Conformal symmetry and representation categories

A conformal primary is specified by its scaling dimension and spin representation, together with any internal charges. Acting with translations creates descendants. The resulting conformal multiplet is generally infinite-dimensional, even when the operator has only one scalar component or four vector components. Compact highest-weight calculators can describe some spin indices; they do not compute this unitary conformal multiplet.

We consider local, unitary CFTs in spacetime dimension $d\geq3$, with a conformally invariant vacuum and the usual state–operator correspondence. For radial quantization assume a reflection-positive Euclidean continuation and a scaling spectrum bounded below, with finite multiplicities at fixed scaling dimension. The vacuum is unique. Spinor operators require spin covers. Two-dimensional Virasoro representation theory and supersymmetric shortening are separate problems, outside the bounds stated below.

Choose the real form and the space

For a flat metric of signature $(p,q)$, $p+q=d\geq3$, the local conformal Lie algebra is $\mathfrak{so}(p+1,q+1)$. In particular:

Spacetime	Conformal real algebra	Compact rotation data used in radial quantization
Euclidean $\mathbb R^d$	$\mathfrak{so}(d+1,1)$	$SO(d)$, or its spin cover
Lorentzian $\mathbb R^{1,d-1}$	$\mathfrak{so}(2,d)$	Spatial rotations of the cylinder $S^{d-1}\times\mathbb R$

The Euclidean generators $P_\mu,K_\mu,D,M_{\mu\nu}$ are translations, special conformal transformations, dilatations and rotations. We use the algebraic convention

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

These conventions and the Euclidean real form are Simmons-Duffin, TASI Lectures on the Conformal Bootstrap, §3.2, equations (32)–(38). The Lorentzian algebra and conformal compactification are discussed in Aharony et al., §2.1.1 and §2.2. A special conformal transformation may leave a chosen flat coordinate patch, so its global action requires a compactification or an appropriate cover. The Lie algebra alone does not specify that choice.

A useful four-dimensional check comes from null rays in $\mathbb R^{2,4}$. A nonzero null vector can be scaled to $(\cos\tau,\sin\tau,\mathbf n)$, where $\mathbf n\in S^3$. Dividing by all nonzero real scalings identifies $(\tau,\mathbf n)$ with $(\tau+\pi,-\mathbf n)$. This projective compactification is $(S^1\times S^3)/\mathbb Z_2$. The matrix $-I_6$ acts trivially on its points, so the faithful connected action is $SO_0(2,4)/\{\pm I_6\}$.

By contrast, the spin map

$$SU(2,2)\longrightarrow SO_0(2,4)$$

has kernel $\{\pm I_4\}$ and still has a source with $\pi_1=\mathbb Z$. It is not the universal cover. Its four-dimensional defining module is a finite-dimensional spinor construction, not a positive-definite unitary CFT state space. The exterior-square map sends $iI_4$ to $-I_6$, so the further projective action kills the full scalar $\mathbb Z_4$ center of $SU(2,2)$. This is a direct kernel check using Garrett, Sporadic isogenies to orthogonal groups, §2.11, pp. 9–10. Compare the group and actual covering map.

For states, one commonly uses the universal covering group of the connected Lorentzian conformal group and unwrapped cylinder time. Returning to a smaller quotient requires its kernel to act trivially; it can impose conditions on spin and energy. Do not infer an integer scaling dimension by silently making the cylinder time periodic. The time choice is also central to the AdS discussion.

Field indices, primaries and descendants

A four-dimensional Lorentzian field may carry finite-dimensional $SL(2,\mathbb C)$ labels $(j_L,j_R)$ on its components. The corresponding conformal-primary label also includes $\Delta$, and the action moves the spacetime point. For example, the statement that a vector has $(j_L,j_R)=(1/2,1/2)$ specifies four components; it does not specify its scaling dimension or make those components a full conformal multiplet. See field versus particle representations.

Radial quantization maps a local operator at the origin to a state on $S^{d-1}$:

$$|\mathcal O_a\rangle=\mathcal O_a(0)|0\rangle, \qquad D|\mathcal O_a\rangle=\Delta|\mathcal O_a\rangle, \qquad K_\mu|\mathcal O_a\rangle=0.$$

The index $a$ lies in a finite-dimensional irrep of $SO(d)$ or $\operatorname{Spin}(d)$. The last condition defines a primary. The states $P_{\mu_1}\cdots P_{\mu_n}|\mathcal O_a\rangle$ have dilatation eigenvalue $\Delta+n$. Null states are removed to obtain the physical module. In the convention $[D,P]=P$, the primary is a lowest-energy state; literature using highest-weight terminology may reverse the choice of raising roots. The construction is Rychkov, §§3.1.2–3.1.4; his generators carry explicit factors of $i$, while the commutators above fix the convention used here.

