Descendant States and Gram Matrices

A descendant Gram matrix is obtained by moving every special conformal generator $K_\mu$ through a monomial of translations $P_\nu$ until it reaches the primary state. Its entries are therefore fixed by the conformal commutators, the primary dimension, and the primary spin representation. Positive semidefiniteness gives unitarity bounds; a kernel identifies null descendants and a reducible module. The calculation must be performed in a complete level basis and then quotiented by the full null submodule—not merely by deleting one small numerical eigenvalue.

Required background. Conjugation and Reflection Positivity supplies $P_\mu^\dagger=K_\mu$ and the positive inner product. The Conformal Algebra and Its Generators fixes the commutators used below. Helpful background. Bilinear and Hermitian Forms, Adjoints, and Isometries explains Gram matrices, null spaces, and changes of basis.

Descendant levels and a complete basis

Let $|\Delta,R,a\rangle$ be a primary state of dimension $\Delta$ in an irreducible rotation representation $R$, with component index $a$. Descendants at level $n$ are spanned by

$$P_{\mu_1}\cdots P_{\mu_n}|\Delta,R,a\rangle.$$

Because $[P_\mu,P_\nu]=0$, only the symmetric tensor product $\operatorname{Sym}^n(V)\otimes R$ is needed before imposing null relations. A practical basis construction has four stages:

1. enumerate symmetric monomials in the $P_\mu$;
2. tensor them with a basis of $R$;
3. decompose the result into irreducible $SO(d)$ or $\operatorname{Spin}(d)$ sectors; and
4. choose one explicit normalization for each projected basis vector.

The decomposition is not optional. Rotational symmetry makes the Gram matrix block diagonal by irreducible spin, and distinct eigenvalue conditions can appear in different blocks. An incomplete component basis can miss the descendant whose norm supplies the sharp bound.

With the local convention

$$[K_\mu,P_\nu]=2\delta_{\mu\nu}D-2M_{\mu\nu}, \qquad [D,P_\mu]=P_\mu, \qquad P_\mu^\dagger=K_\mu,$$

the level-$n$ matrix is

$$G^{(n)}_{IA,JB} = \langle\Delta,R,A| K_{i_n}\cdots K_{i_1} P_{j_1}\cdots P_{j_n} |\Delta,R,B\rangle,$$

where $I=(i_1\ldots i_n)$ and $J=(j_1\ldots j_n)$ label the chosen projected basis. Commute the $K$ operators rightward. Terms with a $K$ acting directly on the primary vanish, $D$ acts by $\Delta$ plus the level of the remaining descendants, and $M_{\mu\nu}$ acts through the known representation matrices on both vector and primary indices.

This is the conformal analogue of a Shapovalov form. Simmons-Duffin 2017, §7.3, pp. 37–39, Open PDF derives the first positivity conditions; the positive-energy representation framework is developed systematically in Dobrev, Mack, Petkova, Petrova, and Todorov 1977, chs. 5–6.

Level one

For a scalar primary, $M_{\mu\nu}|\Delta\rangle=0$, so

$$G^{(1)}_{\mu\nu} =\langle\Delta|K_\mu P_\nu|\Delta\rangle =2\Delta\,\delta_{\mu\nu} \langle\Delta|\Delta\rangle.$$

Level-one positivity alone gives $\Delta\geq0$. It does not give the sharp scalar bound in $d>2$; that information first appears in the scalar block at level two.

For a spinning primary, the $M_{\mu\nu}$ term acts nontrivially. Decomposing $V\otimes R$ gives one scalar eigenvalue condition for each irreducible summand. For a symmetric-traceless primary of spin $\ell>0$, the smallest eigenvalue yields

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

At equality, the spin-$\ell-1$ divergence descendant is null. In local-operator language this is the conservation equation for a symmetric-traceless current. Dimension-specific representations and spinors require their own decomposition; the compact summary is given in Unitarity Bounds and Null States.

The scalar level-two zero

For a scalar primary, the symmetric level-two space decomposes into a traceless tensor and the trace $P^2|\Delta\rangle$. The trace norm is the decisive calculation:

$$\begin{aligned} \|P^2|\Delta\rangle\|^2 &=\langle\Delta|K^2P^2|\Delta\rangle\\ &=8d\,\Delta \left(\Delta-\frac{d-2}{2}\right) \langle\Delta|\Delta\rangle. \end{aligned}$$

Assuming a positive nonzero primary norm and $d>2$, simultaneous positivity at levels one and two gives

$$\Delta=0 \quad\text{or}\quad \Delta\geq\frac{d-2}{2}.$$

The $\Delta=0$ scalar is the identity multiplet under the usual unique-vacuum assumptions. At $\Delta=(d-2)/2$, the trace descendant is null:

$$P^2|\Delta\rangle=0.$$

Away from coincident insertions, the corresponding operator obeys the free scalar equation $\partial^2\mathcal O=0$. This is a representation-theoretic shortening statement; contact terms and the existence of a full local free-field realization are additional questions.

