Unitarity Bounds and Null States

Unitarity bounds are positivity theorems for conformal representations, not dimensional estimates. In a reflection-positive Euclidean CFT, radial conjugation makes translations adjoint to special conformal transformations. Descendant norms can then be reduced to the conformal algebra. For a symmetric traceless primary of spin $\ell\geq1$, level-one positivity gives $\Delta\geq\ell+d-2$; a scalar requires a separate level-two calculation and obeys $\Delta\geq(d-2)/2$ unless it is the identity. Saturation creates a null descendant and hence a short module.

Required background. Primaries, Descendants, and Conformal Multiplets supplies descendant levels and quotient modules. Hilbert Space, Positivity, and Unitary Evolution supplies positive inner products and adjoints. Helpful background. Forms, Adjoints, and Isometries supplies Gram matrices and positive-semidefinite quotients.

Hypotheses behind the bounds

The derivation on this page assumes:

1. a local Euclidean CFT in $d\geq3$ satisfying reflection positivity;
2. radial quantization and the state–operator correspondence;
3. a positive-energy spectrum of $D$ with finite-dimensional rotation eigenspaces;
4. a primary state in an irreducible finite-dimensional representation of $SO(d)$ or $\operatorname{Spin}(d)$;
5. the radial adjoint $D^\dagger=D$, $P_\mu^\dagger=K_\mu$, and $M_{\mu\nu}^\dagger=-M_{\mu\nu}$; and
6. quotienting of zero-norm states so that the physical inner product is positive definite.

After Osterwalder–Schrader continuation these conditions correspond to a unitary positive-energy Lorentzian theory, but that continuation is an additional theorem-level step. Nonunitary, logarithmic, gauge-fixed, defect, or nonlocal sectors can violate one or more assumptions. The standard derivation and its CFT interpretation are given in Simmons-Duffin 2017, §§ 7.1–7.3 and Poland, Rychkov, and Vichi 2019, § III.E.

Level-one positivity for spinning primaries

Let $\lvert\mathcal O,a\rangle$ be a primary of dimension $\Delta$ in a rotation irrep $R$. Choose an orthonormal basis, $\langle\mathcal O,a\vert\mathcal O,b\rangle=\delta_{ab}$. The level-one Gram matrix is

$$\begin{aligned} \langle\mathcal O,a\rvert K_\mu P_\nu \lvert\mathcal O,b\rangle &=\langle\mathcal O,a\rvert[K_\mu,P_\nu] \lvert\mathcal O,b\rangle\\ &=2\left[ \Delta\delta_{\mu\nu}\delta_{ab} -(S_{\mu\nu})_b{}^c\delta_{ac} \right]. \end{aligned}$$

Decompose $V\otimes R$ into irreducible rotation representations $R'$. With the anti-Hermitian rotation matrices used here, define the positive quadratic Casimir by

$$C_2(R)\mathbf1_R=-\frac12S_{\mu\nu}S_{\mu\nu}.$$

The spin term is diagonal on each component, and its eigenvalue gives

$$\lambda_{R'} =2\left[ \Delta+\frac12\bigl(C_2(R')-C_2(R)-C_2(V)\bigr) \right].$$

Every $\lambda_{R'}$ must be nonnegative. For a symmetric traceless rank-$\ell$ representation, $C_2([\ell])=\ell(\ell+d-2)$ and

$$V\otimes[\ell] =[\ell+1]\oplus[\ell,1]\oplus[\ell-1]$$

in generic $d$. The $[\ell-1]$ component is the divergence. Its eigenvalue is

$$\lambda_{[\ell-1]}=2\bigl[\Delta-(\ell+d-2)\bigr],$$

so positivity implies

$$\boxed{\Delta\geq\ell+d-2\qquad(\ell\geq1).}$$

The other level-one channels give weaker inequalities once this bound holds. At saturation, the projected descendant has zero norm. Positivity then makes it orthogonal to the whole physical space, and the quotient imposes

$$P^{\mu_1}\lvert\mathcal O_{\mu_1\cdots\mu_\ell}\rangle=0 \quad\Longleftrightarrow\quad \partial^{\mu_1}\mathcal O_{\mu_1\cdots\mu_\ell}(x)=0.$$

For $\ell=1$, this gives a conserved current with $\Delta=d-1$; for $\ell=2$, a conserved symmetric traceless stress tensor has $\Delta=d$. The level-one calculation supplies the inequality. The classification theorem for positive-energy conformal representations ensures that no higher descendant imposes a stronger bound on these symmetric traceless modules; it is not enough merely to inspect the first level and assume the rest.

