Radial Quantization and the State-Operator Correspondence

Hello! In our last couple of lessons, we've focused on the operators of a conformal field theory. We defined primary operators through their simple transformation properties, and we saw that all other operators, the descendants, are built from them by taking derivatives. We then developed a computational toolkit for determining how any conformal generator acts on any of these operators.

Throughout this, we've kept states and operators as related but separate entities. Today, we will unite them. Our goal is to establish the state-operator correspondence, a profound isomorphism that sits at the very heart of CFT. We will achieve this by introducing a new way of looking at the quantum theory called radial quantization. This correspondence is not just a formal curiosity; it is a powerful dictionary that allows us to translate questions about the operator spectrum into questions about states in a Hilbert space, and vice versa. This will be a key tool in everything that follows.

From Standard to Radial Quantization

In your study of quantum field theory, quantization is typically performed by foliating spacetime into constant-time slices. States are defined on these spatial slices (e.g., at $t = t_{0}$ ), and the Hamiltonian governs their evolution from one slice to the next. In a Euclidean QFT on $R^{d}$ , rotational symmetry means there is no preferred time direction. We could choose any coordinate $x^{i}$ as "time" and quantize on slices perpendicular to it.

Conformal symmetry invites a more natural choice. Instead of parallel planes, we can foliate spacetime with concentric spheres of increasing radius centered at the origin.

States are defined on spheres $S^{d - 1}$ of a fixed radius $r$ .
"Time" evolution corresponds to the progression from an inner sphere to an outer sphere.

What generator corresponds to this radial evolution? It is the dilatation operator, $D$ . This new scheme is called radial quantization.

The Cylinder Picture

This idea of evolving states on expanding spheres becomes exceptionally clear through a conformal mapping. The flat Euclidean space $R^{d}$ is conformally equivalent to a cylinder, $R \times S^{d - 1}$ . This is more than a geometric curiosity; it provides a powerful way to understand the physics.

EPFL Lectures on Conformal Field Theory in D ⩾ 3 ...

To make the idea of radial quantization concrete, we use a conformal map to a more familiar geometry: the cylinder. This is explained clearly in Slava Rychkov's lecture notes.

Please read the short section '3.1.4 Cylinder interpretation' on page 40. Focus on the coordinate change τ = log r and how it maps dilatations in flat space to time translations on the cylinder R × S^(D-1). Note the relationship between the flat space and cylinder metrics, ds²_flat = r² ds²_cyl.

As the reading explains, by defining a logarithmic "time" coordinate $τ = lo g r$ , we find a remarkable dictionary:

Dilatations in flat space ( $x \to λ x$ , which means $r \to λ r$ ) become time translations on the cylinder ( $τ \to τ + lo g λ$ ).
The dilatation operator $D$ in flat space becomes the Hamiltonian on the cylinder, generating time translations.

This means that the seemingly abstract idea of radial quantization is equivalent to standard Hamiltonian quantization on the simpler geometry of the cylinder.

Weyl Transformation from R^D to the Cylinder — This image from Rychkov's notes illustrates the Weyl transformation from D-dimensional flat space (left) to the cylinder $S^{D-1} \times \mathbb{R}$ (right). The concentric circles of constant radius $r$ in $\mathbb{R}^D$ are mapped to the cross-sectional spheres at constant "time" $\tau$ on the cylinder.

The State-Operator Correspondence

Now we have all the ingredients to connect states and operators. The Hilbert space in radial quantization, $H_{cyl}$ , consists of states defined on a sphere of constant radius (a slice of the cylinder at constant $τ$ ). How do we relate these states to the local operators of the theory?

Lectures on the Conformal Bootstrap

With the picture of radial quantization on the cylinder in mind, we can now formally establish the state-operator correspondence. The lecture notes by David Simmons-Duffin provide a concise and rigorous definition.

Please read section 6, 'Radial Quantization and the State-Operator Correspondence,' up to the end of 6.3 'State <=> Operator' (pages 14-15). Pay close attention to how inserting an operator defines a state, how a state can be used to define an operator, and the final identification O(0) <=> |O⟩.

Let's summarize the profound two-way relationship you just read about.

