Local operators, radial quantization, and the OPE

The scaling lesson associated a dimension with the power of a correlation function. Here that same dimension becomes an energy: logarithmic distance from an insertion plays the role of Euclidean time. This gives us a practical way to organize local operators and expand products of nearby insertions.

Work in a unitary conformal field theory in $d=3$ or $4$, continued to Euclidean space. There is no supersymmetry assumption. We use a conformally invariant vacuum, normalized correlation functions, and the usual radial Hilbert space with a discrete spectrum on the sphere. Coincident-point contact terms are excluded unless explicitly discussed. A local operator may be an elementary field, a renormalized composite, or another localized insertion; it need not create one elementary particle.

Turn radius into time

Write a nonzero position as $x=r n$, where $n\cdot n=1$. Introduce a positive length $L$ and a dimensionless coordinate

$$\tau=\log(r/L),\qquad r=L e^\tau.$$

Since $n\cdot dn=0$, differentiating $x$ gives

$$ds_{\rm flat}^2=dr^2+r^2d\Omega_{d-1}^2 =r^2(d\tau^2+d\Omega_{d-1}^2) =\left(\frac rL\right)^2ds_{\rm cyl}^2,$$

where

$$ds_{\rm cyl}^2=dt^2+L^2d\Omega_{d-1}^2,\qquad t=L\tau.$$

Thus a coordinate change followed by a Weyl rescaling produces a cylinder with spatial sphere radius $L$. A dilation $r\mapsto e^s r$ becomes $t\mapsto t+Ls$. The origin and infinity become the two ends $t\to-\infty,+\infty$. Merely changing coordinates would leave the factor $(r/L)^2$ in the metric; conformal covariance is what allows the cylinder description.

Let $D$ be the dimensionless dilation generator, with $D|0\rangle=0$. Evolution through an annulus of radius ratio $r_2/r_1$ is

$$U(r_2,r_1)=e^{-\log(r_2/r_1)D} =e^{-(t_2-t_1)(H_{\rm cyl}-E_{\rm vac})}, \qquad H_{\rm cyl}-E_{\rm vac}=\frac DL.$$

We identify energy gaps. A vacuum-energy contribution from the curved-space definition of the stress tensor, including an even-dimensional Weyl anomaly, is not being set to zero. The cylinder map and normalized Weyl prescription are described in Simmons-Duffin v1, §6.4, (90)–(94).

There is a direct action check. For the ordinary free scalar set $\delta=(d-2)/2$ and $\chi(\tau,n)=r^\delta\phi(rn)$. Its flat action becomes

$$S=\frac12\int d\tau\,d\Omega_{d-1} \left[(\partial_\tau\chi-\delta\chi)^2 +(\nabla_\Omega\chi)^2\right].$$

The powers cancel because $r^d r^{-2}r^{-2\delta}=1$. Expanding the square, the cross term is the boundary contribution $-\delta\int d\Omega\,[\chi^2]/2$. For fields with vanishing endpoint contribution, the remaining cylinder action is

$$S=\frac12\int d\tau\,d\Omega_{d-1} \left[(\partial_\tau\chi)^2+(\nabla_\Omega\chi)^2 +\delta^2\chi^2\right].$$

The constant angular mode therefore has frequency $\delta$ in $\tau$, or energy gap $\delta/L$. The term $\delta^2\chi^2$ is essential: a minimally coupled massless scalar placed arbitrarily on a cylinder would be a different action.

Organize derivatives into a conformal family

Use the Euclidean charge convention $[P_\mu,\mathcal O(x)]=\partial_\mu\mathcal O(x)$, with no factor of $i$. It is a convention for Euclidean conformal charges, not the Hermitian Lorentzian momentum convention of the supersymmetry lessons. In this section measure coordinates in units of $L$.