The scalar bound is a level-two result

For a scalar primary, level one gives only

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

and hence $\Delta\geq0$. The sharper bound comes from the scalar trace at level two. Using the algebra and $M_{\mu\nu}\lvert\mathcal O\rangle=K_\mu\lvert\mathcal O\rangle=0$,

$$K_\mu P^2\lvert\mathcal O\rangle =4\left(\Delta-\frac{d-2}{2}\right)P_\mu\lvert\mathcal O\rangle.$$

Applying $K^\mu$ once more gives the exact norm

$$\lVert P^2\lvert\mathcal O\rangle\rVert^2 =8d\,\Delta \left(\Delta-\frac{d-2}{2}\right) \lVert\mathcal O\rVert^2.$$

Together with $\Delta\geq0$, positivity allows $\Delta=0$ or

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

In an irreducible CFT with a unique invariant vacuum, a scalar primary at $\Delta=0$ is the identity. At the nontrivial saturation value, $P^2\lvert\mathcal O\rangle$ is null and

$$\partial^2\mathcal O=0.$$

This is the free massless scalar equation. The resulting CFT sector has additional consequences—for example, Wick-like structures under standard locality assumptions—but those do not follow from the norm polynomial alone.

Shortening as a quotient

The figure separates the positivity step from the local equation. Inspect the branch at which a descendant becomes both singular and zero norm.

For a symmetric traceless spin-$\ell\geq1$ primary, the level-one divergence becomes null at $\Delta=\ell+d-2$ and yields conservation. For a scalar, the level-two trace becomes null at $\Delta=(d-2)/2$ and yields the free equation. The diagram is schematic; nullness requires the positive radial form and is stronger than algebraic reducibility alone.

Step	Symmetric traceless spin $\ell\geq1$	Scalar	Logical input
Candidate descendant	Level-one divergence $P\mathbin{\cdot}\mathcal O$	Level-two trace $P^2\mathcal O$	Rotation decomposition
Norm factor controlling the threshold	$\Delta-(\ell+d-2)$	$\Delta[\Delta-(d-2)/2]$	Conformal commutators and $P^\dagger=K$
Positivity conclusion	$\Delta\geq\ell+d-2$	$\Delta=0$ or $\Delta\geq(d-2)/2$	Positive radial inner product
Saturation equation	$\partial\mathbin{\cdot}\mathcal O=0$	$\partial^2\mathcal O=0$	Null quotient plus state–operator correspondence
Entire submodule removed	Descendants of the divergence	Descendants of $P^2\mathcal O$	Irreducible short-module quotient

Bounds and equations by representation

The following table is designed for use before interpreting a measured or computed scaling dimension. Its representation labels are irreducible $SO(d)$ or $\operatorname{Spin}(d)$ rotation labels in reflection-positive Euclidean radial quantization; Lorentzian finite-dimensional field labels require continuation to the corresponding real form. Its entries are representation-theoretic constraints; they do not prove that a local CFT realizing the representation exists.

Primary in a reflection-positive $d\geq3$ CFT	Bound or saturation value	First null component	Local conclusion at saturation	Necessary qualifications and failure mode	Proof status and source
Identity-sector scalar	$\Delta=0$	$P_\mu\mathbf1=0$	Constant identity operator	Identifying every dimension-zero scalar with the identity also uses irreducibility and a unique invariant vacuum	Positive-energy representation result under the page hypotheses: Simmons-Duffin 2017, § 7.3
Nonidentity scalar	$\Delta\geq(d-2)/2$	$P^2\mathcal O$ at equality	$\partial^2\mathcal O=0$	The level-one test gives only $\Delta\geq0$; nonunitary scalars may lie below the bound	Level-two Gram theorem plus representation classification: Poland, Rychkov, and Vichi 2019, § III.E
Fundamental spinor	$\Delta\geq(d-1)/2$	Gamma-trace $\gamma^\mu P_\mu\psi$ at equality	Massless Dirac equation	Chirality and reality depend on dimension and signature; the statement is for the appropriate Spin irrep	Positive-energy representation bound: Minwalla 1998, § 2, pp. 792–794
Symmetric traceless rank $\ell\geq1$	$\Delta\geq\ell+d-2$	Divergence at equality	Generalized conservation; ordinary current for $\ell=1$, stress-tensor form for $\ell=2$	Conservation follows only after the zero-norm quotient; a generic operator at the same dimension in a nonunitary theory need not be conserved	Level-one Gram theorem plus sufficiency classification: Simmons-Duffin 2017, § 7.3
Mixed-symmetry highest weight with $\ell_1=\cdots=\ell_p>\lvert\ell_{p+1}\rvert$	$\Delta\geq\ell_1+d-p-1$	Representation-specific generalized divergence	Mixed-symmetry conservation equation	Highest-weight conventions and low-dimensional dualities must be translated before applying the formula	Positive-energy representation bound in the stated highest-weight convention: Dolan 2006, §§ 2–3
Any representation below its applicable bound	Forbidden in a positive-energy unitary module	Negative-norm descendant before quotient	No unitary CFT operator with those labels	It may occur in a nonunitary or gauge-dependent space, where the positive-form hypothesis is absent	Direct consequence of a negative Gram eigenvalue: Dolan 2006, Appendix C