Operator $⟹$ State:
We can prepare a state in $H_{cyl}$ by performing the path integral over the interior of a ball, with a local operator $O (0)$ inserted at the origin. This procedure defines a unique state on the boundary sphere, which we label $∣ O ⟩$ . The path integral over the ball with no operator insertion creates the vacuum state $∣0 ⟩$ .
State $⟹$ Operator:
Conversely, given any state $∣ ψ ⟩ \in H_{cyl}$ , we can define a local operator. We do this by defining its correlation functions: the correlator $⟨ O_{ψ} (x) \dots ⟩$ is computed by performing the path integral with a small hole cut out around $x$ , and imposing boundary conditions corresponding to the state $∣ ψ ⟩$ on the boundary of that hole.

This establishes a one-to-one correspondence, an isomorphism, between the space of local operators at the origin and the Hilbert space of states on the sphere:
$O (0) ⟷ ∣ O ⟩ \in H_{cyl}$

The Correspondence as a Dictionary

This isomorphism is a powerful dictionary that translates the properties of operators into properties of states.

Scaling Dimension $\leftrightarrow$ Energy:
The energy of a state on the cylinder is its eigenvalue under the cylinder Hamiltonian, which is the dilatation operator $D$ . For a state $∣ O ⟩$ created by a primary operator $O (0)$ of dimension $Δ$ , we have:
$D ∣ O ⟩ = D O (0) ∣0 ⟩ = [D, O (0)] ∣0 ⟩ = i Δ O (0) ∣0 ⟩ = i Δ∣ O ⟩$
(Here we used $D ∣0 ⟩ = 0$ for the invariant vacuum and the commutation relation $[D, O (0)] = i Δ O (0)$ ). The scaling dimension of the operator is precisely the energy of the corresponding state.
Primary Operators $\leftrightarrow$ Primary States:
A primary operator is defined by $[K_{μ}, O (0)] = 0$ . This translates directly to the state property:
$K_{μ} ∣ O ⟩ = K_{μ} O (0) ∣0 ⟩ = [K_{μ}, O (0)] ∣0 ⟩ = 0$
States annihilated by all $K_{μ}$ are called primary states. From the conformal algebra, $[D, K_{μ}] = - i K_{μ}$ , which means $K_{μ}$ are "lowering operators" for energy. A primary state is therefore the lowest-energy state in its representation of the conformal group (a "conformal multiplet").
Descendant Operators $\leftrightarrow$ Descendant States:
Descendant operators are derivatives of primaries, like $\partial_{ν} O (0)$ . Since $P_{ν} = i \partial_{ν}$ is the generator of translations, the corresponding state is:
$∣ \partial_{ν} O ⟩ \propto [P_{ν}, O (0)] ∣0 ⟩ = P_{ν} ∣ O ⟩$
From the algebra $[D, P_{μ}] = i P_{μ}$ , the momentum generators $P_{μ}$ are "raising operators" for energy. Acting on a primary state with $P_{μ}$ creates descendant states with higher energy ( $Δ + 1, Δ + 2, \dots$ ).

The entire Hilbert space $H_{cyl}$ can be organized into conformal multiplets, each built upon a primary state by acting with the raising operators $P_{μ}$ . This structure exactly mirrors the classification of local operators.

Test your understanding!

Consider a CFT in $d = 4$ . A conserved vector current $J_{μ} (x)$ is a primary operator with spin $ℓ = 1$ . In a unitary theory, its scaling dimension is fixed to $Δ = d - 2 + ℓ = 4 - 2 + 1 = 3$ .

What is the energy of the corresponding primary state $∣ J_{μ} ⟩$ in radial quantization?
The operator $\partial^{μ} J_{μ} (x)$ is a scalar descendant operator. What is the corresponding state in $H_{cyl}$ ?
Show that this state has zero norm, i.e., $⟨ \partial^{μ} J_{μ} ∣ \partial^{ν} J_{ν} ⟩ = 0$ . This is the Hilbert space reflection of the conservation law $\partial^{μ} J_{μ} = 0$ . You will need the adjoint relations from the conformal algebra: $P_{μ}^{†} = K_{μ}$ .