Besides translations $P_\mu$ and dilations $D$, let $M_{\mu\nu}$ generate rotations and $K_\mu$ generate special conformal transformations. The needed algebra is

$$\begin{aligned} [D,P_\mu]&=P_\mu,& [D,K_\mu]&=-K_\mu,\\ [K_\mu,P_\nu]&=2\delta_{\mu\nu}D-2M_{\mu\nu},& [M_{\mu\nu},P_\rho]&=\delta_{\nu\rho}P_\mu-\delta_{\mu\rho}P_\nu. \end{aligned}$$

A scalar primary at the origin obeys

$$[D,\mathcal O(0)]=\Delta\mathcal O(0),\qquad [K_\mu,\mathcal O(0)]=0,\qquad [M_{\mu\nu},\mathcal O(0)]=0.$$

The last condition is replaced by a spin representation for a nonscalar primary. Primaries and their translation descendants form a conformal family. These definitions and signs follow Simmons-Duffin, (32)–(37), (49)–(53).

Insert $\mathcal O(0)$ in the path integral inside a sphere. The result is a state on that sphere, denoted $|\mathcal O\rangle$. In a vacuum annihilated by the conformal charges, the commutator equations give

$$D|\mathcal O\rangle=\Delta|\mathcal O\rangle,\qquad K_\mu|\mathcal O\rangle=0.$$

Now apply a translation:

$$D P_\mu|\mathcal O\rangle =[D,P_\mu]|\mathcal O\rangle+P_\mu D|\mathcal O\rangle =(\Delta+1)P_\mu|\mathcal O\rangle.$$

Repeating gives dimensions $\Delta+n$ for nonzero level-$n$ descendants. In position space they are derivatives such as $\partial_\mu\mathcal O$ and $\partial_\mu\partial_\nu\mathcal O$. A derivative is generally not another primary:

$$K_\mu P_\nu|\mathcal O\rangle =2\Delta\delta_{\mu\nu}|\mathcal O\rangle.$$

The right side is nonzero for a nonidentity scalar with positive $\Delta$. Derivatives supply excited states of the same conformal family, rather than unrelated scaling data.

Conversely, a sphere state can be evolved inward and expanded in dilation eigenstates, which define insertions at the origin. This is the state–operator correspondence under the Hilbert-space assumptions stated above; see §6.1–§6.3, (84)–(89). It organizes local-operator states, not a massive-particle scattering spectrum.

Check a cylinder correlator

For a scalar primary choose the conformally transported cylinder operator

$$\mathcal O_{\rm cyl}(t,n) =\left(\frac rL\right)^\Delta\mathcal O_{\rm flat}(rn).$$

Take a Hermitian scalar with flat two-point function $C/|x_1-x_2|^{2\Delta}$. Since

$$|r_1n_1-r_2n_2|^2 =2r_1r_2\left[\cosh\!\left(\frac{t_1-t_2}{L}\right)-n_1\cdot n_2\right],$$

we obtain

$$\boxed{ \langle\mathcal O_{\rm cyl}(t_1,n_1) \mathcal O_{\rm cyl}(t_2,n_2)\rangle =\frac{C}{\left\{2L^2[\cosh((t_1-t_2)/L)-n_1\cdot n_2]\right\}^{\Delta}}.}$$

It depends on the time difference, as required by cylinder translations. At positive separation $u=(t_1-t_2)/L$ and $c=n_1\cdot n_2$, factor out $e^u$:

$$\langle\mathcal O_{\rm cyl}\mathcal O_{\rm cyl}\rangle =\frac{C}{L^{2\Delta}}e^{-\Delta u} \left[1+2\Delta c\,e^{-u}+O(e^{-2u})\right].$$

The first exponential measures the gap $\Delta/L$; the next has gap $(\Delta+1)/L$. Angular dependence selects which descendants contribute. At $c=0$, for example, the level-one contribution vanishes without removing those states from the theory. At short separation with $n_1=n_2$, $2L^2(\cosh u-1)\sim(t_1-t_2)^2$, recovering the local flat-space singularity.