The flat Euclidean metric satisfies $dr^2+r^2d\Omega_{d-1}^2=r^2(d\tau^2+d\Omega_{d-1}^2)$ for $\tau=\log r$. Thus radial dilatations become cylinder time translations. Reflection positivity gives the radial adjoints

$$D^\dagger=D,\qquad P_\mu^\dagger=K_\mu, \qquad M_{\mu\nu}^\dagger=-M_{\mu\nu}.$$

These are Simmons-Duffin, §7.2, equation (112). They select the Hilbert-space real structure appropriate to Lorentzian cylinder evolution. One should not interpret the Euclidean spacetime rotation labels as a finite-dimensional unitary representation of the whole noncompact conformal group.

Derive the scalar bound from two descendant levels

Take a scalar primary of unit norm. It obeys $M_{\mu\nu}|\mathcal O\rangle=0$. At level one,

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

so positivity requires $\Delta\geq0$. For a nonidentity scalar this is not yet sufficient. Put $P^2=\sum_\nu P_\nu P_\nu$. Commuting one $K_\mu$ through the two translations gives

$$\begin{aligned} K_\mu P^2|\mathcal O\rangle &=4\left(\Delta+1-\frac d2\right)P_\mu|\mathcal O\rangle,\\ \|P^2|\mathcal O\rangle\|^2 &=8d\Delta\left(\Delta-\frac{d-2}{2}\right). \end{aligned}$$

For the first line, use $DP_\mu|\mathcal O\rangle=(\Delta+1)P_\mu |\mathcal O\rangle$ and $\sum_\nu M_{\mu\nu}P_\nu|\mathcal O\rangle=(d-1)P_\mu |\mathcal O\rangle$. The second line applies $K_\mu$ again and sums $\sum_\mu\|P_\mu|\mathcal O\rangle\|^2=2d\Delta$. For $\Delta>0$, positivity therefore forces

$$\boxed{\Delta\geq\frac{d-2}{2}.}$$

This solves the norm problem posed in Simmons-Duffin, Exercise 7.4; bounds and null states are equations (130)–(131). At saturation, $P^2|\mathcal O\rangle$ is null, giving the free scalar field equation after quotienting null states. The dimension-zero identity is a separate short representation with all descendants null.

In $d=4$, a proposed scalar with $\Delta=3/4$ has level-two norm $-6$ and is excluded by unitarity. At $\Delta=1$ that norm is zero. Its first three physical levels contain $1$, $4$ and $9$ states: the ten symmetric two-derivative components lose one trace. More generally, the free scalar’s level-$n$ descendants form a traceless symmetric rank-$n$ tensor with $(n+1)^2$ components in four dimensions. The first three levels alone already exceed the one scalar component of the primary field.

A conserved vector is another distinct representation

For a symmetric traceless rank-$\ell$ primary, $\ell\geq1$, the unitary bound is

$$\Delta\geq\ell+d-2.$$

At equality, its divergence is a null descendant and the operator is conserved; see Simmons-Duffin, §7.3, equations (129)–(134). In four dimensions, a global-symmetry current therefore has $(\Delta,j_L,j_R)=(3,1/2,1/2)$, while the stress tensor has $(4,1,1)$. These are full conformal labels, unlike a Lorentz label alone. A gauge potential with engineering dimension one is not a gauge-invariant physical vector primary to which this bound applies.

The scalar and symmetric-traceless bounds are not formulas for arbitrary mixed-symmetry tensors or spinors. Satisfying a bound also does not prove that an operator occurs in a particular CFT: its spectrum and operator product coefficients are additional dynamical data. Canonical free-field dimensions need not equal interacting scaling dimensions.

What the current tools can check

The compact Lie tools can check suitable rotation-representation dimensions after the real form and node conventions are fixed. The local-operator workbench counts specified derivative-free Lorentz and gauge contractions with statistics. It does not compute anomalous dimensions, determine conformal primaries, remove general equation-of-motion descendants, solve crossing equations, or classify unitary representations of $SO(2,d)$.

Continue along the spacetime learning path or use the spacetime atlas collection to compare the real groups and their covers. The AdS entry applies the same distinction to bulk fields and positive-energy state representations.