The table deliberately separates “allowed representation” from “realized CFT.” Crossing symmetry, OPE associativity, locality, and the existence of a stress tensor impose additional conditions that a single conformal module does not see.

Counterexamples to careless use

Nonunitary fixed points. The Lee–Yang CFT has operators below unitary bounds because reflection positivity is absent Poland, Rychkov, and Vichi 2019, § VIII. This does not contradict the inequalities; it violates their first hypothesis.

Gauge-fixed fields. A gauge potential or ghost may live in an indefinite auxiliary state space. Apply positivity only to gauge-invariant physical operators after the relevant quotient, not to every gauge-fixed field component.

Logarithmic modules. If $D$ has Jordan blocks, the inner product and module decomposition need not be diagonalizable in the form assumed above. A zero norm does not automatically define a decoupled direct summand.

Dimension alone. A spin-one operator with $\Delta=d-1$ is conserved in the positive irreducible conformal module described above. Without conformal symmetry, radial positivity, and the null quotient, the numerical equality alone is not a conservation proof.

Handoff to radial Gram matrices

This page derives analytic representation bounds. Descendant Gram Matrices constructs the radial inner product in detail, tracks basis and normalization, and checks the scalar and spinning matrices level by level. That page owns the explicit reflection-positive Gram construction; the present result supplies the target eigenvalues and null channels. Conserved Currents and the Stress Tensor then adds Ward identities and normalization data that representation theory alone cannot fix.

Exercises

Derive the scalar level-two norm from the two intermediate identities

$$K_\mu P^2\lvert\mathcal O\rangle =4\left(\Delta-\frac{d-2}{2}\right)P_\mu\lvert\mathcal O\rangle, \qquad K^\mu P_\mu\lvert\mathcal O\rangle=2d\Delta\lvert\mathcal O\rangle.$$

Solution

Using $(P^2)^\dagger=K^2$ and applying the first identity followed by the second,

$$\langle\mathcal O\rvert K^2P^2\lvert\mathcal O\rangle =4\left(\Delta-\frac{d-2}{2}\right) \langle\mathcal O\rvert K^\mu P_\mu\lvert\mathcal O\rangle =8d\Delta\left(\Delta-\frac{d-2}{2}\right) \lVert\mathcal O\rVert^2.$$

Explain why the current bound follows from the $[\ell-1]$ channel rather than the $[\ell+1]$ channel.

Solution

For $R=[\ell]$, the Casimir shift in the $[\ell-1]$ channel is $-(\ell+d-2)$, so its Gram eigenvalue is $2[\Delta-(\ell+d-2)]$. The $[\ell+1]$ shift is $+\ell$, giving $2(\Delta+\ell)$, which is already positive for positive $\Delta$. The divergence channel therefore reaches zero first.

References

• Dolan, F. A. “Character Formulae and Partition Functions in Higher Dimensional Conformal Field Theory.” Journal of Mathematical Physics 47 (2006): 062303. DOI; Open PDF
• Minwalla, Shiraz. “Restrictions Imposed by Superconformal Invariance on Quantum Field Theories.” Advances in Theoretical and Mathematical Physics 2 (1998): 783–851. DOI; Open PDF
• Poland, David, Slava Rychkov, and Alessandro Vichi. “The Conformal Bootstrap: Theory, Numerical Techniques, and Applications.” Reviews of Modern Physics 91 (2019): 015002. DOI; Open PDF
• Simmons-Duffin, David. “TASI Lectures on the Conformal Bootstrap.” In New Frontiers in Fields and Strings, 1–74. Singapore: World Scientific, 2017. DOI; Open PDF