Conformal Ward Identities and Correlation Functions

Hello! Welcome to the second module of our course.

In the first module, we established the kinematic foundation of Conformal Field Theory. We identified the conformal algebra, classified its representations into primary operators and their descendants, and saw how the state-operator correspondence maps this operator structure onto a Hilbert space via radial quantization.

Now, we shift our focus from kinematics to dynamics. The central question is: how does conformal symmetry constrain the dynamics of a quantum theory? The answer lies in the theory's correlation functions, which contain all its dynamical information. In this lesson, we will introduce the primary tool for extracting these constraints: the Ward identities. These identities are the quantum mechanical embodiment of Noether's theorem and provide a powerful, systematic way to see how symmetries are reflected in the observables of the theory.

The Ward Identity from First Principles

In classical field theory, a symmetry of the action leads to a conserved Noether current. In quantum field theory, this principle is elevated to a set of relations between correlation functions, known as Ward identities. They precisely state how a correlation function must transform to respect the underlying symmetry.

Given your familiarity with path integrals from your study of Yang-Mills theory, the most direct way to derive these identities is by considering a symmetry transformation as a change of variables within the path integral. The value of the integral must remain unchanged, and this simple fact leads to powerful constraints.

Introduction to conformal field theory, Lecture 6

To see how this works in detail, let's watch a segment from Tobias Osborne's lecture series. He provides a careful derivation of the general Ward identity by applying an infinitesimal, position-dependent symmetry transformation to the path integral for a generic correlation function.

Please watch the segment from 19:19 to 46:03. Focus on these key steps: Starting with the path integral expression for an n-point correlation function. Performing an infinitesimal change of integration variable \phi'(x) = \phi(x) + \delta\phi(x). Accounting for the change in the action, \delta S, which introduces the Noether current J^\mu, and the change in the operators, \delta X. Expanding the path integral to first order in the transformation and equating the first-order terms to zero.

The derivation you just watched is a cornerstone of modern QFT. Let's distill the main result. By demanding that the path integral is invariant under a change of variables $ϕ \to ϕ^{'}$ , we arrive at the "master" Ward identity. For a transformation parametrized by $ω^{a} (x)$ , this identity states that the expectation value of the variation must vanish:

⟨ δ_{ω} (e^{- S} i \prod O_{i} (x_{i}))⟩ = 0

Expanding this to first order in $ω^{a} (x)$ gives the crucial relation. If we consider a transformation that vanishes at the locations of the operator insertions, the identity simplifies to $⟨ \partial_{μ} J^{μ} (x) \dots ⟩ = 0$ . This is the quantum statement that the Noether current is conserved on average.

The real power comes from considering what happens when the transformation is non-zero at the operator insertion points. This leads to the general form of the Ward identity, which relates the divergence of the current to the transformation properties of the fields themselves.

The Local Form and Operator Commutators

The integral form of the Ward identity can be translated into a more intuitive local operator equation. This form reveals that the divergence of the current operator is non-zero only at the locations of other operator insertions—these contributions are called "contact terms".

Introduction to conformal field theory, Lecture 6

The same lecture segment continues to develop this idea, showing how the integral identity implies a local relation and, remarkably, recovers the familiar commutation relations from canonical quantization.

Please watch from 46:03 to 1:04:55. Pay attention to how the argument proceeds: Since the integral identity must hold for any choice of transformation \omega(x), the integrand itself must obey a local relation. Integrating this local identity over a thin time-slice recovers the commutator of the conserved charge Q with a field operator \phi. The final result: [Q, \phi(x)] = -i \delta\phi(x). This powerful formula connects the abstract path integral derivation to the concrete action of symmetry generators on operators.

This connection is profound. It demonstrates that the path integral formalism, which you've used for gauge theories, contains the same physical information as the canonical operator formalism. The Ward identity is the bridge. It tells us that the action of a symmetry generator $Q$ on an operator $ϕ$ is encoded in the "contact terms" that appear when the associated Noether current $J^{μ}$ gets too close to $ϕ$ inside a correlation function.