Why unequal primary dimensions are orthogonal

Translations, rotations and scaling alone allow

$$G(x)=\langle\mathcal O_1(x)\mathcal O_2(0)\rangle =C_{12}|x|^{-(\Delta_1+\Delta_2)}.$$

For a scalar primary the special conformal action is

$$[K_\mu,\mathcal O(x)] =\left[2x_\mu(x\cdot\partial+\Delta)-x^2\partial_\mu\right]\mathcal O(x).$$

The second insertion at the origin is killed by $K_\mu$. Invariance of the vacuum therefore requires the displayed differential operator with $\Delta=\Delta_1$ to annihilate $G$. Writing $s=\Delta_1+\Delta_2$, use $x\cdot\partial G=-sG$ and $\partial_\mu G=-s x_\mu G/x^2$. The result is

$$0=(2\Delta_1-s)x_\mu G =(\Delta_1-\Delta_2)x_\mu G.$$

Thus either $\Delta_1=\Delta_2$ or $C_{12}=0$. This selection rule uses special conformal symmetry and primarity, not dimensional analysis alone. It is the Ward identity (64).

For equal dimensions, a basis of independent scalar primaries can be chosen orthonormal under their positive two-point matrix. Complex operators are paired with their conjugates, and charges impose additional selection rules.

Positivity is easiest to see radially. In units $L=1$, the radial adjoint of a Hermitian scalar insertion is

$$\mathcal O(x)^{\dagger_{\rm rad}} =|x|^{-2\Delta}\mathcal O\!\left(\frac{x}{x^2}\right).$$

It reflects cylinder time, since $r\mapsto1/r$ sends $\tau\mapsto-\tau$. For a complex operator the right side contains its internal conjugate. Consequently

$$\langle\mathcal O| =\lim_{r\to\infty}r^{2\Delta}\langle0|\mathcal O(rn), \qquad \langle\mathcal O|\mathcal O\rangle=C.$$

Here the dual formula is written for the Hermitian scalar. Reflection positivity gives $C>0$ for a nonzero state. In this radial inner product $D^\dagger=D$, $P_\mu^\dagger=K_\mu$, and $M_{\mu\nu}^\dagger=-M_{\mu\nu}$, as in §7.2, (108)–(116). For a unit-normalized scalar primary,

$$\langle\mathcal O|K_\mu P_\nu|\mathcal O\rangle =2\Delta\delta_{\mu\nu}.$$

This gives a positive descendant norm when $\Delta>0$. A stronger level-two test appears in Exercise 3.

Replace two nearby insertions by local operators

Suppose $\mathcal O_i(x)$ and $\mathcal O_j(0)$ lie inside a sphere centered at $0$ and all other insertions lie outside. Thus $|x|$ is smaller than every other insertion’s distance from $0$. The inside path integral prepares a state on the separating sphere. Expanding that state in primary and descendant states gives

$$\mathcal O_i(x)\mathcal O_j(0) =\sum_k\mathcal C_{ij}{}^k(x,\partial)\mathcal O_k(0).$$

This is the operator product expansion, or OPE. The index $k$ labels primaries; the differential operator $\mathcal C$ packages their descendants. Its validity is inside correlation functions in the separating-sphere domain. In the usual unitary radial setting the expansion converges there; it is not a prescription for multiplying fields at exactly coincident points. See §8, (136)–(142).

For a scalar exchanged primary of dimension $\Delta_k$, scaling fixes its leading coefficient to be proportional to $|x|^{\Delta_k-\Delta_i-\Delta_j}$. It does not fix the proportionality constant. Spinful terms carry angular tensors as well. The remaining constants, after choosing two-point normalizations, are dynamical CFT data.