Show answer

The energy of the state is its scaling dimension. So, $E = Δ = 3$ . More formally, $D ∣ J_{μ} ⟩ = i Δ∣ J_{μ} ⟩ = 3 i ∣ J_{μ} ⟩$ .
The operator $\partial^{μ} J_{μ} (x)$ corresponds to the state $P^{μ} ∣ J_{μ} ⟩$ (up to factors of $i$ ).
Let's compute the norm of this state.
$⟨ \partial^{μ} J_{μ} ∣ \partial^{ν} J_{ν} ⟩ \propto ⟨ P^{μ} J_{μ} ∣ P^{ν} J_{ν} ⟩ = (⟨ J_{μ} ∣ (P^{μ})^{†}) (P^{ν} ∣ J_{ν} ⟩) = ⟨ J_{μ} ∣ K^{μ} P^{ν} ∣ J_{ν} ⟩$
Since $∣ J_{ν} ⟩$ is a primary state, it is annihilated by $K_{λ}$ acting from the right. However, we have $K^{μ}$ acting from the left. The bra state $⟨ J_{μ} ∣$ is created by the adjoint operator $J_{μ} (0)^{†}$ . Using the hermiticity properties of the generators, $P_{μ}^{†} = K_{μ}$ , you can show that $⟨ J_{μ} ∣ P^{μ} = 0$ .
Let's use commutators, which is more direct.
$⟨ J_{μ} ∣ K^{μ} P^{ν} ∣ J_{ν} ⟩ = ⟨ J_{μ} ∣ [K^{μ}, P^{ν}] ∣ J_{ν} ⟩ + ⟨ J_{μ} ∣ P^{ν} K^{μ} ∣ J_{ν} ⟩$
The second term is zero because $K^{μ} ∣ J_{ν} ⟩ = 0$ . For the first term, we use the algebra $[K^{μ}, P^{ν}] = 2 i (δ^{μν} D - M^{μν})$ .
\begin{align*}
\langle J_\mu| [K^\mu, P^\nu] |J_\nu\rangle &= 2i \langle J_\mu| (\delta^{\mu\nu}D - M^{\mu\nu}) |J_\nu\rangle \
&= 2i (\langle J_\mu| D |J_\mu\rangle - \langle J_\mu| M^{\mu\nu} |J_\nu\rangle) \
&= 2i (i\Delta \langle J_\mu|J_\mu\rangle - \langle J_\mu|(-i(\Sigma^{\mu\nu})\nu^\lambda |J\lambda\rangle)) \
&= 2i (i(d-1)\delta_\mu^\nu\langle J_\nu|J_\nu\rangle - (-i(\Sigma^{\mu\nu})\mu^\lambda \langle J\lambda | J_\nu\rangle))
\end{align*}
The conservation condition $\partial^{μ} J_{μ} = 0$ is a "null state" condition. The state $P^{μ} ∣ J_{μ} ⟩$ is a null vector. To show its norm is zero, you need to compute the norm matrix of all level-1 descendants $P_{ν} ∣ J_{μ} ⟩$ and show that one of its eigenvalues is zero. The state corresponding to $\partial^{μ} J_{μ}$ is one of the eigenvectors with zero eigenvalue. The calculation is a bit involved, but the principle is that operator equations of motion (like conservation laws) manifest as null states (states with zero norm) in the Hilbert space.

Conclusion

In this lesson, we have established the state-operator correspondence, one of the most fundamental and elegant concepts in conformal field theory.

Key Takeaways:

Radial quantization provides a natural framework for CFTs, where we quantize the theory on spheres and evolution proceeds outwards in radius.
This framework is equivalent to standard Hamiltonian quantization on a cylinder ( $R \times S^{d - 1}$ ) via a conformal map, where the dilatation operator $D$ plays the role of the Hamiltonian.
There is a one-to-one correspondence between local operators inserted at the origin and states in the Hilbert space on the sphere.
This correspondence acts as a dictionary: the scaling dimension $Δ$ of an operator is the energy of its state, primary operators map to lowest-energy primary states, and descendant operators map to higher-energy descendant states.

We now have a complete and consistent picture of the kinematic structure of a CFT. With this machinery in place, we are ready to see its power in action. In the next lesson, we will begin the next module by applying these symmetry principles to derive powerful constraints on the correlation functions of the theory, starting with the universal form of two- and three-point functions.

Recommended resources

From Standard to Radial Quantization

The Cylinder Picture

The State-Operator Correspondence

The Correspondence as a Dictionary

Conclusion