Ward Identities for the Conformal Group

Now we can apply this general framework to our specific case: conformal symmetry. Each of the generators of the conformal group—translations ( $P_{μ}$ ), Lorentz transformations ( $L_{μν}$ ), dilatations ( $D$ ), and special conformal transformations ( $K_{μ}$ )—has a corresponding conserved Noether current. As you may recall from classical field theory, these currents can all be constructed from a single object: the symmetric, traceless energy-momentum tensor $T_{μν}$ .

Translations: The current is simply $T_{μν}$ itself.
Dilatations: The current is $j_{D}^{μ} = x_{ν} T^{ν μ}$ .
Rotations: The current is $j_{λ ρ}^{μ} = x_{λ} T_{ρ}^{μ} - x_{ρ} T_{λ}^{μ}$ .
SCTs: The current is $j_{K}^{μ} = (2 x_{ν} x_{λ} - δ_{ν λ} x^{2}) T^{λ μ}$ .

For each of these currents, we can write a Ward identity. For example, the dilatation Ward identity, arising from the conservation of $j_{D}^{μ}$ , takes the form:

\partial_{μ} ⟨ x_{ν} T^{ν μ} (x) O_{1} (x_{1}) \dots ⟩ = ⟨ T_{μ}^{μ} (x) O_{1} (x_{1}) \dots ⟩ = - i i \sum δ^{(d)} (x - x_{i}) Δ_{i} ⟨ O_{1} (x_{1}) \dots ⟩

This equation is a precise mathematical statement of what it means for a correlation function to be scale-invariant. It says that the trace of the energy-momentum tensor (which must be zero for classical scale invariance) is zero inside a correlation function except at the points where other fields are inserted. At those points, it produces a delta function spike proportional to the scaling dimension $Δ_{i}$ of the operator at that point.

The Conformal Ward Identity in 2D

In two dimensions, the structure of conformal symmetry becomes infinitely richer, and the Ward identities take on a particularly elegant and powerful form. Using complex coordinates $z = σ^{1} + i σ^{2}$ and $z ˉ = σ^{1} - i σ^{2}$ , the conservation law $\partial_{μ} T^{μν} = 0$ and tracelessness $T_{μ}^{μ} = 0$ imply that the non-zero components of the stress tensor, $T_{z z}$ and $T_{z ˉ z ˉ}$ , are holomorphic and anti-holomorphic functions, respectively: $\partial_{z ˉ} T (z) = 0$ and $\partial_{z} T ˉ (z ˉ) = 0$ , where we define $T (z) \propto T_{z z}$ and $T ˉ (z ˉ) \propto T_{z ˉ z ˉ}$ .

In 2D, the Ward identity can be rephrased using contour integrals in the complex plane. Your background in complex analysis, particularly with the residue theorem, will make this formulation very natural. The variation of an operator $ϕ (w, w ˉ)$ under an infinitesimal conformal transformation $z \to z + ϵ (z)$ is given by a contour integral of the stress tensor $T (z)$ around the point $w$ :

δ_{ϵ} ϕ (w, w ˉ) = \oint_{w} \frac{d z}{2 π i} ϵ (z) T (z) ϕ (w, w ˉ)

This integral just picks up the residue of the product $ϵ (z) T (z) ϕ (w, w ˉ)$ . This means the transformation properties of $ϕ$ are entirely encoded in its Operator Product Expansion (OPE) with the stress tensor.

Conformal Field Theory (Notes by J. Stern)

The following notes provide a concise derivation of the explicit form of the conformal Ward identity in 2D.

Please read Section 2.5, 'Energy-momentum tensor and conformal Ward identities' (pages 25-29). Pay close attention to how the variation of a primary field under a transformation \delta_\epsilon\phi is related to the contour integral involving T(z), and how this leads to the final form of the conformal Ward identity for an N-point function of primary fields.

As you just read, this line of reasoning culminates in the conformal Ward identity. For any set of primary fields $ϕ_{k}$ with conformal weights $(h_{k}, h ˉ_{k})$ inserted at points $w_{k}$ , the correlation function containing the stress tensor is given by:

⟨ T (z) k = 1 \prod N ϕ_{k} (w_{k}, w ˉ_{k})⟩ = j = 1 \sum N (\frac{h _{j}}{( z - w _{j} ) ^{2}} + \frac{1}{z - w _{j}} \frac{\partial}{\partial w _{j}}) ⟨ k = 1 \prod N ϕ_{k} (w_{k}, w ˉ_{k})⟩

An analogous equation holds for $T ˉ (z ˉ)$ . This is a remarkable result. It states that a correlation function involving the complicated stress-tensor operator can be reduced to a sum of simpler terms involving only derivatives of the original correlation function. This equation is the primary tool for computing correlation functions in 2D CFT.