A free scalar supplies a calculation with every coefficient visible. Let

$$\langle\phi(x)\phi(0)\rangle=\frac{A_d}{|x|^{d-2}}, \qquad A_d=\frac{1}{(d-2)\operatorname{vol}(S^{d-1})}.$$

This is the normalization derived in the preceding lesson. Set $\varphi=\phi/\sqrt{A_d}$, so that $\langle\varphi(x)\varphi(0)\rangle=|x|^{-2\delta}$ with $\delta=(d-2)/2$. Normal ordering, denoted by colons, subtracts free self-contractions. Define

$$\mathcal O_2=\frac{:\!\varphi^2\!:}{\sqrt2}.$$

There are two pairings between the fields at $x$ and $0$, hence

$$\langle\mathcal O_2(x)\mathcal O_2(0)\rangle =\frac{2}{2}\frac1{|x|^{4\delta}}.$$

The operator has unit two-point normalization and dimension $2\delta$. It is a scalar primary: the free scalar is primary, and the flat normal-ordered product at the origin has no nonzero special-conformal variation.

Wick contraction separates the singular identity term:

$$\varphi(x)\varphi(0) =\frac1{|x|^{2\delta}}\mathbf1+:\!\varphi(x)\varphi(0)\!:.$$

Taylor expanding the uncontracted first field then gives

$$\boxed{ \varphi(x)\varphi(0) =\frac1{|x|^{2\delta}}\mathbf1 +\sqrt2\,\mathcal O_2(0) +\frac1{\sqrt2}x^\mu\partial_\mu\mathcal O_2(0) +\cdots .}$$

We used $:\!(\partial_\mu\varphi)\varphi\!: =\partial_\mu:\!\varphi^2\!:/2$. The leading OPE coefficient is $\sqrt2$, not one; it remembers the chosen normalization of the composite. At order $x^2$ and beyond, further descendants and other primaries, including operators with spin, must be organized. The ellipsis is not asserted to contain only derivatives of $\mathcal O_2$.

Taking the vacuum expectation value keeps only the identity term. Testing the same expansion against a distant $\mathcal O_2$ instead isolates the composite family, as the next exercise shows.

Exercises

1. Read the free scalar energies from its action

On the unit angular sphere, scalar harmonics obey $-\nabla_\Omega^2Y_j=j(j+d-2)Y_j$ with integer $j\geq0$. Expand the field $\chi$ in those harmonics and find the physical cylinder energy gaps. Check $d=3$ and $d=4$. Explain what would go wrong if the $\delta^2\chi^2$ term were dropped.

Hint 1

Each harmonic coefficient is a Euclidean oscillator in $\tau$. Add its angular eigenvalue to the mass coefficient in the derived action.

Hint 2

Use $d-2=2\delta$ to complete a square. Physical time is $t=L\tau$.

Solution

The squared oscillator frequency is

$$\omega_j^2=j(j+d-2)+\delta^2=(j+\delta)^2.$$

Since $d>2$, the positive frequency is $j+\delta$, so

$$E_j-E_{\rm vac}=\frac{j+(d-2)/2}{L}.$$

The lowest gaps are $1/(2L)$ in three dimensions and $1/L$ in four dimensions. Adding $j$ derivatives gives exactly these higher gaps. For the free field, the equation $\partial^2\varphi=0$ removes trace descendants; symmetric traceless derivatives account for the angular harmonics. They do not represent additional independent scalar primaries.

Dropping $\delta^2$ would give $\sqrt{j(j+d-2)}/L$, including a zero-frequency constant mode. That is not the dilation spectrum of the free flat-space CFT. In positive-sphere curvature conventions the conformal coupling checks the missing coefficient:

$$\xi_c R_{\rm cyl} =\frac{d-2}{4(d-1)} \frac{(d-1)(d-2)}{L^2} =\frac{\delta^2}{L^2}.$$

Here $R_{\rm cyl}$ is positive; it is not the opposite curvature convention used by the three-manifold target paper.

2. Measure the OPE coefficient with a third insertion

Calculate $\langle\varphi(x_1)\varphi(x_2)\mathcal O_2(x_3)\rangle$ by Wick contraction. Then set $x_1=x$, $x_2=0$, $x_3=z$ with $|x|<|z|$ and expand through first order in $x$. Compare with both displayed nonidentity OPE terms.

Hint 1

The two fields inside the normal-ordered composite cannot contract with each other. Count how many ways they can contract with the two external fields.