The level-one and level-two positivity argument, including the scalar bound and its null descendant, is derived in Rychkov 2017, §3.2, pp. 45–47, Open PDF.

In $d=3$, the exact values specified by the accompanying symbolic exercise make the sign change transparent:

$\Delta$	Level-one eigenvalue $2\Delta$	$\lVert P^2\lvert\Delta\rangle\rVert^2/\langle\Delta\lvert\Delta\rangle$	Conclusion
$0.4$	$0.8$	$-0.96$	Level one is positive, but the level-two trace has negative norm
$0.5$	$1$	$0$	The trace descendant is null at the scalar bound
$0.7$	$1.4$	$3.36$	Both displayed tests are positive

These numbers follow from the exact polynomial $24\Delta(\Delta-1/2)$; they are not evidence from a floating-point scan.

Null vectors and the quotient module

If $G^{(n)}$ has an exact kernel, a vector

$$|\chi\rangle=\sum_I c_I|I\rangle, \qquad G^{(n)}c=0,$$

is orthogonal to every state at that level. In a positive-semidefinite representation it has zero norm. Acting with translations produces an entire null submodule. The irreducible conformal multiplet is obtained by quotienting the Verma-like module by that submodule.

This order matters. One should not merely remove $|\chi\rangle$ from level $n$ and retain all of its higher descendants. Nor should a determinant zero be interpreted without checking the kernel and the representation sector: a determinant can vanish because of an exact null relation, a redundant basis, or a normalization singularity.

Characters implement the same quotient combinatorially by subtracting the null submodule and restoring any overlap required by further relations. That subject is developed in Characters and Conformal Multiplet Counting.

Symbolic and numerical checks

A reliable implementation separates exact representation theory from numerical diagnostics.

Stage	Required operation	Independent check
Algebra	Encode $[D,P]$, $[D,K]$, $[K,P]$, and rotation action	Verify representative Jacobi identities before building matrices
Basis	Enumerate symmetric PBW monomials and project spin sectors	Compare the total projected dimension with $\dim[\operatorname{Sym}^n(V)\otimes R]$
Gram form	Move every $K$ through the $P$ monomial exactly	Check Hermiticity and rotational block diagonality
Null point	Compute the symbolic kernel at the predicted $\Delta$	Substitute the value before numerical diagonalization and verify exact annihilation
Quotient	Remove the full descendant submodule	Compare the surviving count with the shortened character
Numerics	Evaluate away from and near the zero at declared precision	Vary basis normalization and precision; the inertia and exact zero locus must agree

Near a null point, the condition number of $G$ diverges. A small eigenvalue at machine precision is therefore not by itself a null vector. Use exact arithmetic when possible, or increase precision and verify the predicted polynomial zero and kernel relations independently. Under a nonsingular basis change $G\mapsto A^\dagger G A$, individual eigenvalues change, but the numbers of positive, negative, and zero directions are invariant.

The $d=3$ scalar test can be reproduced with exact arithmetic, including deliberately nonunitary inputs. The matrix entries and their exact factorization are displayed above.

What a Gram calculation proves

At a fixed level, positive semidefiniteness is a necessary condition for a unitary conformal representation. A negative eigenvalue rules out unitarity for that representation. An exact zero identifies reducibility and, after the quotient is understood, shortening. Finite-level positivity does not prove positivity at every level, and a consistent positive-energy representation does not by itself prove the existence of a complete local CFT with crossing-symmetric correlators.

That separation is the main stop rule: Gram matrices classify representation-theoretic possibilities and null equations. Locality, OPE associativity, and the existence of a full theory enter in later chapters.

References

• Dobrev, Vladimir K., Gerhard Mack, Valentina B. Petkova, Stoyan G. Petrova, and Ivan T. Todorov. Harmonic Analysis on the $n$-Dimensional Lorentz Group and Its Application to Conformal Quantum Field Theory. Lecture Notes in Physics 63. Berlin: Springer, 1977. doi:10.1007/BFb0009678.
• Rychkov, Slava. EPFL Lectures on Conformal Field Theory in $D\geq3$ Dimensions. SpringerBriefs in Physics. Cham: Springer, 2017. doi:10.1007/978-3-319-43626-5. Open PDF.
• Simmons-Duffin, David. “The Conformal Bootstrap.” In New Frontiers in Fields and Strings, 1–74. Singapore: World Scientific, 2017. doi:10.1142/9789813149441_0001. Open PDF.