α ^{'}} \int d^{2} σ \partial_{α} X \partial^{α} X

In complex coordinates $z = σ^{1} + i σ^{2}$ , this becomes $S = \frac{1}{2 π α ^{'}} \int d^{2} z \partial_{z} X \partial ˉ_{z ˉ} X$ . From this action, the fundamental two-point correlation function (or propagator) can be derived. While the correlator for $X$ itself is logarithmic, the correlator for its derivatives, which we will need, is simpler:

⟨ \partial_{z} X (z) \partial_{w} X (w)⟩ = - \frac{α ^{'}}{2} \frac{1}{( z - w ) ^{2}}

For simplicity, we will set the constant $α^{'} = 2$ . This is a common convention that simplifies many expressions. Our fundamental contraction rule is therefore:

⟨ \partial_{z} X (z) \partial_{w} X (w)⟩ = - \frac{1}{( z - w ) ^{2}}

2. The Stress-Energy Tensor

The holomorphic component of the stress-energy tensor is given by $T_{z z} \propto \partial_{z} X \partial_{z} X$ . In the quantum theory, this product of operators at the same point is divergent. We must define it carefully using normal ordering, denoted by colons : :. This procedure subtracts the singular vacuum expectation value.

T (z) = - \frac{1}{2} : \partial_{z} X (z) \partial_{z} X (z) :

The rule for normal ordering a product of two operators is $: A B := A B - ⟨ A B ⟩$ .

3. The OPE via Wick's Theorem

We are now ready to compute the OPE of $T (z)$ with $T (w)$ . This involves applying Wick's theorem to a product of normal-ordered operators. The rule is to sum over all possible contractions of fields between the two normal-ordered groups.

4. Introducing Conformal Field Theory

The following text from David Tong's lecture notes on String Theory provides a clear, step-by-step computation of the T(z)T(w) OPE for a free scalar field. It's a classic calculation that perfectly illustrates the use of Wick's theorem in this context.

Please start reading at the end of Section 4.3.3, beginning with the line 'Finally, lets check to see the OPE of T with itself.' and continue through the end of the first paragraph of Section 4.4, ending before the line 'Before moving on...'. Focus on tracking the application of Wick's theorem: Identify the number of ways to perform zero, one, and two contractions. Follow how the double contraction gives the (z-w)⁻⁴ term. See how the single contraction, combined with a Taylor expansion of the remaining fields, gives the (z-w)⁻² and (z-w)⁻¹ terms.

Let's walk through the key steps of the calculation you just read.

T (z) T (w) = (- \frac{1}{2})^{2} : \partial X (z) \partial X (z) :: \partial X (w) \partial X (w) :

Applying Wick's theorem:

Zero Contractions: This gives the non-singular term $: T (z) T (w) :$ , which we neglect for the singular part of the OPE.
Two Contractions: There are two ways to fully contract the operators between the two groups. Each double contraction yields $(- \frac{1}{( z - w ) ^{2}})^{2}$ . $Two contractions ⟹ 2 \times \frac{1}{4} (\frac{1}{( z - w ) ^{2}})^{2} = \frac{1/2}{( z - w ) ^{4}}$ This is precisely the most singular term, giving us $c /2$ .
One Contraction: There are four ways to make a single contraction. $One contraction ⟹ 4 \times \frac{1}{4} (- \frac{1}{( z - w ) ^{2}}) : \partial X (z) \partial X (w) := - \frac{: \partial X ( z ) \partial X ( w ) :}{( z - w ) ^{2}}$ To get this into the standard OPE form, we Taylor expand the operators at $z$ around the point $w$ : $\partial X (z) = \partial X (w) + (z - w) \partial^{2} X (w) + \dots$ . $- \frac{: ( \partial X ( w ) + ( z - w ) \partial ^{2} X ( w ) + \dots ) \partial X ( w ) :}{( z - w ) ^{2}} = - \frac{: \partial X ( w ) \partial X ( w ) :}{( z - w ) ^{2}} - \frac{: \partial ^{2} X ( w ) \partial X ( w ) :}{z - w} + \dots$ Recalling that $T (w) = - \frac{1}{2} : \partial X (w) \partial X (w) :$ and $\partial_{w} T (w) = - : \partial^{2} X (w) \partial X (w) :$ , this becomes: $\frac{2 T ( w )}{( z - w ) ^{2}} + \frac{\partial _{w} T ( w )}{z - w}$

4. The Result

Combining all the pieces, we arrive at the final OPE:

T (z) T (w) \sim \frac{1/2}{( z - w ) ^{4}} + \frac{2 T ( w )}{( z - w ) ^{2}} + \frac{\partial _{w} T ( w )}{z - w}

Comparing this to the general form, we find that for a single free, massless scalar boson, the central charge is $c = 1$ .

Test your understanding!

Suppose you have a theory of $N$ non-interacting free scalar fields, $X^{i}$ for $i = 1, \dots, N$ . The total stress-energy tensor is the sum of the individual ones: $T_{t o t a l} (z) = \sum_{i = 1}^{N} T_{i} (z)$ . What would the central charge of this theory be?

Show answer

Since the fields are non-interacting, the OPE of $T_{i} (z)$ with $T_{j} (w)$ for $i \neq = j$ will have no singular terms. Therefore, the singular part of the $T_{t o t a l} (z) T_{t o t a l} (w)$ OPE is the sum of the individual OPEs:

T_{t o t a l} (z) T_{t o t a l} (w) \sim i = 1 \sum N (\frac{1/2}{( z - w ) ^{4}} + \dots) = \frac{N /2}{( z - w ) ^{4}} + \dots

This implies that the total central charge is $c_{t o t a l} = N$ . The central charge is an extensive quantity that counts the degrees of freedom.

Conclusion

In this lesson, we performed a pivotal calculation in 2D CFT. By computing the OPE of the stress-energy tensor with itself for a free boson, we saw explicitly how the general structure of this expansion arises, and we made our first determination of a central charge, finding $c = 1$ .

Key Takeaways:

The OPE of the stress-energy tensor with itself has a universal form for any 2D CFT: $T (z) T (w) \sim \frac{c /2}{( z - w ) ^{4}} + \frac{2 T ( w )}{( z - w ) ^{2}} + \frac{\partial _{w} T ( w )}{z - w}$ .
The most singular term, proportional to the identity, defines the central charge $c$ .
The presence of this $c$ -term means that the stress-energy tensor $T (z)$ is not a primary field, which is a fundamental quantum feature of CFTs.
An explicit calculation for a free scalar boson using Wick's theorem yields $c = 1$ .

The $T (z) T (w)$ OPE is the gateway to understanding the full symmetry algebra of a 2D CFT. In our next lesson, we will see how the Laurent expansion of $T (z)$ gives the Virasoro generators, and how this OPE translates directly into the celebrated Virasoro algebra.

Recommended resources

The General Form of the T(z)T(w) OPE

A Concrete Calculation: The Free Boson

Conclusion

The General Form of the $T (z) T (w)$ OPE