Hint 2

Expand $|z-x|^{-2\delta}$, and differentiate $\langle\mathcal O_2(y)\mathcal O_2(z)\rangle=|y-z|^{-4\delta}$ with respect to $y$, setting $y=0$ afterward.

Solution

Two Wick pairings survive, giving

$$\langle\varphi(x_1)\varphi(x_2)\mathcal O_2(x_3)\rangle =\frac{\sqrt2}{|x_1-x_3|^{2\delta}|x_2-x_3|^{2\delta}}.$$

For the chosen positions this becomes

$$\frac{\sqrt2}{|z|^{4\delta}} \left[1+2\delta\frac{x\cdot z}{z^2} +O\!\left(\frac{|x|^2}{|z|^2}\right)\right].$$

The identity OPE term contributes zero because $\langle\mathcal O_2(z)\rangle=0$ in this vacuum. The primary term contributes $\sqrt2/|z|^{4\delta}$. The derivative term contributes

$$\frac{x^\mu}{\sqrt2} \left.\partial_{y^\mu}|y-z|^{-4\delta}\right|_{y=0} =\frac{4\delta}{\sqrt2} \frac{x\cdot z}{|z|^{4\delta+2}},$$

which is precisely the linear term above. Both the normalization and the descendant coefficient have been tested by a different correlator.

The condition $|x|<|z|$ permits a sphere enclosing $0,x$ while excluding $z$. Approaching or crossing the other insertion invalidates this particular expansion domain; it does not invalidate the exact three-point function.

3. Find a null descendant

For a unit-normalized scalar primary use the displayed conformal algebra and $P_\mu^\dagger=K_\mu$ to compute the norm of $P^2|\mathcal O\rangle$, where $P^2=\sum_{\nu=1}^dP_\nu P_\nu$. Derive the scalar unitarity bound for $\Delta>0$, and interpret its saturation in the free example.

Hint 1

Commute one $K_\mu$ through both translations. The rotation generator annihilates the primary but does not annihilate $P_\nu|\mathcal O\rangle$.

Hint 2

Summing the rotation commutator gives $\sum_\nu M_{\mu\nu}P_\nu|\mathcal O\rangle=(d-1)P_\mu|\mathcal O\rangle$. Then apply the remaining $K_\mu$ and sum $\mu$.

Solution

Expanding the first commutator gives

$$\begin{aligned} K_\mu P^2|\mathcal O\rangle &=\sum_\nu\left([K_\mu,P_\nu]P_\nu +P_\nu[K_\mu,P_\nu]\right)|\mathcal O\rangle\\ &=\left[2(\Delta+1)-2(d-1)+2\Delta\right] P_\mu|\mathcal O\rangle\\ &=4\left(\Delta-\frac{d-2}{2}\right)P_\mu|\mathcal O\rangle. \end{aligned}$$

Therefore

$$\boxed{ \|P^2|\mathcal O\rangle\|^2 =8d\Delta\left(\Delta-\frac{d-2}{2}\right).}$$

Nonnegative norms require $\Delta\geq(d-2)/2$ when $\Delta>0$. The separate $\Delta=0$ case has vanishing first descendants; with a unique conformal vacuum it is the identity sector. This level-two test is the scalar case discussed in §7.3, Exercise 7.4 and (131).

At $\Delta=(d-2)/2$, the trace descendant has zero norm. In the positive Hilbert space it is removed as a null state. Under the state–operator map this is the equation $\partial^2\varphi=0$ away from other insertions. It does not erase the contact source in the propagator: the distributional equation for a time-ordered or radially ordered two-point function still includes its delta function.

A local scaling dimension now has three linked meanings: a correlation exponent, an eigenvalue of $D$, and a cylinder energy gap times $L$. Before using an OPE, specify the normalized operators and a sphere centered at the expansion point that separates their insertions from the rest. Before counting a conformal family, check whether any descendants are null. Continue to what BPS and protected mean to use those norms in a supersymmetric shortening argument.