Test your understanding!

In a CFT, the vacuum state $∣0 ⟩$ is invariant under all conformal transformations. This implies that for any operator $O$ , $⟨ O ⟩ \equiv ⟨ 0∣ O ∣0 ⟩ = 0$ , unless $O$ is the identity operator.

Use the 2D conformal Ward identity to prove that for any non-identity primary field $ϕ (w)$ , the one-point function $⟨ ϕ (w)⟩$ must be zero.

Show answer

Let's apply the conformal Ward identity for $N = 1$ :

⟨ T (z) ϕ (w)⟩ = (\frac{h}{( z - w ) ^{2}} + \frac{1}{z - w} \frac{\partial}{\partial w}) ⟨ ϕ (w)⟩

The one-point function $⟨ ϕ (w)⟩$ must be independent of position $w$ due to translational invariance (which is part of the conformal group). Therefore, $\frac{\partial}{\partial w} ⟨ ϕ (w)⟩ = 0$ . The identity simplifies to:

⟨ T (z) ϕ (w)⟩ = \frac{h}{( z - w ) ^{2}} ⟨ ϕ (w)⟩

The left-hand side is a two-point function, which can only depend on the difference $z - w$ . The right-hand side has exactly this dependence. So far, so good.

However, we also need to consider the transformation properties of $T (z)$ . $T (z)$ is a quasi-primary field of weight $(2, 0)$ . Under a scaling $z \to λ z$ , the two-point function must transform as:

⟨ T (λ z) ϕ (λ w)⟩ = λ^{- 2} λ^{- h} ⟨ T (z) ϕ (w)⟩

Let's check our result. On the right-hand side, we have:

\frac{h}{( λ z - λ w ) ^{2}} ⟨ ϕ (λ w)⟩ = \frac{h}{λ ^{2} ( z - w ) ^{2}} λ^{- h} ⟨ ϕ (w)⟩ = λ^{- 2 - h} \frac{h}{( z - w ) ^{2}} ⟨ ϕ (w)⟩

This perfectly matches the required transformation.

So, where is the constraint? The constraint comes from requiring that the identity holds for all conformal transformations, including special conformal transformations. Alternatively, and more simply, the OPE of $T (z) ϕ (w)$ should not contain the identity operator unless $ϕ$ itself is the identity. If $⟨ ϕ (w)⟩ \neq = 0$ , it implies that $⟨ T (z) ϕ (w)⟩ \neq = 0$ , which means the identity operator appears in the $T \times ϕ$ OPE. This can only happen if $ϕ$ is proportional to the identity operator. Thus, for any non-trivial primary operator, its vacuum expectation value must be zero.

Conclusion

In this lesson, we have introduced the Ward identities as the central tool for understanding the dynamical consequences of conformal symmetry.

Key Takeaways:

Ward identities are the quantum mechanical expression of Noether's theorem, derived by requiring the invariance of the path integral under a change of variables.
They provide a local operator equation relating the divergence of the Noether current to the transformation properties of other operators in a correlation function.
For conformal symmetries, the relevant currents are built from the energy-momentum tensor $T_{μν}$ . The resulting identities constrain how correlation functions must behave under translations, rotations, dilatations, and SCTs.
In 2D, these identities take an exceptionally elegant form, relating the insertion of the stress tensor $T (z)$ into a correlator to a sum of differential operators acting on the correlator itself.

We have forged the primary weapon in our analytical arsenal. In the next lesson, we will wield it. We will use the Ward identities to derive differential equations that correlation functions must obey, and by solving them, we will determine the universal, fixed form of all two-point and three-point functions in any CFT.

Recommended resources

The Ward Identity from First Principles

The Local Form and Operator Commutators

Ward Identities for the Conformal Group

The Conformal Ward Identity in 2D

Conclusion